Keywords:Berkovits-Mustonen degree theory Browder degree theory Maximal monotone operator Bounded demicontinuous operator of type (S + ) Let X be a real reflexive Banach space with dual X * . Let L : X ⊃ D(L) → X * be densely defined, linear and maximal monotone. Let T : X ⊃ D(T ) → 2 X * , with 0 ∈ D(T ) and 0 ∈ T (0), be strongly quasibounded and maximal monotone, and C : X ⊃ D(C ) → X * bounded, demicontinuous and of type (S + ) w.r.t. D(L). A new topological degree theory has been developed for the sum L + T + C . This degree theory is an extension of the Berkovits-Mustonen theory (for T = 0) and an improvement of the work of Addou and Mermri (for T : X → 2 X * bounded). Unbounded maximal monotone operators with 0 ∈D(T ) are strongly quasibounded and may be used with the new degree theory.
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.
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