Stability of common fixed points in uniform spaces

Abstract

Stability results for a pair of sequences of mappings and their common fixed points in a Hausdorff uniform space using certain new notions of convergence are proved. The results obtained herein extend and unify several known results.

AMS(MOS) Subject classification 2010: 47H10; 54H25.

1 Introduction

The relationship between the convergence of a sequence of self mappings T _n of a metric (resp. topological space) X and their fixed points, known as the stability (or continuity) of fixed points, has been widely studied in fixed point theory in various settings (cf. [1–18]). The origin of this problem seems into a classical result (see Theorem 1.1) of Bonsall [6] (see also Sonnenshein [18]) for contraction mappings. Recall that a self-mapping f of a metric space (X, d) is called a contraction mapping if there exists a constant k, 0 < k < 1 such that

$$d(f\left(x\right),f\left(y\right)\le kd\left(x,y\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Theorem 1.1. Let (X, d) be a complete metric space and T and T _n (n = 1, 2,...) be contraction mappings of X into itself with the same Lipschitz constant k < 1, and with fixed points u and u _n (n = 1, 2,...), respectively. Suppose that lim _n T _n x = Tx for every x ∈ X. Then, lim _n u _n = u.

Subsequent results by Nadler Jr. [11], and others address mainly the problem of replacing the completeness of the space X by the existence of fixed points (which was ensured otherwise by the completeness of X) and various relaxations on the contraction constant k. In most of these results, pointwise (resp. uniform) convergence plays invariably a vital role. However, if the domain of definition of T _n is different for each n ∈ ℕ (naturals), then these notions do not work. An alternative to this problem has recently been presented by Barbet and Nachi [5] (see also [4]) where some new notions of convergence have been introduced and utilized to obtain stability results in a metric space. For a uniform space version of these results, see Mishra and Kalinde [10]. On the other hand, a result of Jungck [19] on common fixed points of commuting continuous mappings has also been found quite useful. We note that the above-mentioned result of Jungck [19] includes the well-known Banach contraction principle. Using the above ideas of Barbet and Nachi [5] and Jungck [19], we obtain stability results for common fixed points in a uniform space whose uniformity is generated by a family of pseudometrics. These results generalize the recent results obtained by Mishra and Kalinde [10] and which in turn include several known results. Locally convex topological vector spaces being completely regular are uniformizable, where the uniformity of the space is induced by a family of seminorms. Therefore, all the results obtained herein for uniform spaces also apply to locally convex spaces (cf. Remark 4.4).

2 Preliminaries

Let $\left(X,\mathcal{U}\right)$ be a uniform space. A family P = {ρ_α : α∈ I} of pseudometrics on X, where I is an indexing set is called an associated family for the uniformity $\mathcal{U}$ if the family

$$\mathfrak{B}=\left\{V\left(\alpha ,\epsilon \right):\alpha \in I,\epsilon >0\right\},$$

where

$$V\left(\alpha ,\epsilon \right)=\left\{\left(x,y\right)\in X\times X:{\rho }_{\alpha }\left(x,y\right)<\epsilon \right\}$$

is a subbase for the uniformity $\mathcal{U}$. We may assume $\mathfrak{B}$ itself to be a base for $\mathcal{U}$ by adjoining finite intersections of members of $\mathfrak{B}$ if necessary. The corresponding family of pseudometrics is called an augmented associated family for $\mathcal{U}$. An augmented associated family for $\mathcal{U}$ will be denoted by P*. (cf. Mishra [9] and Thron [20]). In view of Kelley [21], we note that each member V (α, ε) of $\mathfrak{B}$ is symmetric and ρ _α is uniformly continuous on X × X for each α ∈ I. Further, the uniformity $\mathcal{U}$ is not necessarily pseudometrizable (resp. metrizable) unless $\mathfrak{B}$ is countable, and in that case, $\mathcal{U}$ may be generated by a single pseudometric (resp. a metric) ρ on X. For an interesting motivation, we refer to Reilly [[22], Example 2] (see also Kelley [[21], Example C, p. 204]). For further details on uniform spaces and a systematic account of fixed point theory there in (including applications), we refer to Kelleyl [21] and Angelov [3] respectively.

Now onwards, unless stated otherwise, $\left(X,\mathcal{U}\right)$ will denote a uniform space defined by P* while $\bar{\mathbb{N}}=\mathbb{N}\cup \left\{\infty \right\}$.

Definition 2.1. [23] Let $\left(X,\mathcal{U}\right)$ be a uniform space and let {ρ_α : α ∈ I} = P*. A mapping T : X → X is called a P*- contraction if for each α ∈ I, there exists a real k(α), 0 < k(α) < 1 such that

$${\rho }_{\alpha }\left(T\left(x\right),T\left(y\right)\right)\le k\left(\alpha \right){\rho }_{\alpha }\left(x,y\right)\mathsf{\text{for all}}x,y\in X.$$

It is well known that T : X → X is P*-contraction if and only if it is P- contraction (see Tarafdar [[23], Remark 1]). Hence, now onwards, we shall simply use the term k-contraction (resp. contraction) to mean either of them. In case the above condition is satisfied for any k = k(α) > 0, T will be called k- Lipschitz (or simply Lipschitz).

The following result due to Tarafdar [[23], Theorem 1.1] (see also Acharya [[24], Theorem 3.1]) presents an exact analog of the well-known Banach contraction principle.

Theorem 2.2. Let $\left(X,\mathcal{U}\right)$ be a Hausdorff complete uniform space and let {ρ_α : α ∈ I}= P*. Let T be a contraction on X. Then, T has a unique fixed point a ∈ X such that Tⁿx → a in τ _u (the uniform topology) for each x ∈ X.

Definition 2.3. Let $\left(X,\mathcal{U}\right)$ be a uniform space, S, T : Y ⊆ X → X. Then, the pair (S, T) will be called J - Lipschitz (Jungck Lipschitz) if for each α ∈ I, there exists a constant μ = μ(α) > 0 such that

$${\rho }_{\alpha }\left(Sx,Sy\right)\le \mu {\rho }_{\alpha }\left(Tx,Ty\right)\mathsf{\text{for all}}x,y\in Y.$$

(2.1)

The pair (S, T) is generally called Jungck contraction (or simply J-contraction) when 0 < μ < 1, and the constant μ in this case is a called Jungck constant (see, for instance, [13]). Indeed, J-contractions and their generalized versions became popular because of the constructive approach of proof adopted by Jungck [19]. Now onwards, a J-Lpschitz map (resp. J-contraction) with Jungck constant μ will be called a J-Lipschitz (resp. J-contraction) with constant μ.

The following example illustrates the generality of J-Lipschitz maps.

Example 2.4. Let X = (0, ∞) with the usual uniformity induced by ρ(x, y) = |x - y| for all x, y ∈ X. Define S : X → X by

$$Sx=\frac{1}{x}\mathsf{\text{for}}\mathsf{\text{all}}x\in X.$$

Then,

$$\rho \left(Sx,Sy\right)=\frac{1}{xy}\rho \left(x,y\right)\mathsf{\text{for}}\mathsf{\text{all}}x,y\in X.$$

Since $\frac{1}{xy}\to \infty$ for small x or y ∈ X, S is not a Lipschitz map. However, if we consider the map T : X → X defined by

$$Tx=\frac{1}{Lx},\mathsf{\text{for}}\mathsf{\text{all}}x\in X\mathsf{\text{and}}\mathsf{\text{some}}L>0,$$

then

$$\rho \left(Sx,Sy\right)=L\rho \left(Tx,Ty\right)$$

and S is Lipscitz with respect to T or the pair (S, T) is J-Lipschitz.

3 G-convergence and stability

Definition 3.1[5, 10]. Let $\left(X,\mathcal{U}\right)$ be a uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a sequence of nonempty subsets of X and ${\left\{S_n:X_n\to X\right\}}_{n\in \bar{\mathbb{N}}}$ a sequence of mappings. Then ${\left\{S_n\right\}}_{n\in \bar{\mathbb{N}}}$ is said to converge G-pointwise to a map S_∞: X_∞ → X, or equivalently ${\left\{S_n\right\}}_{n\in \bar{\mathbb{N}}}$satisfies the property (G), if the following condition holds:

(G) Gr(S_∞) ⊂ lim inf Gr(S _n ): for every x ∈ X_∞, there exists a sequence {x _n } in ${\prod }_{n\in \mathbb{N}}{X}_n$ such that for any α ∈ I,

$$\underset{n}{lim}{\rho }_{\alpha }\left(x_n,x\right)=0\mathsf{\text{and}}\underset{n}{lim}{\rho }_{\alpha }\left(S_n{x}_n,S_{\infty }x\right)=0,$$

where Gr(T) stands for the graph of T.

In view of Barbet and Nachi [5], we note that:

(i) A G-limit need not be unique.

(ii) The property (G) is more general than pointwise convergence. However, the two notions are equivalent provided the sequence {S _n }_n∈ℕis equicontinuous when the domains of definitions are identical.

The following theorem gives a sufficient condition for the existence of a unique G-limit.

Theorem 3.2. Let $\left(X,\mathcal{U}\right)$ be a uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X and ${\left\{S_n:X_n\to X\right\}}_{n\in \bar{\mathbb{N}}}$ a sequence of J-Lipschitz maps relative to a continuous map T : X → X with Lipschitz constant μ. If S_∞ : X_∞ → X is a G-limit of the sequence {S _n }, then S_∞ is unique.

Proof. Let $U\in \mathcal{U}$ be an arbitrary entourage. Then, since $\mathfrak{B}$ is base for $\mathcal{U}$, there exists V (α, ε) ∈ $\mathfrak{B}$, α ∈ I, ε > 0 such that V (α, ε) ⊂ U. Suppose that S_∞ : X_∞ → X and $S_{\infty }^{*}:X_{\infty }\to X$ are G-limit maps of the sequence {S _n }. Then, for every x ∈ X_∞, there exist two sequences {x _n } and {y _n } in ${\prod }_{n\in \mathbb{N}}{X}_n$ such that for any α ∈ I

$$\begin{array}{lll}\hfill \underset{n}{lim}{\rho }_{\alpha }\left(x_n,x\right)& =0\mathsf{\text{and}}\underset{n}{lim}{\rho }_{\alpha }\left(S_n{x}_n,S_{\infty }x\right)=0,& \hfill \text{(1)}\\ \hfill \underset{n}{lim}{\rho }_{\alpha }\left(y_n,x\right)& =0\mathsf{\text{and}}\underset{n}{lim}{\rho }_{\alpha }\left(S_n{y}_n,S_{{}_{\infty }}^{*}x\right)=0.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

Further, since S _n is J-Lipschitz, for any α ∈ I, there exists a constant μ = μ(α) > 0 such that

$${\rho }_{\alpha }\left(S_n{x}_n,S_n{y}_n\right)\le \mu {\rho }_{\alpha }\left(T_n{x}_n,T_n{y}_n\right).$$

Therefore, for any n ∈ ℕ and α ∈ I,

$$\begin{array}{ll}{\rho }_{\alpha }(S_{\infty }x,S_{\infty }^{*}x)\hfill & \le {\rho }_{\alpha }(S_{\infty }x,S_n{x}_n)+{\rho }_{\alpha }(S_n{x}_n,S_n{y}_n)+{\rho }_{\alpha }(S_n{y}_n,S_{\infty }^{*}x)\hfill \\ \le {\rho }_{\alpha }(S_{\infty }x,S_n{x}_n)+\mu {\rho }_{\alpha }(T{x}_n,T{y}_n)+{\rho }_{\alpha }(S_n{y}_n,S_{\infty }^{*}x)\hfill \\ \le {\rho }_{\alpha }(S_{\infty }x,S_n{x}_n)+\mu [{\rho }_{\alpha }(T{x}_n,Tx)+(Tx,T{y}_n)]+{\rho }_{\alpha }(S_n{y}_n,S_{\infty }^{*}x)\hfill \end{array}$$

Since T is continuous and x _n → x and y _n → x as n → ∞, it follows that Tx _n → Tx, Ty _n → Tx. Hence the R.H.S. of the above expression tends to 0 as n → ∞ and so, ${\rho }_{\alpha }\left(S_{\infty }x,S_{\infty }^{*}x\right)<\epsilon$ for all n ≥ N (α, ε). Therefore $\left(S_{\infty }x,S_{\infty }^{*}x\right)\in V\left(\alpha ,\epsilon \right)\subset U$ and since X is Hausdorff, it follows that $S_{\infty }x=S_{\infty }^{*}x$.■

Corollary 3.3. Theorem 3.2 with J-Lipschitz replaced by J-contraction.

Proof. It comes from Theorem 3.2 for μ ∈ (0, 1).■

The following result due to Mishra and Kalinde [[10], Proposition 3.1, see also, Remark 3.2)], which in turn includes a result of Barbet and Nachi [[5], Proposition 1], follows as a corollary of Theorem 3.2.

Corollary 3.4. Let $\left(X,\mathcal{U}\right)$ be a Hausdorff uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X and S _n : X _n → X a k- contraction (resp. k- Lipschitz) mapping for each $n\in \bar{\mathbb{N}}$. If S_∞ : X_∞ → X is a (G) - limit of ${\left\{S_n\right\}}_{n\in \bar{\mathbb{N}}}$ then S_∞ is unique.

Proof. It comes from Theorem 3.2 when T is the identity map and μ ∈ (0, 1) (resp. μ > 0).■

Now, we present our first stability result.

Theorem 3.5. Let $\left(X,\mathcal{U}\right)$ be a uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X and ${\left\{S_n,T_n:X_n\to X\right\}}_{n\in \stackrel{-}{\mathbb{N}}}$ two families of maps each satisfying the property (G) and such that for all $n\in \bar{\mathbb{N}}$, the pair (S _n , T _n ) is J-contraction with constant μ. If for all $n\in \bar{\mathbb{N}}$, z _n is a common fixed point of S _n and T _n , then, the sequence {z _n } converges to z_∞.

Proof. Let $W\in \mathcal{U}$ be arbitrary. Then, there exists $V\left(\lambda ,\epsilon \right)\in \mathfrak{B},\lambda \in I,\epsilon >0$ such that V (λ, ε) ⊂ W. Since z _n is a common fixed point of S _n and T _n for each $n\in \bar{\mathbb{N}}$, and the property (G) holds and z_∞ ∈ X_∞, there exists a sequence {y _n } such that y _n ∈ X _n (for all $n\in \bar{\mathbb{N}}$) such that for any λ ∈ I,

$$\underset{n}{lim}{\rho }_{\lambda }\left(y_n,z_{\infty }\right)=0,\underset{n}{lim}{\rho }_{\lambda }\left(S_n{y}_n,S_{\infty }{z}_{\infty }\right)=0\mathsf{\text{and}}\underset{n}{lim}{\rho }_{\lambda }\left(T_n{y}_n,T_{\infty }{z}_{\infty }\right)=0.$$

Using the fact that the pair (S _n , T _n ) is J-contraction, for any λ ∈ I, we have

$$\begin{array}{ll}{\rho }_{\lambda }(z_n,z_{\infty })\hfill & ={\rho }_{\lambda }(S_n{z}_n,S_{\infty }{z}_{\infty })\hfill \\ \le {\rho }_{\lambda }(S_n{z}_n,S_n{y}_n)+{\rho }_{\lambda }(S_n{y}_n,S_{\infty }{z}_{\infty })\hfill \\ \le \mu (\lambda ){\rho }_{\lambda }(T_n{z}_n,T_n{y}_n)+{\rho }_{\lambda }(S_n{y}_n,S_{\infty }{z}_{\infty })\hfill \\ \le \mu (\lambda ){\rho }_{\lambda }(T_n{z}_n,T_{\infty }{z}_{\infty })+\mu (\lambda ){\rho }_{\lambda }(T_n{y}_n,T_{\infty }{z}_{\infty })+{\rho }_{\lambda }(S_n{y}_n,S_{\infty }{z}_{\infty }).\hfill \end{array}$$

This gives

$${\rho }_{\lambda }\left(z_n,z_{\infty }\right)\le \frac{1}{1-\mu \left(\lambda \right)}\left[\mu \left(\lambda \right){\rho }_{\lambda }\left(T_n{y}_n,T_{\infty }{z}_{\infty }\right)+{\rho }_{\lambda }\left(S_n{y}_n,S_{\infty }{z}_{\infty }\right)\right].$$

Since μ(λ) < 1, it follows that ρ _λ (z _n , z_∞) → 0 as n → ∞. Hence, ρ _λ (z _n , z_∞) < ε for all n ≥ N (λ, ε) and so (z _n , z_∞) ∈ V (λ, ε) ⊂ W and the conclusion follows.■

When for each $n\in \bar{\mathbb{N}}$, T _n is the identity map on X _n in Theorem 3.5, we have the following result due to Mishra and Kalinde [[10], Theorem 3.3], which includes a result of Barbet and Nachi [[5], Theorem 2].

Corollary 3.6. Let $\left(X,\mathcal{U}\right)$ be a Hausdorff uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X and ${\left\{S_n:X_n\to X\right\}}_{n\in \bar{\mathbb{N}}}$ a family of mappings satisfying the property (G) and S _n is a k- contraction for each $n\in \bar{\mathbb{N}}$. If x _n is a fixed point of S _n for each $n\in \bar{\mathbb{N}}$, then the sequence {x _n }_n∈ℕconverges to x_∞.

Again, when X _n = X, for all $n\in \bar{\mathbb{N}}$, we obtain, as a consequence of Theorem 3.5, the following result.

Corollary 3.7. Let $\left(X,\mathcal{U}\right)$ be a uniform space and S _n , T _n : X → X be such that the pair (S _n , T _n ) is J-contraction with constant μ and with at least one common fixed point z _n for all $n\in \bar{\mathbb{N}}$. If the sequences {S _n } and {T _n } converge pointwise respectively to S, T : X → X, then the sequence {z _n } converges to z_∞.

Notice that Corollary 3.7 includes as a special case a result of Singh [[13], Theorem 1] for metric spaces (metrizable spaces).

We remark that under the conditions of Theorem 3.5 the pair (S_∞, T_∞) of G-limit maps is also a J-contraction. Indeed, we have the following stability result.

Theorem 3.8. Let $\left(X,\mathcal{U}\right)$ be a uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X and ${\left\{S_n,T_n:X_n\to X\right\}}_{n\in \stackrel{-}{\mathbb{N}}}$ two families of maps each satisfying the property (G) and such that for all n∈ℕ, the pair (S _n , T _n ) is J-contraction with constant {μ _n }_n∈ℕa bounded (resp. convergent) sequence. Then, the pair (S_∞, T_∞) is J-contraction with constant μ = sup_n∈ℕμ _n (resp. μ = lim _n μ _n ).

Proof. Let x, y ∈ X_∞. Then, by the property (G), there exist two sequences {x _n } and {y _n } in ${\prod }_{n\in \mathbb{N}}{X}_n$ such that the sequences {S _n x _n }, {S _n y _n }, {T _n x _n } and {T _n y _n } converge respectively to S_∞x, S_∞y, T_∞x, and T_∞y.

Therefore, for any n∈N and each α ∈ I,

$$\begin{array}{ll}{\rho }_{\alpha }(S_{\infty }x,S_{\infty }y)\hfill & \le {\rho }_{\alpha }(S_{\infty }x,S_n{x}_n)+{\rho }_{\alpha }(S_n{x}_n,S_n{y}_n)+{\rho }_{\alpha }(S_n{y}_n,S_{\infty }y)\hfill \\ \le {\rho }_{\alpha }(S_{\infty }x,S_n{x}_n)+{\mu }_n{\rho }_{\alpha }(T_n{x}_n,T_n{y}_n)+{\rho }_{\alpha }(S_n{y}_n,S_{\infty }y).\hfill \end{array}$$

Since

$$lim\underset{n}{sup}{\mu }_n{\rho }_{\alpha }\left(T_n{x}_n,T_n{y}_n\right)\le \mu {\rho }_{\alpha }\left(T_{\infty }x,T_{\infty }y\right),$$

the above inequality yields ρ _α (S_∞x, S_∞y) ≤ μ ρ _α (T_∞x, T_∞y) and the conclusion follows.■

Remark 3.9. Theorem 3.8 includes, as a special case, a result of Mishra and Kalinde [[10], Proposition 3.5] for uniform spaces when X _n = X and T _n is an identity mapping for each $n\in \bar{\mathbb{N}}$. Consequently, a result of Barbet and Nachi [[5], Proposition 4] for metric spaces also follows when X is metrizable.

4 H-convergence and stability

Definition 4.1. [5, 10] Let $\left(X,\mathcal{U}\right)$ be a uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X and ${\left\{S_n:X_n\to X\right\}}_{n\in \bar{\mathbb{N}}}$ a family of mappings. Then,

S_∞ is called an (H) - limit of the sequence {S _n }_n∈ℕin or, equivalently ${\left\{S_n\right\}}_{n\in \bar{\mathbb{N}}}$ satisfies the property (H) if the following condition holds:

(H) For all sequences {x _n } in ${\prod }_{n\in \mathbb{N}}{X}_n$, there exists a sequence {y _n } in X_∞ such that for any α ∈ I,

$$\underset{n}{lim}{\rho }_{\alpha }\left(x_n,y_n\right)=0and\underset{n}{lim}{\rho }_{\alpha }\left(S_n{x}_n,S_n{y}_n\right)=0.$$

In case X is a metrizable uniform space (that is the uniformity $\mathcal{U}$ is generated by a metric d), we get the corresponding definitions due to Barbet and Nachi [5].

In view of [5], we note that:

(a) A G-limit map is not necessarily an H-limit.

(b) If ${\left\{S_n:Y\subseteq X\to X\right\}}_{n\in \mathbb{N}}$ converges uniformly to S_∞ on Y, then S_∞ is an H-limit of {S _n }.

(c) The converse of (b) holds only when S_∞ is uniformly continuous on Y.

For details and examples, we refer to Barbet and Nachi [5].

Theorem 4.2. Let $\left(X,\mathcal{U}\right)$ be a uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X. Let ${\left\{S_n,T_n:X_n\to X\right\}}_{n\in \stackrel{-}{\mathbb{N}}}$ be two families of maps each satisfying the property (H). Further, let the pair (S_∞, T_∞) be a J-contraction with constant μ_∞. If, for every $n\in \bar{\mathbb{N}}$, z _n is a common fixed point of S _n and T _n , then the sequence {z _n } converges to z_∞.

Proof. The property (H) implies that there exists a sequence {y _n } in X_∞ such that for any α ∈ I, ρ _α (z _n , y _n ) → 0, ρ _α (S _n z _n , S_∞y _n ) → 0 and ρ _α (T _n z _n , T_∞y _n ) → 0 as n → ∞. Then

$$\begin{array}{ll}{\rho }_{\alpha }(z_n,z_{\infty })\hfill & ={\rho }_{\alpha }(S_n{z}_n,S_{\infty }{z}_{\infty })\hfill \\ \le {\rho }_{\alpha }(S_n{z}_n,S_{\infty }{y}_n)+{\rho }_{\alpha }(S_{\infty }{y}_n,S_{\infty }{z}_{\infty })\hfill \\ \le {\rho }_{\alpha }(S_n{z}_n,S_{\infty }{y}_n)+{\mu }_{\infty }{\rho }_{\alpha }(T_{\infty }{y}_n,T_{\infty }{z}_{\infty })\hfill \\ \le {\rho }_{\alpha }(S_n{z}_n,S_{\infty }{y}_n)+{\mu }_{\infty }[{\rho }_{\alpha }(T_{\infty }{y}_n,T_n{z}_n)+{\rho }_{\alpha }(T_n{z}_n,T_{\infty }{z}_{\infty })].\hfill \end{array}$$

So, we get

$${\rho }_{\alpha }(z_n,z_{\infty })\le \frac{1}{(1-{\mu }_{\infty })}[{\rho }_{\alpha }(S_n{z}_n,S_{\infty }{y}_n)+{\mu }_{\infty }{\rho }_{\alpha }(T_{\infty }{y}_n,T_n{z}_n].$$

Since the right hand side of the above inequality tends to 0 as n → ∞, we deduce that z _n → z_∞ as n → ∞. ■

As a consequence of Theorem 4.2, we have the following result due to Mishra and Kalinde [[10], Theorem 3.13].

Corollary 4.3. Let $\left(X,\mathcal{U}\right)$ be a Hausdorff uniform space, ${\left\{X_n\right\}}_{n\in \bar{\mathbb{N}}}$ a family of nonempty subsets of X and ${\left\{S_n:X_n\to X\right\}}_{n\in \bar{\mathbb{N}}}$ a family of mappings satisfying the property (H) and such that S_∞ is a k_∞ - contraction. If for any $n\in \bar{\mathbb{N}}$, x _n is a fixed point of T _n , then {x _n }_n∈ℕconverges to x_∞.

Proof. It comes from Theorem 4.2 by taking T _n to be the identity mapping for each $n\in \bar{\mathbb{N}}$.■

If X is metrizable, then we get a stability result of Barbet and Nachi [[5], Theorem 11], which in turn includes a result of Nadler [[11], Theorem 1]. Indeed, Nadler's result is a direct consequence of Corollary 4.3 when X _n = X for each n ∈ ℕ with X being metrizable.

Remark 4.4. Every locally convex topological vector space X is uniformizable being completely regular (cf. Kelley [21], Shaefer [25]) where the family of pseudometrics {ρ _α : α ∈ I} is induced by a family of seminorms {ρ _α : α ∈ I} so that ρ _α (x, y) = ρ _α (x - y) for all x, y ∈ X. Therefore, all the results proved previously for uniform spaces also apply to locally convex spaces.