Existence results for a class of second order evolution inclusions and its corresponding first order evolution inclusions
The authors consider the abstract second-order nonlinear evolution inclusion \[ \begin{aligned} &\varepsilon u^{\prime \prime }(t)+A(t)u^{\prime }(t)+Bu(t)+F(t,u(t))\ni g(t)\text{ a.e. on }T=[0,b], \\ & u(0)=u_{0},\;\sqrt{\varepsilon }u^{\prime }(0)=u_{1}\end{aligned}\tag{P} \] in a reflexive Banach space \(V\), where \(0< \varepsilon\ll 1\). \(A\in L^{\infty }\left( T,\mathcal L (V,V^{\ast }\right) )\), \(\mathcal L (V,V^{\ast })\) is the set of linear continuous operators from \(V\) into its dual \(V^{\ast }\) and \(A(t)\) is coercive, \(B\in \) \(\mathcal L (V,V^{\ast })\), \(F\) is closed set valued and satisfies a boundedness condition, and \(g\in L^{q}\left( T,V^{\ast }\right) \) where \(1<q\leq 2\), among other hypotheses. For both of the cases in which \(F\) is convex-valued and \(F\) is not convex-valued, it is first proven that there is a weak solution \(u_{\varepsilon }\) to (P) for every \(\varepsilon \) using the theory of L-pseudomonotone operators. It is then shown, as \(\varepsilon \rightarrow 0\), that \(u_{\varepsilon }\) approaches a solution of the first-order problem \[ \begin{aligned} &A(t)u^{\prime }(t)+Bu(t)+F(t,u(t))\ni g(t)\text{ a.e. on }T=[0,b]\\ &u(0)=u_{0}.\end{aligned} \] In the convex case, \(F(t,x)\) is assumed to be measurable in \(t\) and sequentially closed in \(x\), while in the non-convex case, \(F(t,x)\) is assumed to be lower semicontinuous in \(x\) and graph measurable. Some of the ideas in this article are based on [\textit{S. Migórski} and \textit{A. Ochal}, SIAM J. Math. Anal. 41, No. 4, 1415--1435 (2009; Zbl 1204.35123)].
scientific article; zbMATH DE number 6176739
scientific article