Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter February 10, 2015

New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses

Saïd Abbas, Mouffak Benchohra and Mohamed Abdalla Darwish

Abstract

In this work, we discuss the existence and Ulam's type stability concepts for a class of partial functional differential inclusions with not instantaneous impulses and a nonconvex valued right hand side in Banach spaces. An example is also provided to illustrate our results.

MSC 2010: 26A33; 34A37; 34D10

References

[1] S. Abbas, D. Baleanu and M. Benchohra, Global attractivity for fractional order delay partial integro-differential equations. Adv. Difference Equ. 2012 (2012), 19 pages; doi:10.1186/1687-1847-2012-62.Search in Google Scholar

[2] S. Abbas and M. Benchohra, Fractional order partial hyperbolic differential equations involving Caputo's derivative. Stud. Univ. Babe.s-Bolyai Math, 57, No 4 (2012), 469-479.Search in Google Scholar

[3] S. Abbas and M. Benchohra, Ulam-Hyers stability for the Darboux problem for partial fractional differential and integro-differential equations via Picard operators. Results. Math. 65, No 1-2 (2014), 67-79.Search in Google Scholar

[4] S. Abbas, M. Benchohra and and A. Cabada, Partial neutral functional integro-differential equations of fractional order with delay. Bound. Value Prob. Vol. 2012 (2012), Article No 128, 15 pp.Search in Google Scholar

[5] S. Abbas, M. Benchohra and G. M. N'Gu'er'ekata, Topics in Fractional Differential Equations. Developments in Mathematics, 27, Springer, New York (2012).Search in Google Scholar

[6] S. Abbas, M. Benchohra and G. M. N'Gu'er'ekata, Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2014).Search in Google Scholar

[7] S. Abbas, M. Benchohra and S. Sivasundaram, Ulam stability for partial fractional differential inclusions with multiple delay and impulses via Picard operators. Nonlinear Stud. 20, No 4 (2013), 623-641.Search in Google Scholar

[8] S. Abbas, M. Benchohra and A. N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations. Frac. Calc. Appl. Anal. 15, No 2 (2012), 168-182; DOI: 10.2478/s13540-012-0012-5; http://link.springer.com/article/10.2478/s13540-012-0012-5.Search in Google Scholar

[9] S. Abbas, M. Benchohra and Y. Zhou, Darboux problem for fractional order neutral functional partial hyperbolic differential equations. Int. J. Dyn. Syst. Differ. Equ. 2 (2009), 301-312.Search in Google Scholar

[10] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York (1977).Search in Google Scholar

[11] H. Covitz and S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces. Israel J. Math. 8 (1970), 5-11.Search in Google Scholar

[12] M. A. Darwish, J. Henderson and D. O'Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument. Bull. Korean Math. Soc. 48, No 3 (2011), 539-553.Search in Google Scholar

[13] M. A. Darwish and J. Henderson, Nondecreasing solutions of a quadratic integral equation of Urysohn-Stieltjes type. Rocky Mountain J. Math. 42, No 2 (2012), 545-566.Search in Google Scholar

[14] M. A. Darwish and J. Bana's, Existence and characterization of solutions of nonlinear Volterra-Stieltjes integral equations in two vriables. Abstr. Appl. Anal. 2014 (2014), Article ID 618434, 11 pages.Search in Google Scholar

[15] K. Deimling, Multivalued Differential Equations. Walter De Gruyter, Berlin-New York (1992).Search in Google Scholar

[16] K. Diethelm and N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002), 229-248.Search in Google Scholar

[17] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht (1999).Search in Google Scholar

[18] D. Henry, Geometric theory of Semilinear Parabolic Partial Differential Equations. Springer-Verlag, Berlin-New York (1989).Search in Google Scholar

[19] E. Hern'andez, D. O'Regan, On a new class of abstract impulsive differential equations. Proc. Amer. Math. Soc. 141 (2013), 1641-1649.Search in Google Scholar

[20] A. A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differential Equations41 (2005), 84-89.Search in Google Scholar

[21] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.Search in Google Scholar

[22] A. A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B. V., Amsterdam, 2006.Search in Google Scholar

[23] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.Search in Google Scholar

[24] M. Pierri, D. O'Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous. Appl. Math. Comput. 219 (2013), 6743-6749.Search in Google Scholar

[25] I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 26 (2010), 103-107.Search in Google Scholar

[26] A. N. Vityuk and A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil. 7, No 3 (2004), 318-325.Search in Google Scholar

[27] J. Wang, Y. Zhou and M. Feickan, Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, No 10 (2012), 3389-3405.Search in Google Scholar

[28] J. Wang, M. Feickan and Y. Zhou, Ulam's type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395 (2012), 258-264.Search in Google Scholar

Received: 2014-6-12
Published Online: 2015-2-10

© 2015 Diogenes Co., Sofia