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Papers by Abdul Rahim Khan

Fixed Point Theory and Applications, 2013

Convergence of a new iterative scheme, containing Mann and Ishikawa iterative schemes, for asympt... more Convergence of a new iterative scheme, containing Mann and Ishikawa iterative schemes, for asymptotically nonexpansive mappings on a 2-uniformly convex hyperbolic space is studied. As application, we find a solution of a system of certain nonlinear functional equations in uniformly convex Banach spaces. MSC: 47H09; 47H10

Creative Mathematics and Informatics, 2013

The tripled fixed point is a generalization of the well known concept of ”coupled fixed point”. I... more The tripled fixed point is a generalization of the well known concept of ”coupled fixed point”. In this paper, we establish tripled coincidence and tripled common fixed point theorems for a hybrid pair consisting of a multi-valued and a single valued mapping on a metric space. We give examples to illustrate our results.

Fixed Point Theory and Applications, 2012

Nemeth introduced the notion of order weakly L-Lipschitz mapping and employed this concept to obt... more Nemeth introduced the notion of order weakly L-Lipschitz mapping and employed this concept to obtain nontrivial solutions of nonlinear complementarity problems. In this article, we shall extend this concept to two mappings and obtain the solution of common fixed point equations and hence coincidence point equations in the framework of vector lattices. We present some examples to show that the solution of nonlinear complementarity problems and implicit complementarity problems can be obtained using these results. We also provide an example of a mapping for which the conclusion of Banach contraction principle fails but admits one of our fixed point results. Our proofs are simple and purely order-theoretic in nature.

The main purpose of this paper is to prove some results of uniform boundedness principle type wit... more The main purpose of this paper is to prove some results of uniform boundedness principle type without the use of Baire’s category theorem in certain topological vector spaces; this provides an alternate route and important technique to establish certain basic results of functional analysis. As applications, among other results, versions of the Banach-Steinhaus theorem and the Nikodym boundedness theorem are obtained.

Carpathian Journal of Mathematics, 2017

In the context of a hyperbolic space, we introduce and study convergence of an implicit iterative... more In the context of a hyperbolic space, we introduce and study convergence of an implicit iterative scheme of a finite family of asymptotically nonexpansive mappings without convergence condition. The results presented substantially improve and extend several well-known resullts in uniformly convex Banach spaces.

International Journal of Nonlinear Analysis and Applications, 2020

Many researchers have provided certain interesting results for endpoints of some contractive mult... more Many researchers have provided certain interesting results for endpoints of some contractive multivalued mappings in metric spaces. In this paper, we introduce alphaalphaalpha-$zeta$-contractive multivalued mappings in mathcalFmathcal{F}mathcalF-metric spaces and establish some endpoint results in this framework. An illustrative example is given to elaborate the usability of our main result. In the sequel, we give some endpoint theorems for Suzuki-type contractive multivalued mappings and provide an application to integral equations.

We introduce a new algorithm to approximate a solution of split variational inclusion problems of... more We introduce a new algorithm to approximate a solution of split variational inclusion problems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

Journal of Nonlinear Sciences and Applications, 2015

Javahernia et al. [Fixed Point Theory and Applications 2014, 2014:195] introduced the concept of ... more Javahernia et al. [Fixed Point Theory and Applications 2014, 2014:195] introduced the concept of generalized Mizoguchi-Takahashi type contractions and established some common fixed point results for such contractions. In this paper, we define the notion of generalized α * − Mizoguchi-Takahashi type contractions and obtain some new fixed point results which generalize various results existing in literature. An example is included to show that our results are genuine generalization of the corresponding known results.

Mathematics, 2019

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solu... more The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

Miskolc Mathematical Notes, 2014

TURKISH JOURNAL OF MATHEMATICS, 2016

Fixed Point Theory and Applications, 2015

Annals of Functional Analysis, 2013

Glasgow Mathematical Journal, 1985

In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been p... more In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been proved for group-valued measures. For example, Traynor [4], [6] has established Lebesgue decomposition theorems for exhaustive groupvalued measures on a ring using (1) algebraic and (2) topological notions of continuity and singularity, and generalizations of the Hewitt-Yosida theorem have been given by Drewnowski [2], Traynor [5] and Khurana [3]. In this paper we consider group-valued submeasures and in particular we have established a decomposition theorem from which analogues of the Lebesgue and Hewitt-Yosida decomposition theorems for submeasures may be derived. Our methods are based on those used by Drewnowski in [2] and the main theorem established generalizes Theorem 4.1 of [2].

Fixed Point Theory and Applications, 2011

We establish strong convergence of an implicit algorithm to a common fixed point of a finite fami... more We establish strong convergence of an implicit algorithm to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive maps in CAT 0 spaces. Our work improves and extends several recent results from the current literature.

The Journal of Nonlinear Sciences and Applications, 2017

We introduce and study convergence of a one-step iterative algorithm for a finite family of total... more We introduce and study convergence of a one-step iterative algorithm for a finite family of total asymptotically nonexpansive mappings on a CAT(0) space. Our results are new in Hilbert spaces as well as CAT(0) spaces; in particular, an analogue of Rhoades weak convergence theorem [B. E. Rhoades, Bull. Austral. Math. Soc., 62 (2000), 307-310] is established both for-convergence and strong convergence in CAT(0) spaces.

Journal of Nonlinear and Convex Analysis, 2015

Bulletin of The Iranian Mathematical Society, 2015

The aim of this paper is to establish random coincidence point results for weakly increasing rand... more The aim of this paper is to establish random coincidence point results for weakly increasing random operators in the setting of ordered metric spaces by using generalized altering distance functions. Our results present random versions and extensions of some well-known results in the current literature.

Rendiconti del Circolo Matematico di Palermo Series 2

We study Moudafi’s iterative algorithm for an \alpha α-nonexpansive mapping and a fundamental... more We study Moudafi’s iterative algorithm for an \alpha α-nonexpansive mapping and a fundamentally nonexpansive mapping in the framework of a convex metric space. We prove \triangle ▵-convergence and strong convergence results for the algorithm to a common fixed point of the mappings. Our results are new and are also valid in CAT$$\left( 0\right) 0 spaces and Banach spaces, simultaneously.

Fixed Point Theory and Applications, 2013

Convergence of a new iterative scheme, containing Mann and Ishikawa iterative schemes, for asympt... more Convergence of a new iterative scheme, containing Mann and Ishikawa iterative schemes, for asymptotically nonexpansive mappings on a 2-uniformly convex hyperbolic space is studied. As application, we find a solution of a system of certain nonlinear functional equations in uniformly convex Banach spaces. MSC: 47H09; 47H10

Creative Mathematics and Informatics, 2013

The tripled fixed point is a generalization of the well known concept of ”coupled fixed point”. I... more The tripled fixed point is a generalization of the well known concept of ”coupled fixed point”. In this paper, we establish tripled coincidence and tripled common fixed point theorems for a hybrid pair consisting of a multi-valued and a single valued mapping on a metric space. We give examples to illustrate our results.

Fixed Point Theory and Applications, 2012

Nemeth introduced the notion of order weakly L-Lipschitz mapping and employed this concept to obt... more Nemeth introduced the notion of order weakly L-Lipschitz mapping and employed this concept to obtain nontrivial solutions of nonlinear complementarity problems. In this article, we shall extend this concept to two mappings and obtain the solution of common fixed point equations and hence coincidence point equations in the framework of vector lattices. We present some examples to show that the solution of nonlinear complementarity problems and implicit complementarity problems can be obtained using these results. We also provide an example of a mapping for which the conclusion of Banach contraction principle fails but admits one of our fixed point results. Our proofs are simple and purely order-theoretic in nature.

The main purpose of this paper is to prove some results of uniform boundedness principle type wit... more The main purpose of this paper is to prove some results of uniform boundedness principle type without the use of Baire’s category theorem in certain topological vector spaces; this provides an alternate route and important technique to establish certain basic results of functional analysis. As applications, among other results, versions of the Banach-Steinhaus theorem and the Nikodym boundedness theorem are obtained.

Carpathian Journal of Mathematics, 2017

In the context of a hyperbolic space, we introduce and study convergence of an implicit iterative... more In the context of a hyperbolic space, we introduce and study convergence of an implicit iterative scheme of a finite family of asymptotically nonexpansive mappings without convergence condition. The results presented substantially improve and extend several well-known resullts in uniformly convex Banach spaces.

International Journal of Nonlinear Analysis and Applications, 2020

Many researchers have provided certain interesting results for endpoints of some contractive mult... more Many researchers have provided certain interesting results for endpoints of some contractive multivalued mappings in metric spaces. In this paper, we introduce alphaalphaalpha-$zeta$-contractive multivalued mappings in mathcalFmathcal{F}mathcalF-metric spaces and establish some endpoint results in this framework. An illustrative example is given to elaborate the usability of our main result. In the sequel, we give some endpoint theorems for Suzuki-type contractive multivalued mappings and provide an application to integral equations.

We introduce a new algorithm to approximate a solution of split variational inclusion problems of... more We introduce a new algorithm to approximate a solution of split variational inclusion problems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

Journal of Nonlinear Sciences and Applications, 2015

Javahernia et al. [Fixed Point Theory and Applications 2014, 2014:195] introduced the concept of ... more Javahernia et al. [Fixed Point Theory and Applications 2014, 2014:195] introduced the concept of generalized Mizoguchi-Takahashi type contractions and established some common fixed point results for such contractions. In this paper, we define the notion of generalized α * − Mizoguchi-Takahashi type contractions and obtain some new fixed point results which generalize various results existing in literature. An example is included to show that our results are genuine generalization of the corresponding known results.

Mathematics, 2019

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solu... more The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

Miskolc Mathematical Notes, 2014

TURKISH JOURNAL OF MATHEMATICS, 2016

Fixed Point Theory and Applications, 2015

Annals of Functional Analysis, 2013

Glasgow Mathematical Journal, 1985

In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been p... more In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been proved for group-valued measures. For example, Traynor [4], [6] has established Lebesgue decomposition theorems for exhaustive groupvalued measures on a ring using (1) algebraic and (2) topological notions of continuity and singularity, and generalizations of the Hewitt-Yosida theorem have been given by Drewnowski [2], Traynor [5] and Khurana [3]. In this paper we consider group-valued submeasures and in particular we have established a decomposition theorem from which analogues of the Lebesgue and Hewitt-Yosida decomposition theorems for submeasures may be derived. Our methods are based on those used by Drewnowski in [2] and the main theorem established generalizes Theorem 4.1 of [2].

Fixed Point Theory and Applications, 2011

We establish strong convergence of an implicit algorithm to a common fixed point of a finite fami... more We establish strong convergence of an implicit algorithm to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive maps in CAT 0 spaces. Our work improves and extends several recent results from the current literature.

The Journal of Nonlinear Sciences and Applications, 2017

We introduce and study convergence of a one-step iterative algorithm for a finite family of total... more We introduce and study convergence of a one-step iterative algorithm for a finite family of total asymptotically nonexpansive mappings on a CAT(0) space. Our results are new in Hilbert spaces as well as CAT(0) spaces; in particular, an analogue of Rhoades weak convergence theorem [B. E. Rhoades, Bull. Austral. Math. Soc., 62 (2000), 307-310] is established both for-convergence and strong convergence in CAT(0) spaces.

Journal of Nonlinear and Convex Analysis, 2015

Bulletin of The Iranian Mathematical Society, 2015

The aim of this paper is to establish random coincidence point results for weakly increasing rand... more The aim of this paper is to establish random coincidence point results for weakly increasing random operators in the setting of ordered metric spaces by using generalized altering distance functions. Our results present random versions and extensions of some well-known results in the current literature.

Rendiconti del Circolo Matematico di Palermo Series 2

We study Moudafi’s iterative algorithm for an \alpha α-nonexpansive mapping and a fundamental... more We study Moudafi’s iterative algorithm for an \alpha α-nonexpansive mapping and a fundamentally nonexpansive mapping in the framework of a convex metric space. We prove \triangle ▵-convergence and strong convergence results for the algorithm to a common fixed point of the mappings. Our results are new and are also valid in CAT$$\left( 0\right) 0 spaces and Banach spaces, simultaneously.