Abstract
Existence results for nondensely defined semilinear functional differential inclusions in Fréchet spaces
Highlights
This paper is concerned with an application of a recent Frigon nonlinear alternative for contractive multivalued maps in Frechet spaces [20] to obtain the existence of integral solutions of some classes of initial value problems for first order semilinear functional differential inclusions
For any continuous function y defined on [−r, ∞) and any t ∈ [0, ∞), we denote by yt the element of C([−r, 0], D(A)) defined by yt(θ) = y(t + θ), θ ∈ [−r, 0]
In the case where F is either a single or a multivalued map, and A is a densely defined linear operator generating a C0-semigroup of bounded linear operators, the problems (1)–(2) and (3)–(4) have been investigated on compact intervals in, for instance, the monographs by Ahmed [1], Hale and Lunel [21], Wu [31], Hu and Papageorgiou [22], Kamenskii, Obukhovskii and Zecca [24], and in the papers of Benchohra, Ntouyas [9, 10, 11] for the controllability of differential inclusions with different conditions; see the monograph of Benchohra, Ntouyas, Gorniewicz [13] and the papers of Balachandran and Manimegolai [6], Benchohra et al [7], Benchohra and Ntouyas [8], and Li and Xue [23]and the references cited therein
Summary
This paper is concerned with an application of a recent Frigon nonlinear alternative for contractive multivalued maps in Frechet spaces [20] to obtain the existence of integral solutions of some classes of initial value problems for first order semilinear functional differential inclusions. In the case where F is either a single or a multivalued map, and A is a densely defined linear operator generating a C0-semigroup of bounded linear operators, the problems (1)–(2) and (3)–(4) have been investigated on compact intervals in, for instance, the monographs by Ahmed [1], Hale and Lunel [21], Wu [31], Hu and Papageorgiou [22], Kamenskii, Obukhovskii and Zecca [24], and in the papers of Benchohra, Ntouyas [9, 10, 11] for the controllability of differential inclusions with different conditions; see the monograph of Benchohra, Ntouyas, Gorniewicz [13] and the papers of Balachandran and Manimegolai [6], Benchohra et al [7], Benchohra and Ntouyas [8], and Li and Xue [23]and the references cited therein. These results extend some ones existing in the previous literature in the case of densely defined linear operators
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