Existence result for a nonlinear fractional differential equation with integral boundary conditions at resonance
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In this paper, we establish the existence result for nonlinear fractional differential equations with integral boundary conditions at resonance by means of coincidence degree theory. As applications, three examples are presented to illustrate the main results.
KeywordsFractional Derivative Existence Result Fixed Point Theorem Fractional Differential Equation Integral Boundary
where , , , is the Riemann-Liouville fractional derivative, and is a continuous function.
During the last few years, fractional differential equations have been studied extensively due to their significant applications in various sciences such as physics, mechanics, chemistry, phenomena arising in engineering. See [1, 2, 3] and the references therein. Many researchers paid attention to the existence of boundary value problems for nonlinear fractional differential equations, see [4, 5, 6, 7, 8]. The authors obtained the results by using the classical tools for such problems including the Leray-Schauder nonlinear alternative theorem, etc. Recently, there have been some papers devoted to the theory of fractional differential equations with integral boundary conditions, see [9, 10, 11].
where , , is the Caputo fractional derivative and is a continuous function. The author established sufficient conditions for the existence of nonlinear fractional differential equations with an integral boundary problem.
where λ, μ are parameters and , , , , is the Riemann-Liouville fractional derivative, and are sign-changing continuous functions. Some existence results were given by using the Leray-Schauder nonlinear alternative theorem and the Krasnoselskii fixed point theorem.
where , , , is the Riemann-Liouville fractional derivative, and which may be singular at or (and) , is nonnegative, and is a continuous function. By means of the Guo-Krasnoselskii fixed point theorem, some results on the existence of positive solutions are obtained.
has , as a nontrivial solution.
The rest of this paper is organized as follows. In Section 2, we give some necessary notations, definitions and lemmas. In Section 3, we study the existence of solutions of (1.1) by the coincidence degree theory due to Mawhin . Finally, three examples are given to illustrate our results in Section 4.
Definition 2.1 
provided that the right-hand side is pointwise defined on .
Definition 2.2 
where , provided that the right-hand side is pointwise defined on .
Lemma 2.1 
where , .
Lemma 2.2 
Lemma 2.3 
is valid in the case , , .
is valid in the case , , .
Now let us recall some notation about the coincidence degree continuation theorem. Let Y, Z be real Banach spaces, be a Fredholm map of index zero and , be continuous projectors such that , and , . It follows that is invertible. We denote the inverse of this map by . If Ω is an open bounded subset of Y, the map N will be called L-compact on if is bounded and is compact.
for each ;
for each ;
, where is a continuous projection as above with and is any isomorphism.
Then the equation has at least one solution in .
3 Main results
In this paper, we always assume the following conditions.
Then problem (1.1) can be written by .
Lemma 3.1 The mapping is a Fredholm operator of index zero.
On the other hand, suppose that satisfies . Let , we can easily prove .
where and .
We can see .
For , , one has .
If , so we have .
As a result , we get .
Note that . Then L is a Fredholm mapping of index zero. □
So we have .
We define by . For , we have .
which shows that .
Lemma 3.2 
By Lemma 3.2 and standard arguments, the following lemma holds.
Lemma 3.3 is completely continuous.
- (H1) There exist functions , , such that for all ,
- (H2) There exists a constant such that for every , if , for all , then
- (H3) There exists a constant such that, for each satisfying , we have at least one of the following:or
Lemma 3.4 is bounded.
By (H2), there exists a constant such that .
It is easy to see and are bounded. We can conclude that is bounded. □
Lemma 3.5 is bounded.
It follows that . Here, is bounded. □
Lemma 3.6 is bounded.
Proof Let , so we have , , .
If , then . If , we have .
Thus from the first part of (H3), then . Here, is bounded. □
Remark 3.1 If the other parts of (H3) hold, then the set is bounded.
Theorem 3.1 Suppose (H1)-(H3) hold, then problem (1.1) has at least one solution in Y.
for every ,
for every ,
- (3)Let , where I is the identical operator. Via the homotopy property of degree, we obtain that
Applying Theorem 2.1, we conclude that has at least one solution in .
4 Some examples
To illustrate how our main results can be used in practice, we present three examples.
By the calculation, we can have and (H0)-(H4) are all satisfied. Hence, we obtain that (4.1) has at least one solution.
It is easy to see that (H0)-(H4) hold. By a simple calculation, we can have , . Thus it follows that problem (4.2) has at least one solution.
Corresponding to problem (1.1), we have that , . By a simple calculation, we can get , , then (H0)-(H4) hold. By Theorem 3.1, we obtain that (4.3) has at least one solution.
In this paper, we have obtained the existence of solutions for fractional differential equations with integral boundary conditions at resonance. By using the coincidence degree theory, we have found the existence results. Though the technique applied to establish the existence results for the problem is a standard one, yet its results are new in the context of integral boundary conditions. As applications, three examples are presented to illustrate the main results. In the future, we will consider the uniqueness of solutions for the fractional differential equations at resonance, and also can make further research on fractional differential equations with a singular integral boundary value problem at resonance.
The authors are grateful to the reviewers for their valuable suggestions and detailed comments which have improved the paper very much. This research is supported by the Natural Science Foundation of China (11371364) and 2013 Science and Technology Research Project of Beijing Municipal Education Commission (KM201310016001).
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