Common fixed points for multi-valued mappings by applying inequalities on binomials and trinomials

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.

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 multivalued mappings by applying inequalities on binomials and trinomials.

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 𝐹̃, and define it as follows: 𝐹̃ = 𝐹 ~ = {𝐹̃: 𝐹 ∈ ℱ}. Also define 𝑑̃ on 𝐹̃ with 𝑑̃(𝐹̃, 𝐺̃) = 𝐻(𝑓𝑖𝑥𝐹, 𝑓𝑖𝑥𝐺). (𝐹̃, 𝑑̃) is a metric space. In this article, by considering some conditions on the maps 𝐹 and 𝐺 in complete metric space we conclude that 𝐹̃ = 𝐺̃.

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 multivalued mappings by applying inequalities on binomials and trinomials and using different conditions.