A generalization of the weak convergence theorem in Sobolev spaces with application to differential inclusions in a Banach space. (English) Zbl 0980.34057

Maruyama, Toru

Proc. Japan Acad., Ser. A 77, No. 1, 5-10 (2001).

Let \(E\) be a real reflexive separable Banach space, \(F:[0,T]\times E\to 2^{E}\) a multivalued map satisfying some assumptions that will be specified later and \(u:[0,T]\times E\times E\to \overline {\mathbb{R}}\) a given map. In the first part of the paper, the author considers the multivalued differential equation (1) \(\dot x(t)\in F(t,x(t))\), \(x(0)=a\). It is proved under the conditions \(F(t,x)\) is nonempty compact convex-valued, measurable in \(t\), upper hemi-continuous in \(x\) and \(L^{p}\)-integrably bounded, that problem (1) has a solution \(x\in W^{1,p}([0,T],E)\). Also, if \(A\) is a nonempty compact convex subset of \(E\), the correspondence \(S:A\to W^{1,p}([0,T],E)\) is compact-valued and upper hemi-continuous on \(A\) in the weak topology for \(W^{1,p}\). In the second part, the author studies the following variational problem (2) \(\text{minimize}_{S(a)} \int_{0}^{T}u(t,x(t),\dot x(t)) dt,\) where \(S(a)\) represents the solution set to problem (1). This paper is a continuation of problems (1) and (2) examinated by the author in [Proc. Japan Acad., Ser. A 60, 161-164 (1984; Zbl 0549.49006), ibid. 60, 212-214 (1984; Zbl 0549.49007), ibid. 66, 217-221 (1990; Zbl 0728.49004)].

Reviewer: M.Benchohra (Sidi Bel Abbes)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34A60	Ordinary differential inclusions
46N20	Applications of functional analysis to differential and integral equations
49J24	Optimal control problems with differential inclusions (existence) (MSC2000)

Keywords:

differential inclusion; variational problem; weak convergence; existence; real separable reflexive Banach space

Citations:

Zbl 0549.49006; Zbl 0549.49007; Zbl 0728.49004

References:

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