Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 51, pp. 1–17. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.51 EXISTENCE AND UNIQUENESS RESULTS FOR FOURTH-ORDER FOUR-POINT BVP ARISING IN BRIDGE DESIGN IN THE PRESENCE OF REVERSE ORDERED UPPER AND LOWER SOLUTIONS NAZIA URUS, AMIT K. VERMA Abstract. In this article, we establish the existence of solutions for a fourth- order four-point non-linear boundary value problem (BVP) which arises in bridge design, −y(4)(s)− λy′′(s) = F(s, y(s)), s ∈ (0, 1), y(0) = 0, y(1) = δ1y(η1) + δ2y(η2), y′′(0) = 0, y′′(1) = δ1y ′′(η1) + δ2y ′′(η2), where F ∈ C([0, 1] × R,R), δ1, δ2 > 0, 0 < η1 ≤ η2 < 1, λ = ζ1 + ζ2, where ζ1 and ζ2 are the real constants. We have explored all gathered 0 < ζ1 < ζ2, ζ1 < 0 < ζ2, and ζ1 < ζ2 < 0. We extend the monotone iterative technique and establish the existence results with reverse ordered upper and lower solutions to fourth-order four-point non-linear BVPs. 1. Introduction Higher order boundary value problems (BVP) play a vital role in studying various branches of science and engineering, e.g., suspension bridge [18, 13]. The suspension bridge is identified as a beam of length lb with fixed ends which are supported in equilibrium position and is given as the solution of the steady state equation EIy(4) + ζy+ = W (s), (1.1) y(0) = y(lb) = y′′(0) = y′′(lb) = 0, (1.2) where, E is Youngs’ modulus, I is moment of inertia, ζ is spring constant, W (s) is weight per unit length, and y(s) denotes downward deflection. y+ denotes the y, if y is positive, and zero if y is negative. There have been extensive studies on fourth-order BVP via different techniques such as fixed point theorem [2, 6, 14, 19, 21], upper and lower solutions (UL so- lutions) method [12, 29], monotone iterative (MI) method [20, 22], etc. The fact of sign-constancy of Green’s function for these class of BVPs can be used in the frame of Azbelev W-transform [1, 4, 5] for equation of the forth order (1.1). The 2020 Mathematics Subject Classification. 34B10, 34B15, 34B16, 34B27, 34B60. Key words and phrases. Monotone iterative technique; upper solutions; lower solutions; fourth-order; non-linear; four-point; Green’s function. ©2023. This work is licensed under a CC BY 4.0 license. Submitted March 17, 2023. Published August 4, 2023. 1 2 N. URUS, A. K. VERMA EJDE-2023/51 functional differential equation of second order can be obtained and then analysis of positivity of Green’s function of obtained functional differential equation can be achieved. For an alternative approach one can refer to [3]. The idea and realizations of the monotone iterative technique appeared first in the classical work by Chaplygin [11]. Adventures of monotone iterative technique were explained then by Luzin [24]. For the existence of solutions, method of UL solutions is extensively used to develop MI technique on the second order BVP [16, 17, 30, 33, 35, 36, 37, 38, 39]. There are also several research articles available on higher order two-point BVP with MI technique [7, 8, 26, 32, 34, 42]. To the best of our knowledge only few works are there on fourth-order four-point BVP with monotone iterative technique [12, 25, 31, 43]. By using monotone iterative technique Chen et al. [12] studied the existence of solution of fourth-order four- point BVP with derivative independent non-linear function. The fourth-ordered four-point BVP with derivative dependent non-linear function is also studied in [25, 31, 43]. In this article, we develop MI technique to obtain existence of solution for the four-point non-linear BVP Ly ≡ ( − d(4) ds(4) − λ d (2) ds(2) ) y = F(s, y(s)), 0 < s < 1, B0(y) ≡ (y(0), y′′(0)) = (0, 0), B1(y) ≡ y(1)− δ1y(η1)− δ2y(η2) = 0, B2(y) ≡ y′′(1)− δ1y′′(η1)− δ2y′′(η2) = 0, (1.3) where f ∈ C([0, 1] × R,R), δ1, δ2 > 0, and 0 < η1 ≤ η2 < 1 are constants. The parameter λ = ζ1 + ζ2, where ζ1, ζ2 are any real constants. A function y is said to be a solution of (1.3), if y ∈ C4[0, 1], and satisfies (1.3) for all s ∈ [0, 1]. The above ODE represents the model of the stationary state of the deflection of an elastic beam [15, 10, 23]. To establish the existence of solution we use method of UL solutions and develop MI technique in reverse order case. For this purpose, we obtain Green’s function, solution of corresponding linear problem and its sign. Finding Green’s function and its sign for fourth-order four-point BVP is a very difficult task. To accomplish this, we introduce a linear operator and make this task less difficult. Since the existence of UL solutions ensure the existence of a solution between them. However, even for simple boundary conditions in higher order BVP, the usage of UL solutions is significantly dependent on the sign of the corresponding linear operators [9]. Vrabel [40] considered the following fourth-order two-point BVP y(4)(s) + λy(2)(s) + ζy(s) = F(s, y(s)), 0 < s < 1, (1.4) y(0) = y′′(0) = y(1) = y′′(1) = 0, (1.5) where λ = ζ1 + ζ2 and ζ = ζ1ζ2; ζ1, ζ2 ∈ R such that ζ1 < ζ2 < 0 and F is a continuous and monotone decreasing with respect to y. To establish existence results he formulated method of UL solution. Ma et al. [28] extended this theory for the problem (1.4)-(1.5), where ζ1 < 0 < ζ2 < π2. They established method of UL solution by introducing linear operator. Ma et al. [27] further generalized method of UL solution for fourth-order BVP (1.4)-(1.5), where ζ1, ζ2 ∈ (0, π2). In the above articles Vrabel [40] and Ma et al. [27, 28] have only focused on positivity of Green’s EJDE-2023/51 FOURTH-ORDER FOUR-POINT BVP 3 function. Wang et al. [41] considered BVP (1.4) with the boundary conditions y′(0) = y′′′(0) = y′(1) = y′′′(1) = 0, (1.6) where the non-linear function F(s, y(s)) = βg(s, y(s)), with β > 0. They discussed the existence of positive solutions for more general condition than imposed by Vrabel [40] and Ma et al. [28, 27]. Here, Wang et al. [41] established their result in three gathered, where 0 < ζ1 < ζ2, ζ1 < 0 < ζ2 and ζ1 < ζ2 < 0. Inspired by above articles we study the existence of solution of fourth-order four-point non-linear BVP (1.3). This article generalizes and improves the result of Vrabel [40], Ma et al. [27, 28], and Wang et al. [41] to a class of four-point BVP for all three gathered, when 0 < ζ1 < ζ2, ζ1 < 0 < ζ2 and ζ1 < ζ2 < 0. This article, is divided into 6 sections. In section 2, we develop approximating schemes for corresponding linear BVP and give some important assumptions which we need through out this article. In sections 3, 4, and 5, we study Green’s function, anti-maximum principle and existence theorem for three gathered 0 < ζ1 < ζ2, ζ1 < 0 < ζ2, and ζ1 < ζ2 < 0, respectively. In these sections, we also validate our technique by constructing some numerical illustrations. Section 6, is devoted to conclusions. 2. Preliminaries In this section, we develop iterative scheme and impose conditions which we need through out this article. To study the existence of solution we first linearize BVP (1.3) and consider corresponding nonhomogeneous linear BVP (L− ζ)y(s) = h(s), 0 < s < 1, (2.1) B0(y) = (0, 0), B1(y) = c1, B2(y) = c2, (2.2) where B0, B1, B2 are B.C. defined in (1.3), ζ = ζ1ζ2, h(s) = F(s, y(s)) − ζy(s) ∈ C[0, 1], and c1, c2 are any real constants. Now we define two iterative sequences with initial guesses of l0(s) and u0(s) as follows (L− ζ)ln+1(s) = F(s, ln(s))− ζln(s), 0 < s < 1, n ∈ N, B0(ln+1, l ′′ n+1) = (0, 0), B1(ln+1) = 0, B2(ln+1) = 0. (2.3) (L− ζ)un+1(s) = F(s, un(s))− ζun(s), 0 < s < 1, n ∈ N, B0(un+1, l ′′ n+1) = (0, 0), B1(un+1) = 0, B2(un+1) = 0. (2.4) Assumptions: We assume the following conditions on non-linear term F (A1) Let DF := {(s, y) ∈ [0, 1] × R : u0 ≤ y ≤ l0}, there exists a pair of UL solutions l0(s) and u0(s) such that u0(s) ≤ l0(s). (A2) The function F : DF → R is continuous on DF . (A3) There exists a constant M ≥ 0 in the region DF such that for all (s, yi) ∈ DF , where i = 1, 2, if ζ > 0, y1 ≤ y2 ⇒ F(s, y2)−F(s, y1) ≤M(y2 − y1). (2.5) if ζ < 0, y1 ≤ y2 ⇒ F(s, y2)−F(s, y1) ≤ −M(y2 − y1). (2.6) 4 N. URUS, A. K. VERMA EJDE-2023/51 3. Reverse order MI technique when 0 < ζ1 < ζ2 In this section, first we obtain Green’s function and anti-maximum principle (AMP) for corresponding linear BVP (2.1)-(2.2). Based on AMP we discuss the qualitative properties of linear BVP (2.1)-(2.2) and define UL solutions for non- linear BVP (1.3) when 0 < ζ1 < ζ2. Consequently, we conclude the existence results for the non-linear BVP (1.3). 3.1. Linear BVP. Let E = C[0, 1] be the Banach space of continuous function defined on [0, 1], with its usual normal ‖ · ‖. Denote ζ1 = r2 and ζ2 = m2, (3.1) with some r,m > 0. We substitute the above values of (3.1) in the linear BVP (2.1)- (2.2) and then to obtain Green’s function we consider a corresponding homogeneous linear BVP y(4)(s) + (m2 + r2)y(2)(s) + r2m2y(s) = 0, 0 < s < 1, (3.2) B0(y) = (0, 0), B1(y) = 0, B2(y) = 0. (3.3) We define L∗ : D(L∗)→ E, a linear operator such that L∗y := y(4)(s) + (m2 + r2)y(2)(s) + r2m2y(s), y ∈ D(L∗), (3.4) with domain D(L∗) := {y ∈ C4[0, 1] : B0(y) = (0, 0), B1(y) = 0, B2(y) = 0}. (3.5) To construct G(s, x) for the BVP (3.2)-(3.3), let us define two linear operators L1 and L2 as follows L1y := y′′(t) + r2y(t), y ∈ D(L1), (3.6) L2y := y′′(t) +m2y(t), y ∈ D(L2), (3.7) where D(L1) := {y ∈ C2[0, 1] : y(0) = 0, B1(y) = 0}, (3.8) D(L2) := {y ∈ C2[0, 1] : y(0) = 0 B1(y) = 0}. (3.9) Lemma 3.1. Assume that r,m ∈ (0, π/2) and that the following assumption is fulfilled Dl = δ1 sin(η1l) + δ2 sin(η2l)− sin(l) 6= 0, where l = m or r. (3.10) Then G(s, x) : [0, 1]× [0, 1]→ R of linear fourth-order BVP (3.2)-(3.3) is given by G(s, x) = ∫ 1 0 Gm(s, t)Gr(t, x)dt, (s, x) ∈ [0, 1]× [0, 1], (3.11) EJDE-2023/51 FOURTH-ORDER FOUR-POINT BVP 5 where Gr(t, x) and Gm(t, x) are Green’s function of (3.6)-(3.8) and (3.7)-(3.9), respectively, given as Gl(t, x) = 1 lDl  sin(lt)(δ1 sin(l(x− η1)) + δ2 sin(l(x− η2)) + sin(l − lx)) if 0 ≤ t ≤ x ≤ η1; sin(lx)(δ1 sin(l(t− η1)) + δ2 sin(l(t− η2)) + sin l(1− t)) if 0 ≤ x ≤ t ≤ η1; sin(lt)(δ2 sin(l(x− η2)) + sin l(1− x)) if η1 ≤ t ≤ x ≤ η2; −δ1 sin(η1l) sin(l(x− t)) + sin(lx)(δ2 sin(l(t− η2)) + sin l(1− t)) if η1 ≤ x ≤ t ≤ η2; sin l(1− x) sin(lt) if η2 ≤ t ≤ x ≤ 1; sin(lt)(sin l(1− x) +Dl cos(lx))−Dl sin(lx) cos(lt) if η2 ≤ x ≤ t ≤ 1. (3.12) Proof. Since L∗ is a linear operator, it can be easily proved that L∗y = L2(L1)y and hence the Green’s function of (3.2)-(3.3) is G(s, x) = ∫ 1 0 Gm(s, t)Gr(t, x)dt, (s, x) ∈ [0, 1]× [0, 1]. (3.13) For brevity, we skip the proof of (3.13). For details of the proof we refer the articles of Ma et al. [27] and to Wang et al. [41]. In equation (3.13), Gr(t, x) and Gm(t, x) are Green’s functions of (3.6)-(3.8) and (3.7)-(3.9), respectively, given as Gl(t, x) =  a1 sin(lt) + a2 cos(lt), 0 ≤ t ≤ x ≤ η1, a3 sin(lt) + a4 cos(lt), 0 ≤ x ≤ t ≤ η1, a5 sin(lt) + a6 cos(lt), η1 ≤ t ≤ x ≤ η2, a7 sin(lt) + a8 cos(lt), η1 ≤ x ≤ t ≤ η2, a9 sin(lt) + a10 cos(lt), η2 ≤ t ≤ x ≤ 1, a11 sin(lt) + a12 cos(lt), η2 ≤ x ≤ t ≤ 1, (3.14) where a1 = a5 = a9 = 0, a3 = a7 = a11 = − sin(lx) l , a2 = 1 lDl (δ1 sin(l(x− η1)) + δ2 sin(l(x− η2)) + sin l(1− x)), a4 = −1 lDl (sin(lx)(−δ1 cos(η1l)− δ2 cos(η2l) + cos(l))), a6 = 1 lDl (δ2 sin(l(x− η2)) + sin l(1− x)), a8 = 1 lDl (δ1 sin(η1l) cos(lx) + sin(lx)(δ2 cos(η2l)− cos(l))), a10 = 1 lDl (sin l(1− x)), 6 N. URUS, A. K. VERMA EJDE-2023/51 a12 = 1 lDl (sin l(1− x) +Dl cos(lx)). Substituting the above values of ai, i = 1, 2, . . . , 12 in (3.14), we obtain the Green’s function Gl(t, x) of (3.6) and (3.8) and (3.7) and (3.9) that is given in equation (3.12). � Lemma 3.2. Assume that r,m ∈ (0, π2 ). Then G(s, x) of fourth-order BVP (3.2)- (3.3) given by expression (3.11) is nonnegative on [0, 1] × [0, 1] if and only if 0 < δ1 + δ2 < 1. Proof. From equation (3.11) we observe that sign of G(s, x) of (3.2)-(3.3) follows from sign of Green’s function Gm(s, t) and Gr(t, x) given by equation (3.12). Given that δ1 + δ2 < 1, applying properties of sinx, we have −(δ1 + δ2) sin l(x− η2) ≤ sin l(1− x), sin l(1− x) + δ2 sin l(x− η2) ≥ 0, and Dl < 0. Using the above inequalities in (3.12), we obtain that Gl(t, x) ≤ 0, (t, x) ∈ [0, 1]× [0, 1]. Hence from (3.11), we have G(s, x) ≥ 0 on [0, 1]× [0, 1]. In a similar fashion we can prove that the converse is also true. � Lemma 3.3. Assume that r,m ∈ (0, π/2) such that m2−r2 6= 0 and Dl 6= 0. Then the solution y ∈ C4[0, 1] of linear fourth-order non homogeneous BVP (2.1)-(2.2) is y(s) = Ny(s) (m2 − r2)DmDr − ∫ 1 0 G(s, x)h(x)dx, (3.15) where Ny(s) = Dr(c2 + c1r 2) sin(ms) − Dm(c2 + c1m 2) sin(rs) and the Green’s function G : [0, 1]× [0, 1]→ R is given by equation (3.13). Proof. Let ỹ1(s) ∈ C2[0, 1] be the solution of BVP (2.1)-(2.2), where h(s) = 0 and c1, c2 6= 0. Hence, we have ỹ1(s) = Ny(s) (m2 − r2)DmDr , (3.16) where m2 − r2, Dl 6= 0. Again, let ỹ2(s) ∈ C2[0, 1] be the solution of BVP (2.1)- (2.2), where h(s) 6= 0 and c1 = c2 = 0. Hence, we have ỹ2(s) = − ∫ 1 0 G(s, x)h(x)dx. (3.17) Now, the solution y ∈ C4[0, 1] of linear fourth-order non homogeneous BVP (2.1)- (2.2) can be written as y(s) = ỹ1(s) + ỹ2(s). Substituting the values of ỹ1(s) and ỹ2(s) in the above expression we obtain the final result. � EJDE-2023/51 FOURTH-ORDER FOUR-POINT BVP 7 Anti-maximum principle. Proposition 3.4. Let r,m ∈ (0, π2 ) such that m2 − r2 > 0, Dl 6= 0, and 0 < δ1 + δ2 < 1. Further let h(s) ≥ 0 on s ∈ [0, 1], and the constants c1, c2 ≤ 0, then the solution y ∈ C4[0, 1] of linear fourth-order non homogeneous BVP (2.1)-(2.2) given by (3.15) is nonpositive for all s ∈ [0, 1]. Proof. Given that h(s) ≥ 0, from Lemma 3.2, we have G(s, x) ≥ 0. Now to prove that y(s) ≤ 0, it remains to prove that ỹ1(s) ≤ 0, defined by equation (3.16). Clearly,m2 − r2 > 0 and Dm, Dr < 0, if 0 < δ1 + δ2 < 1, hence it is sufficient to prove that Ny(s) ≤ 0. Using properties of sin s, we deduce that Ny(s) ≤ sin(rs)(c2(Dr −Dm) + c1(r2Dr −m2Dm)). Applying r < m, we obtain (Dr−Dm) ≥ 0, r2Dr−m2Dm ≥ 0, and since c1, c2 ≤ 0, we obtain the desired result, i.e., y(s) ≤ 0. � 3.2. Non-linear BVP. In this subsection, we establish MI technique in RO case to solve the four-point fourth-order non-linear BVP (1.3). To do so, we first introduce the concepts of UL solutions u(s) and l(s), respectively for the BVP (1.3) such that u(s) ≤ l(s). Definition 3.5. A function l(s) ∈ C4[0, 1] is called lower solution of fourth-order non-linear BVP (1.3), if it satisfies Ll(s) ≤ F(s, l(s)), 0 < s < 1, B0(l) = (0, 0), B1(l) ≥ 0, B2(l) ≥ 0. (3.18) Definition 3.6. A function u(s) ∈ C4[0, 1] is called upper solution of fourth-order non-linear BVP (1.3), if it satisfies Lu(s) ≥ F(s, u(s)), 0 < s < 1, B0(u) = (0, 0), B1(u) ≤ 0, B2(u) ≤ 0. (3.19) Theorem 3.7. Assume 0 < ζ1 < ζ2 and 0 < δ1 + δ2 < 1. Let l0(s) and u0(s) ∈ C4[0, 1] exist such that u0(s) ≤ l0(s) satisfy (3.18) and (3.19), respectively. If the non-linear function F is such that it satisfies (A1)–(A3), then (2.1)-(2.2) has at least one solution in the region DF . Further, if there exist a constant ζ > 0 such that ζ −M1 ≥ 0, then the sequences un(s) generated by (2.4), with initial iterate u0(s) converge monotonically non-decreasing and uniformly towards a solution w2(s) of fourth-order BVP (2.1)-(2.2). Similarly, using l0(s) as an initial iterates leads to a non increasing sequence ln(s) generated by (2.3) converging monotonically decreasing and uniformly towards a solution w1(s) of fourth-order BVP (2.1)-(2.2). Every solution y(s) in DF must satisfy w2(s) ≤ y(s) ≤ w1(s). Proof. We prove this theorem in three steps by using the principle of mathematical induction. Step 1: For n = 0, we have that u(s) = u0(s) satisfies inequality (3.19) and from equation (2.4), we have (L− ζ)(u0 − u1)(s) ≥ 0, B0(u0 − u1) = (0, 0), B1(u0 − u1) ≤ 0, B2(u0 − u1) ≤ 0. 8 N. URUS, A. K. VERMA EJDE-2023/51 From the anti-maximum principle 3.4, we obtain u0 ≤ u1. Step 2: Let us assume that un ≤ un+1. Since F(s, y) satisfies (2.5), we have F(s, un+1)−F(s, un) ≤M(un+1 − un). Now from (2.4), in view of ζ −M ≥ 0, we arrive at Lun+1(s) ≥ (ζ −M)(un+1 − un) + F(s, un+1) ≥ F(s, un+1), B0(un+1 − un) = (0, 0), B1(un+1 − un) = 0, B2(un+1 − un) = 0. (3.20) Using (3.20) with n = 0, in (2.4) for n = 1, we obtain (L− ζ)(u1 − u2)(s) ≥ 0, B0(u1 − u2) = (0, 0), B1(u1 − u2) = 0, B2(u1 − u2) = 0. From anti-maximum principle 3.4, we obtain u1 ≤ u2. Step 3: To prove u1 ≤ lo, we use (3.18) and (2.4) for n = 0. Also, in view of u0 ≤ l0 using inequality (2.5) we have (L− ζ)(u1 − l0)(s) ≥ 0, B0(u1 − l0) = (0, 0), B1(u1 − l0) ≤ 0, B2(u1 − l0) ≤ 0. Hence, u1 ≤ l0. Step 4: Now by assuming un+1 ≥ un and un+1 ≤ l0, we show that un+2 ≥ un+1 and un+2 ≤ l0. Using inequality (3.20) in (2.4) for n = n+1 we can easily prove that un+2 ≥ un+1. Now in view of ζ −M ≥ 0, applying inequality (2.5) for un+1 ≤ l0, we have F(s, l0)− ζl0 ≤ F(s, un+1)− ζun+1. (3.21) Using (3.21) in equation (2.4) for n = n+ 1, we obtain (L− ζ)(un+2 − l0)(s) ≥ 0, B0(un+2 − l0) = (0, 0), B1(un+2 − l0) ≤ 0, B2(un+2 − l0) ≤ 0. From the anti-maximum principle 3.4, we obtain un+2 ≤ l0. Hence u(s) = u0 ≤ u1 ≤ · · · ≤ un ≤ un+1 ≤ · · · ≤ l0, (3.22) Step 5: Similarly, we deduce that u0 ≤ · · · ≤ ln+1 ≤ ln ≤ · · · ≤ l1 ≤ l0 = l(s). (3.23) Step 6: Finally, by assuming that un ≤ ln we show that un+1 ≤ ln+1. Subtracting equation (2.3) and (2.4) and applying Lipschitz condition we obtain, (L− ζ)(un+1 − ln+1)(s) ≥ (ζ −M)(un − ln)(s) ≥ 0, B0(un+1 − ln+1) = (0, 0), B1(un+1 − ln+1) = 0, B2(un+1 − ln+1) = 0. Hence, un+1 ≤ ln+1. Thus we arrive at, the sequences ln and un such that u0 ≤ u1 ≤ · · · ≤ un ≤ un+1 ≤ · · · ≤ ln+1 ≤ ln ≤ · · · ≤ l1 ≤ l0. (3.24) EJDE-2023/51 FOURTH-ORDER FOUR-POINT BVP 9 Using Dini’s theorem we prove that the sequences of UL solutions are uniformly convergent. Let w1(s) = lim n→∞ ln(s), and w2(s) = lim n→∞ un(s). (3.25) Taking limit as n→∞ on both sides of solution of (2.3), we obtain lim n→∞ ln(s) = lim n→∞ ( ỹ1(s)− ∫ 1 0 G(s, x)(F(x, ln(x))− ζln(x))dx ) . Then w1(s) = ỹ1(s)− ∫ 1 0 G(s, x)(F(x, l(x))− ζl(x))dx. Similarly, we deduce that w2(s) = ỹ1(s)− ∫ 1 0 G(s, x)(F(x, u(x))− ζu(x))dx, where ỹ1(s) is given by (3.16). These are the solutions of fourth-order linear BVP (2.1)-(2.2). Any solution y(s) in DF can play the role of l0(s) and u0(s), hence we obtain w2(s) ≤ y(s) ≤ w1(s). � Theorem 3.8 (Uniqueness). Let ζ > 0. Suppose that F(s, y) satisfies conditions (A1), (A2) and there is a constant 0 < M3 < π2/4 such that F(s, y1)−F(s, y2) ≥M3(y1 − y2). (3.26) Then the non-linear BVP (1.3) has unique solution. Proof. Let y = y1 − y2. Then y satisfies L(y1 − y2)−M3(y1 − y2) ≥ 0. Applying anti-maximum principle 3.4, we obtain y1 ≤ y2. Similarly we can prove y1 ≥ y2 by taking y = y2−y1. Hence we have the required result as y2 = y1. Hence we obtain unique solution y of non-linear BVP (1.3). � 3.3. Numerical illustrations. Example 3.9 (Reverse order case). Consider the four-point non-linear BVP −y(4)(s)− λy(2)(s)− ζy(s) = es − 1 15 y3 + sin(2s), y(0) = 0, y(1) = 0.5y(0.2) + 0.4y(0.5), y′′(0) = 0, y′′(1) = 0.5y′′(0.2) + 0.4y′′(0.5), (3.27) where λ = ζ1 + ζ2 = 3/2. We define initial lower solution l0(s) = s 2 (1 + 1 3 s3) and initial upper solution u0(s) = −s(2 + 1 2 s2) such that u0(s) ≤ l0(s). With the help of (P3) we obtain Lipschitz constant M = 0.23865. Now using ζ−M ≥ 0 we obtain, ζ ≥ 0.23865, the range for the convergence 10 N. URUS, A. K. VERMA EJDE-2023/51 of iterative sequences of UL solution of non-linear BVP (3.27). In Figure 1, we can see that for ζ = 1/2, where ζ1 = 1/2, ζ2 = 1 such that ζ1 + ζ2 = 3/2, the sequences of UL solution ln(s) and un(s), n = 0, 1, 2, are monotonically converging to the solution of non-linear BVP (3.27) for suitable choices of ζ1, ζ2. l1(s) l2(s) u1(s) u2(s) l0(s) u0(s) 0.2 0.4 0.6 0.8 1.0 s -2.5 -2.0 -1.5 -1.0 -0.5 0.5 (ln,un) Figure 1. Plots of sequences of UL solution ζ = 1/2, n = 3. 4. Reverse order MI technique when ζ1 < 0 < ζ2 In this section, we construct solution of BVP (2.1)-(2.2) and AMP. Further, we establish MI technique in reverse order case to solve the four-point fourth-order non-linear BVP (1.3). Let ζ1 = −r2 and ζ2 = m2, (4.1) with some r,m > 0. We put the above values of (4.1) in the linear BVP (2.1)-(2.2) and then to obtain Green’s function let us consider a corresponding homogeneous linear BVP y(4)(s) + (m2 − r2)y(2)(s)− r2m2y(s) = 0, 0 < s < 1, B0(y) = (0, 0), B1(y) = 0, B2(y) = 0. (4.2) Define L∗ : D(L∗)→ E, such that L∗y := y(4)(s) + (m2 − r2)y(2)(s)− r2m2y(s), y ∈ D(L), (4.3) with domain D(L∗) := {y ∈ C4[0, 1] : B0(y) = (0, 0), B1(y) = 0, B2(y) = 0}. (4.4) To construct G(s, x) for the BVP (4.2) let us first define L1y := y′′(t)− r2y(t), y ∈ D(L1), (4.5) L2y := y′′(t) +m2y(t), y ∈ D(L2), (4.6) where L1, L2 are linear operators and D(L1) := {y ∈ C2[0, 1] : y(0) = 0, B1(y) = 0}, (4.7) D(L2) := {y ∈ C2[0, 1] : y(0) = 0, B1(y) = 0}. (4.8) EJDE-2023/51 FOURTH-ORDER FOUR-POINT BVP 11 Lemma 4.1. Assume that m ∈ (0, π/2) and r ∈ (0,∞). Also assume that D′r = δ1 sinh(η1r) + δ2 sinh(η2r)− sinh(r) 6= 0, and Dm 6= 0, (4.9) where Dm is given by (3.10). Then the G(s, x) : [0, 1]× [0, 1]→ R of linear fourth- order BVP (4.2) is expressed as G(s, x) = ∫ 1 0 Gm(s, t)G′r(t, x)dt, (s, x) ∈ [0, 1]× [0, 1], (4.10) where G′r(t, x) and Gm(t, x) are Green’s functions of (4.5), (4.7) and (4.6), (4.8), respectively. The Green’s function Gm(t, x) is given by (3.12) and G′r(t, x) is G′r(t, x) = 1 rD′r  sinh(rt)(δ1 sinh(r(x− η1)) + δ2 sinh(r(x− η2)) + sinh r(1− x)) quadif 0 ≤ t ≤ x ≤ η1 sinh(rx)(δ1 sinh(r(t− η1)) + δ2 sinh(r(t− η2)) + sinh r(1− t)) if 0 ≤ x ≤ t ≤ η1, sinh(rt)(δ2 sinh(r(x− η2)) + sinh r(1− x)) if η1 ≤ t ≤ x ≤ η2, −δ1 sinh(η1r) sinh(r(x− t)) + sinh(rx)(δ2 sinh(r(t− η2)) + sinh r(1− t)) if η1 ≤ x ≤ t ≤ η2, sinh r(1− x) sinh(rt) if η2 ≤ t ≤ x ≤ 1, sinh r(1− x) sinh(rt)−Dr sinh(r(x− t)) if η2 ≤ x ≤ t ≤ 1. (4.11) For a proof of the above lemma, see the proof of lemma 3.1. Lemma 4.2. Assume that m ∈ (0, π2 ) and r ∈ (0,∞). Then G(s, x) of BVP (4.2) given by expression (4.10) is nonnegative on [0, 1]× [0, 1] if and only if 0 < δ1+δ2 < 1. The proof of the above lemma is similar to the proof of lemma 3.2. We omit it. Lemma 4.3. Assume that r ∈ (0,∞) and m ∈ (0, π2 ) such that DmD ′ r 6= 0. Then the solution y ∈ C4[0, 1] of the linear fourth-order non homogeneous BVP (2.1)- (2.2) is y(s) = −1 (m2 + r2) [ sin(ms)(c1r 2 − c2) Dm + sinh(rs)(c1m 2 + c2) D′r ] − ∫ 1 0 G(s, x)h(x)dx, (4.12) where the G(s, x) is given by (4.10). The above lemma follows from Lemma 3.3. Proposition 4.4 (Anti-maximum principle). Let r ∈ (0,∞) and m ∈ (0, π2 ) such that DmD ′ r 6= 0. Further, assume that h(s) ≥ 0 and the c1 ≤ 0, c2 ≥ 0, and δ1+δ2 < 1. Then the solution y ∈ C4[0, 1] of the linear fourth-order non homogeneous BVP (2.1)-(2.2) given by (4.12) is nonpositive for all s ∈ [0, 1]. Proof. Since δ1 + δ2 < 1, we obtain that Dm and D′r < 0. Also since c1 ≤ 0 and c2 ≥ 0, the result can be concluded easily. � In this case the UL solutions are defined as follows. 12 N. URUS, A. K. VERMA EJDE-2023/51 Definition 4.5 (Lower solution). A function l(s) is known as lower solution of fourth-order non-linear BVP (1.3), if l(s) ∈ C4[0, 1] and it satisfies the following conditions Ll(s) ≤ F(s, l(s)), 0 < s < 1, B0(l) = (0, 0), B1(l) ≥ 0, B2(l) ≤ 0. (4.13) Definition 4.6 (Upper solution). A function u(s) is known as upper solution of the fourth-order non-linear BVP (1.3), if u(s) ∈ C4[0, 1] and it satisfies the following conditions Lu(s) ≥ F(s, u(s)), 0 < s < 1, B0(u) = (0, 0), B1(u) ≤ 0, B2(u) ≥ 0. (4.14) Theorem 4.7. Assume ζ1 < 0 < ζ2 and δ1 + δ2 < 1. Also assume there exist l0(s) and u0(s) ∈ C4[0, 1] such that u0(s) ≤ l0(s) satisfying (4.13) and (4.14), respectively. If the non-linear function F is such that it satisfies (A1)–(A3), then (2.1)-(2.2) has at least one solution in the region DF . Further, if there exists a constant ζ > 0 such that ζ+M ≥ 0, then the monotonically non decreasing sequence un(s) generated by (2.4), with initial iterate u0(s) converges uniformly towards a solution w2(s) of the fourth-order BVP (2.1)-(2.2). Similarly, using l0(s) as an initial iterate leads to a monotonically non-increasing sequence ln(s) generated by (2.3) converging uniformly towards a solution w1(s) of fourth-order BVP (2.1)-(2.2). Any solution y(s) in DF must satisfy w2(s) ≤ y(s) ≤ w1(s). The proof of the above theorem follows from the proof of Theorem 3.7. Theorem 4.8 (Uniqueness). Let ζ < 0. Suppose that F(s, y) satisfies (A1), (A2) and that there is a constant M3 > 0 such that F(s, y1)−F(s, y2) ≥ −M3(y1 − y2), (4.15) then the non-linear BVP (1.3) has unique solution. The proof of the above theorem follows from proof of theorem 3.8. 4.1. Numerical illustrations. Example 4.9 (Reverse order case). Consider the four-point non-linear BVP −y(4)(s)− λy(2)(s)− ζy(s) = −11e+ 25e−y, y(0) = 0, y(1) = 0.4y(0.7) + 0.4y(0.8), y′′(0) = 0, y′′(1) = 0.4y′′(0.7) + 0.4y′′(0.8), (4.16) where λ = ζ1 + ζ2 = −1/2. We define initial lower solution l0(s) = s(4− 3s2 + s3) and initial upper solution u0(s) = s4 40 (s−2) such that u0(s) ≤ l0(s). With the help of (A3) we obtain Lipschitz constants M = 3.126. Now using ζ + M ≥ 0 we obtain, ζ ≥ −3.126, the range for the convergence of iterative sequences of UL solution of non-linear BVP (4.16). We can see in figure 2, for ζ = −1/2, where ζ1 = −1, ζ2 = 1/2 such that ζ1 +ζ2 = −1/2 and n = 6 the sequences of UL solution monotonically converge to the solution for suitable choices of ζ1 and ζ2. EJDE-2023/51 FOURTH-ORDER FOUR-POINT BVP 13 l1(s) l2(s) l3(s) l4(s) l5(s) u1(s) u2(s) u3(s) u4(s) u5(s) l0(s) u0(s) 0.2 0.4 0.6 0.8 1.0 0.5 1.0 1.5 2.0 (ln,un) Figure 2. Plots of sequences of UL solution ζ = −1/2, n = 6. 5. Reverse order MI technique when ζ1 < ζ2 < 0 Let ζ1 = −r2 and ζ2 = −m2, (5.1) with some r,m > 0. Let us consider a corresponding fourth-order four-point linear BVP with non homogeneous boundary conditions y(4)(s)− (m2 + r2)y(2)(s) + r2m2y(s) = 0, 0 < s < 1, B0(y) = (0, 0), B1(y) = 0, B2(y) = 0. (5.2) Define L∗ : D(L∗)→ E, L∗y := y(4)(s)− (m2 + r2)y(2)(s) + r2m2y(s), y ∈ D(L), (5.3) with domain D(L∗) := {y ∈ C4[0, 1] : B0(y) = (0, 0), B1(y) = 0, B2(y) = 0}. (5.4) To construct G(s, t) for the BVP (5.2) let us first define L1y := y′′(t)− r2y(t), y ∈ D(L1), (5.5) L2y := y′′(t)−m2y(t), y ∈ D(L2), (5.6) where D(L1) := {y ∈ C2[0, 1] : y(0) = 0, B1(y) = 0}, (5.7) D(L2) := {y ∈ C2[0, 1] : y(0) = 0, B1(y) = 0}. (5.8) Lemma 5.1. Assume that m, r ∈ (0,∞). Also assume that D′l = δ1 sinh(η1r) + δ2 sinh(η2r)− sinh(r) 6= 0, l = m or r. (5.9) Then the G(s, x) : [0, 1] × [0, 1] → R of linear fourth-order BVP (5.2) is expressed as G(s, x) = ∫ 1 0 G′m(s, t)G′r(t, x)dt, (s, x) ∈ [0, 1]× [0, 1], (5.10) where G′l(t, x), l = m/r, is Green’s function of (5.5) and (5.6) with boundary con- dition (5.7) and (5.8) is given by (4.11), where r = m. 14 N. URUS, A. K. VERMA EJDE-2023/51 Lemma 5.2. Assume that r,m ∈ (0,∞). Then G(s, x) of BVP (5.2) given by (5.10) is nonnegative on [0, 1]× [0, 1] if and only if 0 < δ1 + δ2 < 1. Lemma 5.3. Assume that m, r ∈ (0,∞) such that DmD ′ r(m 2 − r2) 6= 0. Then the solution y ∈ C4([0, 1]) of linear fourth-order non homogeneous BVP (2.1)-(2.2) is y(s) = 1 (m2 − r2) [ sinh(ms)(c1r 2 − c2) D′m + sinh(rs)(c2 − c1m2) D′r ] − ∫ 1 0 G(s, x)h(x)dx, (5.11) where G(s, x) is given by (5.10). Remark 5.4. Since m2 < r2 we have (m2 − r2) < 0. Also since we have D′l < 0, for both l = m and r, we obtain that D′r < D′m. Anti-maximum principle. Proposition 5.5. Let m, r ∈ (0,∞) such that (m2−r2) < 0. Further h(s) ≥ 0 and the constants c1, c2 ≥ 0 and 0 < δ1 + δ2 < 1 then the solution y ∈ C4[0, 1] of linear fourth-order non homogeneous BVP (2.1)-(2.2) given by (5.11) is non-negative for all s ∈ [0, 1]. Proof. Since δ1 + δ2 < 1 and m2 < r2 we deduce that (m2 − r2) < 0, D′l < 0 and D′r < D′m. Using properties of sinh s and c1, c2 ≥ 0 we obtain the desired result. � In this case lower solution l(s) and upper solution u(s) such that u(s) ≤ l(s) are defined as follows Definition 5.6 (Lower solution). A function l(s) is known as lower solution of fourth-order non-linear BVP (1.3), if l(s) ∈ C4[0, 1] and it satisfies the following conditions Ll(s) ≤ F(s, l(s)), 0 < s < 1, B0(l) = (0, 0), B1(l) ≤ 0, B2(l) ≤ 0. (5.12) Definition 5.7 (Upper solution). A function u(s) is known as an upper solution of the fourth-order non-linear BVP (1.3), if u(s) ∈ C4[0, 1] and it satisfies the following conditions Lu(s) ≥ F(s, u(s)), 0 < s < 1, B0(u) = (0, 0), B1(u) ≥ 0, B2(u) ≥ 0. (5.13) Theorem 5.8 (Reverse order). Assume ζ1 < ζ2 < 0 and δ1 + δ2 < 1. Let l0(s) and u0(s) ∈ C4[0, 1] exist such that u0(s) ≤ l0(s) satisfying (5.12) and (5.13), respectively. If the non-linear function F is such that it satisfies (A1)–(A3), then (2.1)-(2.2) has at least one solution in the region DF . Further, if there exists a constant ζ > 0 such that ζ − M ≥ 0. Then the monotonically non decreasing sequence un(s) generated by (2.4), with initial iterate u0(s) converges uniformly towards a solution w2(s) of the fourth-order BVP (2.1)-(2.2). Similarly, using l0(s) as an initial iterates leads to a non increasing sequence ln(s) generated by (2.3) converging monotonically decreasing and uniformly towards a solution w1(s) of the fourth-order BVP (2.1)-(2.2). Every solution y(s) in DF must satisfy w2(s) ≤ y(s) ≤ w1(s). EJDE-2023/51 FOURTH-ORDER FOUR-POINT BVP 15 The proof of the above theorem follows from the proof of Theorem 3.7. Theorem 5.9 (Uniqueness). Let ζ > 0. Suppose that F(s, y) satisfies (A1), (A2) and that there is a constant M3 > 0 such that F(s, y1)−F(s, y2) ≥M3(y1 − y2). (5.14) Then the non-linear BVP (1.3) has unique solution. The proof of the above theorem follows from the proof of Theorem 3.8. 5.1. Numerical illustrations. Example 5.10 (Reverse order case). Consider the four-point non-linear BVP −y(4)(s)− λy(2)(s)− ζy(s) = (e− 1) 50 y2 + 1 25 sin(s), y(0) = 0, y(1) = 0.3y(0.2) + 0.6y(0.5), y′′(0) = 0, y′′(1) = 0.3y′′(0.2) + 0.6y′′(0.5), (5.15) where λ = −3. We define initial lower solution l0(s) = 5s 4 −s 3 and initial upper solution u0(s) = − 5s 4 + s3 such that u0(s) ≤ l0(s). With the help of (P3) we obtain Lipschitz constant M = 0.0369. Now using ζ −M ≥ 0 we obtain, ζ ≥ 0.0369, the range for the convergence of iterative sequences of UL solution of the non-linear BVP (5.15). Hence, we can see in figure 3, for ζ = 2 where ζ1 = −2, ζ2 = −1 such that ζ1 + ζ2 = −3 and n = 3, that the sequences of UL solution converges monotonically to the solution of NLBVP (5.15) for suitable choices of ζ1 and ζ2. l1(s) l2(s) l3(s) u1(s) u2(s) u3(s) l0(s) u0(s) 0.2 0.4 0.6 0.8 1.0 s -0.4 -0.2 0.2 0.4 0.6 (ln,un) Figure 3. Plots of sequences of UL solution ζ = 2, n = 3 6. Conclusion In this article, we developed an MI technique in the reverse order case to establish existence of a unique solution. We have studied existence results in three gathered when 0 < ζ1 < ζ2, ζ1 < 0 < ζ2, and ζ1 < ζ2 < 0. We need to assume one sided Lipschitz condition on the non-linear function F to construct monotone sequences of UL solutions. Based on anti-maximum principle, we observe that for all the three gathered we need to define an appropriate form of UL solutions which paves the way for the establishment of MI technique. To validate our results we have constructed examples in each case. 16 N. 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Vrabel; On the lower and upper solutions method for the problem of elastic beam with hinged ends, Journal of mathematical analysis and applications 421 (2015), no. 2, 1455–1468. [41] J. Wang, C. Gao, Y. Lu; Global structure of positive solutions for semipositone nonlinear euler-bernoulli beam equation with neumann boundary conditions, Quaestiones Mathematicae (2022), 1–29. [42] S. Weng, H. Gao, D. Jiang, X. Hou; Upper and lower solutions method for fourth-order periodic boundary value problems, Journal of Applied Analysis 14 (2008), no. 1, 53–61. [43] Q. Zhang, S. Chen, J. Lü; Upper and lower solution method for fourth-order four-point boundary value problems, Journal of Computational and Applied Mathematics 196 (2006), no. 2, 387–393. Nazia Urus Department of Mathematics, IIT Patna, India Email address: nazia.pma17@iitp.ac.in Amit K. Verma Department of Mathematics, IIT Patna, India Email address: akverma@iitp.ac.in 1. Introduction 2. Preliminaries 3. Reverse order MI technique when 0<1<2 3.1. Linear BVP Anti-maximum principle 3.2. Non-linear BVP 3.3. Numerical illustrations 4. Reverse order MI technique when 1<0<2 4.1. Numerical illustrations 5. Reverse order MI technique when 1<2<0 Anti-maximum principle 5.1. Numerical illustrations 6. Conclusion Acknowledgements References