scholarly journals Some Convergence and Stability Results for the Kirk Multistep and Kirk-SP Fixed Point Iterative Algorithms

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Faik Gürsoy ◽  
Vatan Karakaya ◽  
B. E. Rhoades
Keyword(s):  
Fixed Point ◽  
Iterative Algorithm ◽  
Iterative Algorithms ◽  
Iterative Processes ◽  
Convergence And Stability ◽  
The Stability ◽  
Stability Results

The purpose of this paper is to introduce a new Kirk type iterative algorithm called Kirk multistep iteration and to study its convergence. We also prove some theorems related to the stability results for the Kirk multistep and Kirk-SP iterative processes by employing certain contractive-like operators. Our results generalize and unify some other results in the literature.

scholarly journals Some Convergence and Stability Results for Two New Kirk Type Hybrid Fixed Point Iterative Algorithms

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Faik Gürsoy ◽  
Vatan Karakaya
Keyword(s):  
Fixed Point ◽  
Iterative Algorithms ◽  
Convergence And Stability ◽  
Stability Results

We introduce Kirk-multistep-SP and Kirk-S iterative algorithms and we prove some convergence and stability results for these iterative algorithms. Since these iterative algorithms are more general than some other iterative algorithms in the existing literature, our results generalize and unify some other results in the literature.

scholarly journals Convergence and stability of an iterative algorithm for strongly accretive Lipschitzian operator with applications

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3689-3704
Author(s):  
Vivek Kumar ◽  
Nawab Hussain ◽  
Abdul Khan ◽  
Faik Gürsoy
Keyword(s):  
Iterative Algorithm ◽  
Nonlinear Operator ◽  
Iterative Algorithms ◽  
Convergence Result ◽  
Nonlinear Operator Equation ◽  
Inclusion Problem ◽  
Convergence And Stability ◽  
Variational Inclusion Problem ◽  
Convergence Results ◽  
Stability Results

Using different technique and weaker restrictions on parameters, convergence and stability results of an SP iterative algorithm with errors for a strongly accretive Lipschitzian operator on a Banach space are established. Validity of new convergence results is verified through numerical examples and convergence comparison of various iterative algorithms is depicted. As applications of our convergence result, we solve a nonlinear operator equation and a variational inclusion problem. Our results are refinement and generalization of many classical results.

scholarly journals On the probabilistic stability of some functional equations

2013 ◽  
Vol 29 (1) ◽  
pp. 125-132
Author(s):  
CLAUDIA ZAHARIA ◽  
◽  
DOREL MIHET ◽  
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Quadratic Functional ◽  
Normed Spaces ◽  
Stability Of Functional Equations ◽  
The Stability ◽  
Stability Results

We establish stability results concerning the additive and quadratic functional equations in complete Menger ϕ-normed spaces by using fixed point theory. As particular cases, some theorems regarding the stability of functional equations in β - normed and quasi-normed spaces are obtained.

On the stability of some functional equations in Menger φ-normed spaces

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Dorel Miheţ ◽  
Reza Saadati
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Additive Functional ◽  
Normed Spaces ◽  
Additive Functional Equation ◽  
The Stability ◽  
Stability Results

AbstractRecently, the authors [MIHEŢ, D.—SAADATI, R.—VAEZPOUR, S. M.: The stability of an additive functional equation in Menger probabilistic φ-normed spaces, Math. Slovaca 61 (2011), 817–826] considered the stability of an additive functional in Menger φ-normed spaces. In this paper, we establish some stability results concerning the cubic, quadratic and quartic functional equations in complete Menger φ-normed spaces via fixed point theory.

scholarly journals A ROBUST ITERATIVE APPROACH FOR SOLVING NONLINEAR VOLTERRA DELAY INTEGRO–DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 7 (2) ◽  
pp. 59
Author(s):  
Austine Efut Ofem ◽  
Unwana Effiong Udofia ◽  
Donatus Ikechi Igbokwe
Keyword(s):  
Differential Equations ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Stability Result ◽  
Iterative Approach ◽  
Weak And Strong Convergence ◽  
Mild Conditions ◽  
Convergence Results ◽  
The Stability

This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized \(\alpha\)–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak \(w^2\)–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized \(\alpha\)–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.

scholarly journals Stability for a New Class of GNOVI with (γG,λ)-Weak-GRD Mappings in Positive Hilbert Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Hong Gang Li ◽  
Yongqin Yang ◽  
Mao Ming Jin ◽  
Qinghua Zhang
Keyword(s):  
Fixed Point ◽  
Existence Theorem ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Variational Inclusions ◽  
New Class ◽  
Set Up ◽  
The Stability

By using ordered fixed point theory, we set up a new class of GNOVI structures (general nonlinear ordered variational inclusions) with(γG,λ)-weak-GRD mappings, discuss an existence theorem of solution, consider a perturbed Ishikawa iterative algorithm and the convergence of iterative sequences generated by the algorithm, and show the stability of algorithm for GNOVI structures in positive Hilbert spaces. The results in the instrument are obtained.

scholarly journals Fixed Point Theory and the Ulam Stability

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Janusz Brzdęk ◽  
Liviu Cădariu ◽  
Krzysztof Ciepliński
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theorems ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Stability Of Functional Equations ◽  
Banach’S Fixed Point Theorem ◽  
The Stability ◽  
First Time ◽  
Stability Results

The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach's fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

scholarly journals Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
C. Diop ◽  
M. Sene ◽  
N. Djitté
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Convex Subset ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Uniformly Convex ◽  
Convergence Theorems

Let K be a nonempty closed and convex subset of a uniformly convex real Banach space E and let T1,…,Tm:K→2K be m multivalued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence xn is proved to be an approximating fixed point sequence of each Ti; that is, limdxn;Txn=0. Then, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)).

scholarly journals Hyers–Ulam stability of impulsive Volterra delay integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
D. A. Refaai ◽  
M. M. A. El-Sheikh ◽  
Gamal A. F. Ismail ◽  
Bahaaeldin Abdalla ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Integral ◽  
Finite Interval ◽  
Ulam Stability ◽  
First Order ◽  
Fixed Point Approach ◽  
Different Types ◽  
The Stability ◽  
Stability Results

AbstractThis paper discusses different types of Ulam stability of first-order nonlinear Volterra delay integro-differential equations with impulses. Such types of equations allow the presence of two kinds of memory effects represented by the delay and the kernel of the used fractional integral operator. Our analysis is based on Pachpatte’s inequality and the fixed point approach represented by the Picard operators. Applications are provided to illustrate the stability results obtained in the case of a finite interval.

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