• Nonlinear Functional Analysis and Applications
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• 제27권1호
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• pp.121-140
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• 2022

In this paper, we study the strong convergence of new methods for solving classical variational inequalities problems involving semistrictly quasimonotone and Lipschitz-continuous operators in a real Hilbert space. Three proposed methods are based on Tseng's extragradient method and use a simple self-adaptive step size rule that is independent of the Lipschitz constant. The step size rule is built around two techniques: the monotone and the non-monotone step size rule. We establish strong convergence theorems for the proposed methods that do not require any additional projections or knowledge of an involved operator's Lipschitz constant. Finally, we present some numerical experiments that demonstrate the efficiency and advantages of the proposed methods.

• Nonlinear Functional Analysis and Applications
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• 제27권2호
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• pp.381-403
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• 2022

The main goal of this research is to solve variational inequalities involving quasimonotone operators in infinite-dimensional real Hilbert spaces numerically. The main advantage of these iterative schemes is the ease with which step size rules can be designed based on an operator explanation rather than the Lipschitz constant or another line search method. The proposed iterative schemes use a monotone and non-monotone step size strategy based on mapping (operator) knowledge as a replacement for the Lipschitz constant or another line search method. The strong convergences have been demonstrated to correspond well to the proposed methods and to settle certain control specification conditions. Finally, we propose some numerical experiments to assess the effectiveness and influence of iterative methods.

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

• 한국산업정보학회논문지
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• 제11권5호
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• pp.62-66
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• 2006

We investigate various $\Phi(t)-stability$ of comparison differential equations and we abtain necessary and/or sufficient conditions for the uniform asymptotic and exponential asymptotic stability of the nonlinear differential equation x'=f(t, x).

• 대한수학회지
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• 제37권5호
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• pp.835-847
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• 2000

We investigate various ${\Phi}$(t)-stability of comparison differential equations and we obtain necessary and/or sufficient conditions for the asymptotic and uniform asymptotic stability of the differential equations x'=f(t, x).

• Journal of applied mathematics & informatics
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• 제27권3_4호
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• pp.819-827
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• 2009

In this paper, we introduce new pseudomonotonicity and proper quasimonotonicity with respect to a given function, and show some existence results for strong implicit vector variational inequalities by considering new generalized Minty's lemma. Our results generalize and extend some results in [1].

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제15권4호
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• pp.423-432
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• 2008

In this paper, we consider a new Minty's Lemma for strong implicit vector variational inequality systems and obtain some existence results for systems of strong implicit vector variational inequalities which generalize some results in [1].

• Nonlinear Functional Analysis and Applications
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• 제29권1호
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• pp.131-164
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• 2024

In this research, we study a modified relaxed Tseng method with a single projection approach for solving common solution to a fixed point problem involving finite family of τ-demimetric operators and a quasi-monotone variational inequalities in real Hilbert spaces with alternating inertial extrapolation steps and adaptive non-monotonic step sizes. Under some appropriate conditions that are imposed on the parameters, the weak and linear convergence results of the proposed iterative scheme are established. Furthermore, we present some numerical examples and application of our proposed methods in comparison with other existing iterative methods. In order to show the practical applicability of our method to real word problems, we show that our algorithm has better restoration efficiency than many well known methods in image restoration problem. Our proposed iterative method generalizes and extends many existing methods in the literature.