Existence of solutions for nonlinear fractional differential equations with non-homogenous boundary conditions

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Existence of solutions for nonlinear fractional differential equations with non-homogenous boundary conditions Yanning An1 · Wenjun Liu1 Received: 13 February 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020

Abstract In this paper we consider the existence of solution to systems of nonlinear conformable fractional differential equations with non-homogenous Dirichlet, Neumann, Sturm– Liouville conditions or the periodic condition. We show that the system has at least one solution by using tube solution and Schauder fixed point theorem. Keywords Fractional differential equations · Existence of solution · Conformable fractional calculus · Fixed point theorem Mathematics Subject Classification 26A33 · 34A08 · 34A12 · 34B18 · 47H10

1 Introduction Fractional calculus as a branch of mathematical analysis has been widely used in biology, physics and many other subjects [2,3,6,9,13,14,30,33]. Since the famous mathematician Leibniz [32] proposed the concept of fractional order in 1695, many researchers have studied it and given different definitions of fractional derivative. Khalil et al. [23] recently gave a new definition of fractional derivative, which was called the comfortable fractional derivative. This new definition has the advantage of simplicity of form, and it satisfies many common properties of the integer derivatives [1,7]. By using the comfortable fractional derivative, some problems related to fractional differential equations have been solved. In [28], authors have obtained the existence and uniqueness for a kind of nonlocal fractional evolution equations on the unbounded interval by Knaster’s theorem and the new definition. In [8], by introducing the concept

B 1

Wenjun Liu [email protected] School of Mathematics and Statistics, Nanjing University of Information Science and Technology, 210044 Nanjing, China


Y. An, W. Liu

of tube solution, Bayour and Torres proved existence of solution to a local fractional nonlinear differential equation with initial condition. In this paper, we consider the existence of a positive solution of the following problem.  D β D α (x)(t) = f (t, x(t), x (α) (t)), 0 ≤ t ≤ 1. (1.1) x ∈ BC where f : [0, 1] × R × R → R is a continuous function, x (α) (t) denotes α-order derivatives of x at a point t, and D β D α (x)(t) denotes β-order derivatives of x (α) (t) at t, α, β ∈ (0, 1). BC denotes boundary condition. Here, we consider non-homogenous Dirichlet, Neumann, Sturm–Liouville conditions or the periodic condition:  (S L)  (P)

a0 x(0) − b0 x (α) (0) = r0 , a1 x(1) + b1 x (α) (1) = r1 ;

x(0) = x(1), x (α) (0) = x (α) (1);

where a0 , b0 , r0 , a1 , b1 , r1 are some given constants, and there exists ξi ≥ 0 such that ai ≥ ξi ; bi = 0, 1; ξi + bi > 0 for i = 0, 1. The boundary condition (P) is not a special case of the boundary condition (S L), because it cannot be completely equivalent to a special case of (S L). Fractional boundary value problems have been playing an important role to modeling many physical and natural systems, such as,