Showing 2 results for Common Fixed Point
Professor Hojjat Afshari, Dr. Mohsen Abolhosseinzadeh, ,
Volume 8, Issue 2 (6-2022)
Abstract
Introduction
Fixed point theory has fascinated many researchers since 1922 with the celebrated Banach fixed point theorem. There exists a vast literature on the topic field and this is a very active field of research at present. Fixed point theorems are very important tools for proving the existence and uniqueness of the solutions to various mathematical models (integral and partial equations, variational inequalities, etc). Its core subject is concerned with the conditions for the existence of one or more fixed points of a mapping or multi-valued mapping T from a topological space X into itself that is, we can find x ∈ X such that Tx = x (for mapping) or x ∈ Tx (for multi-valued mapping).
In a wide range of mathematical, computational, economic, modeling, and engineering problems, the existence of a solution to a theoretical or real-world problem is equivalent to the existence of a fixed point for a suitable map or operator. Fixed points are therefore of paramount importance in many areas of mathematics, sciences, and engineering.
In 1922 Stefan Banach proved a famous theorem which under suitable conditions stated the existence and uniqueness of a mapping. The result of the fixed point theorem or Banach contraction principle was obtained by Stefan Banach.
In 1985, V. Popa proved common fixed point theorems for multi-valued mappings that verify rational inequalities, which contain the Hausdorff metric in their expressions.
In 2010, A. Petcu proved other common fixed point theorems for two or more multi-valued mappings without using the Hausdorff metric.
In this paper, by using some completely different conditions, we study the existence of common fixed points for multi-valued mappings with applying inequalities on binomials and trinomials.
Material and methods
The content of this paper is organized as follows. First, we present some definitions, lemmas, and basic results that will be used in the proofs of our theorems. Then, we study the existence of common fixed points for multi-valued mappings by applying inequalities on binomials and trinomials.
Results and discussion
Let F be all multi-valued mappings of X into Pb,cl(X). First, we define an equivalence relation for the elements of F as follows;
F ∼ G if and only if fixF = fixG, (F,G ∈ F).
Where fixF= x∈X:x∈Fx.
Denote the equivalence class of F by F, and define it as follows:
F=F~={F:F∈F}. Also define d on F with dF,G=H(fixF,fixG).
F,d is a metric space. In this article, by considering some conditions on the maps F and G in complete metric space we conclude that F=G.
Conclusion
The well known Banach contraction principle ensures the existence and uniqueness of the fixed point of a contraction on a complete metric space. After this interesting principle, several authors generalized this principle by introducing the various contractions on metric spaces. Thereafter, Popa and Petcu obtained some results in about common fixed points of multi-valued mappings. This paper studies the existence of common fixed points for multi-valued mappings by applying inequalities on binomials and trinomials and using different conditions.
Dr Mohammad Eslamian,
Volume 10, Issue 1 (4-2024)
Abstract
The split common fixed point problem in Banach spaces, is to find a point which belongs to the set of common fixed points of a family of nonlinear mappings in a Banach space such that its image under a linear transformation belongs to the set of common fixed point of another family of nonlinear mappings in a Banach space in the image space. The split common fixed point problem have received much attention due to its applications in many areas of mathematics and engineering sciences such as optimization, image reconstruction, intensity-modulated radiation therapy and image processing. In this paper we present a new algorithm for solving the split common fixed point problem for a finite family of generalized demimetric mappings in uniformly convex and smooth Banach spaces.We establish strong convergence of the sequence generated by the algorithm to a solution of split common fixed point problem which also solves some variational inequality problems. Finally we present some applications of our main result for solving the multiple-set split feasibility problem and the split null point problem.
