Solvability for a coupled system of fractional differential equations at resonance☆
Introduction
The existence of solutions for integer order differential equations at resonance has been studied by many authors (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and references cited therein). Since the extensive applicability of fractional differential equations (see [15], [16]), recently, more and more authors pay their close attention to the boundary value problems of fractional differential equations (see [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]). In papers [27], [28], the existence of solutions to coupled systems of fractional differential equations at nonresonance has been given. In papers [29], [30], the solvability of fractional differential equations at resonance has been investigated. To the best of our knowledge, the existence of solutions to coupled systems of fractional differential equations at resonance has not been studied. We will fill this gap in the literature. In this paper, we discuss the existence of solutions to a coupled system of fractional differential equations at resonance: where , , .
In this paper, we will always suppose that the following conditions hold.
.
satisfy Carathéodory conditions, i.e., and are measurable for each fixed , and are continuous for a.e. , and for each , there exist such that , for all , a.e. .
Section snippets
Preliminaries
For abbreviation, we write instead of .
For convenience, we introduce some notations and a theorem. For more details, see [31].
Let and be real Banach spaces and be a Fredholm operator with index zero, be projectors such that It follows that is invertible. We denote the inverse by .
If is an open bounded subset of , the map will be called -compact on if
Main results
Define operators as follows:
In order to obtain our main results, we first present and prove the following lemmas.
Lemma 3.1 If holds, then is a Fredholm operator of index zero, the linear continuous projectors and can be defined asrespectively,
Example
Let us consider the following coupled system of fractional differential equations at resonance where
Corresponding to the problem (1.1), we have that , . Take
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This work is supported by the Natural Science Foundation of China (11171088), the Natural Science Foundation of Hebei Province (A2009000664), the Doctoral Program Foundation of Hebei University of Science and Technology (QD201020) and the Foundation of Hebei University of Science and Technology (XL201136).