Balwant Singh Thakur

• PhD
• Professor at Pandit Ravishankar Shukla University

The purpose of this paper is to propose an algorithm for solving the proximal split feasibility problems and fixed point problems in Hilbert spaces. The algorithm is motivated by the inertial method and the split proximal algorithm with a self-adaptive stepsize such that their implementation does not need any prior information about the operator no...

The paper analyzes a recently proposed secure authentication and key agreement scheme for roaming service in a ubiquitous network. In 2018, Lee et al. proposed a biometric-based anonymous authentication scheme for roaming in ubiquitous networks. But, we found that Lee et al. scheme is prone to the off-line dictionary attack when a user’s smart devi...

In this paper, we present an implicit iteration scheme with perturbed mapping to approximate fixed points of pseudocontractive mappings. Under certain conditions, strong and weak convergence theorems for this implicit iteration scheme are proved. We use numerical experiments, which show that the algorithm is efficient and not very sensitive to diff...

In this paper, we propose an implicit iteration scheme with perturbed mapping for a finite family of strictly pseudocontractive mappings and establish weak and strong convergence theorems. We report some preliminary computational results related to the influence of parameters of the algorithm. Results in this paper extend and improve recent results...

In this paper, we propose a new hybrid iteration for total asypmp-totically nonexpansive mappings. We establish A-convergence and strong convergence theorems on a CAT(0) space which extend and improve many results in this literature. We also provide numerical examples to illustrate the performance of proposed iteration.

In this paper, we introduce a new three-step iteration scheme and establish convergence results for approximation of fixed points of non expansive mappings in the framework of Banach space. Further, we show that the new iteration process is faster than a number of existing iteration processes. To support the claim, we consider a numerical example a...

We have studied modified absolute Norlund summability of conjugate series of a Fourier series with multipliers and investigated some necessary and sufficient conditions for the generating functions of the conjugate series of a Fourier series to obtain its absolute Norlund summability. In this paper, the best possible summability multipliers in cert...

In this study, we introduce a three step iteration process and prove a convergence result for a countable family of pseudocontractive mappings. Our results improve and generalize most of the results that have been proved for this important class of nonlinear mappings. MSC: 47H09, 47H10.

In this paper, we present best proximity point theorems for new class of K−rational proximal contraction, in the setting of metric spaces. Some illustrative example are also given.

In this paper, using the modified hybrid Picard-Mann iteration process, we establish Δ-convergence and strong convergence theorems for total asymptotically nonexpansive mappings on a C A T ( 0 ) space. Results established in the paper extend and improve a number of results in the literature. A numerical example is also given to examine the fastness...

In this paper, we prove strong and weak convergence theorems for a mapping defined on a bounded, closed and convex subset of a uniformly convex Banach space, satisfying the RCSC condition. This condition was introduced by Karapınar (Dynamical Systems and Methods, 2012). We first establish the demiclosed principle for the mapping satisfying the RCSC...

In this paper, we propose a three-step iteration scheme for nonself asymptotically nonexpansive type mappings in a complete CAT(0) metric space and establish necessary and sufficient conditions for convergence of this process to a fixed point of nonself asymptotically nonexpansive type mappings. We also establish a strong convergence result. These...

In this paper, we study a series associated with Fourier series and conjugate series of a Fourier series by using absolute Riesz summability of positive order and a special but precise type of Riesz mean and obtain a few results which not only include some of the known results due to Bosanquet and Hyslop [1], Mohanty [12] and Nayak [13] but also pr...

In mobile adhoc networks (MANETs) an efficient and secure key management scheme is extremely crucial. Key management schemes for MANETs are mainly based on identity-based public key cryptography (IDPKC) or certificate-based public key cryptography, both of which has their inherit problem. The ID-PKC has the key escrow problem and certificate based...

In this paper, we present best proximity point theorems for new class of K􀀀-rational proximal contraction, in the setting of metric spaces. Some illustrative example are also given.

In this paper, we propose a new hybrid iteration for a finite family of asymptotically strictly pseudocontractive mappings. We also prove that such a sequence converges strongly to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings. Results in the paper extend and improve recent results in the literature....

The purpose of this paper is to introduce a new three step iteration scheme for approximation of fixed points of the nonexpansive mappings. We show that our iteration process is faster than all of the Picard, the Mann, the Agarwal et al., and the Abbas et al. iteration processes. We support our analytic proof by a numerical example in which we appr...

In this paper, a modified general composite implicit iteration process is used to study the convergence of a finite family of asymptotically nonexpansive mappings. Weak and strong convergence theorems have been proved, in the framework of a Banach space. MSC: 47H09, 47H10.

In this work, we propose a proxy-protected proxy multi-signature scheme based on EllipticCurve Digital Signature Algorithm (ECDSA), which aims at providing data authenticity,integrity, and non-repudiation to satisfy the basic properties of partial delegation proxy signaturedescribed by Mambo et al. as well as strong proxy signature properties defin...

In this paper, we consider a generalized nonlinear variational inequality problem involving single valued and multivalued nonlinear operators. We also study criteria of its solvability. Iterative methods for approximate solution are also proposed and a convergence result is established. Further, we study iterative methods for finding common element...

This paper presents a proxy blind signature scheme with forward security mechanism. The proposed digital signature scheme combines the two special-purpose signature schemes, blind signature and proxy signature. In this signature scheme, the original signer gives authority to another entity which is known as a proxy signer, but without having any id...

In this paper, we consider a new class of generalized extended nonlinear quasi-variational inequality problems involving set-valued relaxed monotone operators and establish its equivalence with the fixed point problem. We study criteria for existence of their solutions. Iterative methods for finding approximate solutions are also proposed and analy...

In this article, we establish strong convergence theorems for fixed points of strongly continuous semigroup of uniformly Lipschitzian asymptotically pseudocontractive mappings, in the framework of real Banach space.

Our aim in this paper is to prove a strong convergence theorem by three-step iteration scheme with respect to contractive condition in a Banach space. Our results obtained in this paper is new extension as well as refinement of previous known results.

Viscosity method also called elliptic regularization, provide an efficient approach to a large number of problems coming from different branches of mathe-matical and physical sciences. One of the recent trend in the iterative construction of fixed point of nonlinear mappings is to use viscosity approximation method. In this paper we propose implici...

Using the prox-regularity notion, we introduce a system of nonconvex general variational inequalities and a general three step algorithm for approximate solvability of this system. We establish the convergence of three-step projection method for a general system of nonconvex variational inequality problem. We obtain as a particular case some known...

In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function f such that its image under a bounded linear op-erator A minimizes another convex function g. Based on an idea introduced in [9], we propose a s...

The aim of this work is to study a system of generalized mixed variational inequalities, existence and approximation of its solution using the resolvent operator technique. We further propose an algorithm which converges to its solution and common fixed points of two Lipschitzian mappings. Parallel algorithms are used, which can be used to simultan...

The purpose of this paper is to investigate the demiclosed principle, the existence theorems and convergence theorems in CAT(0) spaces for a class of mappings which is essentially wider than that of asymptotically nonexpansive mappings. The structure of fixed point set of such mappings is also studied. Our results generalize, unify and extend sever...

The purpose of this paper is to prove a strong convergence theorem to a common solution of a finite family of m-accretive operators in strictly convex Banach space using a new viscosity iterative algorithm. Further using this result, strong convergence to a common fixed point of a finite family of pseudocontractive mappings is also discussed.

The aim of this work is to study a class of general strongly mixed variational inequalities. A new iterative algorithm for approximate solvability of general strongly mixed variational inequality is suggested. A convergence result for the iterative sequence generated by the new algorithm is also established. 2000 Mathematics Subject Classification....

In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function f such that its image under a bounded linear operator A minimizes another convex function g. Based on an idea introduced in [10], we propose a s...

We consider the problem of finding a common fixed point of N hemicontractions defined on a compact convex subset of a Hilbert space, an algorithm for solving this problem will be studied. We will prove strong convergence theorem for this algorithm. RESUMEN Consideramos el problema de búsqueda de un punto fijo común de N hemicontracciones definida s...

Using the prox-regularity notion, we introduce and study a generalized mildly nonconvex variational inequality. We study existence and iterative approximation of its solution un-der the partially relaxed cocoercivity. Several consequences of the proposed nonconvex variational inequality are also discussed.

In this paper, we proposed a system of generalized extended nonlinear variational inequality problem (SGENVI) involving three different trivariate mappings and six different nonlinear mappings. We also proposed three steps relaxed parallel projection algorithm for convergence analysis of the approximate solvability of the SGENVI problem. Further, w...

The purpose of this work is to prove a strong convergence theorem for pair of asymptoti-cally Φ−hemicontractive, nearly uniformly L−Lipschitzian mappings in Banach spaces. The main result of this paper is improvement and generalization of well-known corre-sponding results.

Using the prox-regularity notion, we introduce and study a system of general nonconvex variational inequalities. Using the parallel projection technique, we suggest and analyze a three-step iterative method for this system. We establish a convergence result for the proposed iteration method. We obtain some known results as a particular case.

We propose a viscosity iteration process for semigroups of asymptotically pseudocontractive mappings and prove a strong convergence theorem in uniformly convex Banach space for the proposed iteration process.

The purpose of this paper is to prove strong convergence theorems for a finite family of asymptotically nonexpansive mappings in the intermediate sense using an implicit iteration process with error term. The results in this paper extend and generalize corresponding results.

Let E be a real Banach space, {T 1 ,T 2 ,⋯,T n }:E→E be N-asymptotically demicontractive and L i -uniformly Lipschitzian mappings with F(T)=∩ i=1 ∞ F(T i )neφ. We prove necessary and sufficient conditions for the strong convergence of a modified implicit iteration process to a common fixed point in F(T).

A modified general composite implicit random iteration process is proposed and necessary and sufficient conditions for strong convergence of this iteration process to a common random fixed point of a finite family of random asymptotically nonexpansive mappings, are obtained.

A new general composite implicit random iteration scheme with perturbed mapping is proposed and obtain necessary and sufficient conditions for strong convergence of proposed iteration scheme to random fixed point of a finite family of random nonexpansive mappings are obtained.

Strong and weak convergence theorems for multistep iterative scheme with errors for finite family of asymptotically nonexpansive mappings are established in Banach spaces. Our results extend and improve the corresponding results of Chidume and Ali ( 2007), Cho et al. ( 2004), Khan and Fukhar-ud-din ( 2005), Plubtieng et al.( 2006), Xu and Noor ( 20...

In the present paper, we propose a modified composite implicit iteration process for finite family of asymptotically nonexpansive mappings and prove some weak and strong convergence theorems for this family of mappings using proposed iteration process. Our results extend and generalize some well known results.

In the present paper, we first prove a weak convergence theorem for strictly pseudo contractive mapping using modified Mann algorithm. This convergence is in general not strong, therefore as a generalization of modified Mann algorithm we propose a new (CQ) algorithm for strictly pseudo contractive mappings and obtain a strong convergence theorem fo...

In complete convex metric space, the concept of asymptotically quasi-nonexpansive type mapping is defined. Moreover, three-step iteration process with errors is defined for this class of mappings. Some necessary and sufficient condition for convergence of this process to fixed point of asymptotically quasi-nonexpansive type mappings is proved. Thes...

In this paper, we introduce a system of nonlinear variational inequali-ties for fuzzy mappings and develop an iterative algorithm. Further Approximate solvability of general system of nonlinear variational inequalities for fuzzy map-pings, is discussed. Key Words and Phrases. System of nonlinear variational inequality, Fuzzy variational inequality,...

Fixed points for set-valued mappings from a metric space X (not necessarily complete) intoB(X), the collection of nonempty bounded subsets of X are obtained. The result generalizes some known results.

Let K be a nonempty subset of a p-uniformly convex Banach space E, G a left reversible semitopological semigroup, and 𝒮={Tt:t∈G} a generalized Lipschitzian semigroup of K into itself, that is, for s∈G, ‖Tsx−Tsy‖≤as‖x−y‖+bs(‖x−Tsx‖+‖y−Tsy‖)+cs(‖x−Tsy‖+‖y−Tsx‖), for x,y∈K where as,bs,cs>0 such that there exists a t1∈G such that bs+cs<1 for all s≽t1....

In this paper, we study in Banach spaces the existence of fixed points of asymptotically regular mapping T satisfying: for each x, y in the domain and for n=1, 2,..., $$\parallelT^nx-T^ny\parallel\leqa_n\parallelx-y\parallel+b_n (\parallelx-T^nx\parallel+\parallely-T^ny\parallely)$$ where are nonnegative constants satisfying certain conditions. We...

We study the existence of random fixed points of asymptotically (ω,·)-regular mappings in p-uniformly convex Banach spaces. Further we establish some random fixed point theorems in Hilbert spaces, in L p spaces, in Hardy spaces H p and in Sobelev spaces H k,p for 1<p<∞ and k≥0. We extend and randomize the corresponding results of Górnicki.

In the brain, considered as the materialization of a mathematical system of closed spaces [M. Bounias and A. Bonaly, BioSystems 42, 191-205 (1997)], fixed points are needed for the self. Since neuronal chaining is not permanent, continuity may not be provided in some conditions, although the self should be preserved. The existence of fixed points w...

Fixed point theorems for generalized Lipschitzian semigroups are proved in p-uniformly convex Banach spaces and in uniformly convex Banach spaces. As applications, its corollaries are given in a Hilbert space, in Lp spaces, in Hardy space Hp, and in Sobolev spaces Hk,p, for 1<p<∞ and k≥0.

In this paper, we prove a convergence theorem for Passty type asymptotically nonexpansive mappings in a uniformly convex Banach space with Fréchet-differentiable norm.

We prove the following theorem: Let p>1 and let E be a p-uniformly convex Banach space, K a nonempty bounded closed convex subset of E, and A=[t n,k ] n,k≥1 a strongly ergodic matrix. Let T:K→K be a continuous mapping satisfying: for each x, y in K and i=1,2,⋯ ∥T i x-T i y∥≤a i ∥x-y∥+b i (∥x-T i x∥+∥y-T i y∥)+c i (∥x-T i y∥+∥y-T i x∥), where a i ,...

In this paper, it has been shown that the condition of continuity in the fixed point theorem of Tas, Telci and Fisher[2] is not necessary. Also the completeness of (X, d) can be replaced by the completeness of T(X).

Let C be a closed, convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable and let T be an asymptotically nonexpansive mapping from C into itself such that the set F (T) of fixed points of T is nonempty. Let {an} be a sequence of real numbers with $0 \leq a_n \leq 1$, and let x and x0 be elements of C. In thi...

Fixed point theorems for generalized uniformly Lipschitzian semigroups are proved in p-uniformly convex Banach space. As applications, their corollaries are given in a Hilbert space, in L p spaces, in Hardy space H p , and in Sobolev spaces H k,p , for 1<p<∞ and k≥0.

A new type of φ-contraction is defined to prove results on fixed points in metric spaces, which in turn improve several known results [R. M. Tiberio Bianchini, Boll. Unione Mat. Ital., IV. Ser. 5, 103-108 (1972; Zbl 0249.54023); L. B. Ćirić, Proc. Am. Math. Soc. 45, 267-273 (1974; Zbl 0291.54056); B. Fisher, Bull. Calcutta Math. Soc. 68, 265-266 (1...

A fixed point theorem is proved in a Banach space E which has uniformly normal structure for asymptotically regular mapping T satisfying: for each x,y in the domain and for n=1,2,⋯,‖Tnx−Tny‖≤an‖x−y‖+bn(‖x−Tnx‖+‖y−Tny‖)+cn(‖x−Tny‖+‖y−Tny‖), where an,bn,cn are nonnegative constants satisfying certain conditions. This result generalizes a fixed point...

In this paper, some theorems on fixed points of pair of asymptotically regular mappings in p-uniformly convex Banach space are proved. For these mappings some fixed point theorems in a Hilbert space, in Lp spaces, in Hardy spaces Hp and in Sobolev spaces Hpk, for 1 and k0 are also established. Thus, results of Gornicki[9, 10]. Kruppel[11, 12] and o...

We prove in p-uniformly convex space a fixed point theorem for a class of mappings T satisfying: for each x, y in the domain and for n=1,2,3,⋯, ∥T n x-T n y∥≤a∥x-y∥+b(∥x-T n x∥+∥y-T n y∥)+c(∥x-T n y∥+∥y-T n x∥), where a, b, c are nonnegative constants satisfying certain conditions. Further, we establish some fixed point theorems for these mappings...

A foremost general contraction condition is introduced to prove the existence of fixed points for a self-mapping in a complete metric space whose orbital diametral functions are closed. This condition covers not only the Kannan type but also covers Reich, and Hardy and Roger's type contractive conditions. An example is given in its support.

A generalized arbitrary discontinuous quasicontraction is defined on a complete metric space to prove results on fixed points. The results obtained generalize results of L. Nova G. [Pac. J. Math. 123, 189-196 (1986; Zbl 0549.47028)], of W. Walter [Nonlinear Anal., Theory Methods Appl. 5, 21-25 (1981; Zbl 0461.47032)], and of many others.

We give a necessary and sufficient condition for the weak convergence of trajectories of non-Lipschitzian asymptotically nonexpansive mappings.

The iteration process for approximation of fixed points of Lipschitzian local strictly hemi-contractive mappings will remain true in the normed space of type (∪,λ,m+1,m).

The present paper deals with a common fixed point result for compatible mappings in complete metric space, which mainly generalizes results of D. Prasad [Indian J. Pure Appl. Math. 16, 1073-1077 (1985; Zbl 0575.54036); Proc. Natl. Acad. Sci. India, Sect. A 54, 445-451 (1984; Zbl 0599.54057)].