A range and existence theorem for pseudomonotone perturbations of maximal monotone operators

Le, Vy Khoi

doi:10.1090/s0002-9939-2010-10594-4

Cited by 66 publications

(58 citation statements)

References 12 publications

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“…In both existence Theorems 2.1 and 3.2, we obtain the solvability of (1.1) under certain general coercivity condition (condition (C)) that involves all components A, F and J of (1.1), in the spirit of an abstract coercivity condition for variational inequalities introduced in [20] (cf. condition (2.1) in Theorem 2.2, [20]). The proofs of Theorems 2.1 and 3.2 are of topological nature and are based on perturbation-regularization arguments and suitable formulations of our quasi-variational inequalities as fixed point problems.…”

Section: Introductionmentioning

confidence: 90%

“…This quasi-variational inequality is an abstract formulation of a class of nonsmooth second order partial differential inclusions with multivalued coefficients that have been studied recently in e.g. [6,20,21,23,24] and the references therein. The operator A, describing the principal part of the differential operator, is given by (1.2) where A is a multivalued function from Ω × R N into R N .…”

Section: Introductionmentioning

confidence: 99%

“…In variational inequalities, as considered in [20,21,23], J does not depend on the unknown function, while in quasi-variational inequalities, J also depends on the unknown function.…”

Section: Introductionmentioning

confidence: 99%

“…A similar problem for variational inequalities was considered in [20], where some general coercivity conditions on the sum A + F were investigated. We would like to establish here abstract existence results for the quasi-variational inequality (1.1), parallel to those in [20] for variational inequalities, so that such abstract results are applicable to quasi-variational inequalities with operators A and F of the forms (1.2) and (1.3).…”

Section: Introductionmentioning

confidence: 99%

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Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings

2016

Results Math

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This paper is about the existence of solutions of the following abstract quasi-variational inequality:where X is a reflexive Banach space, A is a maximal monotone mapping, F is a multivalued mapping which is compact in certain sense, and Ju is a convex and lower semicontinuous functional depending on u. We are interested in solvability conditions for the above inequality that extends those for a class of boundary value problems containing elliptic operators with multivalued coefficients.Mathematics Subject Classification. 58E35, 47J20, 47J25, 35J87.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

“…In variational inequalities, as considered in [20,21,23], J does not depend on the unknown function, while in quasi-variational inequalities, J also depends on the unknown function.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings

2016

Results Math

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“…For further existence results involving operators of the type + , the reader is referred to Kenmochi [17], Le [18], and Asfaw [19]. For various examples on pseudomonotone and quasimonotone operators, we cite the paper due to Mustonen [20].…”

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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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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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A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

Migórski

Dudek

2019

Z Angew Math Mech

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We study a new class of elliptic variational-hemivariational inequalities in a reflexive Banach space. Based on a surjectivity result for an operator inclusion of Clarke's subdifferential type, we prove existence of solution. Then, we apply this result to a mathematical analysis of the steady Oseen model for a generalized Newtonian incompressible fluid. A variational-hemivariational inequality for the flow problem is derived and sufficient conditions for existence of weak solutions are obtained. The mixed boundary conditions involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier-Fujita slip condition, and the threshold slip and leak condition of frictional type. K E Y W O R D Sfrictional type boundary condition, generalized Newtonian fluid, leak, operator inclusion, Oseen model, slip, unilateral boundary condition 2 0 1 0 M A In this paper we introduce and analyse a new class of operator inclusions of subdifferential type and a corresponding class of elliptic variational-hemivariational inequalities in a reflexive Banach space. Our main goal is to provide existence results to these classes of problems, and apply them to examine the stationary flow Oseen model of generalized Newtonian fluid described by equations − Div + ( ⋅ ∇) + ∇ = in Ω, div = 0 in Ω

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A range and existence theorem for pseudomonotone perturbations of maximal monotone operators

Cited by 66 publications

References 12 publications

Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings

Existence Results for Quasi-Variational Inequalities with Multivalued Perturbations of Maximal Monotone Mappings

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

A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

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