## Recent Development on Nonlinear Methods in Function Spaces and Applications in Nonlinear Fractional Differential Equations

View this Special IssueResearch Article | Open Access

Yumei Zou, Guoping He, "The Existence of Solutions to Integral Boundary Value Problems of Fractional Differential Equations at Resonance", *Journal of Function Spaces*, vol. 2017, Article ID 2785937, 7 pages, 2017. https://doi.org/10.1155/2017/2785937

# The Existence of Solutions to Integral Boundary Value Problems of Fractional Differential Equations at Resonance

**Academic Editor:**Xinguang Zhang

#### Abstract

This paper deals with the integral boundary value problems of fractional differential equations at resonance. By Mawhin’s coincidence degree theory, we present some new results on the existence of solutions for a class of differential equations of fractional order with integral boundary conditions at resonance. An example is also included to illustrate the main results.

#### 1. Introduction

In this paper, we are concerned with the following integral boundary value problem for nonlinear fractional differential equation: where , is the standard Riemann-Liouville differentiation, , and satisfies Carathéodory conditions; is right continuous on and left continuous at ; denotes the Riemann-Stieltjes integrals of with respect to . Our problem are at resonance, in the sense that, under the integral boundary conditions, we study the linear equation , , which has nontrivial solutions.

Recently, fractional differential equations have received considerable attentions not only because of a generalization of ordinary differential equations but also because they have played a significant role in science, engineering, economy, and other fields; see, for example, [1–3].

When , problem (1) is nonresonant. In [4], the authors studied the existence of positive solutions for the nonresonant case by Krasnosel’skii’s fixed point theorem. In [5], the author investigated the uniqueness of solutions for the nonresonant case by use of the -positive operator under a Lipschitz condition on .

In present, many papers are devoted to the integral boundary value problem for fractional differential equation under nonresonance conditions; see [4–20]. On the other hand, there are some papers studying integral boundary value problem for differential equation under resonant conditions; we refer the reader to [21–29].

Motivated by the above results, in this paper, we consider the existence of solutions for the resonance integral boundary value problem (1) under nonlinear growth restriction of . Our method is based upon the coincidence degree theorem of Mawhin.

Now, we recall the essentials of the coincidence degree theory. Let and be real Banach spaces, and let be a Fredholm operator of index zero. If and are continuous projectors such that , , , and , then the inverse operator of exists and is denoted by (generalized inverse operator of ). If is an open bounded subset of such that , the mapping will be called -compact on , if is bounded and is compact.

The abstract equation is shown to be solvable in view of Theorem IV.13 [30].

Theorem 1 (see [30]). *Let be a Fredholm operator of index zero and let be -compact on . Assume the following conditions are satisfied:*(i)* for every .*(ii)* for every .*(iii)*, where is a projector as above with .**Then the equation has at least one solution in .*

Throughout this paper, we always suppose that, ; satisfies the Carathéodory conditions; that is, is measurable for each fixed , is continuous for a.e. , and for each , there exists such that for all , , and and for a.e. .

#### 2. Preliminaries and Lemmas

In this section, first we provide recall some necessary basic definitions and lemmas of the fractional calculus theory, which will be used in this paper. For more details, we refer to books [1–3] for details.

*Definition 2 (see [1]). *The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .

*Definition 3 (see [1]). *The Riemann-Liouville fractional derivative of order of a continuous function is given by where , provided that the right-hand side is pointwise defined on .

Lemma 4 (see [1]). *Assume that . If , , , then *

Lemma 5 (see [1]). *Assume that , . Then one has where .*

Lemma 6 (see [4]). *Let and , then the unique solution ofis given by where is Green’s function given by *

Lemma 7 (see [4]). *The function defined by (8) satisfies and .*

We use the classical Banach space with the norm and with the norm . We also use the Banach space with the norm .

Define and as follows: where .

Then integral boundary value problems (1) can be rewritten as follows:

Lemma 8. *The operator is a Fredholm operator of index zero.*

*Proof. *Firstly, we show that By Lemma 5, means that . It follows from that . That is, .

Now we prove In fact, if and , then by Lemma 5, By the boundary condition, we obtain , The above equalities imply that On the other hand, if satisfies , let By a simple computation, we can obtain that and ; that is, .

Clearly, and is closed. It follows from that where . In fact, for each , we have which shows that . This together with implies that . Note that and thus . Therefore, is a Fredholm operator of index zero. The proof is completed.

Next, define the projections by and byClearly, and .

The generalized inverse operator of , can be defined by

In fact, if , then

For , , we have Then by Lemma 6, we obtain whenever .

By a standard method, we obtain the following lemma.

Lemma 9. * is completely continuous.*

Lemma 10. *For , Moreover, *

*Proof. *It is easy to see that Then by Lemma 7, we obtain It follows that The proof is completed.

#### 3. Main Results

In this section, we will use Theorem 1 to prove the existence of solutions to IBVP (1). To obtain our main theorem, we need the following conditions:There exist functions such that, for all , There exists a constant such that either for each or for each , There exists a constant such that if for all , then .There exists a constant such that if for all , then .

Theorem 11. *Suppose ()–() hold. Then IBVP (1) has at least one solution in , provided *

*Proof. *Set Take . Since , so , , and hence Thus, from (), there exists such that Noticing thatwe obtainObserve that for all . Then by Lemma 10, Using (39) and (41), we have so that that is, for all , Applying (), we have Therefore, is bounded.

Let Then for , for some . So, . By (), we have . Therefore, is bounded.

We define the isomorphism by If (32) holds, then let For , If , then . Otherwise, if , in view of (), one has which contradicts . Thus is bounded.

If (33) holds, then define the set where is as above. Similar to the above argument, we can show that is bounded too.

Next, we will prove that all the assumptions of Theorem 1 are satisfied. Let be given any bounded open subset of such that . By Lemma 9, is compact; thus is -compact on . Clearly, assumptions (i) and (ii) of Theorem 1 are fulfilled.

At last, we will prove that (iii) of Theorem 1 is satisfied. Let . According to above argument, we know Thus, by the homotopy property of degree Then by Theorem 1, has at least one solution in , so that IBVP (1) has a solution. The proof is completed.

Theorem 12. *Suppose ()–() and () hold. Then IBVP (1) has at least one solution in , provided *

*Proof. *As in the proof of Theorem 11, , implies from (), there exist such thatNoticing that we obtainHence, we have In view of , . From this together with Lemma 5, for , we have Thus, by (58), Considering (57), (58), and (60), and applying , we get Therefore, is bounded. The rest of the proof repeats that of Theorem 11.

*Example 13. *Consider the IBVP Let and , then and ; thus () is satisfied. Let so that , , and , which verifies and (34).

Taking , we have , Hence holds. Finally, taking , when , then we obtain ; that is, condition () is satisfied. It follows from Theorem 11 that IBVP (62) has at least one solution.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The project was supported by the National Natural Science Foundation of China (11371221, 11571207, and 51774197).

#### References

- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Amsterdam, Elsevier, 2006. View at: MathSciNet - V. Lakshmikantham, S. Leela, and J. Vasundhara,
*Theory of Fractional Dynamic Systems*, Cambridge Academic Publishers, Cambridge, 2009. - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, Wiley, New York, NY, USA, 1993. View at: MathSciNet - X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1263–1274, 2012. View at: Publisher Site | Google Scholar | MathSciNet - Y. Cui, “Uniqueness of solution for boundary value problems for fractional differential equations,”
*Applied Mathematics Letters*, vol. 51, pp. 48–54, 2016. View at: Publisher Site | Google Scholar | MathSciNet - R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,”
*Acta Applicandae Mathematicae*, vol. 109, no. 3, pp. 973–1033, 2010. View at: Publisher Site | Google Scholar | MathSciNet - B. Ahmad and A. Alsaedi, “Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations,”
*Fixed Point Theory and Applications*, vol. 2010, Article ID 364560, 2010. View at: Publisher Site | Google Scholar - Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 2, pp. 916–924, 2010. View at: Publisher Site | Google Scholar | MathSciNet - H. Li and Y. Liu, “On the uniqueness of the positive solution for a second-order integral boundary value problem with switched nonlinearity,”
*Applied Mathematics Letters*, vol. 24, no. 12, pp. 2201–2205, 2011. View at: Publisher Site | Google Scholar - H. Li and Y. Liu, “On sign-changing solutions for a second-order integral boundary value problem,”
*Computers and Mathematics with Applications*, vol. 62, pp. 651–656, 2011. View at: Publisher Site | Google Scholar | MathSciNet - L. Liu, X. Hao, and Y. Wu, “Positive solutions for singular second order differential equations with integral boundary conditions,”
*Mathematical and Computer Modelling*, vol. 57, no. 3-4, pp. 836–847, 2013. View at: Publisher Site | Google Scholar | MathSciNet - L. Liu, F. Sun, X. Zhang, and Y. Wu, “Bifurcation analysis for a singular differential system with two parameters via to degree theory,”
*Nonlinear Analysis: Modelling and Control*, vol. 22, no. 1, pp. 31–50, 2017. View at: Google Scholar | MathSciNet - L. Liu, H. Li, C. Liu, and Y. Wu, “Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary conditions,”
*Journal of Nonlinear Sciences and Applications. JNSA*, vol. 10, no. 1, pp. 243–262, 2017. View at: Publisher Site | Google Scholar | MathSciNet - M. ur Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,”
*Applied Mathematics Letters*, vol. 23, no. 9, pp. 1038–1044, 2010. View at: Publisher Site | Google Scholar - F. Sun, L. Liu, X. Zhang, and Y. Wu, “Spectral analysis for a singular differential system with integral boundary conditions,”
*Mediterranean Journal of Mathematics*, vol. 13, no. 6, pp. 4763–4782, 2016. View at: Publisher Site | Google Scholar | MathSciNet - S. Zhang and X. Su, “The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order,”
*Comput. Math. Appl*, vol. 62, pp. 1269–1274, 2011. View at: Google Scholar | MathSciNet - X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,”
*Applied Mathematics and Computation*, vol. 218, no. 17, pp. 8526–8536, 2012. View at: Publisher Site | Google Scholar | MathSciNet - X. Zhang, L. Liu, and Y. Wu, “Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations,”
*Journal of Nonlinear Sciences and Applications. JNSA*, vol. 10, no. 7, pp. 3364–3380, 2017. View at: Publisher Site | Google Scholar | MathSciNet - D. Zhao and Y. Liu, “Positive solutions for a class of fractional differential coupled system with integral boundary value conditions,”
*Journal of Nonlinear Sciences and Applications*, vol. 9, pp. 2922–2942, 2016. View at: Google Scholar | MathSciNet - Y. Zou and G. He, “On the uniqueness of solutions for a class of fractional differential equations,”
*Applied Mathematics Letters*, vol. 74, pp. 68–73, 2017. View at: Google Scholar | MathSciNet - Z. Bai and Y. Zhang, “The existence of solutions for a fractional multi-point boundary value problem,”
*Computers and Mathematics with Applications*, vol. 60, no. 8, pp. 2364–2372, 2010. View at: Publisher Site | Google Scholar | MathSciNet - Y. Cui and Y. Zou, “Existence of solutions for second-order integral boundary value problems,”
*Lithuanian Association of Nonlinear Analysts. Nonlinear Analysis: Modelling and Control*, vol. 21, no. 6, pp. 828–838, 2016. View at: Publisher Site | Google Scholar | MathSciNet - Y. Cui, “Existence of solutions for coupled integral boundary value problem at resonance,”
*Publicationes Mathematicae*, vol. 89, no. 1-2, pp. 73–88, 2016. View at: Publisher Site | Google Scholar | MathSciNet - N. Kosmatov and W. Jiang, “Resonant functional problems of fractional order,”
*Chaos, Solitons and Fractals*, vol. 91, pp. 573–579, 2016. View at: Publisher Site | Google Scholar | MathSciNet - Q. Sun and Y. Cui, “Existence results for conjugate boundary-value problems with integral boundary conditions at resonance with dim ker L= 2,”
*Boundary Value Problems*, vol. 2017, no. 1, article no. 29, 2017. View at: Publisher Site | Google Scholar - Q. Sun and Y. Cui, “Solvability of (k, n - k) Conjugate Boundary Value Problems with Integral Boundary Conditions at Resonance,”
*Journal of Function Spaces*, vol. 2016, Article ID 3454879, 2016. View at: Publisher Site | Google Scholar - Z. Zhao and J. Liang, “Existence of solutions to functional boundary value problem of second-order nonlinear differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 373, no. 2, pp. 614–634, 2011. View at: Publisher Site | Google Scholar | MathSciNet - Y. Zou, L. Liu, and Y. Cui, “The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 314083, 2014. View at: Publisher Site | Google Scholar - Y. Zou and Y. Cui, “Existence results for a functional boundary value problem of fractional differential equations,”
*Advances in Difference Equations*, vol. 2013, article no. 233, 2013. View at: Publisher Site | Google Scholar - J. Mawhin, in
*Proceedings of the*, vol. 40 of*NSFCBMS Regional Conference Series in Mathematics*, American Mathematical Society. View at: MathSciNet

#### Copyright

Copyright © 2017 Yumei Zou and Guoping He. 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.