Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 132, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE AND STABILITY FOR FRACTIONAL ORDER PANTOGRAPH EQUATIONS WITH NONLOCAL CONDITIONS ISRAR AHMAD, JUAN JOSE NIETO, GHAUS UR RAHMAN, KAMAL SHAH Abstract. In this article we study the a coupled system of fractional panto- graph differential equations (FPDEs). Using degree theory, we state necessary conditions for the existence of solutions to a coupled system of fractional par- tial differential equations with non-local boundary conditions. Also using tools from non-linear analysis, we establish some stability results. We illustrate our theoretical results with a test problem. 1. Introduction Fractional calculus is gaining prominence in mathematical modeling. For ex- ample in the field of nanotechnology, it has received a lot of attention by re- searchers. This emerging field of mathematics has been used in the modeling of real word problems as well in applications in biological sciences, numerical and physical sciences, dynamical systems, economics and so forth; see for example [9, 12, 13, 15, 16] and the references therein. Looking at the advantage of this field over other fields of mathematics, experts have explored applications in areas including qualitative theory, stability theory, optimization and numerical simula- tions; see [1, 2, 7, 18, 20, 26, 28]. Fractional ordinary ordinary differential equations (FODEs) have been studied in detail including existence theory, stability analysis, and numerical solutions. Similarly, various classes, including impulsive, implicit type FODEs have been con- sidered. On the other hand, an important class known as pantograph differential equations with fractional order derivatives have not been not properly investigated. Ordinary derivatives are local by nature and cannot be used to model some prop- erties like memory and hereditary etc. Therefore, in some cases, FODEs provide an adequate way to model some natural systems. The underlying theory of fractional differential equations has also applied as- pects. The applied aspects in this area include: probability theory, physics, control theory, light absorption and division of cells phenomena [4, 5]. Among the attrac- tive features of fractional differential equations, stability theory is an important and comprehensive aspects of fractional differential equations (FDEs). Stability of FODEs and systems has been considered in several works in last two decades; see for example [3, 19, 24, 25, 27, 29]. Recently, some authors have 2010 Mathematics Subject Classification. 26A33, 34A08, 35R11. Key words and phrases. Coupled system; non-local boundary conditions; stability theory; pantograph equation. c©2020 Texas State University. Submitted December 11, 2019. Published December 26, 2020. 1 2 I. AHMAD, J. J. NIETO, G. U. RAHMAN, K. SHAH EJDE-2020/132 investigated the fractional pantograph differential equation CDγ 0+x(t)− c(t)x(t) = m∑ σ=1 aσ(t)Dγσx(rσt), n− 1 < γ ≤ n, t ∈ [0, S1], x(j−1)(0) = xj−1, j = 1, . . . , n. (1.1) Here rσ ∈ (0, 1) = I1, 0 ≤ γσ < σ ≤ n, while c(t), aσ(t) (σ = 1, 2, . . . ,m), are known functions and D represents Caputo derivative of order γ > 0. In [17] the authors developed numerical solutions based on Bernoulli wavelets. Vignesh et al [23] studied existence theory for the fractional pantograph differential equation Dγx(t) = P1(t, x(t), x(rt)), t ∈ [0, S], γ, r ∈ I1, x(0) = x0. (1.2) Here C[0, S] is the usual Banach-Space of continuous functions on [0, S] and P1 is a non-linear continuous function from [0, S] × R × R into R. Where D represents Riemann-Liouville derivative. Further, the authors in [14] studied stability aspects for the FPDE HDγ,β;ψ 0+ x(t)− P2(t, x(t), x(P3(t))) = 0, t ∈ (0, d] = I2, I1−α;ψ0+ x(0+) = x0 ∈ R, x(t) = φ(t), −h ≤ t ≤ 0. (1.3) Here HDγ,β;ψ 0+ (·) is the ψ-Hilfer of type 0 < γ, β, γ ≤ 1 and I1−α;ψ0+ x(·) is the classical Riemann-Liouville integral with α = γ + β(1 − γ). ψ is a non-linear continuous function and P2 : I2×R2 → R. Afterwards, the authors in [8] formulated a coupled system of fractional order differential equation with ψ Hilfer derivative and explored existence results. To obtain new results for FPDEs with non-local boundary conditions using de- gree theory one needs further explorations. Existence, uniqueness and stability results with coupled system of FPDEs using degree theory has not been considered as far we know. Therefore, we study the coupled system of FPDEs with non-local boundary conditions, CDγ1x1(t) = f1(t, x1(t), x1(rt), x2(t)), CDγ2x2(t) = f2(t, x1(t), x2(t), x2(rt)), g1(x1) = λ1x1(0)− µ1x1(η)− δ1x1(1), g2(x2) = λ2x2(0)− µ2x2(ζ)− δ2x2(1)). (1.4) Here t ∈ I = [0, 1], r ∈ I1, γ1, γ2 ∈ (0, 1], f1, f2 ∈ C(I × R3,R) are non-linear and g1, g2 : C(I,R) → R are given maps. Here, ζ, η ∈ I1, λj 6= δj + µj , λj , δj , µj (j = 1, 2) are real and D is Caputo’s derivative. We use degree theory to obtain existence and uniqueness results, and illustrate our main results through an example. Let X = C(I,R) be the Banach space of all continuous maps with norm ‖x1‖ = max{|x1(t)| : t ∈ I}. The product space X ×X = {(x1, x2) : x1, x2 ∈ X} is also a Banach space with norm ‖(x1, x2)‖ = max{‖x1‖, ‖x2‖}. 2. Preliminaries In this section, we pointed out few important results from degree theory, func- tional analysis and fractional calculus [9, 13, 15, 16]. EJDE-2020/132 FRACTIONAL ORDER PANTOGRAPH EQUATIONS 3 Definition 2.1. Let γ > 0. Assume x(t) ∈ L[0, 1]. The Riemann-Liuville fractional integral of order γ is defined as Iγ0+x(t) = 1 Γ(γ) ∫ t 0 (t− θ)γ−1x(θ) dθ. Definition 2.2. Let k = [γ]+1. If x(t) ∈ AC[0, 1], the Caputo fractional derivative is defined as CDγ 0+x(t) = 1 Γ(k − γ) ∫ t 0 (t− θ)k−γ−1x(k)(θ)dθ. Lemma 2.3. In fractional calculus for a function x(t) ∈ AC[0, 1], we have Iγ0+ CDγ 0+x(t) = x(t) + a0 + a1t+ a2t 2 + · · ·+ ak−1t k−1, for constants aj ∈ R, for j = 0, 1, 2, · · · , k − 1. Definition 2.4 ([22]). The spectral radius of a matrix M ∈ Cn×n is defined by Υ(M ) = max{|β1|, |β2|, . . . , |βn|}, where β1, β2, . . . , βn are the eigenvalues. A matrix M is said to converge to zero if the spectral radius satisfies Υ(M ) < 1. Definition 2.5 ([22]). A system of fractional order pantograph equations (1.4) is Ulam-Hyers stable, if for all C = (Cf1,f2) > 0 such that, for some ordered pair ε = (ε1, ε2) of positive real numbers and for each t ∈ I and if for any solution (x1, x2) ∈ X ×X of the inequalities |CDγ1x1(t)− f1(t, x1(t), x1(rt), x2(t))| < ε1, |CDγ2x2(t)− f2(t, x1(t), x2(t), x2(rt))| < ε2, (2.1) there exist a unique solution (x̄1, x̄2), belonging to X × X, of system (1.4) that satisfy the relation ‖(x̄1, x̄2)− (x1, x2)‖ ≤ Cε . Definition 2.6 ([22]). A system of fractional order pantograph equations (1.4) is generalized Ulam-Hyers stable, if for all C = (Cf1,f2) > 0, ϕ belonging to C(I,R) along with ϕ(0) = 0, for all t ∈ I and a solution (x1, x2) ∈ X ×X of (1.4), there is a unique solution (x̄1, x̄2) ∈ X ×X of (1.4) which satisfies ‖(x̄1, x̄2)− (x1, x2)‖ ≤ C ϕ(ε). Theorem 2.7 ([22]). Assume the operators T1, T2 : X ×X → X such that for the operator system T1(x1, x2) = x1, T2(x1, x2) = x2, (2.2) for all xi, x̄i ∈ X for i = 1, 2, the following system of inequations holds ‖T1(x1, x2)− T1(x̄1, x̄2)‖ ≤ `1‖x1 − x̄1‖+ `2‖x2 − x̄2‖, ‖T2(x1, x2)− T2(x̄1, x̄2)‖ ≤ `3‖x1 − x̄1‖+ `4‖x2 − x̄2‖, (2.3) where (x1, x2), (x̄1, x̄2) ∈ X ×X are exact and approximate solutions, respectively and `1, `2, `3, `4 > 0. Moreover, if the matrix M = [ `1 `2 `3 `4 ] , (2.4) converges to 0, then the fixed points for (2.2) are stable in the sense of Ulam-Hyers. 4 I. AHMAD, J. J. NIETO, G. U. RAHMAN, K. SHAH EJDE-2020/132 Fir our proofs we let P (X) be the power set, and J be the sub collection of P (X) that contain all bounded sets. Also we adopt the terminologies and results from [6]. Definition 2.8. For non-compactness the Kuratowski measure = : J → R+ is defined as =(J) = inf{p1 > 0 : J ∈ J has sets of finite cover of diameter less or equal to p1 t}. Proposition 2.9. The Kuratowski measure = satisfies the following properties: (i) J is relatively compact if and only if =(J) = 0. (ii) J is a semi-norm, i.e. =(lJ) = |l|=(J), l is real and =(J1) + =(J2) ≥ =(J1 + J2). (iii) J1 ⊂ J2 implies =(J1) ≤ =(J2), =(J1 ∪ J2) = max{=(J1),=(J2)}. (iv) =(convJ) = =(J). (v) =(J̄)−=(J) = 0. Definition 2.10. Let Z is a subspace of X, then a function H from Z into X that is bounded and continuous. H is =-Lipschitz if there is an L ≥ 0 such that =(H(J)) ≤ L=(J), ∀J ⊂ Z bounded. H is =-contraction if L< 1. H is =-condensing if =(H(J)) < =(J),∀J ⊂ Z bounded with =(J) > 0. Proposition 2.11. If the functions H,F : Z→ X are =-Lipschitz with real positive constants L1 and L2, respectively, then H + F : Z→ X are =-Lipschitz with L1+L2. Proposition 2.12. If H : Z → X is compact, then H is =-Lipschitz having Lips- chitz constant equal to zero. Proposition 2.13. If H : Z → X is Lipschitz along a constant =, so H is =- Lipschitz with same constant =. For our main result the following theorem due to Isaia [11] plays an important role. Theorem 2.14. Let H : X → X be =-condensing, and S = {y ∈ X : ∃µ ∈ I such that y = µHy}. For a bounded set S ∈ X there exists a positive real number % such that S ⊂ B%(0). If the following equation holds then the degree of H is one and has one fixed point, deg(I − µH, B%(0), 0) = 1, ∀µ ∈ I. The collection of all fixed points of H is denoted by B%(0). Furthermore, to establish our main results, we need the following assumptions: (A1) There exist constants K1,K2 ∈ [0, 1) such that for x1, x̄1, x2 and x̄2 ∈ X, |g1(x̄1)− g1(x1)| ≤ K1‖x̄1 − x1‖, |g2(x̄2)− g2(x2)| ≤ K2‖x̄2 − x2‖. (A2) There exist constants Cg1 , Cg2 ,Mg1 ,Mg2 > 0 such that for x1, x2 ∈ X, |g1(x1)| ≤ Cg1‖x1‖q1 +Mg1 , |g2(x2)| ≤ Cg2‖x2‖q2 +Mg2 . EJDE-2020/132 FRACTIONAL ORDER PANTOGRAPH EQUATIONS 5 (A3) There exist constants C (j) (f1) , C (j) (f2) (j = 1, 2), ,Mf1 ,Mf2 such that for t ∈ I, x1, x2 ∈ X, |f1(t, x1(θ), x1(rθ), x2(θ))| ≤ 2C1 f1‖x1‖ q1 + C2 f1‖x2‖ q2 +Mf1 , |f2(t, x1(θ), x2(θ), x2(rθ))| ≤ C1 f2‖x1‖ q1 + 2C2 f2‖x2‖ q2 +Mf2 . (A4) For arbitrary t ∈ I, x1, x̄1, x2 and x̄2 ∈ X there exist positive constants Lf1 , Lf2 such that |f1(t, x̄1(t), x̄1(rt), x̄2(t))− f1(t, x1(t), x1(rt), x2(t))| ≤ Lf1(2|x̄1 − x1|+ |x̄2 − x2|), |f2(t, x̄1(t), x̄2(t), x̄2(rt))− f2(t, x1(t), x2(t), x2(rt))| ≤ Lf2(|x̄1 − x1|+ 2|x̄2 − x2|). 3. Main results Theorem 3.1. Let x1(t) ∈ C(I,R) and w1(t) ∈ L(I,R) be the solution for the problem Dγ1 0+x1(t) = w1(t), t ∈ I, γ1 ∈ (0, 1], g1(x1) = λ1x1(0)−µ1x1(η)−δ1x1(1), (3.1) is as follow: x1(t) = g1(x1) λ1 − (δ1 + µ1) + δ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ 1 0 (1− θ)γ1−1w1(θ) dθ + µ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ η 0 (η − θ)γ1−1w1(θ) dθ + 1 Γ(γ1) ∫ t 0 (t− θ)γ1−1w1(θ) dθ. (3.2) Proof. Integrating on both sides of (3.1) and with the help of Lemma 2.3, we have x1(t) = Iγ1w1(t) + a0 = 1 Γ(γ1) ∫ t 0 (t− θ)w1(θ)dθ + a0, (3.3) using boundary conditions in (3.1), we obtain x1(0) = a0, x1(η) = 1 Γ(γ1) ∫ η 0 (η − θ)γ1−1w1(θ)dθ + a0, x1(1) = 1 Γ(γ1) ∫ 1 0 (1− θ)γ1−1w1(θ)dθ + a0. (3.4) Putting all the values of (3.4), in the relation given in (3.1), yields g1(t) = a0(λ1 − (δ1 + µ1))− µ1 Γ(γ1) ∫ η 0 (η − θ)γ1−1w1(θ) dθ − δ1 Γ(γ1) ∫ 1 0 (1− θ)γ1−1w1(θ) dθ + 1 Γ(γ1) ∫ t 0 (t− θ)γ1−1w1(θ) dθ. (3.5) The value of a0 from (3.5), is a0(t) = g1(x1) λ1 − (δ1 + µ1) + δ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ 1 0 (1− θ)γ1−1w1(θ) dθ + µ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ η 0 (η − θ)γ1−1w1(θ) dθ. (3.6) 6 I. AHMAD, J. J. NIETO, G. U. RAHMAN, K. SHAH EJDE-2020/132 Putting the value of a0, in (3.3), we obtain the solution x1(t) = g1(x1) λ1 − (δ1 + µ1) + δ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ 1 0 (1− θ)γ1−1w1(θ) dθ + µ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ η 0 (η − θ)γ1−1w1(θ) dθ + 1 Γ(γ1) ∫ t 0 (t− θ)γ1−1w1(θ) dθ. (3.7) Using similar approach for x2(t) ∈ C(I,R) and w2(t) ∈ L(I,R) the solution for the problem Dγ2 0+x2(t) = w2(t), t ∈ I, γ2 ∈ (0, 1], g2(x2) = λ2x2(0)− µ2x2(ζ)− δ2x2(1), (3.8) is x2(t) = g2(x2) λ2 − (δ2 + µ2) + δ2 (λ2 − (δ2 + µ2))Γ(γ2) ∫ 1 0 (1− θ)γ2−1w2(θ) dθ + µ2 (λ2 − (δ2 + µ2))Γ(γ2) ∫ ζ 0 (ζ − θ)γ2−1w1(θ) dθ + 1 Γ(γ2) ∫ t 0 (t− θ)γ2−1w2(θ) dθ. (3.9) � Corollary 3.2. In light of Theorem 3.1, the solution of the problem (1.4) is x1(t) = g1(x1) λ1 − (δ1 + µ1) + δ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ 1 0 (1− θ)γ1−1f1(θ, x1(θ), x1(rθ), x2(θ)) dθ + µ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ η 0 (η − θ)γ1−1f1(θ, x1(θ), x1(rθ), x2(θ)) dθ + 1 Γ(γ1) ∫ t 0 (t− θ)γ1−1f1(θ, x1(θ), x1(rθ), x2(θ)) dθ, x2(t) = g2(x2) λ2 − (δ2 + µ2) + δ2 λ2 − (δ2 + µ2) 1 Γ(γ2) ∫ 1 0 (1− θ)γ2−1f2(θ, x1(θ), x2(θ), x2(rθ)) dθ + µ2 (λ2 − (δ2 + µ2))Γ(γ2) ∫ ζ 0 (ζ − θ)γ2−1f2(θ, x1(θ), x2(θ), x2(rθ)) dθ + 1 Γ(γ2) ∫ t 0 (t− θ)γ2−1f2(θ, x1(θ), x2(θ), x2(rθ))dθ. (3.10) Lemma 3.3. Let f1, f2, f3 be nonlinear continuous the functions. Then (x1, x2) ∈ X×X is the required solution of the integral equations (3.10), if and only if (x1, x2) is a solution of (1.4). EJDE-2020/132 FRACTIONAL ORDER PANTOGRAPH EQUATIONS 7 Proof. If (x1, x2) is the solution of (3.10), then differentiating both sides of (3.10), we obtain (1.4). However, If (x1, x2) is a solution of (1.4), thus (x1, x2) is the solution of (3.10). � It is sufficient for the existence of FPDEs (1.4), to prove that integral system (3.10) has at least one solution (x1, x2) ∈ X×X. The operators H,F, T : X×X → X ×X are defined by H(x1, x2)(t) = (H1(x1(t)), H2(x2(t))), F (x1, x2)(t) = (F1((x1(t)), (x2(t))), F2(x1(t), x2(t)) T (x1, x2) = H(x1, x2) + F (x1, x2), where H1x1(t) = g1(x1) λ1 − δ1 − µ1 , H2x2(t) = g2(x2) λ2 − δ2 − µ2 , F1(x1, x2)(t) = δ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ 1 0 (1− θ)γ1−1f1(θ, x1(θ), x1(rθ), x2(θ))dθ + µ1 (λ1 − (δ1 + µ1))Γ(γ1) ∫ η 0 (−v + η)γ1−1f1(θ, x1(θ), x1(rθ), x2(θ))dθ + 1 Γ(γ1) ∫ t 0 (t− θ)γ1−1f1(θ, x1(θ), x1(rθ), x2(θ))dθ, F2(x1, x2)(t) = δ2 (λ2 − (δ2 + γ2))Γ(γ2) ∫ 1 0 (1− θ)γ2−1f2(θ, x1(θ), x2(θ)), x2(rθ))dθ + µ2 (λ2 − (δ2 + µ2))Γ(γ2) ∫ ζ 0 (ζ − θ)γ2−1f2(θ, x1(θ), x2(θ)), x2(rθ))dθ + 1 Γ(γ2) ∫ t 0 (t− θ)γ2−1f2(θ, x1(θ), x2(θ)), x2(rθ))dθ. The continuity of the functions f1(θ, x1(θ), x1(rθ), x2(θ)), f2(θ, x1(θ), x2(θ)), x2(rθ)) imply that T is well-defined. We can also write (3.10) in operator form as (x1, x2) = T (x1, x2) = H(x1, x2) + F (x1, x2), (3.11) the fixed points of (3.11) are solutions of (3.10). Lemma 3.4. In view of assumptions (A1) and (A2), the operator H : X ×X → X ×X, is Lipschitz and satisfies ‖H(x1, x2)‖ ≤ C‖(x1, x2)‖q +M. (3.12) Proof. For (x1, x2), (x̄1, x̄2) ∈ X ×X, using (A1) and (3.11), we obtain ‖H1x1 −H1x̄1‖ ≤ 1 |λ1 − δ1 − µ1| |g1(x1)− g1(x̄1)| ≤ K1 |λ1 − δ1 − µ1| ‖x1 − x̄1‖ , ‖H2x2 −H2x̄2‖ ≤ K2 |λ2 − (δ2 + µ2)| ‖x2 − x̄2‖ . This implies ‖H(x1, x2)−H(x̄1, x̄2)‖ 8 I. AHMAD, J. J. NIETO, G. U. RAHMAN, K. SHAH EJDE-2020/132 ≤ max ( K1 |λ1 − (δ1 + µ1)| , K2 |λ2 − (δ2 + µ2)| ) ‖(x1, x2)− (x̄1, x̄2)‖ ≤ h‖(x1, x2)− (x̄1, x̄2)‖, where h = max ( K1 |λ1 − δ1 − µ1)| , K2 |λ2 − δ2 − µ2)| ) . So with proposition (2.11), H is =-Lipschitz of positive value h. Using assumption (A2) for growth condition, we have |H1x1(t)| = ‖ g1(x1) λ1 − (δ1 + µ1) ‖ ≤ Cg1‖x1‖q1 |λ1 − (δ1 + µ1)| +Mg1 , |H2x2(t)| = | g1(x2) λ2 − (δ2 + µ2) | ≤ Cg2‖x2‖q2 |λ2 − (δ2 + µ2)| +Mg2 . Consequently ‖H(x1, x2)‖ ≤ C‖(x1, x2)‖q + M, where C = max ( Cg1 |λ1 − δ1 − µ1)| , Cg2 |λ2 − δ2 − µ2)| ) , M = max(Mg1 ,Mg2), and q = max(q1, q2). � Lemma 3.5. If hypothesis (A3) holds, then the operator F from X ×X to X ×X is continuous and satisfies ‖F (x1, x2)‖ ≤ $ ( ‖(x1, x2)‖q +M∗ ) , (x1, x2) ∈ X ×X, (3.13) where $ = max ( Cf1 [|λ1|+ |µ1|+ |δ1|] |λ1 − (δ1 + µ1)|Γ(1 + γ1) , Cf2 [|λ2|+ |γ2|+ |δ2|] |λ2 − (δ2 + µ2)|Γ(1 + γ2) ) , M∗ = max(Mf1 ,Mf2), Cf1 = max{C1 f1 , C 2 f1}, Cf2 = max{C1 f2 , C 2 f2}. Proof. Let Dρ = { (x1, x2) ∈ X ×X : ‖(x1, x2)‖ ≤ ρ } ⊂ X ×X be a bounded set and let {zn = (x1n, x2n)} ∈ Dρ be a sequence such that zn → z = (x1, x2) as n tends to infinity in Dρ. We have to show that ‖Fzn−Fz‖ → 0, as n tends to infinity. From the continuity of f1(t, x1(t), x1(rt), x2(t)), it follows that f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n) → f1(θ, x1(θ), x1(rθ), x2(θ)) as n → ∞. Under assumption (A3), we have (1− θ)γ1−1|(f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n)− f1(θ, x1(θ), x1(rθ), x2(θ)))| ≤ (1− θ)γ1−1[2C1 f1ρ q1 + C2 f1ρ q2 +Mf1 ], (η − θ)γ1−1|(f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n)− f1(θ, x1(θ), x1(rθ), x2(θ)))| ≤ (η − θ)γ1−1[2C1 f1ρ q1 + C2 f1ρ q2 +Mf1 ], (t− θ)γ1−1|(f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n)− f1(θ, x1(θ), x1(rθ), x2(θ)))| ≤ (t− θ)γ1−1[2C1 f1ρ q1 + C2 f1ρ q2 +Mf1 ], EJDE-2020/132 FRACTIONAL ORDER PANTOGRAPH EQUATIONS 9 applying the Lebesgue dominated convergence theorem, for the preceding inequal- ities, we obtain lim n→∞ ∫ 1 0 (1− θ)γ1−1|(f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n) − f1(θ, x1(θ), x1(rθ), x2(θ)))| dθ = 0, lim n→∞ ∫ η 0 (η − θ)γ1−1|f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n) − f1(θ, x1(θ), x1(rθ), x2(θ))|dθ = 0, lim n→∞ ∫ t 0 (t− θ)γ1−1|f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n) − f1(θ, x1(θ), x1(rθ), x2(θ))| dθ = 0. Hence, |F1(x1n, x2n) − F1(x1, x2)| → 0 as n approaches ∞. Similarly, we have |F2(x1n, x2n) − F2(x1, x2)| → 0 as n tends to ∞. This implies |F (x1n, x2n) − F (x1, x2)| → 0 as n approaches ∞, due to the continuity of F . For growth condi- tions we proceed as follows, ‖F1(x1, x2)‖ ≤ 1 |λ1 − (δ1 + µ1)|Γ(γ1) ( |δ1| ∫ 1 0 (1− θ)γ1−1|f1(θ, x1(θ), x1(rθ), x2(θ))|dθ + |γ1| ∫ η 0 (η − θ)γ1−1|f1(θ, x1(θ), x1(rθ), x2(θ))| ) dθ + |µ1| |λ1 − (δ1 + µ1)|Γ(γ1) ∫ t 0 (t− θ)γ1−1|f1(θ, x1(θ), x1(rθ), x2(θ))|dθ ≤ [ |λ1|+ |δ1|+ |µ1| |λ1 − (δ1 + µ1)|Γ(γ1 + 1) ]( 2C1 f1‖x1‖ q2 + C2 f1‖x2‖ q2 +Mf1 ) . (3.14) Similarly, we obtain ‖F2(x1, x2)‖ ≤ [ |λ2|+ |δ2|+ |µ2| |λ2 − (δ2 + µ2)|Γ(γ2 + 1) ]( C1 f2‖x1‖ q2 + 2C2 f2‖x2‖ q2 +Mf2 ) . (3.15) Then growth condition (3.13) follows from (3.14) and (3.15). � Lemma 3.6. The operator F : X ×X → X ×X is compact. Proof. Let the set Z ⊂ Dρ ⊂ X × X be bounded. It is necessary to prove that F (Z) is relatively compact in X×X. For any zn = ((x1)n, (x2)n) ∈ Z ⊂ Dρ, growth condition (3.13) yields ‖F ((x1)n, (x2)n)‖ ≤ $(ρq +M∗), i.e. F (·) is uniformly bounded. To show F is equicontinuous, pick 0 ≤ t ≤ ξ ≤ 1, then ‖F1((x1)n, (x2)n))(t)− F1((x1)n, (x2)n))(ξ)‖ ≤ 1 Γ(γ1) ∫ t 0 [(t− θ)γ1−1 − (ξ − θ)γ1−1]|f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n)|dθ 10 I. AHMAD, J. J. NIETO, G. U. RAHMAN, K. SHAH EJDE-2020/132 + ∫ ξ t (ξ − θ)γ1−1|f1(θ, (x1(t))n, (x1(rt))n, (x2(t))n)|dθ, ≤ 1 Γ(γ1 + 1) [ tγ1 − ξγ1 + (ξ − t)γ1 + (ξ − t)γ1 ] × ( 2C1 f1‖x1‖ q1 + C2 f1‖x2‖ q2 +Mf1 ) , ≤ (ρq(2C1 f1 + C2 f1 ) +Mf1 Γ(γ1 + 1) )[ tγ1 − ξγ1 + (−t+ ξ)γ1 + (−ξ + t)γ1 ] . Similarly, ‖ F2((x1)n, (x2)n)(t)− F2(((x1)n, (x2)n))(ξ)‖ ≤ ( (C1 f2 + 2C2 f2 )ρq +Mf2 Γ(γ2 + 1) )[ tγ2 − ξγ2 + (ξ − t)γ2 + (t− ξ)γ2 ] . (3.16) It follows that ‖F1((x1)n, (x2)n)(t)− F1((x1)n, (x2)n)(ξ)‖ → 0, ‖F2((x1)n, (x2)n)(t)− F2((x1)n, (x2)n)(ξ)‖ → 0 as t → ξ, which implies that F (x1, x2) is equicontinuous. F (x1, x2) is compact by Arzela-Ascoli theorem. Furthermore, F is =-Lipschitzen of constant zero. � Theorem 3.7. Suppose (A1)–(A3) are satisfied. Then (1.4) has at least one solu- tion (x1, x2) ∈ X ×X. Moreover, the solution set is bounded in X ×X. Proof. By Lemma (3.4), H is =-Lipschitz of real constant h, and by Lemma 3.6 F is =-Lipschitz with zero. As a result, T is =-Lipschitz of real h. Let U = { (x1, x2) ∈ X ×X : there exist ρ1 ∈ I, (x1, x2) = ρ1T (x1, x2) } . We need to show U is bounded. For (x1, x2) ∈ U , we have (x1, x2) = ρ1T (x1, x2) = ρ1 ( H(x1, x2) + F (x1, x2) ) , which implies that ‖x1‖ ≤ ρ1[‖H1x1‖+ ‖F1x1‖] ≤ ρ1 [ Cg1 ||x1||q1 |λ1 − (δ1 + µ1)| +Mg1 + |λ1|+ |δ1|+ |µ1| |λ1 − (δ1 + µ1)|Γ(γ1 + 1) (2C1 f1‖x1‖ q1 + C2 f1‖x2‖ q2 +Mf1) ] . (3.17) In the same way we can prove that ‖x2‖ ≤ ρ1 [ Cg2 ||x2||q2 |(δ2 + µ2)− λ2| +Mg2 + |λ2|+ |δ2|+ |µ2| |λ2 − (δ2 + µ2)|Γ(γ2 + 1) ( C1 f2‖x1‖ q1 + 2C2 f2‖x2‖ q2 +Mf2 ) ] . (3.18) The preceding inequalities (3.17) and (3.18) along with q1, q2 ∈ [0, 1) yield that U is bounded in X ×X. If we go against the assertion and divide (3.17) by ‖x1‖ = λ and ‖x1‖ → ∞, we obtain the following result 1 ≤ lim λ→∞ ρ1 λ [ Cf2λ q |(δ1 + µ1)− λ1| +Mg1 EJDE-2020/132 FRACTIONAL ORDER PANTOGRAPH EQUATIONS 11 + |δ1|+ |λ1|+ |µ1| |(δ1 + µ1)− λ1|Γ(γ1 + 1) [(2C1 f1 + C2 f1 ]λq2 +Mf1 ] = 0, which is clearly an absurd result. Similarly, for (3.18) a contradiction occurs. Con- sequently, T has a minimum one fixed point. It results in the existence of at least one solution of (1.4). � Theorem 3.8. The operator T has unique fixed point under the conditions (A1)– (A4) and if max{d1, d2} < 1, where d1 = K1Γ(γ1 + 1) + 4Lf1(|λ1|+ |δ1|+ |µ1|) |λ1 − (δ1 + µ1)|Γ(1 + γ1) , d2 = K2Γ(γ2 + 1) + 4Lf2(|λ2|+ |δ2|+ |µ2|) |λ2 − (δ2 + µ2)|Γ(γ2 + 1) . Proof. For (x1, x2), (x̄1, x̄2) ∈ X ×X, we can obtain the inequality ‖(H1(x1) + F1(x1, x2))− (H1(x̄1) + F1(x̄1, x̄2))‖ ≤ K1‖x1 − x̄1‖ |λ1 − (δ1 + µ1)| + |δ1| |λ1 − (δ1 + µ1)|Γ(γ1) ∫ 1 0 (1− θ)γ1−1|f1(θ, x1(θ), x1(rθ), x2(θ)) − f1(θ, x̄1(θ), x1(rθ), x̄2(θ))|dθ + |µ1| |λ1 − (δ1 + µ1)|Γ(γ1) ∫ η 0 (η − θ)γ1−1|f1(θ, x1(θ), x1(rθ), x2(θ)) − f1(θ, x̄1(θ), x1(rθ), x̄2(θ))|dθ + ∫ t 0 (t− θ)γ1−1 Γ(γ1) |f1(θ, x1(θ), x1(rθ), x2(θ))− f1(θ, x̄1(θ), x1(rθ), x̄2(θ))|dθ, ≤ [ K1 |λ1 − (δ1 + µ1)| + 4Lf1 Γ(γ1 + 1) ( |λ1|+ |δ1|+ |µ1| |λ1 − (δ1 + µ1)| )]( ‖x1 − x̄1‖+ ‖x2 − x̄2‖ ) ≤ d1 ( ‖x1 − x̄1‖+ ‖x2 − x̄2‖ ) . Similarly, we obtain ‖(H2(x2) + F2(x1, x2))− (H2(x̄2) + F2(x̄1, x̄2))‖ ≤ d2 ( ‖x1 − x̄1‖+ ‖x2 − x̄2‖ ) . Hence, the above two expressions yield ‖T (x1, x2)− T (x̄1, x̄2)‖ ≤ max(d1, d2) ( ‖(x1, x2)‖ − ‖(x̄1, x̄2)‖ ) = d ( ‖(x1, x2)‖ − ‖(x̄1, x̄2)‖ ) . Therefore, T is contraction as d < 1 and so T has unique fixed point. � Remark 3.9. It can be seen that the growth conditions in (3.7) are also valid for q1, q2 = 1 whenever, max { Cf1 + |λ1|+ |δ1|+ |µ1| |λ1 − (δ1 + µ1)|Γ(γ1 + 1) (2C1 f1 + C2 f1), Cf2 + |λ2|+ |δ2|+ |µ2| |λ2 − (δ2 + µ2)|Γ(γ2 + 1) (C1 f2 + 2C2 f2) } < 1. 12 I. AHMAD, J. J. NIETO, G. U. RAHMAN, K. SHAH EJDE-2020/132 4. Stability analysis Theorem 4.1. If the matrix M converges to zero and assumptions (A1)–(A4) hold, then system (1.4) is stable in the sense of Ulam-Hyers. Proof. For the proof we consider (x1, x2), (x̄1, x̄2) ∈ X, we write ‖T1(x1, x2)− T1(x̄1, x̄2)‖ ≤ |g1(x1)− g1(x̄1)| |λ1 − (δ1 + µ1)| + |δ1| |λ1 − (δ1 + µ1)|Γ(γ1) × ∣∣∣ ∫ 1 0 (1− θ)γ1−1(f1(θ, x1(θ), x1(rθ), x2(θ))− f1(θ, x̄1(θ), x̄1(rθ), x̄2(θ)))dθ ∣∣∣ + |µ1| |λ1 − (δ1 + µ1)|Γ(γ1) ∣∣ ∫ η 0 (η − θ)γ1−1(f1(θ, x1(θ), x1(rθ), x2(θ)) − f1(θ, x̄1(θ), x̄1(rθ), x̄2(θ)))dθ ∣∣ + 1 Γ(γ1) ∣∣∣ ∫ t 0 (t− θ)γ1−1((f1(θ, x1(θ), x1(rθ), x2(θ)) − f1(θ, x̄1(θ), x̄1(rθ), x̄2(θ)))dθ ∣∣∣ ≤ K1 |λ1 − (δ1 + µ1)| ‖x1 − x̄1‖+ ( |δ1|+ |µ1| |λ1 − (δ1 + µ1)|Γ(γ1 + 1) + 1 Γ(γ1 + 1) ) × Lf1(2||x1 − x̄1||+ ||x2 − x̄2||) = (K1Γ(γ1 + 1) + 4Lf1(|λ1|+ |δ1|+ |µ1|) |λ1 − (δ1 + µ1)|Γ(1 + γ1) ) ||x1 − x̄1|| + ( 2Lf1(|λ1|+ |δ1|+ |µ1|) |λ1 − (δ1 + µ1)|Γ(1 + γ1) ) ‖x2 − x̄2‖ ≤ `1||x1 − x̄1||+ `2‖x2 − x̄2‖ (4.1) where `1 = K1Γ(γ1 + 1) + 4Lf1(|λ1|+ |δ1|+ |µ1|) |λ1 − (δ1 + µ1)|Γ(1 + γ1) , `2 = 2Lf1(|λ1|+ |δ1|+ |µ1|) |λ1 − (δ1 + µ1)|Γ(1 + γ1) are non-negative real numbers. Also with similar approach, we can obtain ‖T2(x1, x2)− T2(x̄1, x̄2)‖ ≤ `3‖x1 − x̄1‖+ `4‖x2 − x̄2‖, (4.2) where `3 = K2Γ(γ2 + 1) + 4Lf2(|λ2|+ |δ2|+ |µ2|) |λ2 − (δ2 + µ2)|Γ(1 + γ2) , `4 = 2Lf2(|λ2|+ |δ2|+ |µ2|) |λ2 − (δ2 + µ2)|Γ(1 + γ2) are real and non-negative. Hence, from (4.1) and (4.2), we have the two inequalities ‖T1(x1, x2)− T1(x̄1, x̄2)‖ ≤ `1‖x1 − x̄1‖+ `2‖x2 − x̄2‖, ‖T2(x1, x2)− T2(x̄1, x̄2)‖ ≤ `3‖x1 − x̄1‖+ `4‖x2 − x̄2‖. (4.3) From (4.3), we have M = [ `1 `2 `3 `4 ] , Since M converges to 0, system (1.4) is UH stable. Consequently it is generalized UH stable. � EJDE-2020/132 FRACTIONAL ORDER PANTOGRAPH EQUATIONS 13 Example 4.2. Consider the coupled system with non-local boundary conditions, CD 9 10x1(t) = 1 3 + 1 eπt(t+ 80) √ |x1(rt)|+ sin √ |x2(t)|+ √ | sinx1(t)|, CD 2 3x2(t) = 1 2 + 1 (t+ 90) (|x1(t)|+ sin √ |x2(rt)|+ sin |x2(t)|, t ∈ [0, 1], x1(0)− x1(1)− x1 (1 2 ) = sinx1(t) 30 , x2(0)− x2(1)− x2 (1 2 ) = sinx2(t) t+ 70 . (4.4) Here f1(t, x1(t), x1(rt), x2(t)) = 1 3 + 1 eπt(t+ 80) (√ |x1(rt)|+ sin √ |x2(t)| + √ | sinx1(t)| ) , f2(t, x1(t), x2(t), x2(rt)) = 1 2 + 1 (t+ 90) (|x1(t)|+ sin √ |x2(rt)|+ sin |x2(t)|, γ1 = 9/10, γ2 = 2/3, λ1 = λ2 = 4, δ1 = δ2 = 1, µ1 = µ2 = 0, η = ζ = 1/2, and r ∈ I. Since λl 6= δl + µl (l = 1, 2), and ‖f1(t, x1(t), x1(rt), x2(t))− f1(t, x̄1(t), x̄1(rt), x̄2(t))‖ ≤ 1 80 [2‖x1 − x̄1‖+ ‖x2 − x̄2‖] , ‖f2(t, x1(t), x2(t), x2(rt)− f2(t, x̄1(t), x̄2(t), x̄2(rt))‖ ≤ 1 90 [‖x1 − x̄1‖+ 2‖x2 − x̄2‖], it follows that K1 = 1/30, K2 = 1/70, Mg1 = 0 = Mg2 , Mf1 = 1/3, Mf1 = 1/2, Lf1 = 1/80, Lf2 = 1/90, C1 f1 = C2 f1 = 1/80, C1 f2 = C2 f2 = 1/90, 1 = q1 = q2. So all assumption of Theorem (3.8) are fulfilled, d1 = 0.0978 < 1, and d2 = 0.0868 < 1. So, system (4.4) has unique solution. Also for stability, the matrix M = [ 0.0978 0.0433 0.0868 0.0410 ] has eigenvalues equal to 0.1370 and 0.0018. Clearly Υ(M ) < 1, and hence, by Theorem 4.1, matrix M converges to zero. Therefore, the system (4.4) is stable in the sense to Ulam-Hyers. Also, it is generalized Ulam-Hyers stable. Conclusion. In this research, we obtained some results regarding the qualitative theory for a delay differential system via degree degree theory from topology and standard functional analysis. Authors contribution: Equal contributions have been done by all the authors. References [1] R. P. Agarwal, M. Belmekki, M. Benchohra; A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Difference Equ., Article ID 981728 2009. [2] B. Ahmad, J. J. Nieto; Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory, Topol. Methods Nonlinear Anal., 35 (2010) (2), 295-304. [3] Z. Ali, A. Zada, K. Shah; On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations, Bull. Malays. Math. Sci. 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Zhang; Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl., 59 (2010), 1300-1309. [29] Y. Zhao, S. Sun, Z. Han, Q. Li; Theory of fractional hybrid differential equations, Comput. Math. Appl., 62 (2011) (3), 1312-1324. EJDE-2020/132 FRACTIONAL ORDER PANTOGRAPH EQUATIONS 15 Addendum posted by the editor on July 26, 2020 After this article was published, a reader indicated that there were some mistakes in this article. The editor contacted the authors and sent them detailed comments, one of the authors tried unsuccessfully to address the concerns several times. The editor therefore decided to post this addendum based on the detailed comments. The paper has some important errors, as well as numerous typos which are not given in this addendum. Definition 2.2, needs at least x(k−1) ∈ AC (Absolutely Continuous), then the Caputo derivative is defined almost everywhere (a.e.), stronger condition is needed if it is wanted to exist for all t. Lemma 2.3, needs (at least) x(k−1) ∈ AC. Theorem 3.1, What is given is correct, but it proves that if x1 ∈ AC is a solution of (3.1) then x1 is a solution of (3.2), it is not proving what is required. It should assume w1 continuous, otherwise x1(0) = a0 could be not valid in the proof. But it is necessary to prove, for Corollary 3.2, that a solution x1 ∈ C[0, 1] of (3.2) is in AC[0, 1] so that the Caputo derivative exists a.e. and it can be a solution of (3.1) for a.e. t. If w is continuous then Iγw is continuous but it does not follow that Iγw ∈ AC without extra conditions, a result shown by G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I., Math. Z. 27 (1928), 565–606. Lemma 3.3 is not valid with given definition of Caputo derivative. It is known to experts that one does not have equivalence (if and only if), which is why the following book uses a different definition. Diethelm, Kai, The analysis of fractional differential equations. An application- oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010 The lack of equivalence has been pointed out explicitly in the recent papers: Cichon, Mieczyslaw; Salem, Hussein A. H., On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems. J. Pseudo- Differ. Oper. Appl. 11 (2020), no. 4, 1869–1895. Webb, Jeffrey R. L., Initial value problems for Caputo fractional equations with singular nonlinearities. Electron. J. Differential Equations 2019, Paper No. 117, 32 pp (addendum) Theorem 4.1. ‘If the matrix M converges to zero’, it is a particular matrix here, which should be specified, and it requires some hypotheses on (4.1), (4.2) to make it true. Example 4.2, some important brackets are omitted; the example is not correct, sin(x1(t))/30 is not a functional g(x1) on X, neither is the other term; probably sin(‖x1‖)/30 is intended. End of addendum. Israr Ahmad Department of Mathematics, University of Malakand, Chakdara, Dir(L), Khyber Pakhtunkhwa, Pakistan Email address: israrahmadjc503@gmail.com Juan Jose Nieto Instituto de Matemeticas, Universidade de Santiago de Compostela, Santiago de Com- postela, 15782, Spain Email address: juanjose.nieto.roig@usc.es 16 I. AHMAD, J. J. NIETO, G. U. RAHMAN, K. SHAH EJDE-2020/132 Ghaus ur Rahman Department of Mathematics and Statistics, University of Swat, Khyber Pakhtunkhwa, Pakistan Email address: ghaus957@yahoo.com Kamal Shah Department of Mathematics, University of Malakand, Chakdara, Dir(L), Khyber Pakhtunkhwa, Pakistan Email address: kamalshah408@gmail.com 1. Introduction 2. Preliminaries 3. Main results 4. Stability analysis Conclusion Authors contribution: References Addendum posted by the editor on July 26, 2020