The degree theory for set-valued compact perturbation of monotone-type mappings with an application

We introduce a topological degree for a class of operators of generalized monotone type in reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we give some surjectivity results for operators of generalized monotone type in reflexive Banach spaces. In the Hilbert space case, this reduces to the celebrated Browder-Minty theorem for monotone operators.

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On the degree theory for densely defined mappings of class(S+)L

Abstract and Applied Analysis ◽

10.1155/s1085337599000111 ◽

1999 ◽

Vol 4 (3) ◽

pp. 141-152 ◽

Cited By ~ 5

Author(s):

Juha Berkovits

Keyword(s):

Topological Degree ◽

Degree Theory ◽

Elliptic Problems ◽

Existence Results ◽

Basic Properties ◽

New Construction ◽

Densely Defined ◽

Monotone Type ◽

Abstract Existence

We introduce a new construction of topological degree for densely defined mappings of monotone type. We also study the structure of the classes of mappings involved. Using the basic properties of the degree, we prove some abstract existence results that can be applied to elliptic problems.

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A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem

Abstract and Applied Analysis ◽

10.1155/2017/7236103 ◽

2017 ◽

Vol 2017 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Teffera M. Asfaw

Keyword(s):

Parabolic Problem ◽

Existence Theorems ◽

Degree Theory ◽

Maximal Monotone ◽

Uniformly Convex ◽

Topological Degree Theory ◽

Nonlinear Parabolic Problem ◽

Compact Perturbations ◽

Nonlinear Parabolic ◽

Monotone Type

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be maximal monotone, S:X→2X⁎ be bounded and of type (S+), and C:D(C)→X⁎ be compact with D(T)⊆D(C) such that C lies in Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈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 Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.

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Degree Theory for Monotone-Type Maps

Topological Degree Theory and Applications - Mathematical Analysis and Applications ◽

10.1201/9781420011487.ch6 ◽

2006 ◽

pp. 127-168

Keyword(s):

Degree Theory ◽

Monotone Type

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Semilinear systems with a multi-valued nonlinear term

Open Mathematics ◽

10.1515/math-2017-0056 ◽

2017 ◽

Vol 15 (1) ◽

pp. 628-644

Author(s):

In-Sook Kim ◽

Suk-Joon Hong

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Nonlinear Term ◽

Degree Theory ◽

Existence Results ◽

Topological Degree Theory ◽

Dirichlet Problems ◽

Semilinear Systems ◽

Densely Defined ◽

Monotone Type

Abstract Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.

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