Some Fixed Point Theorems for $$F(\psi,\varphi)$$ -Contractions and Their Application to Fractional Differential Equatio
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Some Fixed Point Theorems for F (ψ, ϕ)-Contractions and Their Application to Fractional Differential Equations H. M. Srivastava∗,∗∗,∗∗∗,1 , A. Shehata∗∗∗∗,∗∗∗∗∗,2 and S. I. Moustafa∗∗∗∗∗∗,3 ∗
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada ∗∗ Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China ∗∗∗ Department of Mathematics and Informatics, Azerbaijan University 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan, ∗∗∗∗ Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt ∗∗∗∗∗ Department of Mathematics, College of Science and Arts at Unaizah, Qassim University, Qassim, Kingdom of Saudi Arabia ∗∗∗∗∗∗ Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt, E-Mail: 1 [email protected], 2 [email protected], 3 [email protected] Received January 21, 2020; Revised February 14, 2020; Accepted February 15, 2020
Abstract. The main object of this paper is to establish some fixed point results for F (ψ, ϕ)contractions in partially-ordered metric spaces. As an application of one of these fixed point theorems, we discuss the existence of a unique solution for a coupled system of higher-order fractional differential equations with multi-point boundary conditions. The results presented in this paper are shown to extend many recent results appearing in the literature. DOI 10.1134/S1061920820030103
1. INTRODUCTION AND MOTIVATION Existence of fixed points for contractive mappings in partially-ordered metric spaces has been considered in many recent works (see, for example, [3, 8, 13, 15, 28]), in which applications to matrix, ordinary differential and integral equations are also presented. Zhou et al. [30] considered an interesting class A of functions, which includes all bounded functions β : [0, ∞) → [0, K) with upper bound K > 0, in order to refine the Banach contraction principle. Theorem 1. (see [30]) Let (X, , d) be a partially-ordered complete metric space. Suppose that f : X → X is a nondecreasing mapping and that there exists an element x0 ∈ X with x0 f (x0 ). Suppose also that there exist a constant θ ∈ (0, 1/K) and a function h ∈ A such that d f (x), f (y) θh θd(x, y) d(x, y) (for each x, y ∈ X with x y). Assume that either f is continuous or that, if an increasing sequence {xn } tends to x ∈ X then xn x
(∀ n ∈ N).
If, for any x, y ∈ X, there exists z ∈ X, which is comparable to x and y, then f has a unique fixed point. Fractional calculus is a generalization of the ordinary calculus of differentiation and integration to arbitrary noninteger order. It significantly aids in describing various natural phenomena and modelling them more accurately. In particular, fractional differential equations (FDEs) play an important rˆole in many fields of science and engineering (see, for details, [9, 10, 14]). Recently, the existence and uniqueness of a solution of the ini
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