Nonlinear boundary value problems for multivalued differential equations in Banach spaces

Abstract

with F: J x X -0 Xmultivalued map, J = [a, b] compact real interval, X Banach space. A(t, x) is a continuous operator on X for each (t, x) E J x X, L: C(J, X) + X is a linear continuous operator from the space of all continuous functions x: J -+ X, whose range is contained in X, H: C(J, X) + X is a continuous operator, nonlinear in general. Such problems have been studied repeatedly in the literature with different hypotheses on A, F, L, H. The pioneering work with conditions similar to the ones used in the present paper, but for ordinary differential equations, is the note by Scrucca [l], where the operator A was independent from the state variable x E X. Anichini [2] has studied analogous problems (in the context of ordinary differential systems in IR”) in which the operator A also depends on x E IR”. In the case of multivalued differential systems, Kartsatos [3] has also considered boundary value problems for V-valued differential equations, but over an unbounded time interval. Zecca and Zezza [4] have extended the work of Kartsatos to differential equations in Banach spaces; in both [3] and [4] the operator A is dependent only on t E J. Recently, Papageorgiu [5] has established the existence of mild solutions on a compact interval with the operator A depending only on t E J but in general is unbounded. The fundamental tools used in the existence proofs of all above mentioned works are essentially fixed-point theorems: Schauder’s theorem in [ 11; Eilenberg-Montgomery’s theorem in [2]; KY-Fan’s theorem in [3, 4, 5). Here we use a fixed-point theorem due to Martelli [6]. The idea of the present paper has originated from the study of an analogous problem examined by Anichini and Conti [7] for ordinary differential systems in m” and Carbone et al. [8] for multivalued differential systems, again in iR”.