A new topological degree theory for perturbations of the sum of two maximal monotone operators

Adhikari, Dhruba R.; Kartsatos, Athanassios G.

doi:10.1016/j.na.2011.04.023

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…The existing degree theories for operators of the type + cannot be used to treat inclusions involving operators of the type + + because the compact operator is not everywhere defined. For recent degree theories for multivalued bounded ( + ) or bounded pseudomonotone perturbations of arbitrary maximal monotone operators, the reader is referred to the papers by Asfaw and Kartsatos [3], Asfaw [4], Adhikari and Kartsatos [5], and the references therein. In these theories, the maximal monotone operator is arbitrary and ( + ) and/or pseudomonotone operator is everywhere defined.…”

Section: Abstract and Applied Analysismentioning

confidence: 99%

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

Asfaw

2017

Abstract and Applied Analysis

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Let be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space * . Let : ⊇ ( ) → 2 * be maximal monotone, : → 2 * be bounded and of type ( + ), and : ( ) → * be compact with ( ) ⊆ ( ) such that lies in Γ (i.e., there exist ≥ 0 and ≥ 0 such that ‖ ‖ ≤ ‖ ‖ + for all ∈ ( )). A new topological degree theory is developed for operators of the type + + . 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 + + , where 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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Section: Abstract and Applied Analysismentioning

confidence: 99%

“…For a relevant degree mapping for single multivalued operator of type ( + ), we cite the paper of Zhang and Chen [16]. Recent developments on degree theories for perturbations of the sum of two maximal monotone operators can be found in the papers due to Adhikari and Kartsatos [5] and Asfaw [4].…”

Section: Lemma 5 Let Be a Maximal Monotone Set Inmentioning

confidence: 99%

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

Asfaw

2017

Abstract and Applied Analysis

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“…For additional facts and various topological degree theories related to the subject of this paper, the reader is referred to Adhikari and Kartsatos [3,5], Kartsatos and Lin [23], and Kartsatos and Skrypnik [25,27]. For further information on functional analytic tools used herein, the reader is referred to Barbu [10], Browder [16], Pascali and Sburlan [30], Simons [32], Skrypnik [33,34], and Zeidler [36].…”

Section: Introductionmentioning

confidence: 99%

Solvability of Inclusions Involving Perturbations of Positively Homogeneous Maximal Monotone Operators

Adhikari¹,

Aryal²,

Bhatt³

et al. 2021

Preprint

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Let X be a real reflexive Banach space with X * its dual space. Let L : X ⊃ D(L) → X * be a densely defined linear maximal monotone operator, A : X ⊃ D(A) → 2 X * be a maximal monotone and positively homogeneous operator of degree γ > 0, C : X ⊃ D(C) → X * be a bounded demicontinuous operator of type (S + ) w.r.t. D(L), and Q : G 1 → 2 X * be a compact and upper-semicontinuous operator whose values are closed and convex sets in X * . In the case L = 0, we establish the existence of nonzero solutions ofand G 1 is bounded. Otherwise, we assume that A is bounded and establish the existence of nonzero solutions of Lx + Ax + Cx ∋ 0 in the set G 1 \ G 2 as above. We completely remove the restrictions γ ∈ (0, 1] for Ax + Cx + Qx ∋ 0 and γ = 1 for Lx + Ax + Cx ∋ 0 from these previous results established by the first author. Applications to elliptic and parabolic partial differential inclusions in general divergence form that include the p-Laplacian with 1 < p < ∞ and satisfy Dirichlet boundary conditions are also presented.

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“…The theory of topological degrees has progressed significantly in recent years because of its applicability to the analysis of ordinary and partial differential equations and continuation methods in nonlinear analysis in general (e.g. see [1,2,3,4,5,6,7,8,9]). The classical topological degree theory developed by Brouwer [10] in 1912 for continuous functions on finite-dimensional spaces and the Leray-Schauder degree [11] in 1934 for compact displacements of the identity in Banach spaces both assume the boundedness of the domains over which the degrees are defined.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we construct topological degrees on unbounded domains both in finite-dimensional and infinite dimensional spaces and discuss their properties. For the development of degree theories for operators of monotone type that involve (S + )-operators and their generalized and/or multivalued versions, the reader is referred to Kartsatos and Skrypnik [1,2], Berkovits [13], Berkovits and Mustonen [14], Kartsatos and the first author [5,7], Kartsatos and Kerr [8], Hu and Papageorgiou [15], Kittilä [16] and the references therein. For the coincidence degree developed by Mawhin for nonlinear perturbations of certain Fredholm operators in normed spaces, the reader is referred to [17].…”

Section: Introductionmentioning

confidence: 99%

Topological Degrees on Unbounded Domains

Adhikar¹,

Kunwar²

2018

Open j. math. anal.

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Let D be an open subset of R N and f : D → R N a continuous function. The classical topological degree for f demands that D be bounded. The boundedness of domains is also assumed for the topological degrees for compact displacements of the identity and for operators of monotone type in Banach spaces. In this work, we follow the methodology introduced by Nagumo for constructing topological degrees for functions on unbounded domains in finite dimensions and define the degrees for Leray-Schauder operators and (S + )-operators on unbounded domains in infinite dimensions.

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A new topological degree theory for perturbations of the sum of two maximal monotone operators

Cited by 6 publications

References 15 publications

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

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

Solvability of Inclusions Involving Perturbations of Positively Homogeneous Maximal Monotone Operators

Topological Degrees on Unbounded Domains

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