Abstract
In this paper, we study a class of a multivalued perturbations of m-dissipative evolution inclusions with nonlocal initial condition in arbitrary Banach spaces. We prove the existence of solutions when the multivalued right hand side is Lipschitz and admits nonempty closed bounded, but in general case, neither convex nor compact values. Illustrative example is provided.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aizicovici, S., Staicu, V.: Multivalued evolution equations with nonlocal initial conditions in Banach spaces. NoDEA Nonlinear Differ. Equations Appl. 14(3–4), 361–376 (2007)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York (2010)
Benilan, P.: Solutions integrales D’equations D’evolution dans un Espace de Banach. C. R. Acad. Sci. Paris 274, 47–50 (1972)
Bothe, D.: Nonlinear Evolutions in Banach Spaces. Habilitationsschritt, Paderborn (1999)
Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions, Lect. Notes Math., vol. 580. Springer, New York (1977)
Donchev, T.: Multivalued perturbations of m-dissipative differential inclusions in uniformly convex spaces. N. Z. J. Math. 31(1), 19–32 (2002)
Hu, S., Papageorgiou, N.: Handbook of Multivalued Analysis. Volume I: Theory. Kluwer, Dordrecht (1997)
Lakshmikantham, V., Leela, S.: Nonlinear Differential Equations in Abstract Spaces. Pergamon, Oxford (1981)
Paicu, A., Vrabie, I.I.: A class of nonlinear evolution equations subjected to nonlocal initial conditions. Nonlinear Anal. 72(11), 4091–4100 (2010)
Roubiček, T.: Nonlinear Partial Differential Equations with Applications, International Series of Numerical Mathematics 153. Birkhäuser, Basel (2005)
Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys Monographs, vol. 49. AMS, Providence (1997)
Vrabie, I.I.: Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions. J. Funct. Anal. 262(4), 1363–1391 (2012)
Vrabie, I.I.: Global solutions for nonlinear delay evolution inclusions with nonlocal initial conditions. Set Valued Var. Anal. 20(3), 477–497 (2012)
Xue, X.: Semilinear nonlocal differential equations with measure of noncompactness in Banach spaces. J. Nanjing. Univ. Math. Biq. 24(2), 264–276 (2007)
Zhu, L., Huang, Q., Li, G.: Existence and asymptotic properties of solutions of nonlinear multivalued differential inclusions with nonlocal conditions. J. Math. Anal. Appl. 390(2), 523–534 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ahmed, R., Donchev, T. & Lazu, A.I. Nonlocal m-Dissipative Evolution Inclusions in General Banach Spaces. Mediterr. J. Math. 14, 215 (2017). https://doi.org/10.1007/s00009-017-1016-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-017-1016-5