Open Access

Stability of common fixed points in uniform spaces

Fixed Point Theory and Applications20112011:37

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

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

Stabilityfixed pointuniform spaceJ-Lipschitz

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. [118]). 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 ( x ) , f ( y ) k d ( x , y ) for all x , y 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 (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 ( X , U ) be a uniform space. A family P = {ρα : α I} of pseudometrics on X, where I is an indexing set is called an associated family for the uniformity U if the family
B = { V ( α , ε ) : α I , ε > 0 } ,
where
V ( α , ε ) = { ( x , y ) X × X : ρ α ( x , y ) < ε }

is a subbase for the uniformity U . We may assume B itself to be a base for U by adjoining finite intersections of members of B if necessary. The corresponding family of pseudometrics is called an augmented associated family for U . An augmented associated family for 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 B is symmetric and ρ α is uniformly continuous on X × X for each α I. Further, the uniformity U is not necessarily pseudometrizable (resp. metrizable) unless B is countable, and in that case, 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, ( X , U ) will denote a uniform space defined by P* while ̄ = { } .

Definition 2.1. [23] Let ( X , U ) be a uniform space and let {ρα : α I} = P*. A mapping T : XX is called a P*- contraction if for each α I, there exists a real k(α), 0 < k(α) < 1 such that
ρ α T x , T y k ( α ) ρ α ( x , y ) for all x , y X .

It is well known that T : XX 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 ( X , U ) be a Hausdorff complete uniform space and let {ρα : α I}= P*. Let T be a contraction on X. Then, T has a unique fixed point a X such that T n xa in τ u (the uniform topology) for each x X.

Definition 2.3. Let ( X , U ) be a uniform space, S, T : Y XX. 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
ρ α ( S x , S y ) μ ρ α ( T x , T y ) for all x , y 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 : XX by
S x = 1 x for all x X .
Then,
ρ ( S x , S y ) = 1 x y ρ ( x , y ) for all x , y X .
Since 1 x y for small x or y X, S is not a Lipschitz map. However, if we consider the map T : XX defined by
T x = 1 L x , for all x X and some L > 0 ,
then
ρ ( S x , S y ) = L ρ ( T x , T y )

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 ( X , U ) be a uniform space, { X n } n ̄ a sequence of nonempty subsets of X and { S n : X n X } n ̄ a sequence of mappings. Then { S n } n ̄ is said to converge G-pointwise to a map S: XX, or equivalently { S n } 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 n X n such that for any α I,
lim n ρ α ( x n , x ) = 0 and lim n ρ α ( S n x n , S x ) = 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 }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 ( X , U ) be a uniform space, { X n } n ̄ a family of nonempty subsets of X and { S n : X n X } n ̄ a sequence of J-Lipschitz maps relative to a continuous map T : XX with Lipschitz constant μ. If S : XX is a G-limit of the sequence {S n }, then S is unique.

Proof. Let U U be an arbitrary entourage. Then, since B is base for U , there exists V (α, ε) B , α I, ε > 0 such that V (α, ε) U. Suppose that S : XX and S * : X 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 n X n such that for any α I
lim n ρ α ( x n , x ) = 0 and lim n ρ α ( S n x n , S x ) = 0 , (1) lim n ρ α ( y n , x ) = 0 and lim n ρ α ( S n y n , S * x ) = 0 . (2) (3)
Further, since S n is J-Lipschitz, for any α I, there exists a constant μ = μ(α) > 0 such that
ρ α ( S n x n , S n y n ) μ ρ α ( T n x n , T n y n ) .
Therefore, for any n and α I,
ρ α ( S x , S * x ) ρ α ( S x , S n x n ) + ρ α ( S n x n , S n y n ) + ρ α ( S n y n , S * x ) ρ α ( S x , S n x n ) + μ ρ α ( T x n , T y n ) + ρ α ( S n y n , S * x ) ρ α ( S x , S n x n ) + μ [ ρ α ( T x n , T x ) + ( T x , T y n ) ] + ρ α ( S n y n , S * x )

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, ρ α ( S x , S * x ) < ε for all nN (α, ε). Therefore ( S x , S * x ) V ( α , ε ) U and since X is Hausdorff, it follows that S x = S * 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 ( X , U ) be a Hausdorff uniform space, { X n } n ̄ a family of nonempty subsets of X and S n : X n X a k- contraction (resp. k- Lipschitz) mapping for each n ̄ . If S : XX is a (G) - limit of { S n } 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 ( X , U ) be a uniform space, { X n } n ̄ a family of nonempty subsets of X and { S n , T n : X n X } n - two families of maps each satisfying the property (G) and such that for all n ̄ , the pair (S n , T n ) is J-contraction with constant μ. If for all 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 U be arbitrary. Then, there exists V ( λ , ε ) B , λ I , ε > 0 such that V (λ, ε) W. Since z n is a common fixed point of S n and T n for each n ̄ , and the property (G) holds and z X, there exists a sequence {y n } such that y n X n (for all n ̄ ) such that for any λ I,
lim n ρ λ ( y n , z ) = 0 , lim n ρ λ ( S n y n , S z ) = 0 and lim n ρ λ ( T n y n , T z ) = 0 .
Using the fact that the pair (S n , T n ) is J-contraction, for any λ I, we have
ρ λ ( z n , z ) = ρ λ ( S n z n , S z ) ρ λ ( S n z n , S n y n ) + ρ λ ( S n y n , S z ) μ ( λ ) ρ λ ( T n z n , T n y n ) + ρ λ ( S n y n , S z ) μ ( λ ) ρ λ ( T n z n , T z ) + μ ( λ ) ρ λ ( T n y n , T z ) + ρ λ ( S n y n , S z ) .
This gives
ρ λ ( z n , z ) 1 1 - μ ( λ ) [ μ ( λ ) ρ λ ( T n y n , T z ) + ρ λ ( S n y n , S z ) ] .

Since μ(λ) < 1, it follows that ρ λ (z n , z) → 0 as n → ∞. Hence, ρ λ (z n , z) < ε for all nN (λ, ε) and so (z n , z) V (λ, ε) W and the conclusion follows.■

When for each 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 ( X , U ) be a Hausdorff uniform space, { X n } n ̄ a family of nonempty subsets of X and { S n : X n X } n ̄ a family of mappings satisfying the property (G) and S n is a k- contraction for each n ̄ . If x n is a fixed point of S n for each n ̄ , then the sequence {x n }nconverges to x.

Again, when X n = X, for all n ̄ , we obtain, as a consequence of Theorem 3.5, the following result.

Corollary 3.7. Let ( X , U ) be a uniform space and S n , T n : XX 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 ̄ . If the sequences {S n } and {T n } converge pointwise respectively to S, T : XX, 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 ( X , U ) be a uniform space, { X n } n ̄ a family of nonempty subsets of X and { S n , T n : X n X } n - two families of maps each satisfying the property (G) and such that for all n, the pair (S n , T n ) is J-contraction with constant {μ n }na bounded (resp. convergent) sequence. Then, the pair (S, T) is J-contraction with constant μ = supnμ 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 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 Sx, Sy, Tx, and Ty.

Therefore, for any nN and each α I,
ρ α ( S x , S y ) ρ α ( S x , S n x n ) + ρ α ( S n x n , S n y n ) + ρ α ( S n y n , S y ) ρ α ( S x , S n x n ) + μ n ρ α ( T n x n , T n y n ) + ρ α ( S n y n , S y ) .
Since
lim sup n μ n ρ α ( T n x n , T n y n ) μ ρ α ( T x , T y ) ,

the above inequality yields ρ α (Sx, Sy) ≤ μ ρ α (Tx, Ty) 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 ̄ . 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 ( X , U ) be a uniform space, { X n } n ̄ a family of nonempty subsets of X and { S n : X n X } n ̄ a family of mappings. Then,

S is called an (H) - limit of the sequence {S n }nin or, equivalently { S n } n ̄ satisfies the property (H) if the following condition holds:

(H) For all sequences {x n } in n X n , there exists a sequence {y n } in X such that for any α I,
lim n ρ α ( x n , y n ) = 0 a n d lim n ρ α ( S n x n , S n y n ) = 0 .

In case X is a metrizable uniform space (that is the uniformity 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 { S n : Y X X } 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 ( X , U ) be a uniform space, { X n } n ̄ a family of nonempty subsets of X. Let { S n , T n : X n X } 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 ̄ , 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 , Sy n ) → 0 and ρ α (T n z n , Ty n ) → 0 as n → ∞. Then
ρ α ( z n , z ) = ρ α ( S n z n , S z ) ρ α ( S n z n , S y n ) + ρ α ( S y n , S z ) ρ α ( S n z n , S y n ) + μ ρ α ( T y n , T z ) ρ α ( S n z n , S y n ) + μ [ ρ α ( T y n , T n z n ) + ρ α ( T n z n , T z ) ] .
So, we get
ρ α ( z n , z ) 1 ( 1 μ ) [ ρ α ( S n z n , S y n ) + μ ρ α ( T 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 ( X , U ) be a Hausdorff uniform space, { X n } n ̄ a family of nonempty subsets of X and { S n : X n X } n ̄ a family of mappings satisfying the property (H) and such that S is a k - contraction. If for any n ̄ , x n is a fixed point of T n , then {x n }nconverges to x.

Proof. It comes from Theorem 4.2 by taking T n to be the identity mapping for each 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 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

(1)
Department of Mathematics, Walter Sisulu University
(2)

References

  1. Acharya SP: Convergence of a sequence of fixed points in a uniform space. Mat Vesnik 1976,13(28):131–141.MathSciNetGoogle Scholar
  2. Angelov VG: A continuous dependence of fixed points of ϕ -contractive mappings in uniform spaces. Archivum Mathematicum (Brno) 1992,28(3–4):155–162.MathSciNetGoogle Scholar
  3. Angelov VG: Fixed Points in Uniform Spaces and Applications. Babeş-Bolyai University, Cluj University Press; 2009.Google Scholar
  4. Barbet L, Nachi K: Convergence des points fixes de k -contractions (convergence of fixed points of k- contractions). University of Pau 2006. preprintGoogle Scholar
  5. Barbet L, Nachi K: Sequences of contractions and convergence of fixed points. Monografias del Seminario Matemático Garcia de Galdeano 2006, 33: 51–58.MathSciNetGoogle Scholar
  6. Bonsall FF: Lectures on Some Fixed Point Theorems of Functional Analysis. Tata Institute of Fundamental Research, Bombay 1962.Google Scholar
  7. Mishra SN: On sequences of mappings and fixed points. Rend Sem Mat Univers Politecn Torino 1976, 34: 405–410.Google Scholar
  8. Mishra SN: On sequences of mappings and fixed points II. Indian J Pure Appl Math 1979, 10: 699–703.MathSciNetGoogle Scholar
  9. Mishra SN: A note on common fixed points of multivalued mappings in uniform spaces. Math Semin Notes 1981, 9: 341–347.Google Scholar
  10. Mishra SN, Kalinde AK: On certain stability results of Barbet and Nachi. Fixed Point Theory 2011,12(1):137–144.MathSciNetGoogle Scholar
  11. Nadler SB Jr: Sequences of contractions and fixed points. Pacific J Math 1968,27(3):579–585.MathSciNetView ArticleGoogle Scholar
  12. Rhoades BE: Fixed point theorems in a uniform space. Publ L'Institute Mathématique Nouvelle śerie 1979,25(39):153–156.MathSciNetGoogle Scholar
  13. Singh SL: A note on the convergence of a pair of sequences of mappings. Arch Math 1979,15(1):47–52.MathSciNetGoogle Scholar
  14. Singh SL, Mishra SN: Common fixed points and convergence theorems in uniform spaces. Mat Vesnik 1981,5;18(33):403–410.MathSciNetGoogle Scholar
  15. Singh SP: Sequence of mappings and fixed points. Annales Soc Sci Bruxelles 1969,83(2):197–201.Google Scholar
  16. Singh SP: On a theorem of Sonnenshein. Bull de l'Académie Royale de Belgique 1969, 3: 413–414.Google Scholar
  17. Singh SP, Russel W: A note on a sequence of contraction mappings. Can Math Bull 1969, 12: 513–516. 10.4153/CMB-1969-068-2View ArticleGoogle Scholar
  18. Sonnenshein J: Opérateurs de même coefficient de contraction. Bulletin de l' Académie Royale de Belgique 1966, 52: 1078–1082.Google Scholar
  19. Jungck G: Commuting mappings and fixed points. Amer Math Monthly 1976, 83: 261–263. 10.2307/2318216MathSciNetView ArticleGoogle Scholar
  20. Thron WJ: Topological Structures. Holt, Rinehart and Winston, New York; 1966.Google Scholar
  21. Kelley JL: General Topology. Springer, New York; 1955.Google Scholar
  22. Reilly IL: A generalized contraction principle. Bull Aust Math Soc 1974, 10: 359–363. 10.1017/S0004972700041046MathSciNetView ArticleGoogle Scholar
  23. Tarafdar E: An approach to fixed point theorems on uniform spaces. Trans Amer Math Soc 1974, 191: 209–225.MathSciNetView ArticleGoogle Scholar
  24. Acharya SP: Some results on fixed points in uniform spaces. Yokohama Math J 1974,22(1–2):105–116.MathSciNetGoogle Scholar
  25. Shaefer HH: Topological Vector Spaces. Macmillan, New York; 1966.Google Scholar

Copyright

© Mishra et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.