• Bulletin of the Korean Mathematical Society
• /
• v.50 no.2
• /
• pp.589-599
• /
• 2013

In this paper, we propose a sufficient condition for the existence of solutions to general variational inequality problems (GVI(K, F, $g$)). The condition is also necessary when F is a $g-P^M_*$ function. We also investigate the boundedness of the solution set of (GVI(K, F, $g$)). Furthermore, we show that when F is norm-coercive, the general complementarity problems (GCP(K, F, $g$)) has a nonempty compact solution set. Finally, we establish some existence theorems for (GNCP(K, F, $g$)).

• Nonlinear Functional Analysis and Applications
• /
• v.27 no.1
• /
• pp.149-167
• /
• 2022

In this paper, we propose a viscosity iterative algorithm for approximating a common solution of finite family of variational inequality problem and fixed point problem for finite family of multi-valued type-one demicontractive mappings in real Hilbert spaces. A strong convergence result of the aforementioned problems were proved and some consequences of our result was also displayed. In addition, we discuss an application of our main result to convex minimization problem. The result presented in this article complements and extends many recent results in literature.

• Nonlinear Functional Analysis and Applications
• /
• v.26 no.2
• /
• pp.347-371
• /
• 2021

A plethora of applications from mathematical programmings, such as minimax, mathematical programming, penalization and fixed point problems can be framed as variational inequality problems. Most of the methods that used to solve such problems involve iterative methods, that is why, in this paper, we introduce a new extragradient-like method to solve pseudomonotone variational inequalities in a real Hilbert space. The proposed method has the advantage of a variable step size rule that is updated for each iteration based on previous iterations. The main advantage of this method is that it operates without the previous knowledge of the Lipschitz constants of an operator. A strong convergence theorem for the proposed method is proved by letting the mild conditions on an operator 𝒢. Numerical experiments have been studied in order to validate the numerical performance of the proposed method and to compare it with existing methods.

• Journal of applied mathematics & informatics
• /
• v.29 no.5_6
• /
• pp.1453-1462
• /
• 2011

In this paper, we study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obtain some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalize many known results in recent literature.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.553-564
• /
• 1996

Since Giannessi [5] introduced the vector variational inequality in a finite dimensional Euclidean space with further application, Chang et al. [17], Chen et al. [1-4] and Lee et al. [10-16] have considered several kinds of vector variational inequalities in abstract spaces and have obtained existence theorems for their inequalities.

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

• The Pure and Applied Mathematics
• /
• v.11 no.2
• /
• pp.155-166
• /
• 2004

In this paper, we introduce new monotone concepts for set-valued mappings, called generalized C(x)-L-pseudomonotonicity and weakly C(x)-L-pseudomonotonicity. And we obtain generalized Minty-type lemma and the existence of solutions to vector variational inequality problems for weakly C(x)-L-pseudomonotone set-valued mappings, which generalizes and extends some results of Konnov & Yao [11], Yu & Yao [20], and others Chen & Yang [6], Lai & Yao [12], Lee, Kim, Lee & Cho [16] and Lin, Yang & Yao [18].

• Journal of applied mathematics & informatics
• /
• v.29 no.1_2
• /
• pp.61-74
• /
• 2011

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming and the descent-like method for finding a common element of the solution set for a variational inequality and the set of common fixed points of a nonexpansive semigroup in Hilbert spaces. We obtain a strong convergence for the sequence generated by our method in Hilbert spaces. The result in this paper modifies and improves some well-known results in the literature for a more general problem.

• Nonlinear Functional Analysis and Applications
• /
• v.29 no.2
• /
• pp.307-332
• /
• 2024

Two inertial extragradient-type algorithms are introduced for solving convex pseudomonotone variational inequalities with fixed point problems, where the associated mapping for the fixed point is a 𝜌-demicontractive mapping. The algorithm employs variable step sizes that are updated at each iteration, based on certain previous iterates. One notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz-type constants and without necessitating any line search procedures. The iterative sequence constructed demonstrates strong convergence to the common solution of the variational inequality and fixed point problem under standard assumptions. In-depth numerical applications are conducted to illustrate theoretical findings and to compare the proposed algorithms with existing approaches.

• Nonlinear Functional Analysis and Applications
• /
• v.29 no.2
• /
• pp.393-418
• /
• 2024

In this paper, we propose a new inertial subgradient extragradient algorithm with a new linesearch technique that combines the inertial subgradient extragradient algorithm and the KrasnoselskiiMann algorithm. Under some suitable conditions, we prove a weak convergence theorem of the proposed algorithm for finding a common element of the common solution set of a finitely many variational inequality problem and the fixed point set of a nonexpansive mapping in real Hilbert spaces. Moreover, using our main result, we derive some others involving systems of variational inequalities. Finally, we give some numerical examples to support our main result.