Abstract
We consider the existence of mild solutions for fractional semilinear differential inclusions involving a nonconvex set-valued map in Banach spaces. First, we study the continuous property of the solution map for an auxiliary fractional differential equation. Then the main result is obtained by using this solution map, selection theorems from multivalued analysis and Schauder’s fixed point theorem. Finally an example to illustrate the applications of the main result is also given.
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Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 2009, 981728 (2009)
Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)
Liu, Z., Sun, J.: Nonlinear boundary value problems of fractional differential systems. Comput. Math. Appl. 64(4), 463–475 (2012)
Li, C.F., Luo, X.N., Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 59, 1363–1375 (2010)
Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838–1843 (2009)
Lv, L., Wang, J., Wei, W.: Existence and uniqueness results for fractional differential equations with boundary value conditions. Opusc. Math. 31(4), 629–643 (2011)
Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 72(2), 916–924 (2010)
Shu, X.-B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 74, 2003–2011 (2011)
Shu, X.-B., Wang, Q.: The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1<α<2. Comput. Math. Appl. 64, 2100–2110 (2012)
El-Sayed, A.M.A., Ibrahim, A.G.: Multivalued fractional differential equations. Appl. Math. Comput. 68(1), 15–25 (1995)
Ouahab, A.: Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69(11), 3877–3896 (2008)
Chang, Y.-K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009)
Ahmad, B., Ntouyas, S.K.: Some existence results for boundary value problems of fractional differential inclusions with non-separated boundary conditions. Electron. J. Qual. Theory Differ. 71, 1–17 (2010)
Cernea, A.: On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. J. Appl. Math. Comput. 38, 133–143 (2012)
Cernea, A.: A note on the existence of solutions for some boundary value problems of fractional differential inclusions. Fract. Calc. Appl. Anal. 15(2), 183–194 (2012)
Cernea, A.: Some remarks on a fractional differential inclusion with non-separated boundary conditions. Electron. J. Qual. Theory Differ. 45, 1–14 (2011)
Ahmad, B., Ntouyas, S.K.: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15(3), 362–382 (2012)
Agarwal, R.P., Belmekki, M., Benchohra, M.: Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative. DCDIS Ser. A: Math. Anal. 17, 347–361 (2010)
Zhang, Z., Liu, B.: Existence results of nondensely defined fractional evolution differential inclusions. J. Appl. Math. 2012, 316850 (2012)
Wang, J., Zhou, Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal., Real World Appl. 12(6), 3642–3653 (2011)
Hu, S.C., Papageorgiou, N.S.: Handbook of Multivalued Analysis: vol. I: Theory. Kluwer Academic, Dordrecht (1997)
Himmelberg, C.J.: Measurable relations. Fundam. Math. 87, 53–72 (1975)
Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)
Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 4465–4475 (2010)
Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. ZAMM Z. Angew. Math. Mech. 68(11), 535–544 (1986)
Tolstonogov, A.A.: Scorza-Dragoni’s theorem for multi-valued map**s with variable domain of definition. Mat. Zametki 48(5), 109–120 (1990). English transl.: Math. Notes 48, 1151–1158 (1990)
Zhu, J.: On the solution set of differential inclusions in Banach space. J. Differ. Equ. 93(2), 213–237 (1991)
Tolstonogov, A.A.: Relaxation in control systems of subdifferential type. Izv. Math. 70(1), 121–152 (2006)
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The authors are grateful to anonymous referees for their constructive comments and suggestions which led to improvement of the original manuscript.
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Liu, X., Liu, Z. Existence results for fractional semilinear differential inclusions in Banach spaces. J. Appl. Math. Comput. 42, 171–182 (2013). https://doi.org/10.1007/s12190-012-0634-0
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DOI: https://doi.org/10.1007/s12190-012-0634-0