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1

Nonconvex minimization related to quadratic double-well energy - approximation by convex problemsenergy – approximation by convex problems

Naniewicz Z., Puchała P.

Control and Cybernetics

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2012

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Vol. 41, no. 3

525-543

EN

A double-well energy expressed as a minimum of two quadratic functions, called phase energies, is studied taking into account minimization of the corresponding integral functional. Such integral, as being not sequentially weakly lower semicontinuous, does not admit classical minimizers. To derive the relaxation formula for the infimum, the appropriate minimizing sequence is constructed. It consists of solutions of some approximating convex problems involving characteristic functions related to the phase energies. The weak limit of this sequence and the weak limit of the sequence of solutions of dual problems combined with the weak-star limits of the characteristic functions related to the phase energies allow to establish the final relaxation formula. It is also shown that infimum can be expressed by the Young measure associated with constructed minimizing sequence. An explicit form of Young measure in some regions of the involved domain is calculated.

2

Systems of variational inequalities related to economic equilibrium

Naniewicz Z., Nockowska M.

Control and Cybernetics

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2007

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Vol. 36, no 4

889-909

EN

In the paper a new approach to the Walrasian general equilibrium model of economy is presented. The classical market clearing condition is replaced by suitably formulated variational inequality. It states that the market clears for a commodity if its equilibrium price is positive; otherwise, there may be an excess supply of the commodity in equilibrium and then its price is zero. Such approach enables establishing new existence results without assumptions which were fundamental for the currently used methods: (i) Dis-utility functions are not required to be strictly convex and they may attain their minima in the consumption sets (the local nonsatiation of preferences is not required). (ii) The boundary of the positive orthant is allowed for the price vector in equilibrium. It allows for investigation of certain new problems, e.g. bankruptcy conditions.

3

Hemivariational inequalities governed by the p-Laplacian - Neumann problem

Naniewicz Z.

Control and Cybernetics

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2007

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Vol. 36, no 3

727-754

EN

A hemivariational inequality involving p-Laplacian is studied under the hypothesis that the nonlinear part fulfills the unilateral growth condition. The existence of solutions for problems with Neumann boundary conditions is established by making use of Chang's version of the critical point theory for nonsmooth locally Lipschitz functionals, combined with the Galerkin method. The approach is based on the recession technique introduced previously by the author.

4

Hemivariational inequalities governed by the p-Laplacian -Dirichlet problem

Naniewicz Z.

Control and Cybernetics

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2004

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Vol. 33, no 2

181-210

EN

A hemivariational inequality involving p-Laplacian is studied under the hypothesis that the nonlinear part fulfills the unilateral growth condition (Naniewicz, 1994). The existence of solutions for problems with Dirichlet boundary conditions is established by making use of Chang's version of the critical point theory for non-smooth locally Lipschitz functionals (Chang, 1981), combined with the Galerkin method. A class of problems with nonlinear potentials fulfilling the classical growth hypothesis without Ainbrosetti-Rabinowitz type assumption is discussed. The approach is based on the recession technique introduced in Naniewicz (2003).

5

Pseudomonotone semicoercive variational-hemivariational inequalities with unilateral growth condition

Naniewicz Z.

Control and Cybernetics

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2003

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Vol. 32, no 2

223-244

EN

A variational-hemivariational inequality on a vector valued function space is studied with the nonlinear part satisfying the unilateral growth condition. The higher order term is assumed to be pseudo-monotone and semicoercive. The compatibility condition expressed in terms of a recession functional has been proposed and the existence result has been formulated in a form involving the notion of discontinuous subgradient.

6

On some optimization problem related to economic equilibrium

Naniewicz Z.

Control and Cybernetics

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2002

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Vol. 31, no 1

141-165

EN

The paper considers an optimization problem in which the minima of a finite collection of objective functions satisfy some unilateral constraints and are linked together by a certain subdifferential relationship. The governing relations are stated as a variational inequality defined on a nonconvex feasible set. By the reduction to the variational inequality involving nonmonotone multivalued mapping, defined over nonnegative orthant, the existence of solutions is examined. The prototype is the general economic equilibrium problem. The exemplification of the theory for the quadratic multi-objective function is provided.

7

A topological approach to hemivariational inequalities with unilateral growth condition

Motreanu D., Naniewicz Z.

Journal of Applied Analysis

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2001

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Vol. 7, nr 1

23-41

EN

The present paper is devoted to the study of the existence solution problem for a hemivariational inequality on vector-valued function space in the case when the nonlinear nonconvex part satisfies the unilateral growth condition. The critical point theory combined with the Galerkin approximation method have been used to establish the result.