Muhammad aqib Noor - Academia.edu (original) (raw)
Papers by Muhammad aqib Noor
Korean Journal of Computational & Applied Mathematics, 2000
In this paper, we use the auxiliary principle technique to suggest and analyze a class of predict... more In this paper, we use the auxiliary principle technique to suggest and analyze a class of predictor-corrector methods for solving noncoercive mixed variational inequalities. The convergence of the proposed method requires only the partially relaxed strongly monotonicity, which is even weaker than the co-coercivity. As special cases, we obtain a number of new and known results for classical variational inequalities.
Korean Journal of Computational & Applied Mathematics, 2000
In recent years, the theory of Wiener-Hopf equations has emerged as a novel and innovative techni... more In recent years, the theory of Wiener-Hopf equations has emerged as a novel and innovative technique for developing efficient and powerful numerical methods for solving variational inequalities and complementarity problems. In this paper, we provide an account of some of the fundamental aspects of the Wiener-Hopf equations with major emphasis on the formulation, computational algorithms, various generalizations and their applications. We also suggest some open problems for further research with sufficient information and references.
In this paper, we suggest and analyze a class of iterative schemes for solving multivalued quasi ... more In this paper, we suggest and analyze a class of iterative schemes for solving multivalued quasi variational inclusions using the resolvent operator method. As special cases, we obtain a number of known and new iterative schems for solving variational inequalities and related optimization problems. The results obtained in this represent an improvement and a significant refinement of previously known results.
Southeast Asian Bulletin of Mathematics, 2003
In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector ... more In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. From special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems.
Positivity, 2004
Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a cla... more Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a signi cant extension of previously known results in this eld. 2000 AMS Subject Classi cation: 49J40, 90C33.
Nonlinear Analysis: Theory, Methods & Applications, 2009
In this paper, we introduce and consider a new system of general variational inequalities involvi... more In this paper, we introduce and consider a new system of general variational inequalities involving four different operators. Using the projection operator technique, we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving three operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.
Journal of Mathematical Analysis and Applications, 2003
Integral operator, introduced by Noor, is defined by using convolution. Let f n (z) = z/(1 − z) n... more Integral operator, introduced by Noor, is defined by using convolution. Let f n (z) = z/(1 − z) n+1 , n ∈ N 0 , and let f be analytic in the unit disc E. Then I n f = f (−1) n f, where f n f (−1) n = z/(1 − z). Using this operator, certain classes of analytic functions, related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation, are defined and studied in detail. Some basic properties, rate of growth of coefficients, and a radius problem are investigated. It is shown that these classes are closed under convolution with convex functions. Most of the results are best possible in some sense.
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.
Journal of Mathematical Analysis and Applications, 2001
In this paper we establish the equivalence between the general variational inequalities and tange... more In this paper we establish the equivalence between the general variational inequalities and tangent projection equations. This equivalence is used to discuss the local convergence analysis of a wide class of iterative methods for solving the general variational inequalities. We show that some existing methods can identify the optimal face after finitely many iterations under the degenerate assumption.
Journal of Mathematical Analysis and Applications, 2000
In this paper, we suggest and consider a class of new three-step approximation schemes for genera... more In this paper, we suggest and consider a class of new three-step approximation schemes for general variational inequalities. Our results include Ishikawa and Mann iterations as special cases. We also study the convergence criteria of these schemes.
Journal of Mathematical Analysis and Applications, 2003
In this paper, we consider and analyze a new class of extragradient-type methods for solving gene... more In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.
Journal of Mathematical Analysis and Applications, 2002
In this paper, we suggest and analyze a three-step iterative scheme for solving nonlinear strongl... more In this paper, we suggest and analyze a three-step iterative scheme for solving nonlinear strongly accretive operator equation T x = f without continuous condition in a uniformly smooth Banach space. Our results include the Ishikawa, Mann and Noor iterations as special cases. The results presented in this paper improve and extend almost all the current results in the more general setting.
Journal of Mathematical Analysis and Applications, 2003
In this paper, we use the auxiliary principle technique to suggest some new classes of iterative ... more In this paper, we use the auxiliary principle technique to suggest some new classes of iterative algorithms for solving multivalued equilibrium problems. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems. Since multivalued equilibrium problems include equilibrium, variational inequality and complementarity problems as specials cases, our results continue to hold for these problems.
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.
Journal of Computational and Applied Mathematics, 2001
In this paper, we suggest and analyze a number of four-step resolvent splitting algorithms for so... more In this paper, we suggest and analyze a number of four-step resolvent splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods di er from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.
Journal of Applied Mathematics and Stochastic Analysis, 2004
We consider a new class of equilibrium problems, known as hemiequilibrium problems. Using the aux... more We consider a new class of equilibrium problems, known as hemiequilibrium problems. Using the auxiliary principle technique, we suggest and analyze a class of iterative algorithms for solving hemiequilibrium problems, the convergence of which requires either pseudomonotonicity or partially relaxed strong monotonicity. As a special case, we obtain a new method for hemivariational inequalities. Since hemiequilibrium problems include hemivariational inequalities and equilibrium problems as special cases, the results proved in this paper still hold for these problems.
International Journal of Mathematics and Mathematical Sciences, 2004
We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilib... more We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational-like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. As special cases, we also obtained the correct forms of the algorithms for solving variational-like inequalities, which have been considered in the setting of convexity. In fact, our results represent significant and important refinements of the previously known results.
International Journal of Mathematics and Mathematical Sciences, 2003
We use the technique of updating the solution to suggest and analyze a class of new splitting met... more We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known three-step forward-backward splitting of Glowinski, Le Tallec, and M. A. Noor for solving various classes of variational inequalities and complementarity problems. Since general mixed variational inequalities include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.
International Journal of Computer Mathematics, 2000
We use variational inequality theory along with finite difference technique to obtain an approxim... more We use variational inequality theory along with finite difference technique to obtain an approximation for the solution of a class of obstacle problem in elasticity, like those describing the equilibrium configuration of an elastic stretched over an elastic obstacle. ...
Korean Journal of Computational & Applied Mathematics, 2000
In this paper, we use the auxiliary principle technique to suggest and analyze a class of predict... more In this paper, we use the auxiliary principle technique to suggest and analyze a class of predictor-corrector methods for solving noncoercive mixed variational inequalities. The convergence of the proposed method requires only the partially relaxed strongly monotonicity, which is even weaker than the co-coercivity. As special cases, we obtain a number of new and known results for classical variational inequalities.
Korean Journal of Computational & Applied Mathematics, 2000
In recent years, the theory of Wiener-Hopf equations has emerged as a novel and innovative techni... more In recent years, the theory of Wiener-Hopf equations has emerged as a novel and innovative technique for developing efficient and powerful numerical methods for solving variational inequalities and complementarity problems. In this paper, we provide an account of some of the fundamental aspects of the Wiener-Hopf equations with major emphasis on the formulation, computational algorithms, various generalizations and their applications. We also suggest some open problems for further research with sufficient information and references.
In this paper, we suggest and analyze a class of iterative schemes for solving multivalued quasi ... more In this paper, we suggest and analyze a class of iterative schemes for solving multivalued quasi variational inclusions using the resolvent operator method. As special cases, we obtain a number of known and new iterative schems for solving variational inequalities and related optimization problems. The results obtained in this represent an improvement and a significant refinement of previously known results.
Southeast Asian Bulletin of Mathematics, 2003
In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector ... more In this paper, we use the auxiliary principle technique to suggest a class of predictorcorrector methods for solving general mixed variational inequalities. The convergence of the proposed methods only requires the partially relaxed strongly monotonicity of the operator, which is weaker than co-coercivity. From special cases, we obtain various known and new results for solving various classes of variational inequalities and related problems.
Positivity, 2004
Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a cla... more Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a signi cant extension of previously known results in this eld. 2000 AMS Subject Classi cation: 49J40, 90C33.
Nonlinear Analysis: Theory, Methods & Applications, 2009
In this paper, we introduce and consider a new system of general variational inequalities involvi... more In this paper, we introduce and consider a new system of general variational inequalities involving four different operators. Using the projection operator technique, we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving three operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.
Journal of Mathematical Analysis and Applications, 2003
Integral operator, introduced by Noor, is defined by using convolution. Let f n (z) = z/(1 − z) n... more Integral operator, introduced by Noor, is defined by using convolution. Let f n (z) = z/(1 − z) n+1 , n ∈ N 0 , and let f be analytic in the unit disc E. Then I n f = f (−1) n f, where f n f (−1) n = z/(1 − z). Using this operator, certain classes of analytic functions, related with the classes of functions with bounded boundary rotation and bounded boundary radius rotation, are defined and studied in detail. Some basic properties, rate of growth of coefficients, and a radius problem are investigated. It is shown that these classes are closed under convolution with convex functions. Most of the results are best possible in some sense.
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.
Journal of Mathematical Analysis and Applications, 2001
In this paper we establish the equivalence between the general variational inequalities and tange... more In this paper we establish the equivalence between the general variational inequalities and tangent projection equations. This equivalence is used to discuss the local convergence analysis of a wide class of iterative methods for solving the general variational inequalities. We show that some existing methods can identify the optimal face after finitely many iterations under the degenerate assumption.
Journal of Mathematical Analysis and Applications, 2000
In this paper, we suggest and consider a class of new three-step approximation schemes for genera... more In this paper, we suggest and consider a class of new three-step approximation schemes for general variational inequalities. Our results include Ishikawa and Mann iterations as special cases. We also study the convergence criteria of these schemes.
Journal of Mathematical Analysis and Applications, 2003
In this paper, we consider and analyze a new class of extragradient-type methods for solving gene... more In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.
Journal of Mathematical Analysis and Applications, 2002
In this paper, we suggest and analyze a three-step iterative scheme for solving nonlinear strongl... more In this paper, we suggest and analyze a three-step iterative scheme for solving nonlinear strongly accretive operator equation T x = f without continuous condition in a uniformly smooth Banach space. Our results include the Ishikawa, Mann and Noor iterations as special cases. The results presented in this paper improve and extend almost all the current results in the more general setting.
Journal of Mathematical Analysis and Applications, 2003
In this paper, we use the auxiliary principle technique to suggest some new classes of iterative ... more In this paper, we use the auxiliary principle technique to suggest some new classes of iterative algorithms for solving multivalued equilibrium problems. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems. Since multivalued equilibrium problems include equilibrium, variational inequality and complementarity problems as specials cases, our results continue to hold for these problems.
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.
Journal of Computational and Applied Mathematics, 2001
In this paper, we suggest and analyze a number of four-step resolvent splitting algorithms for so... more In this paper, we suggest and analyze a number of four-step resolvent splitting algorithms for solving general mixed variational inequalities by using the updating technique of the solution. The convergence of these new methods requires either monotonicity or pseudomonotonicity of the operator. Proof of convergence is very simple. Our new methods di er from the existing splitting methods for solving variational inequalities and complementarity problems. The new results are versatile and are easy to implement.
Journal of Applied Mathematics and Stochastic Analysis, 2004
We consider a new class of equilibrium problems, known as hemiequilibrium problems. Using the aux... more We consider a new class of equilibrium problems, known as hemiequilibrium problems. Using the auxiliary principle technique, we suggest and analyze a class of iterative algorithms for solving hemiequilibrium problems, the convergence of which requires either pseudomonotonicity or partially relaxed strong monotonicity. As a special case, we obtain a new method for hemivariational inequalities. Since hemiequilibrium problems include hemivariational inequalities and equilibrium problems as special cases, the results proved in this paper still hold for these problems.
International Journal of Mathematics and Mathematical Sciences, 2004
We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilib... more We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational-like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. As special cases, we also obtained the correct forms of the algorithms for solving variational-like inequalities, which have been considered in the setting of convexity. In fact, our results represent significant and important refinements of the previously known results.
International Journal of Mathematics and Mathematical Sciences, 2003
We use the technique of updating the solution to suggest and analyze a class of new splitting met... more We use the technique of updating the solution to suggest and analyze a class of new splitting methods for solving general mixed variational inequalities. It is shown that these modified methods converge for pseudomonotone operators, which is a weaker condition than monotonicity. Our methods differ from the known three-step forward-backward splitting of Glowinski, Le Tallec, and M. A. Noor for solving various classes of variational inequalities and complementarity problems. Since general mixed variational inequalities include variational inequalities and complementarity problems as special cases, our results continue to hold for these problems.
International Journal of Computer Mathematics, 2000
We use variational inequality theory along with finite difference technique to obtain an approxim... more We use variational inequality theory along with finite difference technique to obtain an approximation for the solution of a class of obstacle problem in elasticity, like those describing the equilibrium configuration of an elastic stretched over an elastic obstacle. ...