Solvability of Some Two-Point Fractional Boundary Value Problems under Barrier Strip Conditions
Solvability of Some Two-Point Fractional Boundary Value Problems under Barrier Strip Conditions
Limei He,1 Xiaoyu Dong,1 Zhanbing Bai,1 and Bo Chen2
1College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China
2College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China
Correspondence should be addressed to Zhanbing Bai; moc.361@iabgnibnahz
Received 14 July 2017; Accepted 27 September 2017; Published 26 October 2017
Academic Editor: Manuel De la Sen
Copyright © 2017 Limei He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Topological techniques are used to establish existence results for a class of fractional differential equations , with one of the following boundary value conditions: and or and where is a real number, is the conformable fractional derivative, and is continuous. The main conditions on the nonlinear term are sign conditions (i.e., the barrier strip conditions). The topological arguments are based on the topological transversality theorem.
1. Introduction
Recently, boundary value problems of nonlinear fractional differential equations have been addressed by several researchers. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, control theory, biology, economics, blood flow phenomena, signal and image processing, biophysics, aerodynamics, fitting of experimental data, and so forth. For example, in 2006, by using the fixed point theorem in cones, the existence and multiplicity of solutions to the following problems are obtained [1]:where is a real number, is Caputo fractional derivative, and is a continuous function. In 2010, by the use of the Lipschitz condition and the compression mapping principle, the existence of solutions of the following problem is obtained [2]:where is a real number, , is the standard Riemann-Liouville derivative, and is a continuous function. We refer the readers to other contributions in this line ([1–30], etc.).
The technique of barrier strips has been used by Kelevedjiev and Tersian in [31, 32] to study the solvability of integer order BVPs. Recently, we use it to study the fractional differential equation with Dirichlet boundary value condition [20]. In this paper, by using the topological transversality theorem, we consider the following equation:with one of the following boundary value conditions:where is a real number, is the conformable fractional order derivative, and is a continuous function. The existence results of solutions to the problem are obtained under which satisfies some barrier strip conditions.
2. Definitions and Lemmas about Fractional Calculus
Let .
Definition 1 (see [14]). Suppose is -order differentiable for ; the -order fractional derivative of is defined asprovided the limits of the right side exist.
Lemma 2 (see [12]). Let . Function is -order differentiable if and only if is -order differentiable; moreover, the following relation holds:
Definition 3 (see [14]). The -order fractional integral is defined aswhere is the -order integral.
Lemma 4 (see [20]). For , there holds
Lemma 5 (see [14]). Let be continuous on and -order differentiable on . Then, there exists such that
3. Topological Preliminaries and a New Function Space
We begin with a brief review of the topological results to be used in this paper; see [33]. Let be a convex subset of a Banach space , a metric space, and a continuous map. We say that is compact if is contained in a compact subset of . is completely continuous if it maps bounded subsets in into compact subsets of . A homotopy is said to be compact provided that given by for in is compact.
Let be open in . A compact map is called admissible if it is fixed point free on the boundary, , of . The set of all such maps will be denoted by .
Definition 6. A map in is inessential if there is a fixed point free compact map such that . A map in which is not inessential is called essential.
Lemma 7 (see [33]). Let be an arbitrary value in and be in and be the constant map for in . Then is essential.
Definition 8. Two maps and in are called homotopic if there is a compact homotopy such that and and is admissible for each in .
The following theorem called topological transversality theorem which is very important to our results.
Lemma 9 (see [33]). Let and be in and be homotopic maps, . Then one of these maps is essential if and only if the other is.
Now, we construct a function space. Given , let . Define
By the linearity of integral operator , is a linear space. For , according to Lemma 4, there are . Definewhere . Next, we prove is a norm in the linear space , and (...truncated)