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# Stability of common fixed points in uniform spaces

- Swaminath Mishra
^{1}Email author, - Shyam Lal Singh
^{2}and - Simfumene Stofile
^{1}

**2011**:37

https://doi.org/10.1186/1687-1812-2011-37

© Mishra et al; licensee Springer. 2011

**Received:**10 February 2011**Accepted:**16 August 2011**Published:**16 August 2011

## 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**.

## Keywords

- Stability
- fixed point
- uniform space
- J-Lipschitz

## 1 Introduction

*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

**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

*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

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 $\stackrel{\u0304}{\mathbb{N}}=\mathbb{N}\cup \left\{\infty \right\}$.

**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

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 *T*^{
n
}*x* → *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

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

*x*or

*y*∈

*X*,

*S*is not a Lipschitz map. However, if we consider the map

*T*:

*X*→

*X*defined by

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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a sequence of nonempty subsets of *X* and ${\left\{{S}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a sequence of mappings. Then ${\left\{{S}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ is said to converge *G-pointwise* to a map *S*_{∞}: *X*_{∞} → *X*, or equivalently ${\left\{{S}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$*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*,

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∈ℕ}is 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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X* and ${\left\{{S}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ 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*

*S*

_{ n }is J-Lipschitz, for any

*α*∈

*I*, there exists a constant

*μ*=

*μ*(

*α*) > 0 such that

*n*∈ ℕ and

*α*∈

*I*,

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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X* and *S*_{
n
} : *X*_{
n
} → *X* a *k-* contraction (resp. *k-* Lipschitz) mapping for each $n\in \stackrel{\u0304}{\mathbb{N}}$. If *S*_{∞} : *X*_{∞} → *X* is a (*G*) - limit of ${\left\{{S}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ 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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X* and ${\left\{{S}_{n},\phantom{\rule{2.77695pt}{0ex}}{T}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{-}{\mathbb{N}}}$ two families of maps each satisfying the property (G) and such that for all $n\in \stackrel{\u0304}{\mathbb{N}}$, the pair (*S*_{
n
} , *T*_{
n
} ) is J-contraction with constant *μ*. If for all $n\in \stackrel{\u0304}{\mathbb{N}}$, *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 $n\in \stackrel{\u0304}{\mathbb{N}}$, and the property (G) holds and

*z*

_{∞}∈

*X*

_{∞}, there exists a sequence {

*y*

_{ n }} such that

*y*

_{ n }∈

*X*

_{ n }(for all $n\in \stackrel{\u0304}{\mathbb{N}}$) such that for any

*λ*∈

*I*,

*S*

_{ n },

*T*

_{ n }) is J-contraction, for any

*λ*∈

*I*, we have

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 $n\in \stackrel{\u0304}{\mathbb{N}}$, *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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X* and ${\left\{{S}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of mappings satisfying the property (*G*) and *S*_{
n
} is a *k-* contraction for each $n\in \stackrel{\u0304}{\mathbb{N}}$. If *x*_{
n
} is a fixed point of *S*_{
n
} for each $n\in \stackrel{\u0304}{\mathbb{N}}$, then the sequence {*x*_{
n
} }_{n∈ℕ}converges to *x*_{∞}.

Again, when *X*_{
n
} = *X*, for all $n\in \stackrel{\u0304}{\mathbb{N}}$, 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 $n\in \stackrel{\u0304}{\mathbb{N}}$. 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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X* and ${\left\{{S}_{n},\phantom{\rule{2.77695pt}{0ex}}{T}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{-}{\mathbb{N}}}$ 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∈ℕ}a bounded (resp. convergent) sequence. Then, the pair (*S*_{∞}, *T*_{∞}) is J-contraction with constant *μ* = sup_{n∈ℕ}*μ*_{
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*.

*n*∈N and each

*α*∈

*I*,

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 $n\in \stackrel{\u0304}{\mathbb{N}}$. 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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X* and ${\left\{{S}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of mappings. Then,

*S*_{∞} is called an (*H*) - limit of the sequence {*S*_{
n
} }_{n∈ℕ}in or, equivalently ${\left\{{S}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ 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*,

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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X*. Let ${\left\{{S}_{n},\phantom{\rule{2.77695pt}{0ex}}{T}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{-}{\mathbb{N}}}$ 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 $n\in \stackrel{\u0304}{\mathbb{N}}$, *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

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, ${\left\{{X}_{n}\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of nonempty subsets of *X* and ${\left\{{S}_{n}:{X}_{n}\to X\right\}}_{n\in \stackrel{\u0304}{\mathbb{N}}}$ a family of mappings satisfying the property (*H*) and such that *S*_{∞} is a *k*_{∞} - contraction. If for any $n\in \stackrel{\u0304}{\mathbb{N}}$, *x*_{
n
} is a fixed point of *T*_{
n
} , then {*x*_{
n
} }_{n∈ℕ}converges to *x*_{∞}.

**Proof**. It comes from Theorem 4.2 by taking *T*_{
n
} to be the identity mapping for each $n\in \stackrel{\u0304}{\mathbb{N}}$.■

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.

## Declarations

### Acknowledgements

This research is supported by the Directorate of Research Development, Walter Sisulu University. A special word of thanks is also due to referee for his constructive comments.

## Authors’ Affiliations

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