Solvability of inclusions involving perturbations of positively homogeneous maximal monotone operators

Abstract

Let \(X\) be a real reflexive Banach space and \(X^*\) be its dual space. Let \(G_1\) and \(G_2\) be open subsets of \(X\) such that \(\overline G_2\subset G_1\), \(0\in G_2\), and \(G_1\) is bounded. Let \(L: X\supset D(L)\to X^*\) be a densely defined linear maximal monotone operator, \(A:X\supset D(A)\to 2^{X^*}\) be a maximal monotone and positively homogeneous operator of degree \(\gamma>0\), \(C:X\supset D(C)\to X^*\) be a bounded demicontinuous operator of type \((S_+)\) with respect to \(D(L)\), and \(T:\overline G_1\to 2^{X^*}\) be a compact and upper-semicontinuous operator whose alues are closed and convex sets in \(X^*\). We first take \(L=0\) and establish the existence of nonzero solutions of \(Ax+ Cx+ Tx\ni 0\) in the set \(G_1\setminus G_2\). Secondly, we assume that \(A\) is bounded and establish the existence of nonzero solutions of \(Lx+Ax+Cx\ni 0\) in \(G_1\setminus G_2\). We remove the restrictions \(\gamma\in (0, 1]\) for \(Ax+ Cx+ Tx\ni 0\) and \(\gamma= 1\) for \(Lx+Ax+Cx\ni 0\) from such existing results in the literature. We also present applications to elliptic and parabolic partial differential equations in general divergence form satisfying Dirichlet boundary conditions.