Multivalued differential equations in Banach spaces and their applications (Q698902)

Let H be a separable Hilbert space and V be a dense compactly embedded subspace of H such that V is a separable reflexive Banach space. Let I:=[0,T], \(2\leq p<+\infty ,\) and \(X=L^{p}(I;V).\) Then \(X^{\ast }=L^{p^{^{\prime }}}(I;V^{\ast }).\) Let \(L:D(L)\subset X\rightarrow X^{\ast } \) be defined by \(D(L)=\{v\in X;v^{\prime }\in X^{\ast }and\) \(v(0)=0\}\) and \(L(u)=u^{\prime }\) for \(u\in D(L).\) Then \(L\) is a closed densely defined maximal monotone operator in \(X.\) The authors consider the following initial value problem \[ u^{\prime }+Au+Fu\ni f, \qquad 0<t<T, \quad u(0)=u_{0},\tag{1} \] where \(A:X\rightarrow X^{\ast }\) is demicontinuous and of class \((S_{+})\) with respect to \(D(L),\) and \(F:X\rightarrow 2^{X^{\ast }}\) is a multivalued map with nonempty, bounded, closed and convex values. Under suitable conditions on the operators \(A\) and \(F,\) and by using a surjectivity result of \textit{Z. Liu} and \textit{S. Zhang} [Appl. Math. Mech., Engl. Ed. 19, 1141-1149 (1998; Zbl 0941.47051)] for multivalued \((S_{+})\) type mappings, the authors prove the existence of a solution to (1) for any \(f\in X^{\ast },\) extending previous results by \textit{Z. Liu} [Nonlinear Anal., Theory Methods Appl. 29, 1231-1236; (1997; Zbl 0902.47057)]. Subsequently, they apply their results to prove the existence of solutions to some evolution hemivariational inequalities and parabolic equations with nonmonotone discontinuities, extending previous results by \textit{J. Rauch} [Proc. Am. Math. Soc. 64, 277-282 (1977; Zbl 0413.35031)], \textit{M. Miettinen} [Nonlinear Anal., Theory Methods Appl. 26, 725-734 (1996; Zbl 0858.35072)] and \textit{Z. Liu} [Nonlinear Anal., Theory Methods Appl. 36A, No. 1, 91-100 (1999; Zbl 0920.47056)].