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A degree theory for compact perturbations of monotone type operators and application to nonlinear parabolic problem. (English) Zbl 1470.47046

Summary: Let \(X\) be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space \(X^*\). Let \(T \: X \supseteq D(T) \to 2^{X^*}\) be maximal monotone, \(S \: X\to 2^{X^*}\) be bounded and of type \((S_+)\), and \(C \:D (C) \to X^*\) be compact with \(D(T) \subseteq D(C)\) such that \(C\) lies in \(\Gamma^\tau_\sigma\) (i.e., there exist \(\sigma\geq 0\) and \(\tau\geq 0\) such that \(\|Cx\| \leq \tau \|x\| + \sigma\) for all \(x \in D(C))\). A new topological degree theory is developed for operators of the type \(T+S+C\). The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type \(T+S+C\), where \(C\) is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to T. M. Asfaw and A. G. Kartsatos [Adv. Math. Sci. Appl. 22, No. 1, 91–148 (2012; Zbl 1287.47043)] is improved. The theory is applied to prove existence of weak solution(s) for a nonlinear parabolic problem in appropriate Sobolev spaces.

MSC:

47H11 Degree theory for nonlinear operators
35K55 Nonlinear parabolic equations
47H05 Monotone operators and generalizations

Citations:

Zbl 1287.47043
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References:

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