Existence and controllability results for semilinear neutral functional differential inclusions with nonlocal conditions

Ntouyas, S. ; O'Regan, D.

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 27 (2007) no. 2, pp. 213-264.

Résumé

In this paper, we prove existence and controllability results for first and second order semilinear neutral functional differential inclusions with finite or infinite delay in Banach spaces, with nonlocal conditions. Our theory makes use of analytic semigroups and fractional powers of closed operators, integrated semigroups and cosine families.

Détail

Keywords: semilinear differential inclusions, nonlocal conditions, analytic semigroups, cosine functions, integrated semigroups, fixed point, nonlinear alternative, controllability

Bibliographie

[1] M. Adimy, H. Bouzahir and K. Ezzinbi, Existence and stability for some partial functional differential equations with infinite delay, J. Math. Anal. Appl. 294 (2004), 438-461.

[2] R. Agarwal, L. Górniewicz and D. O'Regan, Aronszain type results for Volterra equations and inclusions, Topol. Methods Nonlinear Anal. 23 (2004), 149-159.

[3] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352.

[4] J.P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Birkhauser, New York, 1984.

[5] M. Benchohra, L. Górniewicz and S.K. Ntouyas, Controllability of Some Nonlinear Systems in Banach Spaces, Pawel Wlodkowic University College in Plock, Plock, 2003.

[6] A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), 69-86.

[7] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.

[8] L. Byszewski, Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem, J. Appl. Math. Stochastic Anal. 12 (1999), 91-97.

[9] G. Da Prato and E. Sinestrari, Differential operators with non-dense domains, Ann. Scuola. Norm. Sup. Pisa Sci. 14 (1987), 285-344.

[10] K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992.

[11] H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, Vol. 108, North-Holland, Amsterdam, 1985.

[12] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999.

[13] J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985.

[14] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.

[15] S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, New York, 1994.

[16] E. Hernandez, Existence results for partial neutral functional integrodifferential equations with unbounded delay, J. Math. Anal. Appl. 292 (2004), 194-210.

[17] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, Boston, London, 1997.

[18] H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), 160-180.

[19] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, vol. I, Academic Press, New York, 1969.

[20] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.

[21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[22] C. Travis and G. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Hungar. 32 (1978), 75-96.

[23] C. Travis and G. Webb, An abstract second order semilinear Volterra integrodifferential equation, SIAM J. Math. Anal. 10 (1979), 412-424.