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1

A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces

Jolaoso Lateef Olakunle

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

527--547

EN

In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.

2

Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

Alakoya Timilehin Opeyemi, Jolaoso Lateef Olakunle, Mewomo Oluwatosin Temitope

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

208--224

EN

In this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge o fthe Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

3

Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces

Abbas Mujahid, Iqbal Hira, de la Sen Manuel

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

347--360

EN

We prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete R-trees without the endpoint condition.

4

A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces

Jolaoso Lateef Olakunle, Taiwo Adeolu, Alakoya Timilehin Opeyemi, Mewomo Oluwatosin Temitope

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

183--203

EN

We consider a new subgradient extragradient iterative algorithm with inertial extrapolation for approximating a common solution of variational inequality problems and fixed point problems of a multivalued demicontractive mapping in a real Hilbert space. We established a strong convergence theorem for our proposed algorithm under some suitable conditions and without prior knowledge of the Lipschitz constant of the underlying operator. We present numerical examples to show that our proposed algorithm performs better than some recent existing algorithms in the literature.

5

Topological sensitivity analysis for a coupled nonlinear problem with an obstacle

Abdelbari M., Nachi K., Sokolowski J.

Control and Cybernetics

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2017

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Vol. 46, no. 1

5--25

EN

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology for several kinds of engineering problems. This derivative provides the sensitivity of the cost functional for a boundary value problem for nucleation of a small hole or a small inclusion at a given point of the domain of integration. In this paper, we present a topological asymptotic analysis with respect to the size of singular domain perturbation for a coupled nonlinear PDEs system with an obstacle on the boundary. The domain decomposition method, referring to the SteklovPoincar´epseudo-diﬀerential operator, is employed for the asymptotic study of boundary value problem with respect to the size of singular domain perturbation. The method is based on the observation that the known expansion of the energy functional in the ring coincides with the expansion of the Steklov-Poincar´e operator on the boundary of the truncated domain with respekt to the small parameter, which measures the size of perturbation. In this way, the singular perturbation of the domain is reduced to the regular perturbation of the Steklov-Poincar´e map ping for the ring. The topological derivative for a tracking type shape functional is evaluated so as to obtain the useful formula for application in the numerical methods of shape and topology optimization.

6

Refined solutions of time inhomogeneous optimal stopping problem and zero-sum game via Dirichlet form

Yang Y.

Probability and Mathematical Statistics

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2014

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Vol. 34, Fasc. 2

253--271

EN

The properties of value functions of time inhomogeneous optimal stopping problem and zero-sum game (Dynkin game) are studied through time dependent Dirichlet form. Under the absolute continuity conditio on the transition function of the underlying process and some other assumptions, the refined solutions without exceptional starting points are proved to exist, and the value functions of the optimal stopping problem and zero-sum game, which belong to certain functional spaces, are characterized as the solutions of some variational inequalities, respectively.

7

A sign preserving mixed finite element approximation for contact problems

Hild P.

International Journal of Applied Mathematics and Computer Science

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2011

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Vol. 21, no 3

487-498

EN

This paper is concerned with the frictionless unilateral contact problem (i.e., a Signorini problem with the elasticity operator). We consider a mixed finite element method in which the unknowns are the displacement field and the contact pressure. The particularity of the method is that it furnishes a normal displacement field and a contact pressure satisfying the sign conditions of the continuous problem. The a priori error analysis of the method is closely linked with the study of a specific positivity preserving operator of averaging type which differs from the one of Chen and Nochetto. We show that this method is convergent and satisfies the same a priori error estimates as the standard approach in which the approximated contact pressure satisfies only a weak sign condition. Finally we perform some computations to illustrate and compare the sign preserving method with the standard approach.

8

Approximation for fixed points of strict pseudo-contractions and solutions of equilibrium and optimization problems

Kang J., Su Y., Zhang H.

Journal of Applied Analysis

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2010

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Vol. 16, nr 1

1-13

EN

In this paper, we introduce some iterative schemes for findin a common element of the set of fixed points of a k-strict pseudo-contractive mapping, the set of solutions of the variational inequality and the set of solutions of an equilibrium problem in a Hilbert space. The authors use the convex combination technique to show that the iterative sequences converge strongly to a common element of the three sets. The results of this paper extend and improve the results of Y. Su et al. [9], S. Plubtieng and R. Punpaeng [7], X. Qin et al. [8] and S. Takahashi and W. Takahashi [10].

9

Topological derivatives for semilinear elliptic equations

Iguernane M., Nazarov S. A., Roche J. R., Sokolowski J., Szulc K.

International Journal of Applied Mathematics and Computer Science

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2009

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Vol. 19, no 2

191-205

EN

The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L [...] norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

10

A level set method in shape and topology optimization for variational inequalities

Fulmański P., Laurain A., Scheid J. F., Sokołowski J.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 3

413-430

EN

The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.

11

Robinson's implicit function theorem

Dontchev A. L.

Control and Cybernetics

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2003

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Vol. 32, no 3

529-541

EN

Robinson's implicit function theorem has played a mayor role in the analysis of stability of optimization problems in the last two decades. In this paper we take a new look at this theorem, and with an updated terminology go back to the roots and present some extensions.

12

A quasistatic contact problem with slip dependent coefficient of friction for elastic materials

Corneschi C., Hoarau-Mantel T.-V. T.-V., Sofonea M.

Journal of Applied Analysis

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2002

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Vol. 8, nr 1

63-82

EN

We consider a mathematical model which describes the frictional contact between a deformable body and an obstacle, say a foundation. The body is assumed to be linear elastic and the contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. The novelty consists here in the fact that we consider a slip dependent coefficient of friction and a quasistatic process. We present two alternative yet equivalent formulations of the problem and establish existence and uniqueness results. The proofs are based on a new result obtained in [10] in the study of evolutionary variational inequalities.

13

On the Solution of a Finite Element Approximation of a Linear Obstacle Plate Problem

Fernandes L. M., Figueiredo I. N., Judice J. J.

International Journal of Applied Mathematics and Computer Science

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2002

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Vol. 12, no 1

27-40

EN

In this paper the solution of a finite element approximation of a linear obstacle plate problem is investigated. A simple version of an interior point method and a block pivoting algorithm have been proposed for the solution of this problem. Special purpose implementations of these procedures are included and have been used in the solution of a set of test problems. The results of these experiences indicate that these procedures are quite efficient to deal with these instances and compare favourably with the path-following PATH and the active-set MINOS codes of the commercial GAMS collection

14

Two projection-type algorithms for pseudo-monotone variational inequalities

Lin G-H., Xia Z-Q.

Archives of Control Sciences

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2000

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Vol. 10, no. 3/4

157-165

EN

Two projection-type algorithms, different from the ones presented by Solodov & Tseng (1996) and He (1997) in the computation of stepsize, for solving pseudo-monotone variational inequality problems are presented. It is shown that the algorithms are globally convergent and, in addition, they are convergent R-linearly or Q-linearly under some mild conditions.