Journal of Function Spaces

Volume 2017, Article ID 1465623, 6 pages

https://doi.org/10.1155/2017/1465623

## Solvability of Some Two-Point Fractional Boundary Value Problems under Barrier Strip Conditions

^{1}College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China^{2}College 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 is complete with this norm.

Theorem 10. * is a Banach space.*

*Proof. *It is easy to verify that satisfies the norm axioms. The following proof is about the completeness of .

Let be a Cauchy sequence in :where and . ThenBecause is a Cauchy sequence of the Banach space ,Thus every term of the above formula converges to 0. Bywe know is a Cauchy sequence in . By the completeness of , there exists such that . The second term isTaking into account that is a Cauchy sequence in , there isThat is to say, is a Cauchy sequence in . Denote its limit as . Continue this process, and we can prove that sequences , all are Cauchy sequences. Denote their limits as , respectively.

Let , and then and . The completeness of is proved.

#### 4. Main Results

Denote

Theorem 11. *Suppose is continuous. If there exist four constants , and , such thatthen Problem (3)-(4) has a solution such that*

*Proof. *Consider the family of boundary value problemsDenoteDefine an operator aswhereThen standard arguments yield that is completely continuous. Clearly, the fixed point of operator is the solution for Problem (22)-(23).

Given , letWe will prove that, for some positive number , the map is admissible.

We claim that all the possible solutions of Problem (22)-(23) on have a priori bound without dependence on . Assume that the setsare not empty.

Choose . Assume there exists such thatTaking into account the fact that is continuous, we can take . The assumption on yields that and for . Consequently, is increasing on . Thuswhich is a contradiction to (29). So, there holdsand, in particular,a contradiction to (23). This shows that . The similar arguments can show that .

By and , there holds for , and then . On the other hand, by Lemma 5, for each , there exists such thatThus, . Finally (22) together with the continuity of and a priori estimations of and show that .

Set , and one hasThe above results indicate that is completely continuous and has no fixed point on and is admissible on for each . By Lemma 7, the constant map for is essential. Combined with Lemma 9, the topological transversality theorem, , is essential. So Problem (3)-(4) has a solution . Moreover, the above arguments show that the solution satisfiesThe proof is completed.

The next theorem can be proved by similar arguments.

Theorem 12. *Let be continuous. Suppose there exist four constants , , such that and :Then Problem (3)-(4) has a solution such that*

*Example 13. *Consider the following fractional BVPs:Let , , , and , and then the simple computation show that , , , andBy the use of Theorem 11, Problem (38) has a solution such that

#### 5. Conclusion

By the use of the topological transversality theorem, some existence results for a class of fractional differential equations with certain boundary value conditions are obtained. The main condition is sign condition which is easy to be satisfied and checked.

#### Conflicts of Interest

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

#### Acknowledgments

This work is supported by NSFC (11571207), the Taishan Scholar Project, and SDUST Graduate Innovation Project SDKDYC170343.

#### References

- S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 252, no. 2, pp. 804–812, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,”
*Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal*, vol. 72, no. 2, pp. 916–924, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - A. Atangana, “A novel model for the lassa hemorrhagic fever: deathly disease for pregnant women,”
*Neural Computing and Applications*, vol. 26, no. 8, pp. 1895–1903, 2015. View at Publisher · View at Google Scholar · View at Scopus - Z. Bai, X. Dong, and C. Yin, “Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions,”
*Boundary Value Problems*, Paper No. 63, 11 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Bai, S. Zhang, S. Sun, and C. Yin, “Monotone iterative method for a class of fractional differential equations,”
*Electronic Journal of Differential Equations*, vol. 6, pp. 1–8, 2016. View at Google Scholar - Z. Bai and Y. Zhang, “Solvability of fractional three-point boundary value problems with nonlinear growth,”
*Applied Mathematics and Computation*, vol. 218, no. 5, pp. 1719–1725, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Bai and Y. Zhang, “The existence of solutions for a fractional multi-point boundary value problem,”
*Computers & Mathematics with Applications*, vol. 60, no. 8, pp. 2364–2372, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. B. Bai and T. T. Qiu, “Existence of positive solution for singular fractional differential equation,”
*Applied Mathematics and Computation*, vol. 215, no. 7, pp. 2761–2767, 2009. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Google Scholar · View at MathSciNet - H. H. Dong, B. Y. Guo, and B. S. Yin, “Generalized fractional supertrace identity for Hamiltonian structure of NLS–MKdV hierarchy with self-consistent sources,”
*Analysis and Mathematical Physics*, vol. 6, no. 2, pp. 199–209, 2016. View at Publisher · View at Google Scholar · View at Scopus - X. Dong, Z. Bai, and S. Zhang, “Positive solutions to boundary value problems of -Laplacian with fractional derivative,”
*Boundary Value Problems*, Paper No. 5, 15 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - X. Dong, Z. Bai, and W. Zhang, “Positive solutions for nonlinear eigenvalue problems with conformable fractional differential derivatives,”
*Journal of Shandong University of Science and Technology (Natural Science)*, vol. 35, pp. 85–90, 2016 (Chinese). View at Google Scholar - Y. Hua and X. Yu, “On the ground state solution for a critical fractional Laplacian equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 87, pp. 116–125, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - U. Katugampola, “A new fractional derivative with classical properties,”
*Mathematics*, vol. 6, no. 4, pp. 1–15, 2014. View at Google Scholar - S. Liu, A. Debbouche, and J. Wang, “On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths,”
*Journal of Computational and Applied Mathematics*, vol. 312, no. 1, pp. 47–57, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Liu, M. Jia, and W. Ge, “The method of lower and upper solutions for mixed fractional four-point boundary value problem with -Laplacian operator,”
*Applied Mathematics Letters*, vol. 65, pp. 56–62, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - X. Liu, M. Jia, and B. Wu, “Existence and uniqueness of solution for fractional differential equations with integral boundary conditions,”
*Electronic Journal of Qualitative Theory of Differential Equations*, pp. 1–10, 2009. View at Publisher · View at Google Scholar · View at Scopus - X. Meng, S. Zhao, T. Feng, and T. Zhang, “Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis,”
*Journal of Mathematical Analysis and Applications*, vol. 433, no. 1, pp. 227–242, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - X. Shang and Z. Du, “Traveling waves in a generalized nonlinear dispersive-dissipative equation,”
*Mathematical Methods in the Applied Sciences*, vol. 39, no. 11, pp. 3035–3042, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Q. Song, X. Dong, Z. Bai, and B. Chen, “Existence for fractional Dirichlet boundary value problem under barrier strip conditions,”
*The Journal of Nonlinear Science and Applications*, vol. 10, pp. 3592–3598, 2017. View at Google Scholar - F. Wang, B. Chen, C. Lin, and X. Li, “Distributed adaptive neural control for stochastic nonlinear multi-agent systems,”
*IEEE Transaction on cybernetics*, vol. 47, no. 7, pp. 1795–1803, 2017. View at Google Scholar - G. Wang, B. Ahmad, L. Zhang, and J. J. Nieto, “Comments on the concept of existence of solution for impulsive fractional differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, no. 3, pp. 401–403, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Wang, Y. Zhou, and M. Feckan, “On recent developments in the theory of boundary value problems for impulsive fractional differential equations,”
*Computers & Mathematics with Applications*, vol. 64, no. 10, pp. 3008–3020, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Wang, “A numerical method for delayed fractional-order differential equations,”
*Journal of Applied Mathematics*, Article ID 256071, 7 pages, 2013. View at Publisher · View at Google Scholar - Z. Wang, X. Huang, and J. Zhou, “A numerical method for delayed fractional-order differential equations: based on G-L definition,”
*Applied Mathematics & Information Sciences*, vol. 7, no. 2, pp. 525–529, 2013. View at Publisher · View at Google Scholar · View at Scopus - Y. Xu, Z. Du, and L. Wei, “Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation,”
*Nonlinear Dynamics*, vol. 83, no. 1-2, pp. 65–73, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Yan, J. Yang, and X. Yu, “Equations involving fractional Laplacian operator: compactness and application,”
*Journal of Functional Analysis*, vol. 269, no. 1, pp. 47–79, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Yu, “Solutions of fractional Laplacian equations and their Morse indices,”
*Journal of Differential Equations*, vol. 260, no. 1, pp. 860–871, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. Zhang, Z. Bai, and S. Sun, “Extremal solutions for some periodic fractional differential equations,”
*Advances in Difference Equations*, Paper No. 179, 8 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Zou and Y. Cui, “Existence results for a functional boundary value problem of fractional differential equations,”
*Advances in Difference Equations*, 2013:233, 25 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - P. Kelevedjiev, “Existence of solutions for two-point boundary value problems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 22, no. 2, pp. 217–224, 1994. View at Publisher · View at Google Scholar · View at MathSciNet - P. S. Kelevedjiev and S. Tersian, “Singular and nonsingular first-order initial value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 366, no. 2, pp. 516–524, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Granas, R. B. Guenther, and J. W. Lee, “Applications of topological transversality to differential equations. I. Some nonlinear diffusion problems,”
*Pacific Journal of Mathematics*, vol. 89, no. 1, pp. 53–67, 1980. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus