In this paper, we shall prove an existence of solution for the constrained evolution variational inequality problem of finding \(\xi\in \mathcal{K}\subseteq Y= L^{p}(0, T; W_{0}^{1,p}(\Omega))\) (where \(\mathcal{K}\) is a nonempty, closed, convex and symmetric subset of \(Y\)) with \(p\geq 2\), \(T>0\), \(\xi^{\prime}\in Y^*\) and \(\xi(0)=\xi(T)\), such that \[\langle \mathcal{A}\xi-\Delta_{p}\xi, v-\xi\rangle + \langle Q\xi, v-\xi\rangle +\Phi(v)-\Phi(\xi) \geq \langle f^*, v-\xi\rangle,\tag{*}\] for all \(v\in \mathcal{K}\), where \(f^*\in Y^*\), \[\begin{align}\langle Q\xi, w\rangle =&\sum\limits_{i=1}^{N}{\int_{\Gamma}{q_{i}(x, t, \xi(x,t), \nabla \xi(x,t))\frac{\partial w(x,t)}{\partial x_i}dxdt}}\\&+\int_{\Gamma}{q_{0}(x,t, \xi(x,t), \nabla \xi(x,t)) w(x,t)dxdt},~w\in Y,~\xi\in Y,\end{align}\tag{**}\] \(\Gamma =[0, T]\times\Omega\), \(\Omega\) is a nonempty, bounded and open subset of \(\mathbf{R}^{N}\) with \(N\in Z_{+}\), \(\Delta_{p}\) is the \(p\)-Laplacian operator, \(q_i: \overline{\Omega}\times [0, T]\times \mathbf{R}\times \mathbf{R}^{N}\to \mathbf{R}\) satisfies mild conditions, \(\mathcal{A}\xi =\xi^{\prime}\) (where \(\xi^{\prime}\) is the derivative of \(\xi\) in the sense of distributions) and \(\Phi: Y\supseteq D(\Phi)\to \mathbf{R}\cup\{\infty\}\) is a proper, convex and lower-semi-continuous function. In order to address problems like (\(\ast\)), we shall establish new maximal monotonicity results for the sum of two maximal monotone operators \(\mathcal{N}\) and \(\mathcal{M}\) defined from reflexive Banach space into its dual, provided that \(\text{int}(D(\psi_{\mathcal{M}})- D(\psi_{\mathcal{N}}))\neq\emptyset\), where \(\psi_{N}\) and \(\psi_{M}\) are corresponding convex functions introduced by Simons. The question of maximal monotonicity of two maximal monotone operators is one of the outstanding problems in monotone operator theory. The significant contribution of Rockafellar gave a foundation in the study of nonlinear problems. In this paper, we give new maximality results, which present generalizations of the existing criteria, and provide a positive solution for Simons problem . With the help of these results, existence of solution for (\(\ast\)) is proved possibly allowing \(D(\mathcal{A}) \cap \text{int}{\mathcal{K}}=\emptyset\).
