S-iteration Process Research Articles

Fractals as Julia and Mandelbrot Sets via S-iteration

To understand the phenomena of expanding symmetries Fractals patterns are an important tool which exhibit similar patterns for different scales. In the present paper, establishing an escape criteria by using S-iteration process to visualize fractals namely Julia and Mandelbrot sets for the function F(w)=aewp+c where c,a∈ C and p≥2. The result obtain is a generalization of the existing algorithm and technique providing fractals for different parameter values. Also, the time taken to obtain fractals for different parameters by using computer software MATLAB is computed in seconds.

S-Iteration inertial subgradient extragradient method for variational inequality and fixed point problems

In developing iterative methods for approximating solutions of optimization problems, one of the major goals of researchers is to construct efficient iterative schemes. Over the years, different techniques have been devised by authors to achieve this goal. In this paper, we study a non-Lipschitz pseudomonotone variational inequality problem with common fixed points constraint of Bregman quasi-nonexpansive mappings. We introduce a new iterative method for approximating the solution of this problem in a more general framework of reflexive Banach spaces. Our method employs several techniques to achieve high level of efficiency. One of the techniques employed is the S-iteration process, a method known to be highly efficient in comparison with several of the well-known methods. Moreover, our proposed algorithm utilizes the inertial technique and a non-monotonic self-adaptive step size to guarantee high rate of convergence and easy implementation. Unlike several of the existing results on variational inequality problem with non-Lipschitz operator, the design of our method does not involve any linesearch technique. We obtain strong convergence result for the proposed algorithm without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when solving pseudomonotone variational inequality problems. Furthermore, we apply our result to study utility-based bandwidth allocation problem and image restoration problem. Finally, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with existing state-of-the-art methods. Our result extends and improves several of the recently announced results in this direction in the literature.

An inertial s-iteration process for a common fixed point of a family of quasi-Bregman nonexpansive mappings

In this paper, an inertial S-iteration iterative process for approximating a common fixed point of a finite family of quasi-Bregman nonexpansive mappings is introduced and studied in a reflexive Banach space. A strong convergence theorem is proved. Some applications of the theorem are presented. The results presented here improve, extend, and generalize some recent results in the literature.

Solving Equilibrium Problem and Fixed Point Problem by Normal S-iteration Process in Hilbert Space

The main purpose of this paper is to find a common element in the solution set of equilibrium problem and fixed point problem of non-expansive mappings in the real Hilbert space with the help of normal S-iteration process. Also, under some acceptable assumptions, we prove the sequences induced by above stated process converge weakly to a point in the solution set of above stated problems. At the end, we give a numerical example to justify our work. The results studied in this work philosophize and boost some contemporary and known results in this direction.

Exploiting the S-Iteration Process for Solving Power Flow Problems: Novel Algorithms and Comprehensive Analysis

In recent studies, the competitiveness of the Newton-S-Iteration-Process (Newton-SIP) techniques to efficiently solve the Power Flow (PF) problems in both well and ill-conditioned systems has been highlighted, concluding that these methods may be suitable for industrial applications. This paper aims to tackle some of the open topics brought for this kind of techniques. Different PF techniques are proposed based on the most recently developed Newton-SIP methods. In addition, convergence analysis and a comparative study of four different Newton-SIP methods PF techniques are presented. To check the features of considered PF techniques, several numerical experiments are carried out. Results show that the considered Newton-SIP techniques can achieve up to an eighth order of convergence and typically are more efficient and robust than the Newton–Raphson (NR) technique. Finally, it is shown that the overall performance of the considered PF techniques is strongly influenced by the values of parameters involved in the iterative procedure.

Convergence analysis of two-step inertial Douglas-Rachford algorithm and application

Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.

Perturbed approximations of fixed points of nonexpansive mappings in CAT_p(0) spaces

Using the concept of the CATp(0) spaces proposed by Khamsi et al. [Khamsi, M. A. and Shukri, S. A., Generalized CAT(0) spaces. Bull. Belg. Math. Soc. Simon Stevin, 24 (2017), No. 3, 417–426], we establish ∆-convergence and strong convergence of a perturbed variant of Agarwal et al. S-iteration process for nonexpansive mapping. Finally, from them we deduce results valid for α-nonexpansive mappings.

Strong and \(\Delta\)-Convergence Results for Generalized Nonexpansive Mapping in Hyperbolic Space

In this paper, we propose a new iteration process which is faster than Picard Normal S-iteration process, Normal S-iteration process, Mann iteration process and Picard iteration process in hyperbolic space for generalized nonexpansive mapping. We also present strong and \(\Delta\)-convergence results for proposed iteration process. An illustrative example with different set of parameters is also given in this paper.

Proximal point algorithms based on S-iterative technique for nearly asymptotically quasi-nonexpansive mappings and applications

In this paper, we combine the S-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal., 8(1), 61–79 2007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim., 14, 877–898 1976) to propose a new modified proximal point algorithm based on the S-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the △-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl., 56, 2572–2579 2008), Khan and Abbas (Comput. Math. Appl., 61, 109–116 2011), Abbas et al. (Math. Comput. Modelling, 55, 1418–1427 2012) and many more.

Power Flow Approach Based on the S-Iteration Process

In this paper, the application of the S-iteration process for solving the Power Flow (PF) problemis comprehensively studied. Thus, a combined approach between the S-iteration processand Newton's method (SIP-NR) is adapted for solving PF equations. Moreover, drawbacksshown by SIP-NRunder heavy loading conditions due to its linear convergence are addressed by developing a simple but reliable technique called Modified SIP-NR(MSIP-NR). MSIP-NR uses anon-repeatedly updating Jacobianscheme with the aim toaccelerate the convergence of SIP-NR. The introduced PF techniques are validated using well and ill-conditioned systemsunder several demand scenarios. The obtained results prove the robustness and efficiency of the introducedPF techniquescompared with other well-known methodologies. On the light of the results obtained, the introduced PF techniques may be used for different applications in power system analysis. Some of these potential applications are enumerated and discussed.

On approximation of fixed point in Busemann space via generalized Picard normal s-iteration process

On approximation of fixed point in Busemann space via generalized Picard normal s-iteration process

A study on fixed point theorems for a class of generalized nonexpansive mappings in Hadamard spaces

In this paper, we study the class of mappings with $$B_{\gamma ,\mu }$$ condition due to Patir et al. (Fixed Point Theory Appl 2018:19, 2018) in the setting of Hadamard spaces. We prove a demiclosedness principle for such mapping in Hadamard spaces. Furthermore, we also prove the $$\Delta $$-convergence of the sequence generated by the S-iteration process for finding fixed points of the class of mappings with $$B_{\gamma ,\mu }$$ condition in Hadamard spaces. Our results extend some known results which appeared in the literature.

An inertial S-iteration process

In this paper, we establish a new iteration method, called an InerSP (an inertial S-iteration process), by combining a modified S-iteration process with the inertial extrapolation. This strategy is for speeding up the convergence of the algorithm. We then prove the convergence theorems of a sequence generated by our new method for finding a common fixed point of nonexpansive mappings in a Banach space. We also present numerical examples to illustrate that the acceleration of our algorithm is effective.

Iterative approximation of attractive points of further generalized hybrid mappings in Hadamard spaces

In this paper, we study the class of further generalized hybrid mappings due to Khan (Fixed Point Theory Appl. 2018:8, 2018) in the setting of Hadamard spaces. We prove a demiclosed principle for such mappings in Hadamard spaces. Furthermore, we also prove the Δ-convergence of the sequence generated by the S-iteration process for finding attractive points of further generalized hybrid mappings in Hadamard spaces satisfying the (mathbb{S}) property and the (overline{Q_{4}}) condition. Moreover, we provide a numerical example to illustrate the convergence behavior of the studied iteration and numerically compare the convergence of the studied iteration scheme with the existing schemes. Our results extend some known results which appeared in the literature.

S-iteration process of Halpern-type for common solutions of nonexpansive mappings and monotone variational inequalities

This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of common solutions of variational inequality problems formed by two inverse strongly monotone mappings in the framework of Hilbert spaces. We also give some numerical examples in support of our main result.

Convergence Rate of Implicit Iteration Process and a Data Dependence Result

The aim of this paper is to introduce an implicit S-iteration processand study its convergence in the framework of W-hyperbolic spaces. We showthat the implicit S-iteration process has higher rate of convergence than implicit Mann type iteration and implicit Ishikawa-type iteration processes. We present a numerical example to support the analytic result proved herein. Finally, we prove a data dependence result for a contractive type mapping using implicit S-iteration process.

Approximating fixed points of generalized nonexpansive mappings in CAT(k) spaces via modified S-iteration process

In this paper, we approximate common fixed points of two generalized nonexpansive mappings under modified S-iteration process in complete CAT(k) spaces and establish some $$\Delta $$ -convergence theorems. Our results generalize and improve the corresponding known results of the existing literature.

A modified S-iteration process for G-nonexpansive mappings in Banach spaces with graphs

In this work, we prove the weak and strong convergence of a sequence generated by a modified S-iteration process for finding a common fixed point of two G-nonexpansive mappings in a uniformly convex Banach space with a directed graph. We also give some numerical examples for supporting our main theorem and compare convergence rate between the studied iteration and the Ishikawa iteration.

Convergence of a Newton-like S-iteration process in R

The aim of this paper is to show that Newton-like S-iteration method converges to the unique solution of the scalar nonlinear equation f(x) = 0 under weaker conditions involving only f and f 0. We also present numerical examples to support our analytical results.

Semilocal Convergence Analysis of S-iteration Process of Newton–Kantorovich Like in Banach Spaces

In the present article, we establish a semilocal convergence theorem for the S-iteration process of Newton---Kantorovich like in Banach space setting for solving nonlinear operator equations and discuss its semilocal convergence analysis. We apply our result to solve the Fredholm-integral equations.