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.

General p-curl systems and duality mappings on Sobolev spaces for Maxwell equations

“…We generalize the duality mapping procedure to general Banach spaces having dual norm which is uniformly Frechét differentiable on the unit sphere; see Section 6. For further details on duality mappings and their applications to the solvability of nonlinear operator equations in Banach spaces, the reader is referred to [2,5,8,26] and the references therein.…”