An existence theorem for a non-autonomous second order nonlocal multivalued problem

Tiziana Cardinali; Serena Gentili

doi:10.24193/subbmath.2017.0008

Authors

Tiziana Cardinali Dipartimento di Matematica e Informatica, Università di Perugia http://orcid.org/0000-0003-3802-8642
Serena Gentili

DOI:

https://doi.org/10.24193/subbmath.2017.0008

Keywords:

nonlocal conditions, semilinear non-autonomous second order differential inclusion, fundamental Cauchy operator, fundamental system

Abstract

In this paper we prove the existence of mild solutions for a nonlocal problem governed by an abstract semilinear non-autonomous second order differential inclusion, where the non-linear part is an upper-Caratheodory semicontinuous multimap. Our existence theorem is obtained thanks to the introduction of a fundamental Cauchy operator. Finally we apply our main result to provide the controllability of a problem involving a non-autonomous wave equation.

References

Ambrosetti, A., Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Univ. Padova, 39 (1967), 349-360.

Bungardi, S., Cardinali, T., Rubbioni, P., Semilinear nonlocal integrodifferential inclusions via vectorial measures of noncompactness. In press on Appl. Anal. (2016), http://dx.doi.org/10.1080/00036811.2016.1227969.

Couchouron, J.F., Kamenski, M., An abstract topological point of view averaging principle in th theory of differential inclusions, Nonlinear Analysis, 42 (2000), 1101-1129.

Cardinali, T., Rubbioni, P., On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl., 308 (2005), No. 2, 620-635.

Denkowski, Z., Migorski, S., Papageorgiou, N.S., An Introduction to Nonlinear Analysis, Theory, Kluwer Acad. Publ. Boston/Dordrecht/London (2003).

Fattorini, H.O., Second Order Linear Differential Equations in Banach Spaces, North-Holland Publishing Co., Amsterdam (1985).

Henriquez, H.R., Existence of solutions of non-autonomous second order differential equations with infinite delay, Nonlinear Anal. 74 (10), (2011), 3333-3352.

Henriquez, H.R., Poblete, V., Pozo, J.C., Mild solutions of nonautonomous

second order problems with nonlocal initial conditions, J. Math. Anal. Appl.,

(2014), 1064-1083.

Kozak, M., A fundamental solution of a second-order differential equation in Banach space, Univ. Iagel. Acta Math. 32 ( 1995), 275-289.

Kamenskii, M., Obukhovskii, V., Zecca, P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl. 7, Walter de Gruyter, Berlin (2001).

Kisynski, J., On cosine operator functions and one-parameter groups of operators, Studia Math. 44 (1972) 93-105.

Serizawa, H., Watanabe, M., Time-dependent perturbation for cosine families in Banach spaces, J. Math. 12, Houston, (1986), 579-586.

Travis, C.C., Webb., G.F., Second order differential equations in Banach space, Nonlinear Equations in Abstract Spaces, Academic Press, New York (1978).

Vasilev, V.V., Piskarev, S.I., Differential equations in Banach spaces, II, Theory of cosine operator functions, J. Math. Sci. (N.Y.) 122 (2004) 7047-7060.

Zvyagin, V., Obukhovskii, V., Zvyagin, A., On inclusions with multivalued operators and their applications to some optimization problem, J. Fixed Point Theory Appl. 16 (2014) 27-82.