In recent years, much attention has been given to develop some analytical methods for solving int... more In recent years, much attention has been given to develop some analytical methods for solving integral equations including the perturbation methods and decomposition method. It is well known that perturbation methods [1, 2] provide the most versatile tools available in ...
Journal of Optimization Theory and Applications, 1995
In this paper, we study some properties of a class of nonconvex functions, called semipreinvex fu... more In this paper, we study some properties of a class of nonconvex functions, called semipreinvex functions, which includes the classes of preinvex functions and arc-connected convex functions. It is shown that the minimum of an arcwise directionally differentiable semi-invex functions on a semi-invex set can be characterized by a class of variational inequalities, known as variational-like inequalities. We use the auxiliary principle technique to prove the existence of a solution of a variational-like inequality and suggest a novel iterative algorithm.
General variational inequalities provide us with a unified, natural, novel and simple framework t... more General variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection, Wiener-Hopf equations, updating the solution, auxiliary principle, inertial proximal, penalty function, dynamical system and well-posedness. We also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection-type algorithms. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, results presented here continue to hold for these problems. Several open problems have been suggested for further research in these areas.
Journal of Computational and Applied Mathematics, 1994
In this paper, we introduce new functions as generalizations of the incomplete gamma functions. T... more In this paper, we introduce new functions as generalizations of the incomplete gamma functions. The functions are found to be useful in heat conduction, probability theory and in the study of Fourier and Laplace transforms. Some important properties of the functions are ...
Journal of Mathematical Analysis and Applications, 1998
In this paper, we establish the equivalence between the generalized set-valued variational inclus... more In this paper, we establish the equivalence between the generalized set-valued variational inclusions, the resolvent equations, and the fixed-point problem, using the resolvent operator technique. This equivalence is used to suggest and analyze some iterative algorithms for solving the generalized set-valued variational inclusions and related optimization problems. ᮊ 1998 Academic Press
Journal of Mathematical Analysis and Applications, 1998
In this paper, we introduce and study a new class of variational inequalities, which is called th... more In this paper, we introduce and study a new class of variational inequalities, which is called the generalized set-valued mixed variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued variational inequalities, fixed point problems, and the generalized setvalued resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.
Journal of Optimization Theory and Applications, 1997
In this paper, we develop a sensitivity analysis framework for quasi-variational inequalities usi... more In this paper, we develop a sensitivity analysis framework for quasi-variational inequalities using the Wiener–Hopf equations technique without assuming the differentiability of the given data.
Journal of Mathematical Analysis and Applications, 2002
In this paper, we suggest and analyze a three-step iterative scheme for asymptotically nonexpansi... more In this paper, we suggest and analyze a three-step iterative scheme for asymptotically nonexpansive mappings in Banach spaces. The new iterative scheme includes Ishikawa-type and Mann-type interations as special cases. The results obtained in this paper represent an extension as well as refinement of previous known results. 2002 Elsevier Science (USA)
Journal of Computational and Applied Mathematics, 1993
In this paper we provide an account of some of the fundamental aspects of variational inequalitie... more In this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence, uniqueness, computational properties, various generalizations, sensitivity analysis and their applications. We also propose some open ...
Journal of Mathematical Analysis and Applications, 2001
In this paper, we suggest and analyze some new classes of three-step iterative algorithms for sol... more In this paper, we suggest and analyze some new classes of three-step iterative algorithms for solving multivalued quasi variational inclusions by using the resolvent equations technique. New iterative algorithms include the Ishikawa, Mann, and Noor iterations for solving variational inclusions (inequalities) and optimization problems as special cases. The results obtained in this paper represent an improvement and a significant refinement of previously known results.
In recent years, much attention has been given to develop some analytical methods for solving int... more In recent years, much attention has been given to develop some analytical methods for solving integral equations including the perturbation methods and decomposition method. It is well known that perturbation methods [1, 2] provide the most versatile tools available in ...
Journal of Optimization Theory and Applications, 1995
In this paper, we study some properties of a class of nonconvex functions, called semipreinvex fu... more In this paper, we study some properties of a class of nonconvex functions, called semipreinvex functions, which includes the classes of preinvex functions and arc-connected convex functions. It is shown that the minimum of an arcwise directionally differentiable semi-invex functions on a semi-invex set can be characterized by a class of variational inequalities, known as variational-like inequalities. We use the auxiliary principle technique to prove the existence of a solution of a variational-like inequality and suggest a novel iterative algorithm.
General variational inequalities provide us with a unified, natural, novel and simple framework t... more General variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection, Wiener-Hopf equations, updating the solution, auxiliary principle, inertial proximal, penalty function, dynamical system and well-posedness. We also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection-type algorithms. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, results presented here continue to hold for these problems. Several open problems have been suggested for further research in these areas.
Journal of Computational and Applied Mathematics, 1994
In this paper, we introduce new functions as generalizations of the incomplete gamma functions. T... more In this paper, we introduce new functions as generalizations of the incomplete gamma functions. The functions are found to be useful in heat conduction, probability theory and in the study of Fourier and Laplace transforms. Some important properties of the functions are ...
Journal of Mathematical Analysis and Applications, 1998
In this paper, we establish the equivalence between the generalized set-valued variational inclus... more In this paper, we establish the equivalence between the generalized set-valued variational inclusions, the resolvent equations, and the fixed-point problem, using the resolvent operator technique. This equivalence is used to suggest and analyze some iterative algorithms for solving the generalized set-valued variational inclusions and related optimization problems. ᮊ 1998 Academic Press
Journal of Mathematical Analysis and Applications, 1998
In this paper, we introduce and study a new class of variational inequalities, which is called th... more In this paper, we introduce and study a new class of variational inequalities, which is called the generalized set-valued mixed variational inequality. The resolvent operator technique is used to establish the equivalence among generalized set-valued variational inequalities, fixed point problems, and the generalized setvalued resolvent equations. This equivalence is used to study the existence of a solution of set-valued variational inequalities and to suggest a number of iterative algorithms for solving variational inequalities and related optimization problems. The results proved in this paper represent a significant refinement and improvement of the previously known results in this area.
Journal of Optimization Theory and Applications, 1997
In this paper, we develop a sensitivity analysis framework for quasi-variational inequalities usi... more In this paper, we develop a sensitivity analysis framework for quasi-variational inequalities using the Wiener–Hopf equations technique without assuming the differentiability of the given data.
Journal of Mathematical Analysis and Applications, 2002
In this paper, we suggest and analyze a three-step iterative scheme for asymptotically nonexpansi... more In this paper, we suggest and analyze a three-step iterative scheme for asymptotically nonexpansive mappings in Banach spaces. The new iterative scheme includes Ishikawa-type and Mann-type interations as special cases. The results obtained in this paper represent an extension as well as refinement of previous known results. 2002 Elsevier Science (USA)
Journal of Computational and Applied Mathematics, 1993
In this paper we provide an account of some of the fundamental aspects of variational inequalitie... more In this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence, uniqueness, computational properties, various generalizations, sensitivity analysis and their applications. We also propose some open ...
Journal of Mathematical Analysis and Applications, 2001
In this paper, we suggest and analyze some new classes of three-step iterative algorithms for sol... more In this paper, we suggest and analyze some new classes of three-step iterative algorithms for solving multivalued quasi variational inclusions by using the resolvent equations technique. New iterative algorithms include the Ishikawa, Mann, and Noor iterations for solving variational inclusions (inequalities) and optimization problems as special cases. The results obtained in this paper represent an improvement and a significant refinement of previously known results.
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