Existence of Solutions for a Nonlinear Fractional Differential Equation with Integral Boundary Condition

Authors: Meng Hu, Lili Wang

Abstract:

This paper deals with a nonlinear fractional differential equation with integral boundary condition of the following form:  Dαt x(t) = f(t, x(t),Dβ t x(t)), t ∈ (0, 1), x(0) = 0, x(1) = 1 0 g(s)x(s)ds, where 1 < α ≤ 2, 0 < β < 1. Our results are based on the Schauder fixed point theorem and the Banach contraction principle.

Keywords: Fractional differential equation, Integral boundary condition, Schauder fixed point theorem, Banach contraction principle.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1335374

References:

[1] D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal. TMA. vol. 69, no. 11, pp. 3692-3705, 2008.
[2] Z. Bai and H. L¨u, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. vol. 311, no. 2, pp. 495-505, 2005.
[3] Y.-K. Chang and J. J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Modelling. vol. 49, no. 3-4, pp. 605-609, 2009.
[4] V. Gafiychuk, B. Datsko, and V. Meleshko, Mathematical modeling of time fractional reactiondiffusion systems, J. Comput. Appl. Math. vol. 220, no. 1-2, pp. 215-225, 2008.
[5] V. Daftardar-Gejji, Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal. Appl. vol. 302, no. 1, pp. 56-64, 2005.
[6] V. Daftardar-Gejji and S. Bhalekar, Boundary value problems for multiterm fractional differential equations, J. Math. Anal. Appl. vol. 345, no. 2, pp. 754-765, 2008.
[7] M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstract Appl. Analysis, vol. 2007, Article ID 10368, 8 pages, 2007.
[8] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands,2006.
[9] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
[10] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
[11] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Elec. J. Diff. Equ. vol. 2006, no. 36, pp. 1-12, 2006.
[12] B. Ahmad, A. Alsaedi, and B. S. Alghamdi, Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear Anal. RWA. vol. 9, no. 4, pp. 1727-1740, 2008.
[13] B. Ahmad and A. Alsaedi, Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions, Nonlinear Anal. RWA. vol. 10, no. 1, pp. 358-367, 2009.
[14] Z. Yang, Existence of nontrivial solutions for a nonlinear Sturm- Liouville problem with integral boundary conditions, Nonlinear Anal. TMA. vol. 68, no. 1, pp. 216-225, 2008.
[15] A. Boucherif, Second-order boundary value problems with integral boundary conditions, Nonlinear Anal. TMA. vol. 70, no. 1, pp. 364-371,2009.