New results on the existence of solutions of boundary value problems for singular fractional differential systems with impulse effects

Open access

Abstract

Results on the existence of solutions to a new class of impulsive singular fractional differential systems with multiple base points are established. The assumptions imposed on the nonlinear- ities, see ((C) and (D) in Theorem 3.1), are weaker than known ones, (i.e., (A) in Introduction section). The analysis relies on the well known fixed point theorems. An example is given to illustrate the efficiency of the main theorems. The investigation shows that these results and methods are helpful for study in the nonlinear area and the numerical simulation, especially for study in the the numerical solution of a fractional differential equation with multiple base points with or without impulse effects. A section "Conclusions" is given with future work research directions.

[1] M. Belmekki, Juan J. Nieto, R. Rodriguez-Lopez, Existence of periodic solution for a nonlin- ear fractional differential equation, Boundary Value Problems, 2009(2009), Article ID 324561, doi:10.1155/2009/324561.

[2] J. J. Nieto, Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Applied Mathematics Letters, 23(2010)1248-1251.

[3] J. J. Nieto, Comparison results for periodic boundary value problems of fractional differential equations, Fractional Differential Equations, 1(2011)99-104.

[4] Z. Wei, W. Dong, J. Che, Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative, Nonlinear Analysis: Theory, Methods and Applications, 73(2010)3232-3238.

[5] Z.Wei, W. Dog, Periodic boundary value problems for Riemann-Liouville fractional differential equations, Electronic Journal of Qualitative Theory of Differential Equations, 87(2011)1-13.

[6] A. A. Kilbas, and J.J. Trujillo, Differential equations of fractional order: methods, results and problems-I, Applicable Analysis, 78(2001)153{192.

[7] A. Arara, M. Benchohra, N. Hamidi, and J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Analysis, 72(2010)580-586.

[8] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of frational differential equations, Elsevier Science B. V. Amsterdam, 2006.

[9] S. Z. Rida, H.M. El-Sherbiny, and A. Arafa, On the solution of the fractional nonlinear Schrodinger equation, Physics Letters A, 372(2008)553-558.

[10] A. M. Nakhushev, The Sturm-Liouville problem for a second order ordinary differential equa- tions with fractional derivatives in the lower terms, Dokl. Akad. Nauk SSSR, 234(1977)308-311.

[11] S. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252(2000)804{812.

[12] E. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear frac- tional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 3(2008)1-11.

[13] R. Dehghant and K. Ghanbari, Triple positive solutions for boundary value problem of a non- linear fractional differential equation, Bulletin of the Iranian Mathematical Society, 33(2007)1- 14.

[14] Y. Liu, Positive solutions for singular FDES, U.P.B. Sci. Series A, 73(2011)89-100.

[15] S. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equation, Electron. J. Diff. Eqns. 36(2006)1-12.

[16] Y. Zhao, S. Sun, Z. Han, M. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Applied Mathematics and Computation, 217(2011)6950-6958.

[17] M. Benchohra, J. Graef, S. Hamani, Existence results for boundary value problems with non- linear frational differential equations, Applicable Analysis, 87(2008)851-863.

[18] X. Wang, C. Bai, Periodic boundary value problems for nonlinear impulsive fractional differen- tial equations, Electronic Journal of Qualitative Theory and Differential Equations, 3(2011)1- 13.

[19] J. Mawhin, Topological degree methods in nonlinear boundary value problems, in: NSFCBMS Regional Conference Series in Math., American Math. Soc. Providence, RI, 1979.

[20] G. L. Karakostas, Positive solutions for the Φ-Laplacian when Φ is a sup-multiplicative-like function, Electron. J. Diff. Eqns., Vol. 68(2004)1-12.

[21] K. S. Miller, S. G. Samko, Completely monotonic functions, Integr. Transf. Spec. Funct., 12(2001)389-402.

[22] M. Belmekki, J. Nieto, R. Rodrguez-Lopez, Existence of solution to a periodic boundary value problem for a nonlinear impulsive fractional differential equation, Electron. J. Qual. Theory Differ. Equ. 2014, No. 16, 1-27.