On the existence of fixed points that belong to the zero set of a certain function

Fixed Point Theory and Applications, Aug 2015

Let \(T: X\to X\) be a given operator and \(F_{T}\) be the set of its fixed points. For a certain function \(\varphi: X\to[0,\infty)\), we say that \(F_{T}\) is φ-admissible if \(F_{T}\) is nonempty and \(F_{T}\subseteq Z_{\varphi}\), where \(Z_{\varphi}\) is the zero set of φ. In this paper, we study the φ-admissibility of a new class of operators. As applications, we establish a new homotopy result and we obtain a partial metric version of the Boyd-Wong fixed point theorem.

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On the existence of fixed points that belong to the zero set of a certain function

Karapinar et al. Fixed Point Theory and Applications On the existence of fixed points that belong to the zero set of a certain function Erdal Karapinar 0 1 2 5 6 Donal O'Regan 0 1 2 4 5 Bessem Samet 0 1 2 3 7 0 11451 , Saudi Arabia 1 University , P.O. Box 2455, Riyadh 2 College of Science , King Saud 3 Department of Mathematics 4 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland , Galway , Ireland 5 Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University , Jeddah , Saudi Arabia 6 Department of Mathematics, Atilim University , Incek, Ankara, 06836 , Turkey 7 Department of Mathematics, College of Science, King Saud University , P.O. Box 2455, Riyadh, 11451 , Saudi Arabia Let T : X → X be a given operator and FT be the set of its fixed points. For a certain function ϕ : X → [0, ∞), we say that FT is ϕ-admissible if FT is nonempty and FT ⊆ Zϕ , where Zϕ is the zero set of ϕ. In this paper, we study the ϕ-admissibility of a new class of operators. As applications, we establish a new homotopy result and we obtain a partial metric version of the Boyd-Wong fixed point theorem. ϕ-admissible; fixed point; homotopy result; partial metric 1 Introduction Let T : X → X be a given operator. The set of fixed points of T is denoted by FT , that Zϕ = x ∈ X : ϕ(x) =  . FT = {x ∈ X : Tx = x}. (F) max{a, b} ≤ F(a, b, c), for all a, b, c ≥ ; (F) F(a, , ) = a, for all a ≥ ; (F) F is continuous. As examples, the following functions belong to F : . F(a, b, c) = a + b + c, . F(a, b, c) = max{a, b} + ln(c + ), . F(a, b, c) = a + b + c(c + ), . F(a, b, c) = (a + b)ec, N . F(a, b, c) = (a + b)(c + )n, n ∈ . © 2015 Karapinar et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. ( ) ψ is upper semi-continuous from the right; ( ) ψ (t) < t, for all t > . For given functions ϕ : X → [, ∞), F ∈ F , and ψ ∈ , we denote by T (ϕ, F, ψ ) the class of operators T : X → X satisfying The aim of this paper is to study the ϕ-admissibility of the set FT , where T belongs to the class of operators T (ϕ, F, ψ ), (F, ψ ) ∈ F × . As applications, we obtain an homotopy result and a partial metric version of the Boyd-Wong fixed point theorem. 2 Main result Theorem . Let (X, d) be a complete metric space and T : X → X be a given operator. Suppose that the following conditions hold: (i) there exist ϕ : X → [, ∞), F ∈ F , and ψ ∈ such that T ∈ T (ϕ, F, ψ ); (ii) ϕ is lower semi-continuous. Then the set FT is ϕ-admissible. Moreover, the operator T has a unique fixed point. F , ϕ(ξ ), ϕ(ξ ) ≤ ψ F , ϕ(ξ ), ϕ(ξ ) . which is impossible from (.). Consequently, we have Using the above equality and (F), we obtain FT ⊆ Zϕ. Consequently, we have Now, we have to prove that FT is a nonempty set. Let x be an arbitrary element of X. Consider the Picard sequence {xn} ⊂ X defined by where T n is the nth iterate of T . If for some N ∈ N we have xN = xN+, then xN will be an element of FT . As a result we can suppose that xn = T nx, n ∈ N = {, , , . . .}, d(xn, xn+) > , n ∈ N. Using (.), we have F d(Txn, Txn–), ϕ(Txn), ϕ(Txn–) ≤ ψ F d(xn, xn–), ϕ(xn), ϕ(xn–) , n ∈ N∗; here N∗ = {, , . . .}. If for some N ∈ N∗, we have F d(xN , xN–), ϕ(xN ), ϕ(xN–) = , then property (F) yields d(xN , xN–) ≤ F d(xN , xN–), ϕ(xN ), ϕ(xN–) = , which is a contradiction with (.). Thus F d(xn, xn–), ϕ(xn), ϕ(xn–) > , n ∈ N∗. Using (.), (.), the definition of the sequence {xn}, and ( ), we have ⎨⎧ F(d(xn+, xn), ϕ(xn+), ϕ(xn)) ≤ ψ (F(d(xn, xn–), ϕ(xn), ϕ(xn–))), n ∈ N∗. Suppose now that c > . Using the properties ( )-( ), we deduce from (.) that Using (F) and (.), we get Next, we show that {xn} is a Cauchy sequence in the metric space (X, d). Suppose that {xn} is not a Cauchy sequence. Then there exists ε >  for which we can find two sequences of positive integers {m(k)} and {n(k)} such that, for all k ∈ N, n(k) > m(k) > k, Using (.), for all k ∈ N we have ≤ d(xm(k), xn(k)–) + d(xn(k)–, xn(k)) ε ≤ d(xm(k), xn(k)) < ε + d(xn(k)–, xn(k)), k ∈ N. d(xm(k), xn(k)) ≥ for k ∈ N. Using the properties (F)-(F), (.), and (.), we get Using the above limit and ( ), we obtain On the other hand, using (.) and (F), for all k ∈ N we have which is a contradiction. As a consequence, {xn} is a Cauchy sequence. Since (X, d) is a complete metric space, there is a z ∈ X such that  ≤ ϕ(z) ≤ linm→i∞nf ϕ(xn) = , z ∈ Zϕ. Since ϕ is lower semi-continuous, it follows from (.) and (.) that d(xn+, Tz) ≤ ψ F d(xn, z), ϕ(xn),  , n ∈ N. Also using the continuity of F , (F), (.), and (. (...truncated)


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Erdal Karapinar, Donal O’Regan, Bessem Samet. On the existence of fixed points that belong to the zero set of a certain function, Fixed Point Theory and Applications, 2015, pp. 152, Volume 2015, Issue 1, DOI: 10.1186/s13663-015-0401-7