Nonlocal boundary value problems for nonlinear toppled system of fractional differential equations

Abstract

The aim of this paper is to study multiplicity results for the solutions of a coupled system of fractional differential equations. The problem under consideration is subjected to nonlocal boundary conditions involving Riemann-Liouville integrals and derivatives of fractional order. Necessary and sufficient conditions are established for the existence of at least one and more solutions by using various fixed point theorems of cone type. Moreover sufficient conditions for uniqueness is also discussed by using a concave type operator for the considered problem. Further, the conditions are also provided under which the considered system has no positive solution. The results are demonstrated by providing several examples.

Keywords

• Fractional differential equations,Coupled system,Concave and contractions operator

References

1. [1] Y. Cui, and Y. Zou, Existence results and the monotone iterative technique for nonlinear fractional differential systems with coupled four-point Boundary Value Problems, Abst. Appl. Anal., Art. ID 242591, 6 pp, 2014.
2. [2] K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985.
3. [3] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
4. [4] X. Han and T. Wang, The existence of solutions for a nonlinear fractional multipoint boundary value problem at resonance, Int. J. Differ. Equ. Art., ID 401803, 14 pp., 2011.
5. [5] M. Jleli and B. Samet, Existence of positive solutions to a coupled system of fractional differential equation, Math. Methods Appl. Sci. 38 (6), 1014-1031, 2015.
6. [6] R.A.Khan and K.Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value problems, Commun. Appl. Anal. 19, 515−526, 2015.
7. [7] R.A. Khan, Existence and approximation of solutions to three-point boundary value problems for fractional differential equations, Electron. J. Qual. Theory Differ. Equ. 58, 25−35, 2011.
8. [8] R.A. Khan and M. Rehman, Existence of multiple positive solutions for a general system of fractional differential equations, Commun. Appl. Nonlinear Anal. 18, (2011) 25−35.

9. [9] A.A. Kilbas, H. Srivastava and J. Trujillo, Theory and application of fractional differential equations, North Holland Mathematics Studies, vol. 204, Elseveir, Amsterdam, 2006.
10. [10] R.W. Leggett and L.R Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28, 673−688, 1979.
11. [11] Y. Liu, Solvability of multi-point boundary value problems for multiple term Riemann- Liouville fractional differential equations, Comput. Math. Appl. 64, 413−431, 2012.
12. [12] R. Ma, A survey On nonlocal boundary value problems, Applied Mathematics E-Notes. 7, 257−279, 2007.
13. [13] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
14. [14] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
15. [15] I. Podlubny, Fractional differential equations, Academic press, New York, 1993.
16. [16] P.H. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9, 536−548, 1971.
17. [17] B.P. Rynne, Spectral properties and nodal solutions for second-order, m-point, boundary value problems, Nonlinear Anal. 67(12), 3318−3327, 2007.
18. [18] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, (Springer- Verlag, Berlin, 2007).
19. [19] K. Shah and R.A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions, Differ. Equ. Appl. 7(2), 245−262, 2015.
20. [20] K. Shah, A. Ali and R.A. Khan, Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems, Boundary Value Problems, 2016:43, 12 pages, 2016.
21. [21] K. Shah, H. Khalil and R.A. Khan, Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Prog. Frac. Differ. Equ. Appl. 2(1), 1−10, 2016.
22. [22] Y. Shang, Vulnerability of networks: Fractional percolation on random graphs, Physical Review E. 89, 234−245, 2014.
23. [23] H. Wang, Y.G. Yu, G.G. Wen, S. Zhang, and J.Z. Yu, Global stability analysis of fractional-order Hopfield neural networks, Neurocomputing, 154, 15−23, 2015.
24. [24] X. Xu, D. Jiang, and C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear. Anal. 71, 4676−4688, 2009.
25. [25] E. Zeidler, Non linear fruntional analysis and its applications, Springer, New York, 1986.
26. [26] Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. 11, 4465−4475, 2010.
27. [27] Y. Zhou, F. Jiao, and J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal. 71, 3249−3256, 2009.