Nonlinear parametric evolution inclusions (Q2785235)

Let \(T=[0,b]\) and \((X,H,X^*)\) be an evolution triple of spaces with the embedding of \(X\) into \(H\) being compact; moreover, let \(E\) be a complete metric space. NEWLINENEWLINENEWLINEThe authors study nonlinear evolution inclusions parametrized by \(\lambda\in E\): NEWLINE\[NEWLINE\dot x(t)+A(t,x(t),\lambda)\in F(t,x(t),\lambda) \text{ a.e. on } T, \qquad x(0)=x_0(\lambda),NEWLINE\]NEWLINE where \(A: T\times X\times E \to X^*\) is a nonlinear maximal monotone coercive operator and \(F\) is a nonlinear multifunction which takes values in \(H\). NEWLINENEWLINENEWLINEUsing the concept of \(G\)-convergence for nonlinear maximal monotone operators, the authors obtain results of upper semicontinuity, lower semicontinuity and \(h\)-continuity on the solution set \(S(\lambda)\). They also solve a relevant minimax problem and present two examples of parabolic distributed parameter systems which illustrate the applicability of their work.