Electronic Journal of Differential Equations, Vol. 2025 (2025), No. ??, pp. 1–35. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2025.?? EXISTENCE OF SOLUTIONS FOR A n-DIMENSIONAL SYSTEMS OF NONLOCAL BOUNDARY VALUE PROBLEMS GEORGE L. KARAKOSTAS Abstract. We show the existence of nontrivial solutions of a system of nonlocal boundary value problems for a second order n-dimensional ordinary differential equation with a Caratheodory type response. To do this, we apply the Krasnosel’skii (contraction+compact) fixed point theo- rem. Examples in two and three dimensional cases illustrate the results, where the best condi- tions are suggested. 1. Introduction We study the existence of solutions of a system of n-dimensional second-order ordinary differen- tial equations subject to some nonlocal boundary value conditions. More specifically, our subject is a differential equation of the form (Θ(t)x′(t))′ = (Nx)(t), t ∈ [0, 1] (1.1) where, for each t ∈ [0, 1], the symbol Θ(t) stands for a n × n-square nonsingular matrix and the function t → Θ(t) is assumed to be Lebesgue measurable. N is an operator acting on the space C(I,Rn) into itself and it satisfies some conditions to be specified below. We associate equation (1.1) with the two nonlocal boundary conditions A0x(0)−B0x ′(0) = ψ0[x] + ζ0, (1.2) A1x(1) +B1x ′(1) = ψ1[x] + ζ1. (1.3) Here the coefficients A0, A1, B0, B1 are n×n square matrices with real entries and the items ψ0, ψ1 are linear functions expressed in the usual Riemann-Stieltjes integral form ψi[x] := ∫ 1 0 dΨi(s)x(s), i = 0, 1, where Ψ0,Ψ1 are n × n matrix valued functions defined on the interval [0, 1] and they are of bounded variation. Our purpose is to provide sufficient conditions which guarantee the existence of solutions of the boundary value problem (1.1), (1.2), (1.3). This will be done by applying the well known Krasnosel’skii’s (contr.+comp.) fixed point theorem. In the previous four decades, boundary value problems with nonlocal boundary value conditions have appeared in a great number of scientific works. This is because of the many theoretical problems which study physical phenomena in life sciences. Such related works appear in the literature (see, e.g., [7, 15, 25, 34, 35, 43, 45, 48] and the references therein). It is well known that the study of such problems when (1.1) is linear, was initiated in [4] and then a variety of articles followed them, (see, e.g. [3, 4, 8, 9, 14, 16, 19, 20, 22, 38, 39, 43]) where m-point boundary value problems for nonlinear ordinary differential equations with Θ = 1, were investigated. Apart of the use of the Green’s function for the linear case and the upper-lower solutions method for the nonlinear case (developed mainly by Lakshmikantham and Leela in their fruitful work on boundary value problems, published elsewhere; see, also, [7]), many techniques have recently been developed 2020 Mathematics Subject Classification. 34B15, 34B18, 34K40. Key words and phrases. Nonlocal boundary value problems; Krasnoselskii’s fixed point theorem; Caratheodory functions; Riemann-Stieltjes integral; contractions; compact operators; positive solutions. ©2025. This work is licensed under a CC BY 4.0 license. Submitted July 29, 2025. Published November 25, 2025. 1 2 G. L. KARAKOSTAS EJDE-2025/?? to give existence results. These techniques lie in the use of fixed points theorems applied to an appropriate (usually completely continuous) operator. More often the Leray-Schauder’s fixed point theorem (see, e.g. [11]) via index theory is applied. Other methods are exhibited elsewhere, see, e.g. the S-type method operators in [10]. A good survey of recent existence results for solutions of first and second order nonlinear differential systems with nonlocal boundary conditions by using methods based on convexity, topological degree and maximum-principle like techniques, is presented in [37]. The motivation of this paper is the work done in [1], where a system of scalar differential equations of the form x′′(t) + f(t, x(t), y(t)) = 0, y′′(t)− g(t, x(t), y(t)) = 0, t ∈ I (1.4) is investigated, under the pair of the nonlocal boundary conditions ã0x(0)− b̃0x ′(0) = ϕ̃0[x], a0y(0)− b0y ′(0) = ϕ0[y] + c0, ã1x(1) + b̃1x ′(1) = ϕ̃1[x], a1y(1) + b1y ′(1) = ϕ1[y] + c1, (1.5) where a0, ã0, a1, ã1, b0, b̃0, b1, b̃1 are positive reals and the functionals ϕ0[y], etc, are defined by the type ϕ0[y] := ∫ 1 0 y(s)dΦ0(s), etc. The author proves the existence of positive solutions of the system and, in order to succeed it, suitable conditions are given so that a hybrid-type Krasnosel’skii-Schauder fixed point theorem suggested by Infante, Mascali and Rodriguez-Lopez [26] to be applicable. According to this fixed point theorem, the Krasnoselskii’s fixed point theorem on cones in a coordinate and the Schauder’s fixed point in the second is applied. Boundary value problems of the same form concerning two-dimensional systems are also studied elsewhere, as e.g. in [23, 47, 52] and the references therein. In the one-dimensional case such problems are studied in many papers, some of which are [2, 5, 6, 9, 12, 13, 17, 18, 21, 27, 28, 31, 32, 33, 36, 38, 39, 40, 41, 42, 44, 45, 46, 49, 50, 51]. BVPs for such equations, in an abstract setting, have been investigated in the literature, see, e.g. [8, 30]. We especially mention to [30], which is concerned with the existence of solutions of boundary value problems for nonlinear second order ordinary differential equations of the type x′′ = H(t, x, x), 0 < t < 1 with the conditions ax(0) − bx′(0) = x0 and cx(1) + dx′(1) = x1 in a general Banach space. We want to elaborate a little on the form of the boundary value problem (1.1), (1.2), (1.3). The operator N includes the Nemytskii type operators, as well as Volterra integral operators. Also, the case of delay and/or advanced values of the solution is included, see section 10. The coefficient Θ in the left side comes from the Sturm Liouville form. Obviously, the problem (1.1), (1.2), (1.3) is essentially more general than the problem (1.4), (1.5). So, boundary value conditions such as (1.2), (1.3) include the so-called nonlocal and multi-point boundary value problems. Many of the results obtained in the literature for scalar type equations refer to positive solutions and these results are succeeded by application of the so called Krasnoselskii’s fixed point theorem on cones, or some variants of that. The process is quite simple. Formulate a positive cone on a space of functions and seek for a solution in this cone. In this work we shall use the following so called Krasnoselskii’s contr+comp fixed point theorem which states as follows: Theorem 1.1 (Krasnoselskii [29]). Suppose A is a closed bounded convex subset of a Banach space X. If T : A → X is a contraction, C : A → X is compact and T (A) + C(A) := {z = T (x) + C(y), x, y ∈ A} ⊆ A, then T + C has a fixed point in A. Recall that an operator C : A→ X is said to be compact if it is continuous and maps bounded sets into precompact sets. Concerning the terminology by Krasnoselskii used in his survey [29] (p. 370), an operator C : A → X is compact in case it is continuous and the set C(A) belongs to a compact set, where A need not be bounded. In this work we shall give several conditions for the existence of solutions of the problem (1.1), (1.2), (1.3) depending on the regularity of the four matrices A0, B0, A1, B1. The standard way EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 3 we shall follow in any case is to transform the boundary value problem in an operator equation written as the sum of two operators T and C, where the first is a contraction and the second one is compact. This work is organized as follows: Section 2 deals with some preliminaries needed for the reformulation of the problem. In Sections 3, 4, 5, 6 we discuss the cases when respectively, the matrices A0, B0, A1, B1 are nonsingular. In the final section 7, we shall present the case where A0 is nonsingular and the two matrices A0, A1 commute. In all cases we use the same symbols with a subindex 1, 2, 3, 4, 5, just to distinguish the conditions given in each section. In section 8 we discuss the problem (1.4)-(1.5), some applications to 2 and 3-dimensional systems are given in sections 9 and 10, while in the last section 11, we present a result concerning the positivity of the solutions and give an example which illustrates the results. 2. Preliminaries Let Rn be the n-dimensional real space endowed with its usual euclidean topology and with the euclidean norm. We shall use the same norm | · | as for the real numbers, without confusion. Also, on the space of all n× n matrices we consider the Euclidean norm, namely, the norm of a matrix A := (aij) is given by ∥A∥E = (∑n i=1 ∑n j=0 a 2 ij )1/2 . Thus the norm of the identity matrix In×n is equal to √ n. This norm agrees with the euclidean norm of a vector in the sense that it holds |Ax| ≤ ∥A∥E |x|. Any two n× n matrices A,B satisfy ∥AB∥E ≤ ∥A∥E∥B∥E . We shall work on the space C(I,Rn) of all n-dimensional continuous functions defined on the interval I. We furnish the space with the usual sup-norm ∥ · ∥∞. Notice that the total variations of the matrix valued functions appeared in (1.2), (1.3) are defined by V (Ψi) := ∫ 1 0 ∥dΨi(s)∥E , i = 0, 1 (2.1) and, obviously, in case Ψi is differentiable, then V (Ψi) := ∫ 1 0 ∥Ψ′ i(s)∥Eds. The constants ζ0, ζ1 are such that |ζ0|+ |ζ1| ≠ 0, since, otherwise, zero is a solution of the problem. Assume that x is a solution of the problem (1.1)-(1.2)-(1.3). From (1.1) we obtain Θ(t)x′(t) = Θ(0)x′(0) + ∫ t 0 (Nx)(u)du and by the nonsingularity of Θ(t) we can multiply both sides by its inverse to obtain x′(t) = Θ(t)−1Θ(0)x′(0) + Θ(t)−1 ∫ t 0 (Nx)(u)du. (2.2) Thus we have x′(1) = Θ(1)−1Θ(0)x′(0) + Θ(1)−1 ∫ 1 0 (Nx)(u)du. (2.3) One more integration of (2.2) gives x(t) = x(0) + ∫ t 0 Θ(s)−1dsΘ(0)x′(0) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds, (2.4) which will be used to express the solution as a fixed point of an operator equation. In the sequel we shall assume that the operator N is defined on the set C(I,Rn) and it is such that for all x ∈ C(I,Rn) the item (Nx)(·) is a measurable n-dimensional function defined on I. Moreover assume that there is function m : (0,+∞) → (0,+∞), which for all r > 0 satisfies the condition ∥x∥∞ < r =⇒ |(Nx)(t)| ≤ m(r), t ∈ I. (2.5) (H0) Given the function m and three positive constants ρ,M,K, which will be specified below, we shall assume that there exists a positive real number r such that K + ρr +Mm(r) < r. (2.6) 4 G. L. KARAKOSTAS EJDE-2025/?? Notice that such a condition can be implied if, for instance, it holds: lim inf r→+∞ m(r) r < 1− ρ M . (2.7) Indeed, if such a relation is true, then we have lim inf r→+∞ [K r + ρ+M m(r) r ] = lim r→+∞ K r + ρ+M lim inf r→+∞ m(r) r = ρ+M lim inf r→+∞ m(r) r < ρ+M 1− ρ M = 1. The latter fact implies that there exists r > 0 satisfying (2.6). As we shall see, in some cases the first part of (2.7) is zero, so this is satisfied for any value of the factorM , provided that 0 < ρ < 1. Now we proceed to the investigation of the problem by taking into account when the matrices A0, A1, B0, B1 are nonsingular. To do that we have to solve the system of equations (1.2), (1.3) and (2.4). So, we distinguish the following cases: 3. Case(1): detA0 ̸= 0 In this section we shall prove the following theorem. Theorem 3.1. Consider the nonlocal boundary value problem (1.1), (1.2), (1.3), where N satisfies (2.5). Moreover we make the following assumptions: (H1) detA0 ̸= 0. (H2) The operator P1 defined by P1 := A1A −1 0 B0 +A1 ∫ 1 0 Θ(s)−1dsΘ(0) +B1Θ(1)−1Θ(0), is nonsingular. (H3) The quantity ρ1 := V (A−1 0 Ψ0) + ∥(A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0))P−1 1 ∥E ( V (A0Ψ1) + V (A1Ψ0) ) satisfies the condition ρ1 < 1 and moreover condition (H0) is satisfied with ρ1 and the constants M1 := ∥(A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1( ∫ 1 u Θ(s)−1ds+B1Θ(1)−1)∥E + ∫ 1 0 ∥Θ(s)−1∥Eds, K1 := |A−1 0 ζ0|+ ∥A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0)P−1 1 ∥E |A0ζ1 −A1ζ0|. Then there is a solution of the problem (1.1), (1.2), (1.3). Proof. In (2.4) we set t = 1 and obtain the equality x(1) = x(0) + ∫ 1 0 Θ(s)−1dsΘ(0)x′(0) + ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds (3.1) From (3.1), (2.3) and (1.3) we take A1x(0) +A1 ∫ 1 0 Θ(s)−1dsΘ(0)x′(0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds +B1Θ(1)−1Θ(0)x′(0) +B1Θ(1)−1 ∫ 1 0 (Nx)(u)du) = ψ1[x] + ζ1, EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 5 which can be written as A1x(0) + ( A1 ∫ 1 0 Θ(s)−1dsΘ(0) +B1Θ(1)−1Θ(0) ) x′(0) ×A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B1Θ(1)−1 ∫ 1 0 (Nx)(u) du = ψ1[x] + ζ1. (3.2) Next our goal is to eliminate the values x(0) and x′(0) in (2.4) and to express it in terms of the rest items. To do that, from (1.2) we work as follows: From (1.2) we obtain x(0) = A−1 0 B0x ′(0) +A−1 0 (ψ0[x] + ζ0) (3.3) and substitute it to (3.2). Then we obtain A1A −1 0 B0x ′(0) +A1A −1 0 (ψ0[x] + ζ0) + ( A1 ∫ 1 0 Θ(s)−1dsΘ(0) +B1Θ(1)−1Θ(0) ) x′(0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B1Θ(1)−1 ∫ 1 0 (Nx)(u)du = ψ1[x] + ζ1. To solve this equation with respect to x′(0) we write it in the form P1x ′(0) = −A1A −1 0 (ψ0[x]+ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ψ1[x]+ζ1, where P1 is the nonsingular matrix defined as in assumption (H3). Next, we solve it with respect to x′(0) as x′(0) = P−1 1 ( ψ1[x]+ζ1−A1A −1 0 (ψ0[x]+ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) . Then from (3.3) we obtain x(0) = A−1 0 B0P −1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) +A−1 0 (ψ0[x] + ζ0). Setting these two values in (2.4) we obtain the expression of the solution as x(t) = A−1 0 B0P −1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) +A−1 0 (ψ0[x] + ζ0) + ∫ t 0 Θ(s)−1dsΘ(0)P−1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds (3.4) namely, x(t) = A−1 0 (ψ0[x] + ζ0) + ( A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0) ) × P−1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. (3.5) 6 G. L. KARAKOSTAS EJDE-2025/?? Inversely: If x is a function satisfying relation (3.5), then it is a solution of the boundary value problem (1.1), (1.2), (1.3). Indeed, the fact that x is differentiable and it satisfies equation (1.1) is obvious. Next, we observe that x(0) = A−1 0 (ψ0[x] + ζ0) +A−1 0 B0P −1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) and x′(0) = P−1 1 (ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du). Hence, we have A0x(0)−B0x ′(0) = ψ0[x] + ζ0 +B0P −1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) −B0P −1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) & = ψ0[x] + ζ0, and therefore condition (1.2) is satisfied. Also, we have x(1) = A−1 0 (ψ0[x] + ζ0) + ( A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0) ) P−1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) + ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds, x′(1) = Θ(1)−1Θ(0)P−1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) +Θ(1)−1 ∫ 1 0 (Nx)(u)du. So the first part of relation (1.3) becomes A1x(1) +B1x ′(1) = A1A −1 0 (ψ0[x] + ζ0) + ( A1A −1 0 B0 +A1 ∫ 1 0 Θ(s)−1dsC(0) ) × P−1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B1Θ(1)−1 ×Θ(0)P−1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) +B1Θ(1)−1 ∫ 1 0 (Nx)(u)du namely A1x(1) +B1x ′(1) = A1A −1 0 (ψ0[x] + ζ0) + ( A1A −1 0 B0 +A1 ∫ 1 0 Θ(s)−1dsΘ(0) EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 7 +B1Θ(1)−1Θ(0) ) P−1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B1Θ(1)−1 ∫ 1 0 (Nx)(u)du = A1A −1 0 (ψ0[x] + ζ0) + ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B1Θ(1)−1 ∫ 1 0 (Nx)(u)du +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B1Θ(1)−1 ∫ 1 0 (Nx)(u)du = ψ1[x] + ζ1. Therefore the function x satisfies both boundary conditions (1.2) and (1.3). To proceed we apply Fubini’s theorem and obtain∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds = ∫ t 0 ∫ t u Θ(s)−1ds(Nx)(u)du. (3.6) Next we write relation (3.5) in the form x(t) = (T1x)(t) + (C1x)(t), where (T1x)(t) := A−1 0 (ψ0[x] + ζ0) + ( A−1 0 B0 − ∫ t 0 Θ(s)−1dsΘ(0) ) P−1 1 ( ψ1[x] + ζ1 −A1A −1 0 (ψ0[x] + ζ0) ) (3.7) and (C1x)(t) := − ( A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0) ) P−1 1 (∫ 1 0 ∫ 1 u Θ(s)−1ds +B1Θ(1)−1 ) (Nx)(u)du+ ∫ t 0 ∫ t u Θ(s)−1ds(Nx)(u)du. The latter can be written in the integral form (C1x)(t) = ∫ 1 0 G1(t, u)(Nx)(u)du, where the Green’s function G1 is the matrix G1(t, u) := − ( A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0) ) P−1 1 (∫ 1 u Θ(s)−1ds+B1Θ(1)−1 ) + χ[0,t](u) ∫ t u Θ(s)−1ds. So far we have shown that x is a solution of the original problem if and only it is a fixed point of the operator equation T1x+C1x = x. Hence the existence of a solution of the problem is equivalent to the existence of a fixed point of this operator equation. In the sequel we shall work on the ball Br(0) of the space C([0, 1],Rn), where r is a positive real number defined later. To proceed, first we obtain an upper bound of the quantity G1(t, u) as follows ∥G1(t, u)∥E ≤ ∥(A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1( ∫ 1 u Θ(s)−1ds+B1Θ(1)−1)∥E + ∫ 1 u ∥Θ(s)−1∥Eds =:M1. and therefore it holds ∥C1y∥∞ ≤M1m(r), y ∈ Br(0). (3.8) Let x, y ∈ Br(0) be given. Then we have |(T1x)(t)− (T1y)(t)| 8 G. L. KARAKOSTAS EJDE-2025/?? ≤ |A−1 0 (ψ0(x− y))|+ ∥(A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1∥E |(A0 ( ψ1(x− y))−A1(ψ0(x− y)))| ≤ [ V (A−1 0 Ψ0) + ∥(A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0))P−1∥E ( V (A0Ψ1) + V (A1Ψ0) )] ∥x− y∥∞, namely ∥T1x− T1y∥∞ ≤ ρ1∥x− y∥∞, (3.9) where ρ1 is the quantity defined in (H3) and satisfies ρ1 < 1. Now, let x ∈ Br(0). Then from (3.7) we have |(T1x)(t)| ≤ |A−1 0 (ψ0[x] + ζ0)|+ ∥(A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0))P−1 1 ∥E |A0 ( ψ1[x] + ζ1)−A1(ψ0[x] + ζ0 ) | ≤ V (A−1 0 Ψ0)∥x∥∞ + |A−1 0 ζ0|+ ∥(A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0))P−1 1 ∥E ( V (A0Ψ1 −A1Ψ0)∥x∥∞ + |A0ζ1 −A1ζ0| ) , and therefore |(T1x)(t)| ≤ K1 + ρ1r, where K1 and ρ1 are defined in (H3). From this and (3.8) we see that ∥x∥∞, ∥y∥∞ ≤ r =⇒ |(T1x)(t) + (C1y)(t)| ≤ K1 + ρ1r +M1m(r). Because of condition (H0) there exists r1 > 0 such that K1 + ρ1r1 +M1m(r1) ≤ r1, which means that T1x+ C1y ∈ Br1(0), for all x, y ∈ Br1(0). We shall show that the operator C1 is compact. First of all it is easy to prove that C1 is a continuous operator. Let W be a bounded subset of Br1(0). We shall show that C1W is compact. Boundedness of this set is obvious, by (2.5) and (3.8). To show equicontinuity, we consider an x ∈W and two points 0 ≤ t1 ≤ t2 ≤ 1 and observe that |(C1x)(t1)− (C1x)(t2)| ≤ ∣∣∣ ∫ 1 0 G1(t1, u)(Nx)(u)du− ∫ 1 0 G1(t2, u)(Nx)(u)du| = | − ( A−1 0 B0 + ∫ t1 0 Θ(s)−1dsΘ(0) ) P−1 1 (∫ 1 0 ∫ 1 u Θ(s)−1ds+B1Θ(1)−1 ) (Nx)(u)du + ∫ t1 0 ∫ t1 u Θ(s)−1ds(Nx)(u)du+ ( A−1 0 B0 + ∫ t2 0 Θ(s)−1dsΘ(0) ) × P−1 1 (∫ 1 0 ∫ 1 u Θ(s)−1ds+B1Θ(1)−1 ) (Nx)(u)du− ∫ t2 0 ∫ t2 u Θ(s)−1ds(Nx)(u)du ∣∣∣ ≤ ∣∣∣ ∫ t2 t1 Θ(s)−1dsΘ(0)P−1 1 (∫ 1 0 ∫ 1 u Θ(s)−1ds+B1Θ(1)−1 ) (Nx)(u)du ∣∣∣ + ∣∣∣ ∫ t2 0 ∫ s 0 Θ(s)−1(Nx)(u) du ds− ∫ t1 0 ∫ s 0 Θ(s)−1(Nx)(u) du ds ∣∣∣ ≤ ∫ t2 t1 ∥Θ(s)−1dsΘ(0)P−1 1 ∥E (∫ 1 0 ∫ 1 0 ∥Θ(s)−1∥Eds+ ∥B1Θ(1)−1∥E ) |(Nx)(u)du| + ∣∣ ∫ t2 t1 ∫ s 0 Θ(s)−1(Nx)(u) du ds ∣∣ ≤ m(r1) ∫ t2 t1 ( ∥Θ(s)−1dsΘ(0)P−1 1 ∥E (∫ 1 0 ∫ 1 u ∥Θ(s)−1∥Eds+B1Θ(1)−1∥E ) +m(r1) ∫ t2 t1 ∥Θ(s)−1∥Eds. EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 9 This relation proves the (uniform) equicontinuity of the family {C1x : x ∈ Br1(0)} and so C1 is a compact mapping. Also, from relation (3.9) and condition (H3a) it follows that the operator T is a contraction. By the Krasnosel’skii fixed point theorem (1.1), the mapping T1 + C1 admits a fixed point x̄ in Br1(0), which is a solution of the original problem and the proof is complete. □ 4. Case(2): detB0 ̸= 0 The result in this section is given in the following theorem: Theorem 4.1. Consider the nonlocal boundary value problem (1.1), (1.2), (1.3), where N satisfies (2.5). Moreover we make the following assumptions: (H4) detB0 ̸= 0. (H5) The operator P2 defined by P2 := A1 +A1 ∫ 1 0 Θ(s)−1dsΘ(0)B−1 0 A0 +B1Θ(1)−1Θ(0)B−1 0 A0 is nonsingular. (H5) The quantity ρ2 := {√ n+ ∫ 1 0 ∥Θ(s)−1dsΘ(0)B−1 0 A0∥E } ∥P−1 2 ∥E × [ ∥A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1∥E∥Θ(0)B−1 0 ∥EV (Ψ0) + V (Ψ1) ] satisfies the condition ρ2 < 1 and moreover condition (H0) is satisfied with ρ2, M2 := ( √ n+ ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)B−1 0 A0)P −1 2 ∥E [ ∥A1∥E ∫ 1 0 ∥Θ(s)−1∥Eds + ∥B1Θ(1)−1∥E ] + ∫ 1 0 ∥Θ(s)−1∥Eds and K2 := ∥ [ In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0 ] ∥E∥P−1 2 ∥E [ ∥ ( A1 ∫ 1 0 Θ(s)−1ds +B1Θ(1)−1 ) ∥E∥Θ(0)B−1 0 ∥E |ζ0|+ |ζ1| ] . Then there is a solution of the problem (1.1), (1.2), (1.3). Proof. Let x be a solution of the problem. From (1.2) we have x′(0) = B−1 0 A0x(0)−B−1 0 (Ψ0[x] + ζ0) (4.1) and so (2.4) becomes x(t) = x(0) + ∫ t 0 Θ(s)−1dsΘ(0) [ B−1 0 A0x(0)−B−1 0 (Ψ0[x] + ζ0) ] + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. Putting t = 1 we obtain x(1) = x(0) + ∫ 1 0 Θ(s)−1dsΘ(0) [ B−1 0 A0x(0)−B−1 0 (Ψ0[x] + ζ0) ] + ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds, while from (2.3) and (4.1) we obtain x′(1) = Θ(1)−1Θ(0) [ B−1 0 A0x(0)−B−1 0 (Ψ0[x] + ζ0) ] +Θ(1)−1 ∫ 1 0 (Nx)(u) du ds. Then from (1.3) we obtain that ψ1[x] + ζ1 = A1x(1) +B1x ′(1) 10 G. L. KARAKOSTAS EJDE-2025/?? = A1x(0) +A1 ∫ 1 0 Θ(s)−1dsΘ(0) [ B−1 0 A0x(0)−B−1 0 (ψ0[x] + ζ0) ] +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B1Θ(1)−1Θ(0) [ B−1 0 A0x(0) −B−1 0 (ψ0[x] + ζ0) ] +B1Θ(1)−1 ∫ 1 0 (Nx)(u) du ds, which can be written as P2x(0) = (A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1)Θ(0)B−1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1, where P2 is defined in (H5). Thus x(0) = P−1 2 [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) Θ(0)B−1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] . (4.2) Substituting this value to (4.1) we obtain x′(0) = B−1 0 A0P −1 2 [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) Θ(0)B−1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] −B−1 0 (Ψ0[x] + ζ0) (4.3) and so from (2.4) we obtain the expression of the solution as x(t) = { In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0 } P−1 2 [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) ×Θ(0)B−1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] − ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 (ψ0[x] + ζ0) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. (4.4) Until now we have proved that if x is a solution of the original problem, then it satisfies the operator equation (4.4). We shall show the inverse, namely, any function satisfying equation (4.4) is a solution of the boundary value problem (1.1), (1.2), (1.3). Indeed, let x be such a function. Then from (4.3) and (4.4) we obtain A0x(0)−B0x ′(0) = A0P −1 2 [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) Θ(0)B−1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u)du−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] + ψ1[x] + ζ1 ] −A0P −1 2 [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) Θ(0)B−1 0 (ψ0[x] + ζ0) +B0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds+B0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] + ψ0[x] + ζ0 = ψ0[x] + ζ0, (4.5) EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 11 namely condition (1.2) is satisfied. Also we have x(1) = ( In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0 ) P−1 2 [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) ×Θ(0)B−1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] − ∫ 1 0 Θ(s)−1dsΘ(0)B−1 0 (ψ0[x] + ζ0) + ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. and x′(1) = Θ(1)−1Θ(0)B−1 0 A0P −1 2 [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) ×Θ(0)B−1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u)du −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] −Θ(1)−1Θ(0)B−1 0 (ψ0[x] + ζ0) + Θ(1)−1 ∫ 1 0 (Nx)(u) du ds. Then we see that relation (1.3) is satisfied, because it holds that A1x(1) +B1x ′(1) = A1 {( In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0 ) P−1 2 × [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) Θ(0)B−1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u))du+ ψ1[x] + ζ1 ] − ∫ 1 0 Θ(s)−1dsΘ(0)B−1 0 (ψ0[x] + ζ0) + ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds } +B1 { Θ(1)−1Θ(0)B−1 0 A0P −1 2 [( A1 ∫ 1 0 Θ(s)−1ds +B1Θ(1)−1 ) Θ(0)B−1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] −Θ(1)−1Θ(0)B−1 0 (ψ0[x] + ζ0) + Θ(1)−1 ∫ 1 0 (Nx)(u) du ds } = [ A1 + (A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1)Θ(0)B−1 0 A0 ] × P−1 2 [ A1 ∫ 1 0 Θ(s)−1dsΘ(0)B−1 0 (ψ0[x] + ζ0) +B1Θ(1)−1Θ(0)B−1 0 (ψ0[x] + ζ0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ ψ1[x] + ζ1 ] −A1 ∫ 1 0 Θ(s)−1ds 12 G. L. KARAKOSTAS EJDE-2025/?? ×Θ(0)B−1 0 (ψ0[x] + ζ0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −B1Θ(1)−1Θ(0)B−1 0 (ψ0[x] + ζ0) +B1Θ(1)−1 ∫ 1 0 (Nx)(u)du = A1 ∫ 1 0 Θ(s)−1dsΘ(0)B−1 0 (ψ0[x] + ζ0) +B1Θ(1)−1Θ(0)B−1 0 (ψ0[x] + ζ0) −A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du + ψ1[x] + ζ1 −A1 ∫ 1 0 Θ(s)−1dsΘ(0)B−1 0 (ψ0[x] + ζ0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1Θ(0)B−1 0 (ψ0[x] + ζ0) +B1Θ(1)−1 ∫ 1 0 (Nx)(u)du = ψ1[x] + ζ1. That a function x satisfying (4.4) satisfies (1.1), also, is obvious. Now we write equation (4.4) in the form x(t) = (T2x)(t) + (C2x)(t), where (T2x)(t) = { In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0 } P−1 2 × [( A1 ∫ 1 0 Θ(s)−1ds+B1Θ(1)−1 ) Θ(0)B−1 0 (ψ0[x] + ζ0) + ψ1[x] + ζ1 ] . (4.6) and (C2x)(t) = { In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0 } P−1 2 [ −A1 ∫ 1 0 Θ(s)−1 × ∫ s 0 (Nx)(u) du ds−B1Θ(1)−1 ∫ 1 0 (Nx)(u))du ] + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u)du)) du ds. By (3.6), the latter can be written in the form (C2x)(t) = ∫ 1 0 G2(t, u)(Nx)(u)du, where G2(t, u) := −(In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0)P −1 2 [ A1 ∫ 1 u Θ(s)−1ds+B1Θ(1)−1 ] + χ0,t](u) ∫ t u Θ(s)−1ds. Hence the existence of a solution of the problem is equivalent to the existence of a fixed point of the operator equation x = T2x+ C2x. To proceed we observe that ∥G2(t, u)∥E ≤ ( √ n+ ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)B−1 0 A0∥E)∥P−1 2 ∥E × [ ∥A1∥E ∫ 1 0 ∥Θ(s)−1∥Eds+ ∥B1Θ(1)−1∥E ] + ∫ 1 0 ∥Θ(s)−1∥Eds =:M2 and therefore, for any x ∈ Br(0), it holds ∥C2x∥∞ ≤M2m(r). Let x, y ∈ Br(0). Then we have |(T2x)(t)− (T2y)(t)| ≤ {√ n+ ∫ 1 0 ∥Θ(s)−1dsΘ(0)B−1 0 A0∥E } ∥P−1 2 ∥E [ ∥A1 ∫ 1 0 Θ(s)−1ds ×+B1Θ(1)−1∥E∥Θ(0)B−1 0 ∥E |ψ0(x− y)|+ |ψ1(x− y)| ] . and so ∥T2x− T2y∥∞ ≤ ρ2∥x− y∥∞, (4.7) EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 13 where ρ2 is defined in (H6) which, as we have assumed is smaller than 1. Also for any x ∈ Br(0), we have |(T2x)(t)| ≤ ∥(In×n + ∫ t 0 Θ(s)−1dsΘ(0)B−1 0 A0)∥E∥P−1 2 ∥E [ ∥ ( A1 ∫ 1 0 Θ(s)−1ds +B1Θ(1)−1∥E ) ∥Θ(0)B−1 0 ∥E(V (Ψ0)|x|+ |ζ0|) + V (Ψ1)|x|+ |ζ1| ] , namely |(T2x)(t) ≤ K2 + ρ2r, where K2 and ρ2 are defined in (H6). Finally, we see that for all x, y ∈ Br(0) it holds ∥T2x + C2y∥∞ ≤ K2 + L2r +M2m(r). Now, from condition (H0) we can conclude that there exists r2 > 0 such that K2+ρ2r2+M2m(r2) ≤ r2. Then, from the previous arguments we conclude that T2x+C2y ∈ Br2(0), for all x, y ∈ Br2(0). The relations we have found so far prove the (uniform) equicontinuity of the family {C2x : x ∈ Br2(0)} and so C2 is a compact mapping. Also, from relation (4.7) and condition (H6) it follows that the operator T2 is a contraction. By the Krasnosel’skii fixed point theorem the mapping T2 + C2 admits a fixed point in Br2(0), which is a solution of the original problem. □ 5. Case(3): detA1 ̸= 0 Here we prove the following theorem. Theorem 5.1. Consider the nonlocal boundary value problem (1.1), (1.2), (1.3), where N satisfies (2.5). Moreover we make the following Assumptions: (H7) detA1 ̸= 0. (H8) The operator P3 defined by P3 := A0 ∫ 1 0 Θ(s)−1dsΘ(0) +A0A −1 1 B1Θ(1)−1Θ(0) +B0 is nonsingular. (H9) The quantity ρ3 := (∫ 1 0 ∥Θ(s)−1dsΘ(0)∥E + ∥A−1 1 B1Θ(1)−1Θ(0)∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E ) ∥P−1 3 ∥E(V (A0A −1 1 Ψ1) + V (Ψ0)) + V (A−1 1 Ψ1) satisfies the inequality ρ3 < 1 and moreover condition (H0), where M3 := (∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E + ∥A−1 1 B1Θ(1)−1Θ(0)∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E ) ∥P−1 3 [ A0A −1 1 B1Θ(1)−1 +A0 ∫ 1 0 Θ(s)−1ds ] ∥E + ∥A−1 1 B1Θ(1)−1∥E + ∫ 1 0 ∥Θ(s)−1∥Eds+ ∥ ∫ 1 0 E−1(s)∥Eds and K3 := (∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E + ∥A−1 1 B1Θ(1)−1Θ(0)∥E + ∫ t 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E ) ∥∥P−1 3 ∥E(|A0A −1 1 ζ1|+ |ζ0|) + |A−1 1 ζ1|. Then there is a solution of the problem (1.1), (1.2), (1.3). Proof. Let x be a solution of the problem. From (1.1) we have x′(t) = Θ(t)−1 ( Θ(0)x′(0) + ∫ t 0 (Nx)(u)du ) . 14 G. L. KARAKOSTAS EJDE-2025/?? This and (1.3) give x(1) = −A−1 1 B1x ′(1) +A−1 1 (ψ1[x] + ζ1) = −A−1 1 B1Θ(1)−1 ( Θ(0)x′(0) + ∫ 1 0 (Nx)(u)du ) +A−1 1 (ψ1[x] + ζ1). (5.1) Then relations (5.1) and (3.1) give x(0) + (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0) ) x′(0) = −A−1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A−1 1 (ψ1[x] + ζ1)− ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. (5.2) The unique solution of the system of equations (1.2)-(5.2) is given by x′(0) = −P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u))du +A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] (5.3) and x(0) = (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0) ) P−1 3 × [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] −A−1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du +A−1 1 (ψ1[x] + ζ1)− ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. (5.4) Therefore, by (2.4) it follows that the solution x can be expressed in the form x(t) = (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0) ) × P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] −A−1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du +A−1 1 (ψ1[x] + ζ1)− ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds − ∫ t 0 Θ(s)−1dsΘ(0)P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du +A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0A −1 1 (Ψ1[x] + ζ1) + ψ0[x] + ζ0 ] + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds, (5.5) EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 15 namely x(t) = (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0)− ∫ t 0 Θ(s)−1dsΘ(0) ) × P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] −A−1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du +A−1 1 (ψ1[x] + ζ1)− ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. (5.6) So, we proved that if x is a solution of the original problem, then it satisfies the operator equation (5.6). We shall show that the inverse, is also true, namely, we shall show that if a function satisfies equation (5.6) then it is a solution of the boundary value problem (1.1), (1.2), (1.3). To do that we observe that if x is a function satisfying (5.6), then the values x(0) and x′(0) given in (5.3) and (5.4) satisfy the boundary condition (1.2). This is true since these values are obtained as the solutions of the system (1.2)-(5.2). To see that x satisfies (1.3), too, we obtain the values x(1) and x′(1) as follows x(1) = A−1 1 B1Θ(1)−1Θ(0)P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du +A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] −A−1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A−1 1 (ψ1[x] + ζ1). and x′(1) = −Θ(1)−1Θ(0)P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A0 ∫ 1 0 Θ(s)−1 × ∫ s 0 (Nx)(u) du ds−A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] +Θ(1)−1 ∫ 1 0 (Nx)(u) du ds. Therefore, A1x(1) +B1x ′(1) = B1Θ(1)−1Θ(0)P−1 3 ( A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ +A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ) −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+ (ψ1[x] + ζ1) +B1Θ(1)−1Θ(0) × P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] +B1Θ(1)−1 ∫ 1 0 (Nx)(u) du ds = ψ1[x] + ζ1). 16 G. L. KARAKOSTAS EJDE-2025/?? Next we write equation (5.6) in the form x(t) = T3x+ C3x, where T3x(t) := (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0)− ∫ t 0 Θ(s)−1dsΘ(0) ) × P−1 3 [ −A0A −1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] +A−1 1 (ψ1[x] + ζ1) (5.7) and C3x(t) := (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0)− ∫ t 0 Θ(s)−1dsΘ(0) ) × P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A0 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds ] −A−1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du− ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. By relation (3.6), the operator C3 can be written as C3x(t) = (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0)− ∫ t 0 Θ(s)−1dsΘ(0) ) × P−1 3 [ A0A −1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du+A0 ∫ 1 0 ∫ 1 u Θ(s)−1ds(Nx)(u)du ] −A−1 1 B1Θ(1)−1 ∫ 1 0 (Nx)(u)du− ∫ 1 0 ∫ 1 u Θ(s)−1ds(Nx)(u)du + ∫ t 0 ∫ t u E−1(s)ds(Nx)(u)du = ∫ 1 0 G3(t, u)(Nx)(u)du, (5.8) where the Greens’ function G3 is defined by G3(t, u) = (∫ 1 0 Θ(s)−1dsΘ(0) +A−1 1 B1Θ(1)−1Θ(0)− ∫ t 0 Θ(s)−1dsΘ(0) ) × P−1 3 [ A0A −1 1 B1Θ(1)−1 +A0 ∫ 1 u Θ(s)−1ds ] −A−1 1 B1Θ(1)−1 − ∫ 1 u Θ(s)−1ds+ χ[0,t](u) ∫ t u E−1(s)ds. To proceed we observe that ∥G3(t, u)∥E ≤ (∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E + ∥A−1 1 B1Θ(1)−1Θ(0)∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E ) ∥P−1 3 [ A0A −1 1 B1Θ(1)−1 +A0 ∫ 1 u Θ(s)−1ds ] ∥E + ∥A−1 1 B1Θ(1)−1∥E + ∫ 1 0 ∥Θ(s)−1∥Eds+ ∥ ∫ 1 0 E−1(s)ds∥E =M3 and therefore for any x ∈ Br(0) it holds that ∥C3x∥∞ ≤M3m(r). EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 17 r Also, we can see that, for all fixed x, y ∈ Br(0) we have |T3x(t)− T3y(t)| = {(∫ 1 0 ∥Θ(s)−1dsΘ(0)∥E + ∥A−1 1 B1Θ(1)−1Θ(0)∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E ) ∥P−1 3 ∥E ( V (A0A −1 1 Ψ1) + V (Ψ0) ) + V (A−1 1 Ψ1) } ∥x− y∥∞ = ρ3∥x− y∥∞, (5.9) where ρ3 is defined in (H9) and satisfies ρ3 < 1. On the other hand, given any x ∈ Br(0), we have |T3x(t)| ≤ (∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E + ∥A−1 1 B1Θ(1)−1Θ(0)∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E ) ∥P−1 3 ∥ [ V (A0A −1 1 Ψ1)r + |A0A −1 1 ζ1| ] + V (Ψ0)r + |ζ0|) + V (A−1 1 Ψ1)r + |A−1 1 ζ1| namely |T3x(t)| ≤ K3 + ρ3r, where K3 and ρ3 is defined in (H9). Finally, we see that for all x, y ∈ Br(0) it holds |(T3x)(t) + (C3y)(t)| ≤ K3 + ρ3r +M3m(r). Now, from condition (H8) there exists r3 > 0 such that K3+ρ3r3+M3m(r3) ≤ r3. Then, from the previous arguments we conclude that T3x + C3y ∈ Br3(0) for all x, y ∈ Br3(0). The relations we have found so far prove the (uniform) equicontinuity of the family {C3x : x ∈ Br3(0)} and so C3 is a compact mapping. Also, from relation (5.9) and condition (H9a) it follows that the operator T3 is a contraction. By the Krasnosel’skii fixed point theorem the mapping T3 + C3 admits a fixed point in Br3(0), which is a solution of the original problem. □ 6. Case(4): detB1 ̸= 0 In this section we prove the following theorem. Theorem 6.1. Consider the nonlocal boundary value problem (1.1), (1.2), (1.3), where N satisfies (2.5). Moreover we make the following assumptions: (H10) detB1 ̸= 0. (H11) The operator P4 defined by P4 := A0 +A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 A1 +B0Θ(0)−1Θ(1)B−1 1 A1. is nonsingular. (H12) The quantity ρ4 := (√ n+ ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 A1∥E ) ∥P−1 4 ∥E [ ∥A0 ∫ 1 0 Θ(s)−1dsΘ(1) ×B−1 1 ∥EV (Ψ1) + ∥B0Θ(0)−1Θ(1)B−1 1 ∥E(V (Ψ1) + V (Ψ0)) ] + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 ∥EV (Ψ1). satisfies the the inequality ρ4 < 1 and condition (H0), where M4 := (∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 A1∥E + √ n ) ∥P−1 4 ∥E [ ∫ 1 0 ∥A0Θ(s)−1∥Eds + ∥B0Θ(0)−1∥E ] + ∫ 1 0 ∥Θ(s)−1∥Eds, and K4 := (√ n+ ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 A1∥E ) ∥P−1 4 ∥E 18 G. L. KARAKOSTAS EJDE-2025/?? × [ ∥A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 ∥E |ζ1|+ ∥B0Θ(0)−1Θ(1)B−1 1 ∥E |ζ1|+ |ζ0| ] + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 ∥E |ζ1|. Then there is a solution of the problem (1.1), (1.2), (1.3). Proof. In this section we assume that the matrix B1 is nonsingular. From equation (1.1) we obtain x′(t) = Θ(t)−1Θ(1)x′(1)−Θ(t)−1 ∫ 1 t (Nx)(u)du and x(t) = x(1)− ∫ 1 t Θ(s)−1dsΘ(1)x′(1) + ∫ 1 t Θ(s)−1 ∫ 1 s (Nx)(u) du ds. (6.1) From the first of these relations we take x′(0) = Θ(0)−1Θ(1)x′(1)−Θ(0)−1 ∫ 1 0 (Nx)(u)du (6.2) and from the second one, x(0) = x(1)− ∫ 1 0 Θ(s)−1dsΘ(1)x′(1) + ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds. By using (1.3) and (6.2) we have x′(0) = Θ(0)−1Θ(1)[B−1 1 (ψ1[x] + ζ1)−B−1 1 A1x(1)]−Θ(0)−1 ∫ 1 0 (Nx)(u)du and x(0) = x(1)− ∫ 1 0 Θ(s)−1dsΘ(1)[B−1 1 (ψ1[x] + ζ1)−B−1 1 A1x(1)] + ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds = − ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) + [In×n + ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 A1]x(1) + ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds. We put these values to the boundary condition (1.2) and obtain A0 { − ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) + [In×n + ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 A1]x(1) + ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds } −B0 ( Θ(0)−1Θ(1)[B−1 1 (ψ1[x] + ζ1)−B−1 1 A1x(1)]−Θ(0)−1 ∫ 1 0 (Nx)(u)du } = ψ0[x] + ζ0. The latter relation can be written in the form P4x(1) = A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0, where P4 is defined in (H11). So x(1) = P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] . (6.3) EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 19 Then from (1.3) we obtain x′(1) = B−1 1 (ψ1[x] + ζ1)−B−1 1 A1P −1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) −A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) −B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] . (6.4) Finally, from equation (6.1) we see that the solution x can be expressed in the form x(t) = P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] − ∫ 1 t Θ(s)−1dsΘ(1) { B−1 1 (ψ1[x] + ζ1)−B−1 1 A1P −1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1) ×B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ]} + ∫ 1 t Θ(s)−1 ∫ 1 s (Nx)(u) du ds. (6.5) We proved that if x is a solution of the original problem, then it satisfies relation (6.5) Now, we shall show that the inverse, is true, namely, we shall show that if a function satisfies equation (6.5), then it is a solution of the boundary value problem (1.1), (1.2), (1.3). To do that we can easily see that if x is such a function then the values x(1) and x′(1) given in (6.3) and (6.4) satisfy the boundary condition (1.3). To check relation (1.2), from (6.5) we obtain x(0) = P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] − ∫ 1 0 Θ(s)−1dsΘ(1) { B−1 1 (ψ1[x] + ζ1)−B−1 1 A1P −1 4 [ A0 ∫ 1 0 Θ(s)−1ds ×Θ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) −B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 } + ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds = [ In×n + ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 A1 ] P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1) ×B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1Θ(1) ×B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] − ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) + ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds (6.6) 20 G. L. KARAKOSTAS EJDE-2025/?? and x′(0) = Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−Θ(0)−1Θ(1)B−1 1 A1P −1 4 [ A0 ∫ 1 0 Θ(s)−1ds ×Θ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] −Θ(0)−1 ∫ 1 0 (Nx)(u) du ds. (6.7) Hence it holds that A0x(0)−B0x ′(0) = A0 [ In×n + ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 A1 ] P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1) ×B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) −B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] −A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) +A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds−B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) +B0Θ(0)−1Θ(1)B−1 1 A1P −1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) −A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) −B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] +B0Θ(0)−1 ∫ 1 0 (Nx)(u) du ds = [ A0 +A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 A1 +B0Θ(0)−1Θ(1)B−1 1 A1 ] × P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 ] −A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) +A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds −B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) +B0Θ(0)−1 ∫ 1 0 (Nx)(u) du ds = A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1)−A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1)−B0Θ(0)−1 ∫ 1 0 (Nx)(u)du+ ψ0[x] + ζ0 −A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) +A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u) du ds −B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) +B0Θ(0)−1 ∫ 1 0 (Nx)(u) du ds = ψ0[x] + ζ0. Therefore the condition (1.2) is satisfied. EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 21 Next write equation (6.5) in the form x(t) = T4x(t) + C4x(t), where T4x(t) := P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] − ∫ 1 t Θ(s)−1dsΘ(1) { B−1 1 (ψ1[x] + ζ1) −B−1 1 A1P −1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1) +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ]} = ( In×n + ∫ 1 t Θ(s)−1dsΘ(1)B−1 1 A1 ) P−1 4 [ A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 × (ψ1[x] + ζ1) +B0Θ(0)−1Θ(1)B−1 1 (ψ1[x] + ζ1) + ψ0[x] + ζ0 ] − ∫ 1 t Θ(s)−1dsΘ(1)B−1 1 (ψ1[x] + ζ1). (6.8) and C4x(t) := −P−1 4 [ A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1 ∫ 1 0 (Nx)(u)du ] + ∫ 1 t Θ(s)−1dsΘ(1) { B−1 1 A1P −1 4 [ A0 ∫ 1 0 Θ(s)−1 ∫ 1 s (Nx)(u)du+B0Θ(0)−1 × ∫ 1 0 F (u, x(u))du ]} + ∫ 1 t Θ(s)−1 ∫ 1 s (Nx)(u) du ds = ∫ 1 0 G4(t, u)(Nx)(u)du. Here the kernel G4 is defined by G4(t, u) := −P−1 4 [ A0 ∫ u 0 Θ(s)−1ds+B0Θ(0)−1 ] + ∫ 1 t Θ(s)−1dsΘ(1)B−1 1 A1 P−1 4 [ A0 ∫ u 0 Θ(s)−1ds+B0Θ(0)−1 ] + χ[t,1](u) ∫ u 0 Θ(s)−1ds = (∫ 1 t Θ(s)−1dsΘ(1)B−1 1 A1 − In×n ) P−1 4 [ A0 ∫ u 0 Θ(s)−1ds+B0Θ(0)−1 ] + χ[t,1](u) ∫ u 0 Θ(s)−1ds, (6.9) where we have applied the Fubini’s Theorem twice. For the kernel we observe that ∥G4(t, u)∥E ≤ (∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 A1∥E + √ n ) ∥P−1 4 ∥E × [ ∫ 1 0 ∥A0Θ(s)−1∥Eds+ ∥B0Θ(0)−1∥E ] + ∫ 1 0 ∥Θ(s)−1∥Eds =M4 and therefore, for any x ∈ Br(0), ∥C4x∥∞ ≤M4m(r). Also, for all x, y ∈ Br(0), we have |T4x(t)− T4y(t)| = ∣∣∣(In×n + ∫ 1 t Θ(s)−1dsΘ(1)B−1 1 A1 ) P−1 4 { A0 ∫ 1 0 Θ(s)−1ds ×Θ(1)B−1 1 Ψ1[x− y] +B0Θ(0)−1Θ(1)B−1 1 (ψ1(x− y) + ψ0(x− y)) } − ∫ 1 t Θ(s)−1dsΘ(1)B−1 1 Ψ1[x− y] ∣∣∣ 22 G. L. KARAKOSTAS EJDE-2025/?? ≤ (√ n+ ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 A1∥E ) × ∥P−1 4 ∥E [ ∥A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 ∥EV (Ψ1)∥x− y∥∞ + ∥B0Θ(0)−1Θ(1)B−1 1 ∥E(V (Ψ1)∥x− y∥∞ + V (Ψ0)∥x− y∥∞) ] + ∥ ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 ∥EV (Ψ1)∥x− y∥∞, which implies that ∥T4x− T4y∥∞ ≤ ρ4∥x− y∥∞, (6.10) where ρ4 is defined in (H12) and it satisfies the inequality ρ4 < 1. On the other hand, for any x ∈ Br(0) and t ∈ I, it holds that |T4x(t)| ≤ (√ n+ ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 A1∥E ) ∥P−1 4 ∥E × [ ∥A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 ∥E |ζ1|+ ∥B0Θ(0)−1Θ(1)B−1 1 ∥E |ζ1|+ |ζ0| ] + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 ∥E |ζ1| (√ n+ ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 A1∥E ) × ∥P−1 4 ∥E [ ∥A0 ∫ 1 0 Θ(s)−1dsΘ(1)B−1 1 ∥EV (Ψ1)r + ∥B0Θ(0)−1Θ(1)B−1 1 ∥EV (Ψ1)r + V (Ψ0)r ] + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(1)B−1 1 ∥EV (ψ1)r, and so ∥T4x∥∞ ≤ K4 + ρ4r, where K4 and ρ4 are defined in (H12). Finally, we see that for all x, y ∈ Br(0) it holds ∥T4x + C4y∥∞ ≤ K4 + ρ4r +M4m(r). Now, from condition (H0) there exists r4 > 0 such that K4 + ρ4r4 +M4m(r4) ≤ r4. Then, from the previous arguments, we conclude that T4x+C4y ∈ Br4(0) for all x, y ∈ Br4(0). The relations we have found so far prove the (uniform) equicontinuity of the family {C4x : x ∈ Br4(0)} and so C4 is a compact mapping. Also, from relation (6.10) and condition (H12) it follows that the operator T4 is a contraction. By the Krasnosel’skii fixed point theorem the mapping T4 + C4 admits a fixed point x̄ in Br4(0), which is a solution of the original problem. □ 7. Case(5): detA0 ̸= 0 and A0A1 = A1A0 In this section we prove the following theorem. Theorem 7.1. Consider the nonlocal boundary value problem (1.1), (1.2), (1.3), where N satisfies (2.5). Moreover we make the following conditions: (H13) detA0 ̸= 0, detA1 ̸= 0 and A0A1 = A1A0. (H14) The operator P5 defined by P5 := A0B1Θ(1)−1Θ(0) +A0A1 ∫ 1 0 Θ(s)−1dsΘ(0) +A1B0 is nonsingular. (H15) The quantity ρ5 := (∥A−1 0 B0∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E)∥P−1 5 ∥E [ V (A0Ψ1) + V (A1Ψ0) ] + V (A−1 0 Ψ0) EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 23 satisfies the inequality ρ5 < 1 and condition (H0) where M5 := (∥A−1 0 B0∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E)∥P−1 5 ∥E × ( ∥A0A1∥E ∫ 1 0 ∥Θ(s)−1∥Eds+ ∥A0B1Θ(1)−1∥E ) + ∫ 1 0 ∥Θ(s)−1∥ds and K5 := (∥A−1 0 B0∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E)∥P−1 5 ∥E [ |A0ζ1|+ |A1ζ0|) ] + |A−1 0 ζ0|). Then there is a solution of the problem. Proof. From (1.1) we have the expression of x′ as in (2.2) and so (2.3) holds. Thus B1x ′(1) = B1Θ(1)−1Θ(0)x′(0) +B1Θ(1)−1 ∫ 1 0 (Nx)(u)du. Then from (1.3) it follows that Ψ1[x] + ζ1 −A1x(1) = B1Θ(1)−1Θ(0)x′(0) +B1Θ(1)−1 ∫ 1 0 (Nx)(u)du, which implies that A1x(1) = ψ1[x] + ζ1 −B1Θ(1)−1Θ(0)x′(0)−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du. From (2.4) we obtain A1x(1) = A1x(0) +A1 ∫ 1 0 Θ(s)−1dsΘ(0)x′(0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds and therefore Ψ1[x] + ζ1 −B1Θ(1)−1Θ(0)x′(0)−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du = A1x(0) +A1 ∫ 1 0 Θ(s)−1dsΘ(0)x′(0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. Hence A1x(0) = ψ1[x] + ζ1 −B1Θ(1)−1Θ(0)x′(0)−B1Θ(1)−1 ∫ 1 0 (Nx)(u)du −A1 ∫ 1 0 Θ(s)−1dsΘ(0)x′(0)−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds = ψ1[x] + ζ1 − ( B1Θ(1)−1 +A1 ∫ 1 0 Θ(s)−1ds ) Θ(0)x′(0) −B1Θ(1)−1 ∫ 1 0 (Nx)(u)du−A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. (7.1) By condition (H13), the matrices A0, A1 satisfy the condition A0A1 = A1A0. Notice that in this case we have A0A1A −1 0 = A1 =⇒ A1A −1 0 = A−1 0 A1, (7.2) Then from (1.2) we have A0A1x(0) = A1B0x ′(0) +A1(ψ0[x] + ζ0), which, due to (7.1), gives P5x ′(0) = A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u))du, 24 G. L. KARAKOSTAS EJDE-2025/?? where P5 is defined in H14. From (H14) we obtain x′(0) = P−1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] . From (1.2) and (2.4) it follows that x(t) = x(0) + ∫ t 0 Θ(s)−1dsΘ(0)x′(0) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds = (A−1 0 B0+ + ∫ t 0 Θ(s)−1dsΘ(0))x′(0) +A−1 0 (Ψ0[x] + ζ0) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds = (A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] +A−1 0 (ψ0[x] + ζ0) + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. (7.3) We proved that if x is a solution of the original problem, then it satisfies relation (7.3). Now, we shall show that the inverse is true, namely, we shall show that if a function satisfies equation (7.3), then it is a solution of the boundary value problem (1.1), (1.2), (1.3). Indeed, from (7.3) we obtain x(0) = A−1 0 B0P −1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] +A−1 0 (ψ0[x] + ζ0) and x′(0) = P−1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] . Then we can easily see that condition (1.2) is satisfied. Also, we have x(1) = (A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0))P−1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] +A−1 0 (ψ0[x] + ζ0) + ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds and x′(1) = Θ(1)−1Θ(0)P−1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] +Θ(1)−1 ∫ 1 0 (Nx)(u)du. Therefore, by (7.2), it holds A1x(1) +B1x ′(1) EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 25 = A1 { (A−1 0 B0 + ∫ 1 0 Θ(s)−1dsΘ(0))P−1 5 [ A0(ψ1[x] + ζ1) −A1(ψ0[x] + ζ0)−A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u)) du ds−A0B1Θ(1)−1 × ∫ 1 0 (Nx)(u)du ] +A−1 0 (ψ0[x] + ζ0) + ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds } +B1 { Θ(1)−1 ×Θ(0)P−1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0)−A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] +Θ(1)−1 ∫ 1 0 (Nx)(u)du = [ A1A −1 0 B0 +A1 ∫ 1 0 Θ(s)−1dsΘ(0) +B1Θ(1)−1Θ(0) ] P−1 5 [ A0(ψ1[x] + ζ1) −A1(ψ0[x] + ζ0)−A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] +A1A −1 0 (ψ0[x] + ζ0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u)du. The first factor in the big parenthesis can be written as A−1 0 [ A1B0 +A0A1 ∫ 1 0 Θ(s)−1dsΘ(0) +A0B1Θ(1)−1Θ(0) ] , which is equal to A−1 0 P5. Therefore, A1x(1) +B1x ′(1) = A−1 0 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds−A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] +A1A −1 0 (ψ0[x] + ζ0) +A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u)du = ψ1[x] + ζ1. Thus conditions (1.2) and (1.3) are satisfied. The fact that (1.1) is, also, satisfied, is obvious. Now write equation (7.3) in the form x = T5x+ C5, (7.4) where the operators T5 and C5 are defined as follows: T5x(t) = (A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1 5 [ A0(ψ1[x] + ζ1)−A1(ψ0[x] + ζ0) ] +A−1 0 (ψ0[x] + ζ0), and C5x(t) = (A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1 5 [ −A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds −A0B1Θ(1)−1 ∫ 1 0 (Nx)(u)du ] + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds. Hence, to show the existence of solutions of the original problem is sufficient to seek for the existence of fixed points of equation (7.4). By Fubini’s theorem the operator C5 can be written in the form C5x(t) = ∫ 1 0 G5(t, u)(Nx)(u)du, 26 G. L. KARAKOSTAS EJDE-2025/?? where the kernel is defined by G5(t, u) = (A−1 0 B0+ ∫ t 0 Θ(s)−1dsΘ(0))P−1 5 [ −A0A1 ∫ 1 u Θ(s)−1ds−A0B1Θ(1)−1 ] + ∫ t u Θ(s)−1ds. For this function we have ∥G5(t, u)∥E ≤ (∥A−1 0 B0∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E)∥P−1 5 ∥E × ( ∥A0A1∥E ∫ 1 0 ∥Θ(s)−1∥Eds+ ∥A0B1Θ(1)−1∥E ) + ∫ 1 0 ∥Θ(s)−1∥Eds =M5. Then by (2.5), for all r > 0 and x ∈ Br(0), we obtain ∥C5x∥∞ ≤M5m(r). (7.5) Also, for all x, y we have T5x(t)−T5y(t) = (A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1 5 [ A0ψ1(x− y)−A1ψ0(x− y) ] +A−1 0 ψ0(x− y), and therefore |T5x(t)− T5y(t)| ≤ { (∥A−1 0 B0∥E + ∫ 1 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E)∥P−1 5 ∥E × [ V (A0Ψ1) + V (A1Ψ0) ] + V (A−1 0 Ψ0) } ∥x− y∥∞, namely ∥T5x− T5y∥∞ ≤ ρ5∥x− y∥∞, (7.6) where ρ5 is defined in (H15). Also, for any x ∈ Br(0) we have |T5x(t) + C5y(t)| ≤ (∥A−1 0 B0∥E + ∫ t 0 ∥Θ(s)−1∥Eds∥Θ(0)∥E)∥P−1 5 ∥E ∣∣∣[A0(ψ1[x] + ζ1) −A1(ψ0[x] + ζ0) ]∣∣∣+ |A−1 0 (ψ0[x] + ζ0)|+ |C5y(t)| ≤ K5 + ρ5r +M5m(r), (7.7) where K5 and ρ5 are defined in (H15). From condition (H0) it follows that there exists r5 > 0 such that K5 + ρ5r5 +M5m(r5) ≤ r5. This means that T5x + C5y ∈ Br5(0) for all x, y ∈ Br5(0). The relations we have found so far prove the (uniform) equicontinuity of the family {C5x : x ∈ Br5(0)} and so C5 is a compact mapping. Also, by condition (H15) the operator T5 is a contraction. By the Krasnosel’skii fixed point theorem the mapping T5 + C5 admits a fixed point in Br5(0), which is a solution of the original problem. □ 8. Existence of solutions of the problem (1.4), (1.5) In this section we shall apply the results of section 7 to the boundary value problem (1.4), (1.5). First, we write the problem in terms of (1.1), (1.2), (1.3), where, notice that, Θ is the identity 2× 2-matrix. To do that we consider the vectors x(t) := ( x1(t) x2(t) ) , ζi := ( 0 ci ) , i = 0, 1, (Nx)(t) := ( −f(t, x1(t), x2(t)) g(t, x1(t), x2(t)) ) . as well as the matrices Ai := ( ãi 0 0 ai ) , Bi := ( b̃i 0 0 bi ) , i = 0, 1, ψi(x) := ( ϕ̃i[x1] 0 0 ϕi[x2] ) , Ψi(s) := ( Φ̃i(s) 0 0 Φi(s) ) , i = 0, 1, EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 27 with ã0a0ã1a1 ̸= 0. Since the matrices A0, A1 are diagonal, they commute, so Theorem 7.1 may be applicable. To proceed we obtain P5 := ( ã0b̃1 + ã0ã1 + ã1b̃0 0 0 a0b1 + a0a1 + a1b0 ) P−1 5 := ( (ã0b̃1 + ã0ã1 + ã1b̃0) −1 0 0 (a0b1 + a0a1 + a1b0) −1 ) . Also, we have ∥P−1 5 ∥E = [( ã0b̃1 + ã0ã1 + ã1b̃0) −2 + ( a0b1 + a0a1 + a1b0) −2 )]1/2 , ∥A−1 0 B0∥E = [ ã−2 0 b̃20 + a−2 0 b20 ]1/2 , ∥A0A1∥E = [( ã0ã1 )2 + ( a0a1 )2]1/2 , ∥A0B1∥E = [( ã0b̃1 )2 + ( a0b1 )2]1/2 , V (A0Ψ1) = ∫ 1 0 d ([( ã0Φ̃1(s) )2 + ( a0Φ1(s) )2]1/2) , V (A1Ψ0) = ∫ 1 0 d ([( ã1Φ̃0(s) )2 + ( a1Φ0(s) )2]1/2) , V (A−1 0 Ψ0) = ∫ 1 0 d ([( ã−1 0 Φ̃0(s) )2 + ( a−1 0 Φ1(s) )2]1/2) , |A0ζ1| = |a0c1|, |A1ζ0| = |a1c0|, |A−1 0 ζ0| = |a−1 0 c0|. Therefore, ρ5 = [[( ã0ã1 )2 + ( a0a1 )2]1/2 + 1 ][( ã0b̃1 + ã0ã1 + ã1b̃0) −2 + ( a0b1 + a0a1 + a1b0) −2 )]1/2[ ∫ 1 0 d ([( ã0Φ̃1(s) )2 + ( a0Φ1(s) )2]1/2) + ∫ 1 0 d ([( ã1Φ̃0(s) )2 + ( a1Φ0(s) )2]1/2)] + ∫ 1 0 d ([( ã−1 0 Φ̃0(s) )2 + ( a−1 0 Φ1(s) )2]1/2) . Now we assume that ρ5 < 1 and moreover, that the response functions f, g satisfy |f(t, x1, x2)| ≤ α1|x1|µ + β1|x2|ν , |g(t, x1, x2)| ≤ α2|x1|ν + β2|x2|µ, (8.1) where the coefficients satisfy αi, βi > 0 and 0 < µ, ν < 1. Obviously, we have m(r) = ( (α1 + β1) 2 + (α2 + β2) 2 )1/2 rξ, where ξ := max{µ, ν}(< 1). Therefore lim inf r→+∞ m(r) r = 0. (8.2) Thus, assumption (2.7) is true and so all conditions of Theorem 7.1 are satisfied. This implies that there is a solution of the problem (1.4), (1.5). From the previous special case, we see that, actually, the result depends on the value ρ5. Thus a question arises: Which value of ρ is better for such a decision, namely, which of the theorems we have presented requires less restrictions. In the sequel we shall show that such a fact depends on the parameters involved in the boundary conditions. 9. Application to a 2-dimensional system Consider the system of equations (1.4) with f, g satisfying (2.5) with condition (8.2), associated with the boundary conditions A0x(0)−B0x ′(0) = λψ0[x] + ζ0, (9.1) 28 G. L. KARAKOSTAS EJDE-2025/?? A1x(1) +B1x ′(1) = λψ1[x] + ζ1 (9.2) where λ is a real parameter playing an important role for condition ρ < 1. Assume that the 2× 2 square matrices A0, B0, A1, B1 are defined by A0 := ( µ 0 0 ν ) , A1 = B0 = B1 = I2×2, where µ, ν are nonzero real numbers with µ, ν ̸= − 1 2 . Then we obtain P1 := ( 2µ+1 µ 0 0 2ν+1 ν ) , P2 = P3 = P4 = P5 = ( 2µ+ 1 0 0 2ν + 1 ) , ρi = Zi|λ|, i = 1, 2, 3, 4, 5, where Z1 := V (A−1 0 Ψ0) + [( µ+ 1 2µ+ 1 )2 + ( ν + 1 2ν + 1 )2]1/2( V (A0Ψ1) + V (Ψ0) ) , Z2 := ( √ 2 + √ µ2 + ν2) [ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2( 4V (Ψ0) + V (Ψ1) ) , Z3 := 2( √ 2 + 1) [ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2( V (A0Ψ1) + V (Ψ0) ) + V (Ψ1), Z4 := 2 √ 3 [ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2( ( √ µ2 + ν2 + √ 3)V (Ψ1) + √ 3V (Ψ0) ) + 3V (Ψ1), Z5 := ( 2 + [ 1 µ2 + 1 ν2 ]1/2)[ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2( V (A0Ψ1) + V (Ψ0) ) + V (A−1 0 Ψ0). Once we have found the quantities Zi, we shall give a comparison of the conditions appeared in theorems 3.1, 4.1, 5.1, 6.1 and 7.1, to see which of them gives better results. And, indeed, in case the condition (8.1) holds, the only requirement which we need to check is the inequality ρj < 1, for some j ∈ {1, 2, 3, 4, 5}. For instance, comparing ρ1 with ρ5 we see that the inequality ρ1 > ρ5 may hold independently of the items Ψi, i = 0, 1. Indeed, this inequality holds if and only if Z1 > Z5, which is equivalent to the inequality[( µ+ 1 2µ+ 1 )2 + ( ν + 1 2ν + 1 )2]1/2 > ( 2 + [ 1 µ2 + 1 ν2 ]1/2)[ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2 . If we set Y (µ, ν) := ( µ+ 1 2µ+ 1 )2 + ( ν + 1 2ν + 1 )2 − ( 2 + [ 1 µ2 + 1 ν2 ]1/2)2[ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ] we observe that it satisfies lim (µ,ν)→(0,0) Y (µ, ν) = −∞, lim (µ,ν)→(±∞,±∞) Y (µ, ν) = 1 2 > 0. This means that there are values of the parameters µ, ν for which the corresponding quantity ρ5 is greater than ρ1. Then we have (0, Z−1 5 ) ⊆ (0, Z−1 1 ). In this case, from Theorem 3.1, for any λ with |λ| ∈ [0, Z−1 1 ), we result the existence of solutions and this interval is larger than the one obtained from the quantity ρ5. Also, there are values of the parameters µ, ν for which the corresponding quantity ρ5 is less than ρ1. In this case, if |λ| ∈ [0, Z−1 5 ), there are solutions and this interval is larger than the corresponding ρ1, see Figure (1). To obtain a more clear picture about the values of the parameter λ we consider the case when Ψi(s) := s. Then we obtain V (Ψi) = 1, V (A−1 0 Ψi) = √ 1 µ2 + 1 ν2 , V (A0Ψi) := √ µ2 + ν2, i = 0, 1 and so Z1 := √ 1 µ2 + 1 ν2 + [( µ+ 1 2µ+ 1 )2 + ( ν + 1 2ν + 1 )2]1/2(√ µ2 + ν2 + 1 ) , EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 29 Z2 := 5( √ 2 + √ µ2 + ν2) [ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2 , Z3 := 2( √ 2 + 1) [ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2(√ µ2 + ν2 + 1 ) + 1, Z4 := (2 + √ 2) [ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2(√ µ2 + ν2 + 2 √ 2 ) + 2, Z5 := ( 2 + [ 1 µ2 + 1 ν2 ]1/2)[ 1 (2µ+ 1)2 + 1 (2ν + 1)2 ]1/2(√ µ2 + ν2 + 1 ) + √ 1 µ2 + 1 ν2 . Therefore the existence of solutions is guaranteed if we assume that |λ| ∈ [0, Z−1 1 ) in case we apply Theorem 3.1, |λ| ∈ [0, Z−1 2 ) in case we apply Theorem 3.1, λ ∈ [0, Z−1 3 ) in case we apply Theorem 5.1, |λ| ∈ [0, Z−1 4 ) in case we apply Theorem 6.1 and |λ| ∈ [0, Z−1 5 ) in case we apply Theorem 7.1. Figure 1. The shadow area represent the set of values of (µ, ν) for which the inequality ρ1 > ρ5 holds. 9.1. A special numerical case. Assume that (µ, ν) = (1, 1) and consider the cases ∥A0A1∥E = ∥A0B1∥E = ∥A−1 0 B0∥E = √ 2, Ψ0(s) := Ψ1(s) := ( s 2 2 + s)I2×2. Then we obtain V (A0Ψ1) = V (A1Ψ0) = V (A−1 0 Ψ0) = 3 √ 2 2 λ, Pi = 3I2×2, for i = 1, 2, 3, 4, 5, as well as Z1 = 3 √ 2 + 8 2 , Z2 = 20 3 , Z3 = 11 √ 2 + 4 2 , Z4 = 4 + √ 2, Z5 = 7 √ 2 + 8 2 . These facts imply that for |λ| < 2 3 √ 2+8 ≈ 0.16336, we have ρ1 < 1, for |λ| < 3 20 ≈ 0.15, we have ρ2 < 1, for |λ| < 2 11 √ 2+4 ≈ 0.10226, we have ρ3 < 1, for |λ| < 1 4+ √ 2 ≈ 0.18469903, we have ρ4 < 1, and for |λ| < 2 7 √ 2+8 ≈ 0.11173, we have ρ5 < 1. Obviously, the best upper bound of the parameter |λ| is 0.18469903 and so for such values of |λ| in the interval [0, 0.18469903), the conditions of Theorem 6.1 are satisfied, and so for all these values of the parameter λ there is a solution of the boundary value problem (1.4)-(1.5). 10. Application to a 3-dimensional system Consider the system of differential equations x′′1(t) = f1(t, x1, x2, x3), x′′2(t) = f2(t, x1, x2, x3), x′′3(t) = f3(t, x1, x2, x3), (10.1) 30 G. L. KARAKOSTAS EJDE-2025/?? for t ∈ I, where for each i = 1, 2, 3 the function fi maps the set I × C2(I,R)3 into C(I,R). Also, for any x ∈ C(I,R3) we assume that the operator (Nx)(t) := f1(t, x1, x2, x3)f2(t, x1, x2, x3) f3(t, x1, x2, x3)  satisfies a condition specified latter. We associate system (10.1) with the following nonlocal bound- ary value conditions: 2x1(0) + x3(0)− x′1(0) = λ ∫ 1 0 x1(s)ds+ 1, x1(0) + 2x2(0)− x′2(0) = λ ∫ 1 0 (x1(s) + x3(s))ds, x2(0) + x3(0)− x′3(0) = λ ∫ 1 0 x2(s)ds− 1 x1(1) + x3(0) + 2x3(1) + x′1(1) = λ ∫ 1 0 x2(s)ds+ 1, 3x1(1) + x2(1) + x3(1) + x′2(1) = λ ∫ 1 0 x1(s)ds, x1(1) + 2x2(1)− x3(1) + x′3(1) = λ ∫ 1 0 x3(s)ds− 1. (10.2) The parameter λ is a real number which plays an important role in the existence of solutions. To write the problem in the general form, we formulate the nonsingular matrices A0 = 2 0 1 1 2 0 0 1 1  , A1 = 1 1 2 3 1 1 1 2 −1  , B0 = B1 = Θ(s) = I3×3, s ∈ I and ζ0 :=  1 0 −1  , ζ1 := 1 1 1  , Ψ0(s) := λ s 0 1 s 1 s 0 s 1  , Ψ1(s) := λ 1 s 0 s 0 1 2 0 s  . It is obvious that due to the values of ζ0, ζ1, the zero function is not a solution. Since the matrices satisfy det(A0) = 5, det(A1) = 11 and det(B0) = det(B1) = 1, they are nonsingular, and since A0A1 = 3 4 3 7 3 4 4 3 0  = A1A0, the matrices A0, A1 commute. These facts mean that all Theorems 3.1-7.1 are applicable. Before we continue we must calculate some items needed in the sequel. Now we have A−1 0 = 1 5  2 1 −2 −1 2 1 1 −2 4  , A−1 1 = 1 11 −3 5 1 4 −3 5 5 −1 −2  , with norms ∥Θ(t)∥E = √ 3, t ∈ I, ∥A0∥E = √ 12, ∥A1∥E = √ 22, ∥A−1 1 ∥E = 1 11 √ 115, and ∥A0 −1B0∥ = ∥A0 −1∥ = 6 5 , ∥A0A1∥ = √ 133. Also, we have V (Ψ0) = 1 11 |λ| ∫ 1 0 ∥∥∥d s 0 1 s 1 s 0 s 1 ∥∥∥ = 2|λ|, V (Ψ1) = 1 11 |λ| ∫ 1 0 ∥∥∥d 1 s 0 s 0 1 2 0 s ∥∥∥ = √ 3|λ|, EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 31 V (A0Ψ1) = |λ| ∫ 1 0 ∥∥∥d 2 0 1 1 2 0 0 1 1 1 s 0 s 0 1 2 0 s ∥∥∥ = √ 12|λ|, V (A1Ψ0) = |λ| ∫ 1 0 ∥∥∥d 1 1 2 3 1 1 1 2 −1 s 0 1 s 1 s 0 s 1 ∥∥∥ = √ 41|λ|, V (A−1 0 Ψ0) = |λ|1 5 ∫ 1 0 ∥∥∥d  2 1 −2 −1 2 1 1 −2 4 s 0 1 s 1 s 0 s 1 ∥∥∥ = 1 5 √ 41|λ|, V (A−1 1 Ψ1) = 1 11 |λ| ∫ 1 0 ∥∥∥d −3 5 1 4 −3 5 5 −1 −2 1 s 0 s 0 1 2 0 s ∥∥∥ = 1 11 √ 134|λ|, V (A0A −1 1 Ψ1) = 1 11 |λ| ∫ 1 0 ∥∥∥d = −1 9 0 5 −1 11 9 −4 3 1 s 0 s 0 1 2 0 s ∥∥∥ = 1 11 √ 335|λ|. Then we obtain P1 = 1 5 13 4 17 21 13 4 4 13 −4  , P−1 1 = 1 14025 −104 237 −205 100 −80 305 −136 −153 85  . Also we have (A−1 0 + I3×3)P −1 1 = 1 70125  −356 1885 −1300 668 −950 2425 −1528 −980 −50  and therefore ∥(A−1 0 + I3×3)P −1 1 ∥E = 3987, 10345991. To apply Theorem 4.1 we need the quantity P2 = A1 +A1A0 +A0 =  5 5 6 11 6 5 5 6 0  , P−1 2 = 1 191 −30 36 −11 25 −30 41 36 −5 −25  and so ∥P−1 2 ∥E = 0, 45226633. To apply Theorem 5.1 we need P3 = A0 +A0A −1 1 + I = 1 11 −3 5 1 4 −3 5 5 −1 −2  , P−1 3 = 1 771  42 −30 97 −81 −3 −16 36 87 −50  with norm ∥P−1 3 ∥E = 0, 17830562. To apply Theorem 6.1 we need P4 which equals to P2 and so we have ∥P−1 4 ∥E = 0, 45226633. For Theorem 7.1 we have P5 =  6 5 6 11 6 5 5 8 0  , P−1 5 = 1 233 −40 48 −11 25 −30 36 58 −23 19  , with norm ∥P−1 5 ∥ = 0, 4078. Now we assume that the response function satisfies a condition like |fi(t, x1(t), x2(t), x3(t))| ≤ αi|x1( t 2 )|µ1 + βi|x2(t)|µ2 + γi|x3(sin(t))|µ3 , i = 1, 2, 3, (10.3) where αi, βi, γi are positive real numbers and µi are such that µ := maxi µi < 1. In this case the truth of the Theorems 3.1–7.1 depends on the values of the parameters ρ1, ρ2, ρ3, ρ4, ρ5, only, since conditions (2.5) and (8.2) are satisfied. So, in order to apply the previous theorems we must obtain the basic parameters ρ1, . . . , ρ5, which correspond to the five cases above. We can easily obtain estimates of these parameters as ρ1 = 1, 9858304|λ|, ρ2 = 32, 27093226|λ|, ρ3 = 4, 78066942|λ|, ρ4 = 12.85258214|λ|, ρ5 = 22, 52419992|λ|. Therefore, the best value for λ, 32 G. L. KARAKOSTAS EJDE-2025/?? which satisfies ρ < 1, is given by ρ1 and it is |λ| < 0, 50356768. Hence for any such λ, Theorem 3.1 guarantees the existence of solutions of the problem, in the ball B(0, r) ⊆ R3, where r is any large positive real number satisfying the inequality K1 r + ρ1 +M1 m(r) r < 1, where m(r) := √∑ i α2 i + β2 i + γ2i r µ. Now we assume that the response function satisfies the condition |F (t, x)| ≤ α|x|+ β, (10.4) for some nonnegative real numbers α, β. In this case condition (2.6) becomes K + ρr +Mar +Mb < r, which is satisfied only if 1− ρ−Ma > 0. (10.5) To see what happens when applying all Theorems 3.1-7.1, we need to calculate the parameters M1,M2,M3,M4,M5. Indeed we calculate these numbers and obtain the following estimated values: M1 = 1, 92816175, M2 = 32, 93510197, M3 = 6, 57065738, M4 = 23, 17394972, M5 = 41, 8775797. Then in the various cases inequality (10.5) becomes σ1(λ, a) := 1− 1, 9858304∥λ| − 1, 92816175a > 0, σ2(λ, a) := 1− 32, 27093226|λ| − 38, 32050810a > 0, σ3(λ, a) := 1− 4, 78066942|λ| − 6, 57065738a > 0, σ4(λ, a) := 1− 31, 87116522|λ| − 12.85258214a > 0, σ5(λ, a) := 1− 22, 52419992|λ| − 41, 8775797a > 0. Each of these relations guarantee the existence of solutions, for instance when apply Theorem 7.1, we conclude that a solution of the problem exists in the ball B(0, r5) ⊆ R3, where r5 satisfies r5 > K5 +M5b σ5(λ, a) = 14, 08835163 + 41, 8775797b 1− 22, 52419992|λ| − 41, 8775797a , where K5 = 14, 08835163. Notice that between these five cases the first one gives better results, because, as we can, easily, see it holds {(λ, a) : σi(λ, a) > 0} ⊆ {(λ, a) : σ3(λ, a) > 0} ⊆ {(λ, a) : σ1(λ, a) > 0}, i = 2, 4, 5. 11. Positive solutions We say that a vector y := (y1, y2, . . . , yn) T is positive (nonnegative) if all its coordinates are positive (nonnegative) real numbers. Then we write y > 0(y ≥ 0), or 0 < y (y ≤ 0). Also, we write y < w (y ≤ w), if y−w < 0, (y−w ≤ 0). Analogous things we have for matrices. It is clear that for any vector x > 0 and matrices A,B > 0 we have Ax > 0, A+B > 0 and AB > 0. In this section we shall be concerned with the existence of positive solutions of the original problem. To this direction we will show the following results. Theorem 11.1. Assume that rm (H13) and the following conditions hold: (H16) For each t ∈ I the matrix (A−1 0 B0 + ∫ t 0 Θ(s)dsΘ(0))P−1 5 is positive. (H17) Ψ0(s) = Ψ1(s) = 0, s ∈ I. (H18) The operator N maps the set C(I,Rn) into C(I,Rn −). (H19) The inequality A0ζ1 −A1ζ0 > 0 holds. (H20) For any x ∈ C(I,Rn) and t ∈ I it holds A−1 0 ζ0 + ∫ t 0 Θ−1(s) ∫ s 0 (N(x)(u) du ds > 0. Then there is a solution of the problem and such a solution is positive. EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 33 Proof. Since Ψ0 = Ψ1 = 0, it follows that ρ5 = 0(< 1). Therefore Theorem 7.1 is applicable. From relation (7.3) we see that any solution is expressed in the form x(t) = (A−1 0 B0 + ∫ t 0 Θ(s)−1dsΘ(0))P−1 5 [ [A0ζ1 −A1ζ0] +A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (−Nx)(u) du ds+A0B1Θ(1)−1 ∫ 1 0 (−Nx)(u)du ] +A−1 0 ζ0 − ∫ t 0 Θ(s)−1 ∫ s 0 (−Nx)(u) du ds. (11.1) Here, all the factors are positive, so x(t) > 0, for all t ∈ I.. □ 11.1. An application. Consider the system of equations x′′i (t) = fi(t, x1, x2, . . . , xn)), i = 1, 2, . . . , n where x1, x2, . . . xn ∈ C2(I,R2), associated with the boundary value conditions κixi(0)− x′i(0) = ζi0, i = 1, 2, . . . , n, xi(1) + x′i(1) = ζ1i , i = 1, 2, . . . , n, where κi are positive real numbers, ζi0, ζ i 1, are nonzero reals and such that κiζ i 1 − ζi0 > 0, i = 1, 2, . . . , n. Assume, also, that the functions fi satisfy the conditions −min { νi, n∑ j=1 αi,j |xj(γi,j(t))|µi,j } ≤ fi(t, x1, x2, . . . , n) ≤ 0, i = 1, 2, . . . , n for all xi ∈ C2(I,R2). The coefficients νi, µi,j , are positive reals with µi,j < 1 and 2ζi0 ≥ κiνi, i = 1, 2, . . . , n. The arguments γi,j : I → I are continuous functions. Then conditions (2.5) and (8.2) are satisfied. Here we have A1 = B0 = B1 = In×n and A0 :=  κ1 0 0 . . . 0 0 κ2 0 . . . 0 ... 0 0 0 . . . κn  , P5 :=  2κ1 + 1 0 0 . . . 0 0 2κ2 + 1 0 . . . 0 ... 0 0 0 . . . 2κn + 1  and so the matrix in condition (H16) is equal to 1 k1 + 1 + 1 2k1+1 0 0 . . . 0 0 1 k2 + 1 + 1 2k2+1 0 . . . 0 ... 0 0 0 . . . 1 kn + 1 + 1 2kn+1  =: Ξ, which is positive. Also we have (Nx)(t) :=  f1(t, x1, x2, . . . , xn) f2(t, x1, x2, . . . , xn) ... fn(t, x1, x2, . . . , xn)  Now, a solution of the problem exists according to Theorem (7.1) and it has an expression of the form (11.1), which is positive, because it can be written as x(t) = Ξ [ k1ζ 1 1 − ζ10 k2ζ 2 1 − ζ20 ... knζ n 1 − ζn0 +A0A1 ∫ 1 0 Θ(s)−1 ∫ s 0 (−Nx)(u) du ds 34 G. L. KARAKOSTAS EJDE-2025/?? +A0B1Θ(1)−1 ∫ 1 0 (−Nx)(u)du ] +A−1 0 ζ0 + ∫ t 0 Θ(s)−1 ∫ s 0 (Nx)(u) du ds ≥ Ξ [ k1ζ 1 1 − ζ10 k2ζ 2 1 − ζ20 ... knζ n 1 − ζn0 + (positive quantity) ] +  ζ1 0 κ1 − ν1 2 ζ2 0 κ2 − ν2 2 ... ζn 0 κn − νn 2  > 0, t ∈ I. The proof is complete. References [1] Aleksejs Antonuks; Existence of a positive solution with concave and convex components for a system of bound- ary value problems, Non. Analysis: Modelling and Control, Vol. 30(2), (2025), 333-345. zbl DE080095489. [2] G. E. Abduragimov; On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Integral Boundary Conditions for a Second-Order Nonlinear Functional Differential Equation, Russian Mathematics, 2024, Vol. 68, No. 12, pp. 33–39. [3] A. V. Bitsadze; On the theory of nonlocal boundary value problems, Soviet Math. Dock. 30 (1964), 8-10. [4] A. V. Bitsadze, A. A. Samarskii; Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR 185 (1969), 739-740. [5] Gabriele Bonanno, Giuseppa Riccobono; Multiplicity results for Sturm-Liouville boundary value problems, Applied Mathematics and Computation, 210 (2009), 294-297. [6] Abdelkader Boucherif; Second-order boundary value problems with integral boundary conditions, Nonlinear Analysis, 70 (2006), 364-371. [7] J. Chandra, V. Lakshmikantham, S. Leela; Comparison principle and theory of nonlinear boundary value problems, in Proc. International Conf. on Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982, pp. 241-248. [8] J. Chandra, V. Lakshmikantham, A. R. Mitchell; Existence of solutions of boundary value problems for non- linear second order systems in a Banach space, Nonlinear Analysis, Theory, Methods and Applications 2(2) (1978) 157-168. [9] Donald S. Cohen; Singular perturbation of nonlinear two-point boundary-value problems, J. Mathematical Analysis and Applications, 43 (1973), 151-160. [10] Julio G. Dix, George L. Karakostas,; A fixed-point theorem for S-type operators on Banach spaces and its applications to boundary-value problems, Nonlinear Analysis 71 (2009) ,3872-3880. [11] J. Dugundji, A. Granas; Fixed Point Theory, Vol. I, Monographie Matematyczne, Warsaw, 1982. [12] L. H. Erbe, S. Hu, H. Wang; Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184 (1994), 640-648. [13] L. H. Erbe, H. Wang; On existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120 (1994), 743-748. [14] W. Feng; On a m-point nonlinear boundary-value problem, Nonlinear Anal., 30 (1997) 5369-5374. [15] W. Feng, J. R. L. Webb; Solvability of three point boundary value problems at resonance, Nonlinear Anal., 30 (1997), 3227-3238. [16] G. N. Galanis, A. P. Palamides; Positive solutions of three-point boundary-value problems for p-Laplacian singular differential equations, Electron. J. Differential Equations, 2005 (2005) No. 106, 1-18. [17] J. A. Gatica, Vladimir Oliker, Paul Waltman; Singular nonlinear boundary value problems for second order ordinary differential equations, J. Differential equations 79, (1989), 62-78. [18] X. He, W. Ge; Triple solutions for second-order three-point boundary-value problems, J. Math. Anal. Appl., 268 (2002) 256-265. [19] D. Guo, V. Lakshmikantham; Nonlinear Problems in Abstract Cone, Academic Press, New York, 1988. [20] C. Gupta; Solvability of a generalized multi-point boundary value problem of mixed type for second order ordinary differential equations, Proc. Dynamic Systems and Applications, 2 (1995), 215-222. [21] C. Gupta; Solvability of a three-point boundary-value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168 (1992), 540-551. [22] C. Gupta, S. N. Ntouyas, P. Ch. Tsamatos; Existence results for m-point boundary value problems, Diff. Eq. Dynam. Systems, 2 (1994), 289-298. [23] Johnny Henderson, Rodica Luca; On a system of second-order multi-point boundary value problems, Applied Mathematics Letters, (2012), DOI: 10.1016/j.aml.2012.05.005. [24] V. Il’in,d E. Moiseev; Nonlocal boundary value problems of the second kind for a Sturm-Liouville operator, Differential Equations, 23 (1987), 979-987. [25] V. Il’in, E. Moiseev; Nonlocal boundary value problems of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations, 23 (1988), 803 810. [26] G. Infante, G. Mascali, J. Rodriguez-Lopez; A hybrid Krasnoselskii-Schauder fixed point theorem for systems, Nonlinear Anal., Real World Appl., Vol. 80:104165 (2024), DOI: 10.1016/j.nonrwa.2024.104165. EJDE-2025/?? EXISTENCE OF SOLUTIONS FOR n-DIMENSIONAL SYSTEMS OF NONLOCAL BVP 35 [27] Jiqiang Jiang, Lishan Liu; Positive solutions for nonlinear second-order m-point boundary value problems, Electronic Journal of Differential Equations, Vol. 2009(2009), No. 110, pp. 1–12. [28] R. A. Khan; The generalized quasilinearization technique for a second order differential equation with separate boundary conditions, Math. Comput. Modelling, 43(2006) 727-742. [29] M. A. Krasnoselskii; Some Problems of Nonlinear Analysis, AMS Translations, Ser. 2 Vol. 10, Providence, RI, 1958. [30] V. Lakshmikantham; Abstract boundary value problems, in Nonlinear Equations in Abstract Spaces, Academic Press, New York, 1978. [31] L. Liu, B. Liu, Y. Wu; Positive solutions of singular boundary-value problems for nonlinear differential systems, Appl. Math. Comput., 186 (2007) 1163-1172. [32] Zhaoli Liu, Fuyi Li; Multiple positive solutions of nonlinear two-point boundary value problems, Journal of Mathematical Analysis and Applications, 203 (1996), 620-625. [33] Hongyu Li, Jingxian Sun; Positive solutions of sublinear Sturm-Liouville problems with changing sign nonlin- earity, Computers and Mathematics with Applications, 58 (2009), 1808-1815. [34] A. G. Lomtatidze; A nonlocal boundary value problem for second-order linear ordinary differential equations, Differential Equations, 31 (1995), 411-420. [35] R. Ma; Positive solutions of a nonlinear three-point boundary-value problem, Electron. J. Differential Equa- tions 1999 (1999) No. 34, 1-8. [36] R. Ma; Multiple positive solutions for nonlinear m-point boundary-value problems, Appl. Math. Comput. 148 (2004) 249-262. [37] Jean Mawhin, Katarzyna Szymanska-Debowska; Convexity topology and nonlinear differential systems with nonlocal boundary conditions: a survey, Rend. Istit. Mat. Univ. Trieste, Volume 51 (2019), 125-166. DOI: 10.13137/2464-8728/29333. [38] Klaus Schmitt; A nonlinear Boundary value problem, Journal of Differential equations, 7(1970), 527-537. [39] B. L. Shekhter; On the existence and zeros of solutions of a nonlinear two-point boundary value problem, Journal of Mathematical Analysis and Applications, 97 (1983), 1-20. [40] Szymanska-Debowska, Katarzyna; On two systems of non-resonant nonlocal boundary value problems, An. St. Univ. Ovidius Constanta , Vol. 21(3), 2013, 257-267. [41] Szymanska-Debowska, Katarzyna; On the existence of solutions for nonlocal boundary value problems, Geor- gian Mathematical Journal, Vol. 22, no. 2, 2015, pp. 273-279. DOI: 10.1515/gmj-2015-0005. [42] Y. Sun; Positive solutions of nonlinear second-order m-point boundary-value problem, Nonlinear Anal., 61 (2005), 1283-1294. [43] J. R. Ward; Semilinear boundary value problems in Banach Spaces, in Nonlinear Equations in Abstract Spaces, Academic Press, New York, 1978. [44] J. R. L. Webb, Gennaro Infante; Positive solutions of nonlocal boundary value problems: A unified approach, J. London Math. Soc., 74(2), (2006), 673-693. [45] J. R. L. Webb; A class of positive linear operators and applications to nonlinear boundary value problems, Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder University Centre, 39(2012), 221-242. [46] F. H. Wong, T. G. Chen, S. P. Wang; Existence of positive solutions for various boundary value problems, Computers and Mathematics with Applications 56 (2006), 953-958. [47] Shouliang Xi, Mei Jia, Huipeng Ji; Multiple nonnegative solutions for second-order boundary value problems with sign-changing nonlinearities, Electron. J. Differential Equations, 2009(2009), No. 66. pp. 1-10. [48] D. Yuyun; On solvability of second-order Sturm-Liouville boundary value problems at resonance, Proc. Amer. Math. Soc., 126 (1998), 145-152. [49] G. Zhang, J. Sun; Multiple positive solutions of singular second-order m-point boundary-value problems, J. Math. Anal. Appl., 317 (2006) 442-447. [50] Xuemei Zhang, Weigao Ge; Existence of positive solutions for a class of m-point boundary value problems, Advances in Difference Equations, Vol. 2008, Art. ID 845121, 9 pages. [51] Xuemei Zhang, Meiqiang Feng, Weigao Ge; Multiple positive solutions for a class of m-point boundary value problems, Applied Mathematics Letters, 22(2009), 12-18. [52] Xinguang Zhang, Lishan Liu; A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions, Applied Mathematics and Computation 215 (2010) 3501-3508. George L. Karakostas Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece Email address: gkarako@uoi.gr, gkarako@hotmail.com 1. Introduction 2. Preliminaries 3. Case(1): A0=0 4. Case(2): B0=0 5. Case(3): A1=0 6. Case(4): B1=0 7. Case(5): A0=0 and A0A1=A1A0 8. Existence of solutions of the problem (??), (??) 9. Application to a 2-dimensional system 9.1. A special numerical case 10. Application to a 3-dimensional system 11. Positive solutions 11.1. An application References