# SOLUTIONS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEM WITH MIXED NONLINEARITIES

• Zhang, Ziheng (Department of Mathematics Tianjin Polytechnic University)
• Published : 2016.09.30

#### Abstract

In this paper we investigate the existence of nontrivial solutions for the following fractional boundary value problem (FBVP) $$\{_tD_T^{\alpha}(_0D_t^{\alpha}u(t))={\nabla}W(t,u(t)),\;t{\in}[0,T],\\u(0)=u(T)=0,$$ where ${\alpha}{\in}(1/2,1)$, $u{\in}{\mathbb{R}}^n$, $W{\in}C^1([0,T]{\times}{\mathbb{R}}^n,{\mathbb{R}})$ and ${\nabla}W(t,u)$ is the gradient of W(t, u) at u. The novelty of this paper is that, when the nonlinearity W(t, u) involves a combination of superquadratic and subquadratic terms, under some suitable assumptions we show that (FBVP) possesses at least two nontrivial solutions. Recent results in the literature are generalized and significantly improved.

#### Acknowledgement

Supported by : National Natural Science Foundation of China

#### References

1. R. Agarwal, M. Benchohra, and S. Hamani, Boundary value problems for fractional differential equations, Georgian Math. J. 16 (2009), no. 3, 401-411.
2. O. Agrawal, J. Tenreiro Machado, and J. Sabatier, Fractional Derivatives and Their Application: Nonlinear dynamics, Springer-Verlag, Berlin, 2004.
3. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal. 14 (1973), no. 4, 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
4. Z. B. Bai and H. S. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2005), no. 2, 495-505. https://doi.org/10.1016/j.jmaa.2005.02.052
5. D. G. de Figueiredo, J. P. Gossez, and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Functional Anal. 199 (2003), no. 2, 452-467. https://doi.org/10.1016/S0022-1236(02)00060-5
6. R. Hilfer, Applications of Fractional Calculus in Physics, World Science, Singapore, 2000.
7. M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems, J. Differential Equations 219 (2005), no. 2, 375-389. https://doi.org/10.1016/j.jde.2005.06.029
8. W. H. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal. 74 (2011), no. 5, 1987-1994. https://doi.org/10.1016/j.na.2010.11.005
9. F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2011), no. 3, 1181-1199. https://doi.org/10.1016/j.camwa.2011.03.086
10. F. Jiao and Y. Zhou, Existence results for fractional boundary value problem via critical point theory, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), no. 4, 1250086, 17 pp.
11. A. Kilbas, H. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol 204, Singapore, 2006.
12. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
13. K. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
14. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
15. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in: CBMS Reg. Conf. Ser. in. Math., vol. 65, American Mathematical Society, Provodence, RI, 1986.
16. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, PA, 1993.
17. M. Schechter, Linking Methods in Critical Point Theory, Birkhauser, Boston, 1999.
18. C. Torres, Mountain pass solutions for a fractional boundary value problem, J. Fract. Cal. Appl. 5 (2014), no. 1, 1-10.
19. W. Z. Xie, J. Xiao, and Z. G. Luo, Existence of solutions for fractional boundary value problem with nonlinear derivative dependence, Abstr. Appl. Anal. 2014 (2014), Art. ID 812910, 8 pp.
20. S. Q. Zhang, Existence of a solution for the fractional differential equation with nonlinear boundary conditions, Comput. Math. Appl. 61 (2011), no. 4, 1202-1208. https://doi.org/10.1016/j.camwa.2010.12.071
21. Z. H. Zhang and J. Li, Variational approach to solutions for a class of fractional boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2015 (2015), no. 11, 10 pp. https://doi.org/10.1186/s13662-014-0345-y