• Journal of the Korean Mathematical Society
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• v.38 no.3
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• pp.683-695
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• 2001

In this paper we are concerned with theoretical properties of gap functions for the extended variational inequality problem (EVI) in a general Banach space. We will present a correction of a recent result of Chen et. al. without assuming the convexity of the given function Ω. Using this correction, we will discuss the continuity and the differentiability of a gap function, and compute its gradient formula under tow particular situations in a general Banach space. Our results can be regarded as infinite dimensional generalizations of the well-known results of Fukushima, and Zhu and Marcotte with soem modifications.

• Transactions of the Korean Society of Mechanical Engineers
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• v.7 no.1
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• pp.52-63
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• 1983

The purpose of the study is to find development of contact area, contact pressure and friction forces occurring at joints or connection areas inbetween structural members or mechanical parts. The problem has a pair of difficulties intrinsically; a constraint of displacement due to contact, and presence of work term by nonconservative friction force in the variational principle of the problem. Because of these difficulties, the variational principle remains in the form of inequality. It is resolved by penalty method and perturbation method making the inequality to an equality which is proper for computational purposes. A contact problem without friction is solved to find contact area and contact pressure, which are to be used as data for the analysis of the friction problem using perturbed variational principle. For numerical experiments, a Hertz problem, a rigid punch problem, and the latter one with friction effects are solved using $Q_2$-finite elements.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.991-1002
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• 2011

An iterative algorithm is provided to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of some variational inequality in a Hilbert space. Using this result, we consider a strong convergence result for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping. Our results include the previous results as special cases and can be viewed as an improvement and refinement of the previously known results.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.377-392
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• 2008

Let X be a real reflexive Banach space with a uniformly $G\hat{a}teaux$ differentiable norm, C a nonempty closed convex subset of X, T : C $\rightarrow$ X a continuous pseudocontractive mapping, and A : C $\rightarrow$ C a continuous strongly pseudocontractive mapping. We show the existence of a path ${x_t}$ satisfying $x_t=tAx_t+(1- t)Tx_t$, t $\in$ (0,1) and prove that ${x_t}$ converges strongly to a fixed point of T, which solves the variational inequality involving the mapping A. As an application, we give strong convergence of the path ${x_t}$ defined by $x_t=tAx_t+(1-t)(2I-T)x_t$ to a fixed point of firmly pseudocontractive mapping T.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.1-22
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• 2022

The purpose of this research is to formulate a new proximal-type algorithm to solve the equilibrium problem in a real Hilbert space. A new algorithm is analogous to the famous two-step extragradient algorithm that was used to solve variational inequalities in the Hilbert spaces previously. The proposed iterative scheme uses a new step size rule based on local bifunction details instead of Lipschitz constants or any line search scheme. The strong convergence theorem for the proposed algorithm is well-proven by letting mild assumptions about the bifunction. Applications of these results are presented to solve the fixed point problems and the variational inequality problems. Finally, we discuss two test problems and computational performance is explicating to show the efficiency and effectiveness of the proposed algorithm.

• Journal of applied mathematics & informatics
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• v.41 no.3
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• pp.487-510
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• 2023

We propose a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The behavior of the material is described with a linearly electro-viscoelastic constitutive law with long term memory. The mechanical process is dynamic and the electrical conductivity coefficient depends on the total slip rate, the friction is modeled with Tresca's law which the friction bound depends on the total slip rate with taking into account the electrical conductivity of the foundation both. The main results of this paper concern the existence and uniqueness of the weak solution of the model; the proof is based on results for second order evolution variational inequalities with a time-dependent hemivariational inequality in Banach spaces.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.227-245
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• 2011

In this paper, we introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a ${\beta}$inverse-strongly monotone mapping and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Ces$\`{a}$ro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang [A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mapping, Nonlinear Analysis: Hybrid Systems, 3(2009), 475-86], Peng and Yao [Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Mathematical and Computer Modelling, 49(2009), 1816-828], Shimizu and Takahashi [Strong convergence to common fixed points of families of nonexpansive mappings, Journal of Mathematical Analysis and Applications, 211(1) (1997), 71-83] and some authors.

• East Asian mathematical journal
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• v.28 no.5
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• pp.561-567
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• 2012

The approximate solvability of a generalized system for relaxed cocoercive extended general variational inequalities is studied by using the project operator technique. The results presented in this paper are more general and include many previously known results as special cases.

• Journal of applied mathematics & informatics
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• v.4 no.2
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• pp.433-446
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• 1997

In this paper we study an existence and the approxi-mation of the solution of the solution of the elliptic variational inequality from an abstract axiomatic point of view. We discuss convergence results and give an error estimate for the difference of the two solutions in an appropriate norm Also we present some computational results by using fixed point method.

• Korean Journal of Mathematics
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• v.20 no.2
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• pp.225-237
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• 2012

In this paper, we introduce a new iterative scheme to investigate the problem of nding a common element of nonexpansive mappings and the set of solutions of generalized variational inequalities for a $k$-strict pseudo-contraction by relaxed extra-gradient methods. Strong convergence theorems are established in $q$-uniformly smooth Banach spaces.