• Bulletin of the Korean Mathematical Society
• /
• 제47권4호
• /
• pp.675-685
• /
• 2010

In this paper, we consider a new class of regularized (nonconvex) generalized mixed variational inequality problems in real Hilbert space. We give the concepts of partially relaxed strongly mixed monotone and partially relaxed strongly $\theta$-pseudomonotone mappings, which are extension of the concepts given by Xia and Ding [19], Noor [13] and Kazmi et al. [9]. Further we use the auxiliary principle technique to suggest a two-step iterative algorithm for solving regularized (nonconvex) generalized mixed variational inequality problem. We prove that the convergence of the iterative algorithm requires only the continuity, partially relaxed strongly mixed monotonicity and partially relaxed strongly $\theta$-pseudomonotonicity. The theorems presented in this paper represent improvement and generalization of the previously known results for solving equilibrium problems and variational inequality problems involving the nonconvex (convex) sets, see for example Noor [13], Pang et al. [14], and Xia and Ding [19].

• Communications of the Korean Mathematical Society
• /
• 제22권3호
• /
• pp.401-410
• /
• 2007

Some existence theorems of solutions of a new class of generalized vector F-implicit complementarity problems with the corresponding generalized vector F-implicit variational inequality problems were established.

• Honam Mathematical Journal
• /
• 제42권1호
• /
• pp.17-35
• /
• 2020

In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the fixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

• Communications of the Korean Mathematical Society
• /
• 제24권3호
• /
• pp.425-432
• /
• 2009

This paper introduces a local non-positivity of two set-valued mappings (F,Q) and considers the existences and properties of solutions for set-valued mixed vector FQ-implicit variational inequality problems and set-valued mixed vector FQ-complementarity problems in the neighborhood of a point belonging to an underlined domain K of the set-valued mappings, where the neighborhood is contained in K. This paper generalizes and extends many results in [1, 3-7].

• Journal of applied mathematics & informatics
• /
• 제27권1_2호
• /
• pp.161-173
• /
• 2009

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extend and improve the corresponding results announced by many others.

• East Asian mathematical journal
• /
• 제26권3호
• /
• pp.371-378
• /
• 2010

In this paper, we study generalized F-implicit multivalued variational inequality problems on a real normed vector space setting. As an application, we study generalized F-implicit multivalued complementarity problems.

• Korean Journal of Mathematics
• /
• 제27권2호
• /
• pp.297-327
• /
• 2019

In this paper, we introduce and study a general iterative algorithm to approximate a common solution of split generalized equilibrium problem, variational inequality problem and fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. Further, we prove a strong convergence theorem for the sequences generated by the proposed iterative scheme. Finally, we derive some consequences from our main result. The results presented in this paper extended and unify many of the previously known results in this area.

• Bulletin of the Korean Mathematical Society
• /
• 제33권1호
• /
• pp.45-55
• /
• 1996

Recently, Giannessi [1] introduced a vector variational inequalityy for vector-valued functions in an Euclidean space. Since then, Chen et al. [2-6], Lee et al. [7], and Yang [8] have intensively studied vector variational inequalities for vector-valued functions in abstract spaces.

• Journal of Information Processing Systems
• /
• 제13권2호
• /
• pp.417-427
• /
• 2017

Power allocation is an important factor for cognitive radio networks to achieve higher communication capacity and faster equilibrium. This paper considers power allocation problem to each cognitive user to maximize capacity of the cognitive systems subject to the constraints on the total power of each cognitive user and the interference levels of the primary user. Since this power control problem can be formulated as a mixed-integer nonlinear programming (NP) equivalent to variational inequality (VI) problem in convex polyhedron which can be transformed into complementary problem (CP), we utilize modified projection method to solve this CP problem instead of finding NP solution and give a power control allocation algorithm with a subcarrier allocation scheme. Simulation results show that the proposed algorithm performs well and effectively reduces the system power consumption with almost maximum capacity while achieve Nash equilibrium.

• Nonlinear Functional Analysis and Applications
• /
• 제28권1호
• /
• pp.205-235
• /
• 2023

In this paper, we introduce and study an iterative technique for solving quasimonotone split variational inequality problems and fixed point problem in the framework of real Hilbert spaces. Our proposed iterative technique is self adaptive, and easy to implement. We establish that the proposed iterative technique converges strongly to a minimum-norm solution of the problem and give some numerical illustrations in comparison with other methods in the literature to support our strong convergence result.