Solvability of inclusions involving perturbations of positively homogeneous maximal monotone operators

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