Open Access

Remarks on some recent fixed point theorems

Fixed Point Theory and Applications20122012:76

https://doi.org/10.1186/1687-1812-2012-76

Received: 1 February 2012

Accepted: 8 May 2012

Published: 8 May 2012

Abstract

The purpose of this article is to show that some recent fixed point theorems are particular results of previous existing theorems in the literature.

Mathematics Subject Classification 2000: 54H25; 47H10.

Keywords

fixed pointcoupled fixed pointordered setmetric spacecone metric space

1 Introduction

In this section, we recall some known results on fixed point theory.

We start with the well-known Banach contraction principle [1].

Theorem 1. Let (X, d) be a complete metric space and let F: X → X be a mapping such that for each pair of points x, y X,
d ( F x , F y ) k d ( x , y ) ,

where k is a constant in [0, 1). Then F has a unique fixed point.

In 2008, Dutta and Choudhury [2] obtained the following generalization of the Banach contraction principle.

Theorem 2. Let (X, d) be a complete metric space and let F: X → X be a mapping such that for each pair of points x, y X,
ψ ( d ( F x , F y ) ) ψ ( d ( x , y ) ) - φ ( d ( x , y ) ) ,

where ψ, φ: [0, ) [0, ) are continuous, non-decreasing and ψ -1({0}) = φ 1({0}) = {0}. Then F has a unique fixed.

Remark 3. Note that Theorem 2 remains true if φ satisfies only the following assumptions: φ is lower semi-continuous and φ-1({0}) = {0} (see, for example, Abbas and Dorić [3]and Dorić [4]).

Using the above remark, we have also

Theorem 4. Let (X, d) be a complete metric space and let F: X → X be a mapping such that for each pair of points x, y X,
ψ ( d ( F x , F y ) ) ψ ( d ( x , y ) ) - φ ( d ( x , y ) ) ,
(1)

where ψ: [0, ) [0, ) is continuous, non-decreasing, ψ -1({0}) = {0}, and φ: [0, ) [0, ) is lower semi-continuous and φ -1({0}) = {0}. Then F has a unique fixed.

We have also an ordered version of Theorem 4 (see [35]).

Theorem 5. Let (X,) be a partially ordered set and suppose that there is a metric d on X such that (X, d) is a complete metric space. Let F: X → X be a continuous non-decreasing mapping such that
ψ ( d ( F x , F y ) ) ψ ( d ( x , y ) ) - φ ( d ( x , y ) ) ,
(2)

for all x, y X with x y, where ψ: [0, ) [0, ) is continuous, non-decreasing, ψ -1({0}) = {0}, and φ: [0, ) [0, ) is lower semi-continuous and φ -1({0}) = {0}. If there exists x0 X such that x0 Fx0, then F has a fixed point.

The following result was obtained by Olaleru [6].

Theorem 6. Let (X, d) be a cone metric space with a cone P having non-empty interior. Let f, g: X → X be mappings such that
d ( f x , f u ) α 1 d ( f x , g x ) + α 2 d ( f u , g u ) + α 3 d ( f u , g x ) + α 4 d ( f x , g u ) + α 5 d ( g x , g u )
(3)

for all x, u X, where α1, α2, α3, α4, α5 [0, 1) and α1 + α2 + α3 + α4 + α5 < 1. Suppose that f (X) g(X) and g(X) is a complete subspace of X. Then f and g have a coincidence point. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.

The purpose of this article is to show that some recent fixed point theorems are particular cases of the above mentioned results. This article can be considered as a continuation of the recent work of Haghi et al. [7].

2 Main results

Beiranvand et al. [8] introduced a new class of mappings T: X → X as follows.

Definition 7. The mapping T: X → X is said to be sequentially convergent, if the sequence {y n } in X is convergent whenever {T y n } is convergent.

In the same article, the authors established the following result.

Theorem 8. Let (X, d) be a complete metric space and T, f: X → X be two mappings satisfying
d ( T f x , T f y ) k d ( T x , T y ) ,
(4)

for all x, y X, where k is a constant in [0, 1) and T is continuous, injective and sequentially convergent. Then f has a unique fixed point.

Theorem 8 has attracted the attention of many authors, see, for example, [915], where extensions and generalizations of Theorem 8 were considered.

We shall prove the following result.

Theorem 9. Theorem 1 and Theorem 8 are equivalent.

Proof. Clearly, if T is the identity mapping, Theorem 8 reduces to Theorem 1. Now, we shall prove that Theorem 8 can be deduced for the Banach contraction principle. Define the mapping δ: X × X → [0, ) by
δ ( x , y ) = d ( T x , T y ) , x , y X .
For all x, y, z X, we have δ(x, y) = δ(y, x), δ(x, y) ≤ δ(x, y) + δ(y, z) and
δ ( x , y ) = 0 d ( T x , T y ) = 0 T x = T y x = y ( since  T  is injective ) .
Then δ is a metric on X. Moreover, (X, δ) is a complete metric space. Indeed, let {x n } be a Cauchy sequence in (X, δ). From the definition of δ, this implies that {T x n } is a Cauchy sequence in (X, d). Since (X, d) is complete, there exists y X such that d(T x n , y) 0 as n → ∞. But T is sequentially convergent, then there exists x X, such that d(x n , x) 0 as n → ∞. Since T is continuous, this implies that d(T x n , Tx) 0 as n → ∞, that is, δ(x n , x) 0 as n → ∞. This proves that (X, δ) is complete. Now, condition (4) reduces to
δ ( f x , f y ) k δ ( x , y )

for all x, y X. Thus Theorem 8 follows immediately from the Banach contraction principle (Theorem 1). □

Recently, Eslamian and Abkar [16] etablished the following result.

Theorem 10. Let (X, d) be a complete metric space and f: XX be such that
ψ ( d ( f x , f y ) ) α ( d ( x , y ) ) - β ( d ( x , y ) ) ,
(5)
for all x, y X, where ψ, α, β: [0, ) → [0, ) are such that ψ is continuous and non-decreasing, α is continuous, β is lower semi-continuous,
ψ t = 0 i f a n d o n l y i f t = 0 , α 0 = β 0 = 0 ,
(6)
a n d ψ t - α t + β t > 0 f o r a l l t > 0 .
(7)

Then f has a unique fixed point.

We shall prove the following result.

Theorem 11. Theorem 10 and Theorem 4 are equivalent.

Proof. Taking α = ψ in Theorem 10, we obtain immediately Theorem 4. Now, we shall prove that Theorem 10 can be deduced from Theorem 4. Indeed, let f: X → X be a mapping satisfying (5) with ψ, α, β: [0, ) [0, ) satisfy conditions (6) and (7). From (5), for all x, y X, we have
ψ d f x , f y α d x , y - β d x , y = ψ d x , y - β d x , y - α d x , y + ψ d x , y .
Define θ: [0, ) [0, ) by
θ t = β t - α t + ψ t , t 0 .
Then, we have
ψ d f x , f y ψ d x , y - θ d x , y ,

for all x, y X. Clearly, from (6) and (7), θ is lower semi-continuous and θ 1({0}) = {0}. Now, Theorem 10 follows immediately from Theorem 4. □

Binayak et al. [17] extended Theorem 10 to the ordered case.

Theorem 12. Let (X, ) be a partially ordered set and suppose that there is a metric d on X such that (X, d) is a complete metric space. Let f: X → X be a continuous non-decreasing mapping such that
ψ ( d ( f x , f y ) ) α ( d ( x , y ) ) - β ( d ( x , y ) ) ,
for all x, y X with x y, where ψ, α, β: [0, ) [0, ) are such that ψ is continuous and non-decreasing, α is continuous, β is lower semi-continuous,
ψ t = 0 i f a n d o n l y i f t = 0 , α 0 = β 0 = 0 ,
a n d ψ t - α t + β t > 0 f o r a l l t > 0 .

If there exists x0 X such that x0 fx0, then f has a fixed point.

Following similar arguments as in the proof of Theorem 11, we obtain

Theorem 13. Theorem 5 and Theorem 12 are equivalent.

Abbas et al. [18] introduced the concept of w-compatibility for a pair of mappings F: X × XX and g: XX.

Definition 14. The mappings F: X × XX and g: XX are called w-compatible if g(F (x, y)) = F(gx, gy) whenever gx = F(x, y) and gy = F(y, x).

In the same article, the authors established the following result.

Theorem 15. Let (X, d) be a cone metric space with a cone P having non-empty interior, F: X × XX and g: XX be mappings satisfying
d ( F ( x , y ) , F ( u , v ) ) a 1 d ( g x , g u ) + a 2 d ( F ( x , y ) , g x ) + a 3 d ( g y , g u )
(8)
+ a 4 d F ( u , v ) , g u + α 5 d F ( x , y ) , g u + a 6 d F ( u , v ) , g x ,
(9)

for all x, y, u, v X, where a i , i = 1, 2,..., 6 are nonnegative real numbers such that i = 1 6 a i < 1 . If F (X × X) g(X) and g(X) is complete subset of X, then F and g have a coupled coincidence point in X, that is, there exists (x, y) X × X such that gx = F(x, y) and gy = F(y, x). Moreover, if F and g are w-compatible, then there exists a unique u X such that u = gu = F(u, u).

We shall prove the following result.

Theorem 16. Theorem 6 and Theorem 15 are equivalent.

Proof. (i) Theorem 15 Theorem 6. Let f, g: XX be mappings satisfying the hypotheses of Theorem 6. Define the mapping F: X × XX by
F x , y = f x , x , y X .
From (3), we get
d F x , y , F u , v < α 1 d F x , y , g x + α 2 d F u , v , g u + α 3 d F u , v , g x + α 4 d F x , y , g u + α 5 d g x , g u ,
for all x, y, u, v X. Then condition (8) is satisfied with (a1, a2, a3, a4, a5, a6) = (α5, α1, 0, α2, α4, α3). On the other hand, from the definition of F, we have F(X × X) = f(X) g(X). Also, g(X) is a complete subspace of (X, d). Now, applying Theorem 15, we obtain that F and g have a coupled coincidence point in X, that is, there exists (x, y) X × X such that gx = F (x, y) and gy = F(y, x). From the definition of F, this implies that gx = fx, that is, x is a coincidence point of f and g. Suppose now that f and g are weakly compatible. Let x, y X such that gx = F (x, y). This implies that gx = fx. Since f and g are weakly compatible, we get fgx = gfx, that is, F(gx, gy) = g(F(x, y)). This implies that F and g are w-compatible. From Theorem 15, there exists a unique u X such that u = gu = F (u, u), that is, there exists a unique u X such that u = gu = fu. Then f and g have a unique common fixed point. Thus we proved Theorem 6.
  1. (ii)

    Theorem 6 Theorem 15.

     
Let F: X × XX and g: XX be mappings satisfying the hypotheses of Theorem 15. Define the mapping f: XX by
f x = F x , x , x X .
From (8), we have
d f x , f u α 1 d f x , g x + α 2 d f u , g u + α 3 d f u , g x + α 4 d f x , g u + α 5 d g x , g u ,

for all x, u X, where (α1, α2, α3, α4, α5) = (a2, a4, a6, a5, a1 + a3). Then condition (3) of Theorem 6 is satisfied. On the other hand, we have f(X) F(X × X) g(X) and g(X) is a complete subspace of (X, d). Applying Theorem 6, we obtain that there exists x X (a coincidence point) such that fx = gx, that is, F(x, x) = gx. Moreover, if F and g are w-compatible, then f and g are weakly compatible. Applying again Theorem 6, we obtain that f and g have a unique common fixed point, that is, there exists a unique u X such that u = gu = fu = F (u, u). Thus we proved Theorem 15. □

Declarations

Acknowledgements

The authors thank the referees for their helpful remarks and suggestions which improved the final presentation of this article.

Authors’ Affiliations

(1)
Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse
(2)
Department of Mathematics, Atılım University
(3)
Department of Mathematics, King Saud University

References

  1. Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund Math 1922, 3: 133–181.Google Scholar
  2. Dutta PN, Choudhury BS: A generalization of contraction principle in metric spaces. Fixed Point Theory Appl 2008, 2008: 8. (Article ID 406368)MathSciNetView ArticleGoogle Scholar
  3. Abbas M, Dorić D: Common fixed point theorem for four mappings satisfying generalized weak contractive condition. Filomat 2010, 24(2):1–10. 10.2298/FIL1002001AMathSciNetView ArticleGoogle Scholar
  4. Dorić D: Common fixed point for generalized ( ψ, ϕ )-weak contractions. Appl Math Lett 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001MathSciNetView ArticleGoogle Scholar
  5. Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003MathSciNetView ArticleGoogle Scholar
  6. Olaleru JO: Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory Appl 2009, 2009: 10. (Article ID 6579140)MathSciNetView ArticleGoogle Scholar
  7. Haghi RH, Rezapour Sh, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052MathSciNetView ArticleGoogle Scholar
  8. Beiranvand A, Moradi S, Omid M, Pazandeh H: Two fixed point theorems for special mappings. arxiv:0903.1504v1 math FA 2009.Google Scholar
  9. Satisfying a contractive condition depended on an another function Lobachevskii J Math 2009, 30(4):289–291. 10.1134/S1995080209040076Google Scholar
  10. Chi KP, Thuy HT: A fixed point theorem in 2-metric spaces for a class of maps that satisfy a contractive condition dependent on an another function. Lobachevskii J Math 2010, 31(4):338–346. 10.1134/S1995080210040050MathSciNetView ArticleGoogle Scholar
  11. Luong NV, Thuan NX, Hai TT: Coupled fixed point theorems in partially ordered metric spaces depended on another function. Bull Math Anal Appl 2011, 3(3):129–140.MathSciNetGoogle Scholar
  12. Moradi S, Alimmohammadi D: New extensions of Kannan fixed-point theorem on complete metric and generalized metric spaces. Int J Math Anal 2011, 5(47):2313–2320.Google Scholar
  13. Moradi S, Omid M: A fixed-point theorem for integral type inequality depending on another function. Int J Math Anal 2010, 4(29–32):1491–1499.MathSciNetGoogle Scholar
  14. Morales JR, Rojas E: Cone metric spaces and fixed point theorems of T-Kannan contractive mappings. Int J Math Anal 2010, 4(4):175–184.MathSciNetGoogle Scholar
  15. Sumitra R, Uthariaraj VR, Hemavathy R: Common fixed point theorem for T -Hardy-Rogers contraction mapping in a cone metric space. Int Math Forum 2010, 5(30):1495–1506.MathSciNetGoogle Scholar
  16. Eslamian M, Abkar A: A fixed point theorems for generalized weakly contractive mappings in complete metric space. Ital J Pure Appl Math, in press.Google Scholar
  17. Choudhury BS, Kundu A: ( ψ, α, β )-weak contractions in partially ordered metric spaces. Appl Math Lett 2012, 25: 6–10. 10.1016/j.aml.2011.06.028MathSciNetView ArticleGoogle Scholar
  18. Abbas M, Ali Khan M, Radenović S: Common coupled fixed point theorems in cone metric spaces for w-compatible mappings. Appl Math Comput 2010, 217(1):195–202. 10.1016/j.amc.2010.05.042MathSciNetView ArticleGoogle Scholar

Copyright

© Aydi et al; licensee Springer. 2012

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