Existence results for second-order nonlinear evolution inclusions
The paper concerns a second-order evolution system: \[ x''(t)+A(t,x'(t))+Bx(t)\in F(t,x(t),x'(t)); \quad x(0)=x_0\in V; \quad x'(0)=x_1\in H. \] Here, \(V\subset H\subset V^*\) is a Gelfand triple, \(A:[0,T]\times V\to V^*\) is a nonlinear operator, \(B:V\to V^*\) is a bounded linear operator, and \(F:[0,T]\times H\times H\to H\) is a set-valued operator. Sufficient conditions are given for existence, local existence, and uniqueness of solutions. The proofs make use of a new Wirtinger inequality. The theory is illustrated through an example where \(H=L^2(\Omega)\), \(V=W^{m,2}_0(\Omega)\), and \(V^*=W^{-m,2}(\Omega)\), \(\Omega\) being a bounded domain in \(\mathbb{R}^n\) with smooth boundary.
scientific article; zbMATH DE number 1283841