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Erschienen in: Journal of Applied Mathematics and Computing 1-2/2013

01.07.2013 | Applied mathematics

Existence results for fractional semilinear differential inclusions in Banach spaces

verfasst von: Xiaoyou Liu, Zhenhai Liu

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2013

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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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Literatur
1.
Zurück zum Zitat 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) MATHCrossRef 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) MATHCrossRef
2.
Zurück zum Zitat Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) MATH Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) MATH
3.
Zurück zum Zitat 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) MathSciNet 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) MathSciNet
4.
Zurück zum Zitat 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) MathSciNetMATHCrossRef 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) MathSciNetMATHCrossRef
5.
Zurück zum Zitat Liu, Z., Sun, J.: Nonlinear boundary value problems of fractional differential systems. Comput. Math. Appl. 64(4), 463–475 (2012) MathSciNetMATHCrossRef Liu, Z., Sun, J.: Nonlinear boundary value problems of fractional differential systems. Comput. Math. Appl. 64(4), 463–475 (2012) MathSciNetMATHCrossRef
6.
Zurück zum Zitat 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) MathSciNetMATHCrossRef 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) MathSciNetMATHCrossRef
7.
Zurück zum Zitat 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) MathSciNetMATHCrossRef 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) MathSciNetMATHCrossRef
8.
Zurück zum Zitat 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) MathSciNetMATHCrossRef 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) MathSciNetMATHCrossRef
9.
10.
Zurück zum Zitat Shu, X.-B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 74, 2003–2011 (2011) MathSciNetMATHCrossRef Shu, X.-B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 74, 2003–2011 (2011) MathSciNetMATHCrossRef
11.
Zurück zum Zitat 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) MathSciNetCrossRef 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) MathSciNetCrossRef
12.
13.
Zurück zum Zitat Ouahab, A.: Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69(11), 3877–3896 (2008) MathSciNetMATHCrossRef Ouahab, A.: Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69(11), 3877–3896 (2008) MathSciNetMATHCrossRef
14.
Zurück zum Zitat Chang, Y.-K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009) MathSciNetMATHCrossRef Chang, Y.-K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009) MathSciNetMATHCrossRef
15.
Zurück zum Zitat 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) CrossRef 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) CrossRef
16.
Zurück zum Zitat Cernea, A.: On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. J. Appl. Math. Comput. 38, 133–143 (2012) MathSciNetCrossRef Cernea, A.: On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. J. Appl. Math. Comput. 38, 133–143 (2012) MathSciNetCrossRef
17.
Zurück zum Zitat 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) MathSciNet 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) MathSciNet
18.
Zurück zum Zitat Cernea, A.: Some remarks on a fractional differential inclusion with non-separated boundary conditions. Electron. J. Qual. Theory Differ. 45, 1–14 (2011) Cernea, A.: Some remarks on a fractional differential inclusion with non-separated boundary conditions. Electron. J. Qual. Theory Differ. 45, 1–14 (2011)
19.
Zurück zum Zitat Ahmad, B., Ntouyas, S.K.: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15(3), 362–382 (2012) MathSciNet Ahmad, B., Ntouyas, S.K.: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15(3), 362–382 (2012) MathSciNet
20.
Zurück zum Zitat 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) MathSciNetMATH 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) MathSciNetMATH
21.
Zurück zum Zitat Zhang, Z., Liu, B.: Existence results of nondensely defined fractional evolution differential inclusions. J. Appl. Math. 2012, 316850 (2012) Zhang, Z., Liu, B.: Existence results of nondensely defined fractional evolution differential inclusions. J. Appl. Math. 2012, 316850 (2012)
22.
Zurück zum Zitat Wang, J., Zhou, Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal., Real World Appl. 12(6), 3642–3653 (2011) MathSciNetMATHCrossRef Wang, J., Zhou, Y.: Existence and controllability results for fractional semilinear differential inclusions. Nonlinear Anal., Real World Appl. 12(6), 3642–3653 (2011) MathSciNetMATHCrossRef
23.
Zurück zum Zitat Hu, S.C., Papageorgiou, N.S.: Handbook of Multivalued Analysis: vol. I: Theory. Kluwer Academic, Dordrecht (1997) Hu, S.C., Papageorgiou, N.S.: Handbook of Multivalued Analysis: vol. I: Theory. Kluwer Academic, Dordrecht (1997)
25.
Zurück zum Zitat Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010) MathSciNetMATHCrossRef Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010) MathSciNetMATHCrossRef
26.
27.
Zurück zum Zitat Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. ZAMM Z. Angew. Math. Mech. 68(11), 535–544 (1986) MathSciNet Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. ZAMM Z. Angew. Math. Mech. 68(11), 535–544 (1986) MathSciNet
28.
Zurück zum Zitat Tolstonogov, A.A.: Scorza-Dragoni’s theorem for multi-valued mappings with variable domain of definition. Mat. Zametki 48(5), 109–120 (1990). English transl.: Math. Notes 48, 1151–1158 (1990) MathSciNet Tolstonogov, A.A.: Scorza-Dragoni’s theorem for multi-valued mappings with variable domain of definition. Mat. Zametki 48(5), 109–120 (1990). English transl.: Math. Notes 48, 1151–1158 (1990) MathSciNet
29.
Zurück zum Zitat Zhu, J.: On the solution set of differential inclusions in Banach space. J. Differ. Equ. 93(2), 213–237 (1991) MATHCrossRef Zhu, J.: On the solution set of differential inclusions in Banach space. J. Differ. Equ. 93(2), 213–237 (1991) MATHCrossRef
Metadaten
Titel
Existence results for fractional semilinear differential inclusions in Banach spaces
verfasst von
Xiaoyou Liu
Zhenhai Liu
Publikationsdatum
01.07.2013
Verlag
Springer-Verlag
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2013
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-012-0634-0

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