Existence, Error estimation, Rate of convergence, Ulam-Hyers stability, Well-posedness and Limit Shadowing Property Related to a Fixed Point Problem

abstract: In this paper we consider a fixed point problem where the mapping is supposed to satisfy a generalized contractive inequality involving rational terms. We first prove the existence of a fixed point of such mappings. Then we show that the fixed point is unique under some additional assumptions. We investigate four aspects of the problem, namely, error estimation and rate of convergence of the fixed point iteration, Ulam-Hyers stability, well-psoedness and limit shadowing property. In the existence theorem we use an admissibility condition. Two illustrations are given. The research is in the line with developing fixed point approaches relevant to applied mathematics.


Introduction
Our primary purpose in this paper is to establish a fixed point theorem for a mapping which satisfies a contractive inequality involving rational terms and also some other conditions which are conceptual extension of admissibility conditions. The latter has been used quite extensively in the recent developments of fixed point theory [1,8,10,19,20,23]. It is shown that the fixed point is unique if some additional conditions are imposed. We investigate some aspects of the fixed point problem considered here. We make an error estimation of the fixed point iteration which we construct in this paper. We also investigate the rate of convergence of the iteration process. Such considerations have appeared in the fixed point theory through works like [2,7,14,25].
Next we investigate the Ulam-Hyers stability of the fixed point problem. It is a type of stability which was initiated by a mathematical question by Ulam [24] and subsequent partial answers by Hyers [9] and Rassias [16]. The investigation of such stability has been performed in various contexts of mathematics like functional equations [5,6], isometries [12,17], etc.
Finally we investigate the well-posedness and limit shadowing property of the problem. These are two related properties of the fixed point problem. The study of well-posedness has appeared in several recent works related to fixed point theory as for instances in [4,10,11,13,15,22].
The relevance of the present study lies in the theoretical development of fixed point methodologies applicable to different domains of applied mathematics like differential equations, functional equations etc. [3,21].

Mathematical background
For the purpose of the following three definitions we formally state the following fixed point problem to which they are related. Problem (P): Let (X, d) be a metric space and F : X → X be a mapping. We consider the problem of finding a fixed point of F , that is, the problem of finding x ∈ X such that (2.1) Definition 2.1 ( [10,18]). The problem (P) is called Ulam-Hyers stable if there exists c > 0 such that, for any ǫ > 0 and y ∈ X with d(y, F y) ≤ ǫ there exists a solution x * ∈ X of x = F x such that d(y, x * ) ≤ c ǫ.
Definition 2.2 ( [22]). The problem (P) is called generalized Ulam-Hyers stable if there exists a function φ : [0, ∞) → [0, ∞), which is monotone increasing, continuous at 0 with φ(0) = 0, such that for each ǫ > 0 and for each solution u * ∈ X of the inequality d( Definition 2.5 ( [22]). The problem (P) has the limit shadowing property in X if, for any sequence {x n } ∈ X for which d(x n , F x n ) → 0 as n → ∞, it follows that there exists z ∈ X such that d(x n , F n z) → 0 as n → ∞.
Definition 2.6. Let X be a nonempty set and α : The above definition is illustrated though the following example.
Definition 2.9 ( [19]). Let (X, d) be a metric space and α : X × X → [0, ∞). Then X is said to have regular property with respect to α (or α-regular property) if for every sequence {x n } in X converging to x ∈ X, α(x n , x n+1 ) ≥ 1, for all n =⇒ α(x n , x) ≥ 1, for all n.
Remark 2.10. For the metric space X and the mapping α as in Example 2.7, it can be easily verified that α has triangular property and X is regular with respect to α.

Main results
In this section we establish a fixed point result. We discuss its uniqueness under some additional assumptions. We deduce some corollaries of the main result and illustrate it with an example.
Also, suppose that the property (A1) holds. Then F has a fixed point in X.
Proof. Let x 0 ∈ X be arbitrary. We construct a sequence {x n } in X such that Suppose that 0 ≤ r n < r n+1 . From (3.4) and (3.5), we have which is a contradiction. Therefore, r n+1 ≤ r n , for all n ≥ 0. Then from (3.5), we have By repeated application of (3.6), we have With the help of (3.7), we have which implies that {x n } is a Cauchy sequence in X. As X is complete, there exists x ∈ X such that lim n→∞ x n = x.
Taking limit as n → ∞ in (3.9) and using (3.8), we have which implies that d(x, F x) = 0, that is, x = F x, that is, x is a fixed point of F . Proof. By Theorem 3.1, the set of fixed points of F is nonempty. If possible, let x and x * be two fixed points of F . Then x = F x and x * = F x * . Our aim is to show that x = x * . By the assumption (A3), there exists u ∈ X such that α(x, u) ≥ 1 and α(x * , u) ≥ 1. Put u 0 = u. Then α(x, u 0 ) ≥ 1. Let u 1 = F u 0 . Similarly, as in the proof of Theorem 3.1, we inductively define a sequence {u n } such that u n+1 = F u n , for all n ≥ 0. (3.10) As F is α-dominated, we have α(u n , u n+1 ) ≥ 1, for all n ≥ 0. (3.11) Arguing similarly as in proof of Theorem 3.1, we prove that {u n } is a Cauchy sequence in X and there exists p ∈ X such that lim n→∞ u n = p. (3.12) We claim that In fact, we shall use mathematical induction. As α(x, u 0 ) ≥ 1 and α(u 0 , u 1 ) ≥ 1, by the assumption (A2), we have α(x, u 1 ) ≥ 1. Therefore, our claim is true for n = 1. We assume that α(x, u m ) ≥ 1 holds for some m > 1. Now by (3.11), α(u m , u m+1 ) ≥ 1. Then applying the assumption (A2), we have α(x, u m+1 ) ≥ 1 and this proves our claim. By (3.1) and (3.13) we have, for all n ≥ 0 (3.14) Taking limit as n → ∞ in (3.14) and using (3.12), we have which is a contradiction unless d(x, p) = 0, that is, x = p. Similarly, we can prove that x * = p. Hence we have x = x * , that is, the fixed point of F is unique.
We present the following illustrative examples in support of Theorems 3.1 .
Example 3.3. Using the metric space X, mappings α and F as in Example 2.7, we see that α has triangular property and X is regular with respect to α (see Remark 2.10) and F is a α-dominated mapping. Take k = 1 4 .
Let x, y ∈ X with α(x, y) ≥ 1. Then x ∈ [0, 1] and y ∈ [0, 1 8 ]. Therefore, it is required to verify the inequality in Theorem 3.1 for x ∈ [0, 1] and y ∈ [0, 1 8 ]. Now, d(x, y) =| x − y | and Then it follows that the inequality in Theorem 3.1 is satisfied for all x, y ∈ X with α(x, y) ≥ 1. Hence all the conditions of Theorem 3.2 are satisfied and 0 is the unique fixed point of F .

Error estimation and rate of convergence
We now study the rate at which the iteration method of finding the fixed point of the problem (P) converges if the initial approximation to the fixed point is sufficiently close to the desired fixed point.
Definition 4.1. The problem (P) is said to be of order r or has the rate of convergence r if (i) F has a unique fixed point x, (ii) r is the positive real number for which there exists a finite constant C > 0 such that R k+1 ≤ C R r k , where R k = d(x, x k ) is the error in k th iterate. The constant C is called the asymptotic error.
When r = 1 we say that the problem (P) is linearly convergent.

Suppose that F satisfies all the assumptions of Theorem 3.2. Then
Proof. By Theorem 3.2, F has a unique fixed point x ∈ X. Let x 0 ∈ X be the initial approximation of x and x 1 = F x 0 . Similarly, as in the proof of Theorem 3.1, we define a sequence {x n } such that x n+1 = F x n , for all n ≥ 0. Then arguing similarly as in proof of Theorem 3.1, we can show that (a) (3.3) and (3.7) are satisfied; (b) {x n } is a Cauchy sequence in X and converges to a fixed point of F in X. As, we consider that x is the unique fixed point of F , we have lim n→∞ x n = x. By (3.3), (3.8) and the assumption (A1), we have α(x n , x) ≥ 1 for all n ≥ 0. Then we have Suppose that R n+1 > R n ≥ 0. Then we have R n+1 ≤ k R n+1 . As 0 < k < 1 and R n+1 > 0, it leads to a contradiction. Therefore, R n+1 ≤ R n . Hence it follows from (4.1) that that is, which, by (3.7), implies that  Remark 4.5. By Theorem 3.2, F has unique fixed point x. The inequality (4.2), that is, R n+1 ≤ k R n shows that the fixed point problem (P) is linearly convergent with asymptotic error k.

Ulam-Hyers stablity
In this section we discuss Ulam-Hyers stablity of fixed problem (P) via α− dominated mapping. For this purpose we consider the fixed point problem (P) and the inequality We consider the following assumption which we use in the theorem of this section.