Yeni bir iterasyon yöntemi için hemen-hemen büzülme dönüşümleri altında bazı sabit nokta teoremleri
Yükleniyor...
Tarih
2018
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Marmara Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
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.
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.
Açıklama
Anahtar Kelimeler
Yeni İterasyon Yöntemi, Hemen Hemen Büzülme Dönüşümleri, Yakınsaklık Hızı, Veri Bağlılığı, New Iteration Method, Almost Contraction Mappings, Rate of Convergence, Data Dependence
Kaynak
Marmara Fen Bilimleri Dergisi
WoS Q Değeri
Scopus Q Değeri
Cilt
30
Sayı
3
