CONVERGENCE OF S-ITERATIVE METHOD TO A SOLUTION OF FREDHOLM INTEGRAL EQUATION AND DATA DEPENDENCY

The convergence of normal S-iterative method to solution of a nonlinearFredholm integral equation with modied argument is established. The correspondingdata dependence result has also been proved. An example in support of the established results is included in our analysis

The local and semilocal convergence analysis of new Newton-like iteration methods

The aim of this paper is to find new iterative Newton-like schemes inspired by the modified Newton iterative algorithm and prove that these iterations are faster than the existing ones in the literature. We further investigate their behavior and finally illustrate the results by numerical examples

Existence of Tripled Fixed Points for a Class of Condensing Operators in Banach Spaces

We give some results concerning the existence of tripled fixed points for a class of condensing operators in Banach spaces. Further, as an application, we study the existence of solutions for a general system of nonlinear integral equations

Some fixed point results for a new three steps iteration process in banach spaces

In this paper, we introduce a three step iteration method and show that this method can be used to approximate fixed point of weak contraction mappings. Furthermore, we prove that this iteration method is equivalent to Mann iterative scheme and converges faster than Picard-S iterative scheme for the class of weak contraction mappings. We also present tables and three graphics to support this result. Finally, we prove a data dependence result for weak contraction mappings using this three step iterative scheme

Some fixed point theorems for a new iteration method under almost contraction mappings

Bu makalede (1) ile verilen iterasyon yönteminden daha sade olan yeni bir iterasyon yöntemi tanımlanmıştır. Bu iterasyon yönteminin hemen hemen büzülme dönüşümü şartını sağlayan iki operatörün ortak sabit noktasına yakınsak olduğu ispatlanmıştır. Ayrıca yeni iterasyon yönteminin (1) ile verilen iterasyon yönteminden daha hızlı olduğu gösterilmiştir ve bu sonucu destekleyen bir nümerik örnek verilmiştir. Son olarak, hemen hemen büzülme dönüşümü şartını sağlayan iki operatör için yeni tanımlanan iterasyon kullanılarak veri bağlılığı sonucu elde edilmiştir.Throughout history, the emergence of scientific knowledge in real life has been associated with fields such as Physics, Chemistry, Biology, Medicine, Economics, Computer. Under these names, each field contains many abstract or practical problems to implement in itself, or as a result of the association with one of the others. Mathematical models of such problems are either an equation type or an equation system. The methods that can be used for solving the obtained equation systems can be listed as differential-integral equations or operator-functional equations. Generally, in research on the existence of solutions of many problems which are belong to integral equations, differential equations, partial differential equations, dynamic programming, system analysis and fractal modeling, fixed point theory emerges as a useful method. This theory can also be applied to problems encountered in approach theory, game theory, mathematical economics and applied sciences. The mathematical modeling of the real-life problem in historical development first began with Isaac Newton's idea of modeling the movements of planets with mechanical laws. In differential calculus found by Newton and Leibniz at the same time, the Euler equation for dynamic systems, the Lagrange equation for motion, the Fourier equation for heat diffusion, the Navier-Stokes equation for viscosity and the movement of fluids, the Maxwell equation for electromagnetic field and Schrodinger and Dirac equations for quantum mechanics were solved with the help of differential equations; thus, many scientific and technological developments were opened up. With this rapid development of the differential calculus, many equations could be solved in closed form. However, qualitative and quantitative details which belong to the problem, along with initial and boundary values that are important for some equations, have become apparent with the application of the iterative method developed by Picard for the solution of differential equations. With the help of the iteration method used in the integral or differential equations to be solved, the limit of obtained sequence gives the solution of equation. For this reason, the iteration methods are gradually developed from several equations of the type () f x x = whose solutions are fixed points of the f function. However, the largest share in the placement of applications, which are outside of the basic differential and integral equations, into an abstract framework belongs to Stefan Banach. Banach Contraction Principle (BCP), which is proved by Banach in his doctoral thesis, from Hilbert spaces to metric spaces, has given a new direction to the study of the existence of fixed points in any space. BCP, which is very useful in solving differential and integral equations of different kinds, is also used as an effective tool to solve nonlinear problems. In addition to having a wide applicability, researchers have generalized BCP by putting new conditions on mapping or space. With the process that started with Picard, fixed point iteration methods have attracted the attention of many researchers because they have wide application areas in science and they have come up to these days as a large working area. In this process, a number of iteration methods have been developed for certain classes of mappings to investigate their strong convergence, equivalence of convergence, rate of convergence and whether fixed points of these mappings are data dependent. The equivalence of convergence between two iterations is expressed as follows: When an iteration method for a given mapping converges to the fixed point of this mapping, does the other method converge to the same point? Based on this problem, many researchers have studied the equivalence of convergence of iteration methods for various classes of mappings, and a large literature has been created in this sense. For two iterative methods that are equivalent in the sense of convergence, the knowledge of which method converges faster than the other is of great importance in applied mathematics. In this context, the rate of convergence of the iteration methods, which are in literature and newly defined, has been compared for the different classes of mappings by many researchers. When constructing an iteration method, another mapping can be used that is close enough to the mapping chosen and is called the approach operator. Moving from this approach operator accepting that it has a different fixed point, the questions of how close the fixed point of the chosen mapping and fixed point of this approach operator to each other is and how the distance between them will be calculated, have revealed the concept of data dependence of these fixed points. In this work, we show that the new iteration method, which is simpler than the iteration method (1), converges strongly to the common fixed point of two operators satisfying almost contraction condition . Also, we prove that the new iteration method is faster than the iteration method (1) and in order to show the validity of this result we give a numerical example. Finally, we obtain a data dependence result for two operators satisfying almost contraction mappings condition using new iteration method

Genelleştirilmiş banach-büzülme prensibi kullanılarak yeni sabit nokta teoremlerinin elde edilmesi

In this study, a new three step iterative algorithm was introduced with the help of Jungck-contraction principle which is one of the remerkable generalizations of Banach-contraction principle. Also, the convergence and stability results were obtained for the pair of nonself mappings which satisfy a certain contractive condition by using this iterative algorithm in any Banach space. In addition, it was shown that the new iterative algorithm has a better convergence speed when compared the other Jungck-type iterative algorithms in the current literature, and to support this result, numerical examples were given.Bu çalışmada, Banach-büzülme prensibinin dikkate değer genellemelerinden biri olan Jungck-büzülme prensibi yardımıyla yeni üç adımlı iterasyon algoritması tanımlanmıştır. Ayrıca bu iterasyon algoritması kullanılarak, kendi üzerine olmayan ve belirli bir büzülme şartını sağlayan dönüşüm çifti için herhangi bir Banach uzayında yakınsaklık ve kararlılık sonuçları elde edilmiştir. Ek olarak, tanımlanan yeni algoritmanın literatürde bulunan diğer Jungck-tipindeki algoritmalarla kıyaslandığında yakınsama anlamında daha hızlı olduğu gösterilmiş ve bu sonucu destekleyen nümerik örnekler verilmiştir

Common fixed point theorems for complex-valued mappings with applications

The aim of this paper is to obtain some results which belong to fixed point theory such as strong convergence, rate of convergence, stability, and data dependence by using the new Jungck-type iteration method for a mapping defined in complex-valued Banach spaces. In addition, some of these results are supported by nontrivial numerical examples. Finally, it is shown that the sequence obtained from the new iteration method converges to the solution of the functional integral equation in complex-valued Banach spaces. The results obtained in this paper may be interpreted as a generalization and improvement of the previously known results

An example of data dependence result for the class of almost contraction mappings

Atalan, Yunus ( Aksaray, Yazar )In the present paper, we show that S* iteration method can be used to approximate fixed point of almost contraction mappings. Furthermore, we prove that this iteration method is equivalent to CR iteration method and it produces a slow convergence rate compared to the CR iteration method for the class of almost contraction mappings. We also present table and graphic to support this result. Finally, we obtain a data dependence result for almost contraction mappings by using S* iteration method and in order to show validity of this result we give an example