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

Asfaw, Teffera M.

doi:10.1016/j.jmaa.2015.09.035

Cited by 9 publications

(21 citation statements)

References 37 publications

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“…Similarly, it is not difficult to see that (III) implies (8). Therefore, Theorem 5 improves the well-known maximality result due to Rockafellar [5].…”

Section: Preliminariessupporting

confidence: 77%

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Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

Asfaw

2016

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 * and : ⊇ ( ) → 2 * be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for + under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition ∘ ( ) ∩ ( ) ̸ = 0 and Browder and Hess who used the quasiboundedness of and condition 0 ∈ ( ) ∩ ( ). In particular, the maximality of + is proved provided that ∘ ( ) ∩ ( ) ̸ = 0, where : → (−∞, ∞] is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator. PreliminariesIn what follows, the norm of spaces and * will be denoted by ‖ ⋅ ‖. For ∈ and * ∈ * , pairing ⟨ * , ⟩ denotes value * ( ). Let and be real Banach spaces. For operator : → 2 , we define domain ( ) of by ( ) = { ∈ : ̸ = 0} and range ( ) of by ( ) = ⋃ ∈ ( ) . We also use symbol ( ) for the graph of : ( ) = {( , * ) : ∈ ( ), * ∈ }. A single-valued operator : ⊃ ( ) → is "demicontinuous," if it is continuous from the strong topology of ( ) to the weak topology of . It is "compact," if it is strongly continuous and maps bounded subsets of ( ) to relatively compact subsets of . A multivalued operator : ⊃ ( ) → 2 is "bounded," if it maps each bounded subset of ( ) into a bounded subset of . It is "finitely continuous," if it is upper semicontinuous from each finite dimensional subspace of to the weak topology of . Throughout the paper, we use notations ⇀ 0 and → 0 in to denote the weak and strong convergence of sequence { }, respectively. Analogous notations are used for convergence of a sequence in * . Let : [0, ∞) → (−∞, ∞) be a continuous and strictly increasing function such that ( ) → ∞ as → ∞. The mapping : → 2 * defined byis called the "generalized duality mapping" associated with . If ( ) = for all ≥ 0, is denoted by and is called "the normalized duality mapping." As a consequence of the Hahn-Banach theorem, it is well-known that ( ) ̸ = 0 for all ∈ . Since and * are locally uniformly convex, is single valued, bounded, monotone, and bicontinuous. Definition 1. An operator : ⊃ ( ) → 2 * is said to be (i) "monotone" if for every ∈ ( ), ∈ ( ), * ∈ , and V * ∈ , one has ⟨ * − V * , − ⟩ ≥ 0;(ii) "maximal monotone" if is monotone and ( + ) = * for every > 0; that is, is maximal monotone if and only if is monotone and then ( , * ) ∈ and ⟨ * , ⟩ → ⟨ * , ⟩ as → ∞. Browder and Hess [2] introduced the following definitions. The original definition of single valued pseudomonotone operator is due to Brézis [3].Definition 3. An operator : ⊃ ( ) → 2 * is said to be "pseudomonotone" if the following conditions are satisfied:is nonempty, closed, convex, and bounded subset of * .(ii) is finitely continuous; that is, for every 0 ∈ ( )∩ and every...

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“…Similarly, it is not difficult to see that (III) implies (8). Therefore, Theorem 5 improves the well-known maximality result due to Rockafellar [5].…”

Section: Preliminariessupporting

confidence: 77%

“…In addition, Asfaw [8] used the degree theory developed by himself to prove maximality of sum + , where is arbitrary and is densely defined which satisfies Γ condition; that is, there exists a continuous strictly increasing function : [0, ∞) → [0, ∞) and, for each ∈ , there exists a number ( ) such that…”

Section: Abstract and Applied Analysismentioning

confidence: 99%

Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

Asfaw

2016

Abstract and Applied Analysis

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“…In view of these, our work in developing a degree theory for operators of the type + + , where is a compact operator with ( ) ⊆ ( ), is essential. It is worth mentioning that the theory associated with (i) is a generalization of the previous degree theories for bounded ( + ) perturbations of maximal monotone operators due to Browder [6], Kobayashi and Otani [8], Hu and Papageorgiou [7], Asfaw and Kartsatos [3], and the references therein. The most general degree theory currently available which is due to Asfaw [9] is for pseudomonotone perturbations of the sum of two maximal monotone operators with one of the maximal monotone operators which is of type Γ ;…”

Section: Abstract and Applied Analysismentioning

confidence: 99%

“…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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Topological degree for quasibounded multivalued (S̃)₊-perturbations of maximal monotone operators

Adhikari

2019

Applicable Analysis

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

Cited by 9 publications

References 37 publications

Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

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

Topological degree for quasibounded multivalued (S̃)₊-perturbations of maximal monotone operators

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

Cited by 9 publications

References 37 publications

Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

Maximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problems

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

Topological degree for quasibounded multivalued (S̃)+-perturbations of maximal monotone operators

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Topological degree for quasibounded multivalued (S̃)₊-perturbations of maximal monotone operators