Pseudocontractive Operators Research Articles

Constructive approach and randomization of a two-parameter chaos system for securing data

Secure communication techniques are important due to the increase in the number of technology users across the world. Likewise, a more random encryption algorithm suitable to secure data from unauthorised users is highly expected. This paper proposes a two-parameter nonlinear chaos map that is sensitive to the trio seed (s0, \alpha, \lambda) and has better information encryption. We introduce the parameter \alpha to linearise the conventional chaos system, which in turn brings a delay in the cryptosystems. The delay is a phenomenon that changes the chaotic features of a system. A small delay in the system leads to more aperiodicity and the unpredictability of the chaotic attractions. We normalise the new chaos map and use the Lipschitz and pseudo-contractive operators to obtain its irregularity region in Hilbert spaces. We also analyse the chaos map in terms of trajectory, Lyapunov exponent, complexity, and information entropy. Results obtained show that the new chaos map has a wide chaotic range and better statistical properties. It also maintains low complexity due to its linearity and produces more key spaces than most existing chaotic maps.

On monotone pseudocontractive operators and Krasnoselskij iterations in an ordered Hilbert space

The aim of this work is to establish fixed point results in ordered Hilbert spaces for monotone operators with a pseudocontractive property. We state monotone versions of Theorem 12 in [F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228] and Theorem 2.1 in [Berinde, Vasile. Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators, Annals of West University of Timisoara-Mathematics and Computer Science, vol. 56, no. 2, 2018, pp. 13–27], as well as, several related results. Further results, in Hilbert spaces without a partial order, are stated too.

Some new convergence and stability results for Jungck generalized pseudo-contractive and Lipschitzian type operators using hybrid iterative techniques in the Hilbert space

The convergence of generalized pseudo-contractive operators were studied with regard to a unique fixed point in the article (Verma in Proc Roy Irish Acad Soct A, 97(1):83–86, 1997). In this present article, we introduce Jungck generalized pseudo-contractive and Lipschitzian type operators in the prehilbertian space and in the Hilbert space settings, establish the existence and the uniqueness of a common fixed point for the operator S and a sequence of operators {T_i:Drightarrow D}_{i=1}^k using a new Jungck–Kirk–Mann type fixed point iterative algorithm as well as the general Kirk–Mann type iterative algorithm. The strong convergence of these fixed point iterative algorithms to a unique common fixed point is also investigated for the same set of operators considered. Moreover, some stability results are established for the Jungck–Kirk–Mann type iterative algorithm as well as the general Kirk–Mann type iterative algorithm. In addition, some examples are given to support our arguments. The results are new, original as well as generalizing some existing ones in the literature.

An Iterative Algorithm for Solving Fixed Point Problems and Quasimonotone Variational Inequalities

In this paper, we survey a common problem of the fixed point problem and the quasimonotone variational inequality problem in Hilbert spaces. We suggest an iterative algorithm for finding a common element of the solution of a quasimonotone variational inequality and the fixed point of a pseudocontractive operator. Convergence theorems are shown under some mild conditions. Several corollaries are also obtained.

Strong convergence of an iterative procedure for pseudomonotone variational inequalities and fixed point problems

Pseudomonotone variational inequalities have been investigated by many authors, a common assumption ?weak sequential continuity? being imposed on pseudomonotone operators. In this paper, we propose an iterative procedure for solving pseudomonotone variational inequalities and fixed point problems of asymptotically pseudocontractive operators by using self-adaptive techniques. Under a weaker assumption than weak sequential continuity imposed on pseudomonotone operators, we prove that the suggested procedure has strong convergence.

Algorithmic and analytical approach to the proximal split feasibility problem and fixed point problem

In this paper, we investigate the proximal split feasibility algorithm and fixed point problem in Hilbert spaces. We propose an iterative algorithm for finding a common element of the solution of the proximal split feasibility algorithm and fixed point of an L-Lipschitz pseudocontractive operator. We demonstrate that the considered algorithm converges strongly to a common point of the investigated problems under some mild conditions.

A split common fixed point and null point problem for Lipschitzian $J-$quasi pseudocontractive mappings in Banach spaces

A split common fixed point and null point problem (SCFPNPP) which includes the split common fixed point problem, the split common null point problem and other problems related to the fixed point problem and the null point problem is studied. We introduce a Halpern--Ishikawa type algorithm for studying the split common fixed point and null point problem for Lipschitzian $J-$quasi pseudocontractive operators and maximal monotone operators in real Banach spaces. Moreover, we establish a strong convergence results under some suitable conditions and reduce our main result to the above-mentioned problems. Finally, we applied the study to split feasibility problem (FEP), split equilibrium problem (SEP), split variational inequality problem (SVIP) and split optimization problem (SOP).

Approximation-solvability of population biology systems based on p-Laplacian elliptic inequalities with demicontinuous strongly pseudo-contractive operators

The aim of this paper is to investigate existence, uniqueness and convergence of approximants of nonzero positive weak solutions for a class of population biology systems, which are models of one species based on p-Laplacian elliptic inequalities with demicontinuous strongly pseudo-contractive operators and involved logistic growth and harvesting rates. Toward this end, we foremost develop a kind of new general variational inequality principles with generalized duality mappings in reflexive Banach spaces. Then, we employ the new principle to obtain our main results for a general p-Laplacian elliptic inequality and the population biology systems. We note that the results are different from those relevant work of p-Laplacian elliptic inequalities in the literature, it is because a pseudo-contractive operator may not be an S-contractive operator and vice versa.

Convergence of Tseng-type self-adaptive algorithms for variational inequalities and fixed point problems

In this paper, we present a Tseng-type self-adaptive algorithm for solving a variational inequality and a fixed point problem involving pseudomonotone and pseudocontractive operators in Hilbert spaces. A weak convergent result for such algorithm is proved under a weaker assumption than sequentially weakly continuous imposed on the pseudomonotone operator. Some corollaries are also included.

A Tseng-Type Algorithm with Self-Adaptive Techniques for Solving the Split Problem of Fixed Points and Pseudomonotone Variational Inequalities in Hilbert Spaces

In this paper, we survey the split problem of fixed points of two pseudocontractive operators and variational inequalities of two pseudomonotone operators in Hilbert spaces. We present a Tseng-type iterative algorithm for solving the split problem by using self-adaptive techniques. Under certain assumptions, we show that the proposed algorithm converges weakly to a solution of the split problem. An application is included.

Strong Convergence Analysis of Iterative Algorithms for Solving Variational Inclusions and Fixed-Point Problems of Pseudocontractive Operators

Iterative methods for solving variational inclusions and fixed-point problems have been considered and investigated by many scholars. In this paper, we use the Halpern-type method for finding a common solution of variational inclusions and fixed-point problems of pseudocontractive operators. We show that the proposed algorithm has strong convergence under some mild conditions.

Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators

The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature.

Modified extragradient algorithms for solving monotone variational inequalities and fixed point problems

In this paper, we investigate the monotone variational inequalities and fixed point problems in Hilbert spaces. Two modified extragradient algorithms are presented for finding a common element of the set of fixed points of a pseudocontractive operator and the set of solutions of the variational inequality problem. Weak and strong convergence of the suggested algorithms are proved.

Extragradient methods with CQ technique for fixed point problems and equilibrium problems

In this paper, we study iterative algorithms for solving fixed point problems and equilibrium problems in Hilbert spaces. We present an extragradient algorithm with CQ technique for finding a common element of the fixed points of pseudocontractive operators and the solutions of pseudomonotone equilibrium problems. Strong convergence result of the proposed algorithm is proved.

Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators

In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included.

Iterative Algorithms for Split Common Fixed Point Problem Involved in Pseudo-Contractive Operators without Lipschitz Assumption

Two iterative algorithms are suggested for approximating a solution of the split common fixed point problem involved in pseudo-contractive operators without Lipschitz assumption. We prove that the sequence generated by the first algorithm converges weakly to a solution of the split common fixed point problem and the second one converges strongly. Moreover, the sequence { x n } generated by Algorithm 3 strongly converges to z = proj S 0 , which is the minimum-norm solution of problem (1). Numerical examples are included.

Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators

Abstract In this paper, we introduce and study the class of enriched strictly pseudocontractive mappings in Hilbert spaces and extend some convergence theorems, i.e., Theorem 12 in [Brow-der, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228] and Theorem 3.1 in [Marino, G., Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336–346], from the class of strictly pseudocontractive mappings to that of enriched strictly pseudocontractive mappings and thus include many other important related results from literature as particular cases.

System of multivariate pseudo-contractive operator equations and the existence of solutions

The purpose of this paper is to study the following system of nonlinear operator equations for N-variable pseudo-contractive mappings $$T_1, T_2,\ldots ,T_N$$ in a Banach space: $$\begin{aligned} {\left\{ \begin{array}{ll} T_1(x_{1},x_{2},\ldots ,x_{N})=x_1,\\ T_2(x_{1},x_{2},\ldots ,x_{N})=x_2,\\ T_3(x_{1},x_{2},\ldots ,x_{N})=x_3,\\ \ \ \ \ \ \ \ \cdots \\ T_N(x_{1},x_{2},\ldots ,x_{N})=x_N.\\ \end{array}\right. } \end{aligned}$$ The existence theorems of solutions are proved using a comprehensive analytical method. To get the expected results, the reflexivity of the product of Banach space and Opial’s condition of the product of Banach spaces are discussed. Meanwhile, the weak topology of the product of Banach spaces and normal structure of the product of Banach spaces are also discussed. The results of this article improve and extend the previous classical results given in the literature.

On Some Relations between Accretive, Positive, and Pseudocontractive Operators and Passivity Results in Hilbert Spaces and Nonlinear Dynamic Systems

This paper investigates some parallel relations between the operators I-G and G in Hilbert spaces in such a way that the pseudocontractivity, asymptotic pseudocontractivity, and asymptotic pseudocontractivity in the intermediate sense of one of them are equivalent to the accretivity, asymptotic accretivity, and asymptotic accretivity in the intermediate sense of the other operator. If the operators are self-adjoint then the obtained accretivity-type properties are also passivity-type properties. Such properties are very relevant in stability theory since they refer to global stability properties of passive feed-forward, in general, nonlinear, and time-varying controlled systems controlled via feedback by elements in a very general class of passive, in general, nonlinear, and time-varying controllers. These results allow the direct generalization of passivity results in controlled dynamic systems to wide classes of tandems of controlled systems and their controllers, described by G-operators, and their parallel interpretations as pseudocontractive properties of their counterpart I-G-operators. Some of the obtained results are also directly related to input-passivity, output-passivity, and hyperstability properties in controlled dynamic systems. Some illustrative examples are also given in the framework of dynamic systems described by extended square-integrable input and output signals.

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.