Open Access

Fixed point theorems for contraction mappings in modular metric spaces

  • Chirasak Mongkolkeha1,
  • Wutiphol Sintunavarat1 and
  • Poom Kumam1Email author
Fixed Point Theory and Applications20112011:93

https://doi.org/10.1186/1687-1812-2011-93

Received: 20 June 2011

Accepted: 2 December 2011

Published: 2 December 2011

The Erratum to this article has been published in Fixed Point Theory and Applications 2012 2012:103

Abstract

In this article, we study and prove the new existence theorems of fixed points for contraction mappings in modular metric spaces.

AMS: 47H09; 47H10.

Keywords

modular metric spacesmodular spacescontraction mappingsfixed points

1 Introduction

Let (X, d) be a metric space. A mapping T : XX is a contraction if
d ( T ( x ) , T ( y ) ) k d ( x , y ) ,
(1.1)

for all x, y X, where 0 ≤ k < 1. The Banach Contraction Mapping Principle appeared in explicit form in Banach's thesis in 1922 [1]. Since its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions, see [210]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [1113] and others. Further and the most complete development of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [1418] and their collaborators. A lot of mathematicians are interested fixed points of Modular spaces, for example [4, 1926].

In 2008, Chistyakov [27] introduced the notion of modular metric spaces generated by F-modular and develop the theory of this spaces, on the same idea he was defined the notion of a modular on an arbitrary set and develop the theory of metric spaces generated by modular such that called the modular metric spaces in 2010 [28].

In this article, we study and prove the existence of fixed point theorems for contraction mappings in modular metric spaces.

2 Preliminaries

We will start with a brief recollection of basic concepts and facts in modular spaces and modular metric spaces (see [14, 15, 2729] for more details).

Definition 2.1. Let X be a vector space over (or ). A functional ρ : X → [0, ∞] is called a modular if for arbitrary x and y, elements of X satisfies the following three conditions :

(A.1) ρ(x) = 0 if and only if x = 0;

(A.2) ρ(αx) = ρ(x) for all scalar α with |α| = 1;

(A.3) ρ(αx + βy) ≤ ρ(x) + ρ(y), whenever α, β ≥ 0 and α + β = 1.

If we replace (A.3) by

(A.4) ρ(αx + βy) ≤ α s ρ(x) + β s ρ(y), for α, β ≥ 0, α s + β s = 1 with an s (0, 1], then the modular ρ is called s-convex modular, and if s = 1, ρ is called a convex modular.

If ρ is modular in X, then the set defined by
X ρ = { x X : ρ ( λ x ) 0 as λ 0 + }
(2.1)
is called a modular space. X ρ is a vector subspace of X it can be equipped with an F-norm defined by setting
x ρ = inf { λ > 0 : ρ x λ λ } , x X ρ .
(2.2)
In addition, if ρ is convex, then the modular space X ρ coincides with
X ρ * = { x X : λ = λ ( x ) > 0 such that ρ ( λ x ) < }
(2.3)

and the functional x ρ * = inf { λ > 0 : ρ x λ 1 } is an ordinary norm on X ρ * which is equivalence to x ρ (see [16]).

Let X be a nonempty set, λ (0, ∞) and due to the disparity of the arguments, function w : (0, ∞) × X × X → [0, ∞] will be written as wλ(x, y) = w(λ, x, y) for all λ > 0 and x, y X.

Definition 2.2. [[28], Definition 2.1] Let X be a nonempty set. A function w : (0, ∞) × X × X → [0, ∞] is said to be a metric modular on X if satisfying, for all x, y, z X the following condition holds:

(i) w λ (x, y) = 0 for all λ > 0 if and only if x = y;

(ii) w λ (x, y) = w λ (y, x) for all λ > 0;

(iii) wλ + μ(x, y) ≤ w λ (x, z) + w μ (z, y) for all λ, μ > 0.

If instead of (i), we have only the condition

(i') w λ (x, x) = 0 for all λ > 0, then w is said to be a (metric) pseudomodular on X.

The main property of a (pseudo) modular w on a set X is a following: given x, y X, the function 0 < λ w λ (x, y) [0, ∞] is a nonincreasing on (0, ∞).

In fact, if 0 < μ < λ, then (iii), (i') and (ii) imply
w λ ( x , y ) w λ - μ ( x , x ) + w μ ( x , y ) = w μ ( x , y ) .
(2.4)
It follows that at each point λ > 0 the right limit w λ + 0 ( x , y ) : = lim ε + 0 w λ + ε ( x , y ) and the left limit w λ - 0 ( x , y ) : = lim ε + 0 w λ - ε ( x , y ) exists in [0, ∞] and the following two inequalities hold :
w λ + 0 ( x , y ) w λ ( x , y ) w λ - 0 ( x , y ) .
(2.5)

Definition 2.3. [[28], Definition 3.3] A function w : (0, ∞) × X × X → [0, ∞] is said to be a convex (metric) modular on X if it is satisfies the conditions (i) and (ii) from Definition 2.2 as well as this condition holds;

(iv) w λ + μ ( x , y ) = λ λ + μ w λ ( x , z ) + μ λ + μ w μ ( z , y ) f o r a l l λ , μ > 0 a n d x , y , z X .

If instead of (i), we have only the condition (i') from Definition 2.2, then w is called a convex(metric) pseudomodular on X.

From [27, 28], we know that, if x0 X, the set X w = { x X : lim λ w λ ( x , x 0 ) = 0 } is a metric space, called a modular space, whose metric is given by d w ( x , y ) = inf { λ > 0 : w λ ( x , y ) λ } for all x, y X w . Moreover, if w is convex, the modular set X w is equal to X w * = { x X : λ = λ ( x ) > 0 such that w λ (x, x0) <∞} and metrizable by d w * ( x , y ) = inf { λ > 0 : w λ ( x , y ) 1 } for all x , y X w * . We know that (see [[28], Theorem 3.11]) if X is a real linear space, ρ : X → [0, ∞] and
w λ ( x , y ) = ρ x - y λ for all  λ > 0 and  x , y X ,
(2.6)
then ρ is modular (convex modular) on X in the sense of (A.1)-(A.4) if and only if w is metric modular (convex metric modular, respectively) on X. On the other hand, if w satisfy the following two conditions (i) w λ (μx, 0) = w λ/μ (x, 0) for all λ, μ > 0 and x X, (ii) w λ (x + z, y + z) = w λ (x, y) for all λ > 0 and x, y, z X, if we set ρ(x) = w1(x, 0) with (2.6) holds, where x X, then
  1. (i)

    X ρ = X w is a linear subspace of X and the functional x ρ = d w ( x , 0 ) , x X ρ , is an F-norm on X ρ ;

     
  2. (ii)

    if w is convex, X ρ * X w * ( 0 ) = X ρ is a linear subspace of X and the functional x ρ = d w * ( x , 0 ) , x X ρ * , is an norm on X ρ * .

     

Similar assertions hold if replace the word modular by pseudomodular. If w is metric modular in X, we called the set X w is modular metric space.

By the idea of property in metric spaces and modular spaces, we defined the following:

Definition 2.4. Let X w be a modular metric space.

(1) The sequence (x n )n in X w is said to be convergent to x X w if w λ (x n , x) → 0, as n →for all λ > 0.

(2) The sequence (x n ) n in X w is said to be Cauchy if w λ ( x m , x n ) → 0, as m, n →for all λ > 0.

(3) A subset C of X w is said to be closed if the limit of a convergent sequence of C always belong to C.

(4) A subset C of X w is said to be complete if any Cauchy sequence in C is a convergent sequence and its limit is in C.

(5) A subset C of X w is said to be bounded if for all λ > 0 δ w (C) = sup{w λ (x, y); x, y C} <∞.

3 Main results

In this section, we prove the existence of fixed points theorems for contraction mapping in modular metric spaces.

Definition 3.1. Let w be a metric modular on X and X w be a modular metric space induced by w and T : X w → X w be an arbitrary mapping. A mapping T is called a contraction if for each x, y X w and for all λ > 0 there exists 0 ≤ k < 1 such that
w λ ( T x , T y ) k w λ ( x , y ) .
(3.1)

Theorem 3.2. Let w be a metric modular on X and X w be a modular metric space induced by w. If X w is a complete modular metric space and T : X w → X w is a contraction mapping, then T has a unique fixed point in X w . Moreover, for any x X w , iterative sequence {T n x} converges to the fixed point.

Proof. Let x0 ba an arbitrary point in X w and we write x1 = Tx0, x2 = Tx1 = T2x0, and in general, x n = Txn-1= T n x0 for all n . Then,
w λ ( x n + 1 , x n ) = w λ ( T x n , T x n - 1 ) k w λ ( x n , x n - 1 ) = k w λ ( T x n - 1 , T x n - 2 ) k 2 w λ ( x n - 1 , x n - 2 ) k n w λ ( x 1 , x 0 )
for all λ > 0 and for each n . Therefore, lim n w λ ( x n + 1 , x n ) = 0 for all λ > 0. So for each λ > 0, we have for all > 0 there exists n0 such that w λ (x n , xn+1) < for all n with n ≥ n0. Without loss of generality, suppose m, n and m > n. Observe that, for λ m - n > 0 , there exists nλ/(m-n) such that
w λ m - n ( x n , x n + 1 ) < ε m - n
for all n ≥ nλ/(m-n). Now, we have
w λ ( x n , x m ) w λ m - n ( x n , x n + 1 ) + w λ m - n ( x n + 1 , x n + 2 ) + + w λ m - n ( x m - 1 , x m ) < ε m - n + ε m - n + + ε m - n = ε

for all m, n ≥ nλ/(m-n). This implies {x n }nis a Cauchy sequence. By the completeness of X w , there exists a point x X w such that x n → × as n → ∞.

By the notion of metric modular w and the contraction of T, we get
w λ ( T x , x ) w λ 2 ( T x , T x n ) + w λ 2 ( T x n , x ) k w λ 2 ( x , x n ) + w λ 2 ( x n + 1 , x )
(3.2)
for all λ > 0 and for each n . Taking n → ∞ in (3.2) implies that w λ (Tx, x) = 0 for all λ > 0 and thus Tx = x. Hence, x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z is another fixed point of T. We see that
w λ ( x , z ) = w λ ( T x , T z ) k w λ ( x , z )

for all λ > 0. Since 0 ≤ k < 1, we get w λ (x, z) = 0 for all λ > 0 this implies that x = z. Therefore, x is a unique fixed point of T and the proof is complete.   □

Theorem 3.3. Let w be a metric modular on X and X w be a modular metric space induced by w. If X w is a complete modular metric space and T : X w → X w is a contraction mapping. Suppose x* X w is a fixed point of T, {ε n } is a sequence of positive numbers for which lim n ε n = 0 , and {y n } X w satisfies
w λ ( y n + 1 , T y n ) ε n

for all λ > 0. Then, lim n y n = x * .

Proof. For each m , we observe that
w λ ( T m + 1 x , y m + 1 ) = w λ m m ( T m + 1 x , y m + 1 ) w λ ( m - 1 ) m ( T m + 1 x , T y m ) + w λ m ( T y m , y m + 1 ) k w λ ( m - 1 ) m ( T m x , y m ) + ε m k w λ ( m - 2 ) m ( T m x , T y m - 1 ) + k w λ m ( T y m - 1 x , y m ) + ε m k 2 w λ ( m - 2 ) m ( T m - 1 x , y m - 1 ) + k ε m - 1 + ε m i = 0 m k m - i ε i
(3.3)
for all λ > 0. Thus, we get
w λ ( y m + 1 , x * ) w λ 2 ( y m + 1 , T m + 1 x ) + w λ 2 ( T m + 1 x , x * ) i = 0 m k m - i ε i + w λ 2 ( T m + 1 x , x * ) .
(3.4)

Next, we claimed that lim m w λ ( y m + 1 , x * ) = 0 for all λ > 0.

Now let ε > 0. Since lim n ε n = 0 , there exists N such that for m ≥ N, ε m ε. Thus,
i = 0 m k m - i ε i = i = 0 N k m - i ε i + i = N + 1 m k m - i ε i k m - N i = 0 N k N - i ε i + ε i = N + 1 m k m - i .
(3.5)
Taking limit as m → ∞ in (3.5), we have
lim m i = 0 m k m - i ε i = 0 .
(3.6)
Since x0 is a fixed point of T and using result of Theorem 3.2, we get the sequence {T n x} converge to x*. This implies that
lim m w λ 2 ( T m + 1 x , x * ) = 0
(3.7)
for all λ > 0. From (3.4), (3.6) and (3.7), we have
lim m w λ ( y m + 1 , x * ) = 0
(3.8)

for all λ > 0 which implies that lim n y n = x * .   □

Theorem 3.4. Let w be a metric modular on X and X w be a modular metric space induced by w. If X w is a complete modular metric space and T : X w X w is a mapping, which T N is a contraction mapping for some positive integer N. Then, T has a unique fixed point in X w .

Proof. By Theorem 3.2 , T N has a unique fixed point u X w . From T N (T u ) = TN+1u = T(T N u) = Tu, so Tu is a fixed point of T N . By the uniqueness of fixed point of T N , we have Tu = u. Thus, u is a fixed point of T. Since fixed point of T is also fixed point of T N , we can conclude that T has a unique fixed point in X w .   □

Theorem 3.5. Let w be metric modular on X, X w be a complete modular metric space induced by w and for x* X w we define
B w ( x * , γ ) : = { x X w | w λ ( x , x * ) γ f o r a l l λ > 0 } .
If T : B w (x*, γ) → X w is a contraction mapping with
w λ 2 ( T x * , x * ) ( 1 - k ) γ
(3.9)

for all λ > 0, where 0 ≤ k < 1. Then, T has a unique fixed point in B w (x*, γ).

Proof. By Theorem 3.2 , we only prove that B w (x*, γ) is complete and Tx B w (x*, γ), for all x B w (x*, γ). Suppose that {x n } is a Cauchy sequence in B w (x*, γ), also {x n } is a Cauchy sequence in X w . Since X w is complete, there exists x X w such that
lim n w λ 2 ( x n , x ) = 0
(3.10)
for all λ > 0. Since for each n , x n B w (x*, γ), using the property of metric modular, we get
w λ ( x * , x ) w λ 2 ( x * , x n ) + w λ 2 ( x n , x ) γ + w λ 2 ( x n , x * )
(3.11)

for all λ > 0. It follows the inequalities (3.10) and (3.11), we have w λ (x*, x) ≤ γ which implies that x B w (x*, γ). Therefore, {x n } is convergent sequence in B w (x*, γ) and also B w (x*, γ) is complete.

Next, we prove that Tx B w (x*, γ) for all x B w (x*, γ). Let x B w (x*, γ). From the inequalities (3.9), using the contraction of T and the notion of metric modular, we have
w λ ( x * , T x ) w λ 2 ( x * , T x * ) + w λ 2 ( T x * , T x ) ( 1 - k ) γ + k w λ 2 ( x * , x ) ( 1 - k ) γ + k γ = γ .

Therefore, Tx B w (x*, γ) and the proof is complete.

Theorem 3.6. Let w be a metric modular on X, X w be a complete modular metric space induced by w and T : X w X w . If
w λ ( T x , T y ) k ( w 2 λ ( T x , x ) + w 2 λ ( T y , y ) )
(3.12)

for all x, y X w and for all λ > 0, where k [ 0 , 1 2 ) , then T has a unique fixed point in X w . Moreover, for any x X w , iterative sequence {T n x} converges to the fixed point.

Proof. Let x0 be an arbitrary point in X w and we write x1 = Tx0, x2 = Tx1 = T2x0, and in general, x n = Txn-1= T n x0 for all n . If T x n 0 - 1 = T x n 0 for some n0 , then T x n 0 = x n 0 . Thus, x n 0 is a fixed point of T. Suppose that Txn-1Tx n for all n . For k [ 0 , 1 2 ) , we have
w λ ( x n + 1 , x n ) = w λ ( T x n , T x n - 1 ) k ( w 2 λ ( T x n , x n ) + w 2 λ ( T x n - 1 , x n - 1 ) ) k ( w λ ( x n + 1 , x n ) + w λ ( x n , x n - 1 ) )
(3.13)
for all λ > 0 and for all n . Hence,
w λ ( x n + 1 , x n ) k 1 - k w λ ( x n , x n - 1 )
(3.14)
for all λ > 0 and for all n . Put β : = k 1 - k , since k [ 0 , 1 2 ) , we get β [0, 1) and hence
w λ ( x n + 1 , x n ) β w λ ( x n , x n - 1 ) β 2 w λ ( x n - 1 , x n - 2 ) β n w λ ( x 1 , x 0 )
(3.15)
for all λ > 0 and for all n . Similar to the proof of Theorem 3.2, we can conclude that {x n } is a Cauchy sequence and by the completeness of X w there exists a point x X w such that x n x as n → ∞. By the property of metric modular and the inequality (3.12), we have
w λ ( T x , x ) w λ 2 ( T x , T x n ) + w λ 2 ( T x n , x ) k ( w λ ( T x , x ) + w λ ( T x n , x n ) ) + w λ 2 ( T x n , x ) k ( w λ ( T x , x ) + w λ 2 ( T x n , x ) + w λ 2 ( x , x n ) ) + w λ 2 ( T x n , x ) = k ( w λ ( T x , x ) + w λ 2 ( x n + 1 , x ) + w λ 2 ( x , x n ) ) + w λ 2 ( x n + 1 , x )
(3.16)
for all λ > 0 and for all n . Taking n → ∞ in the inequality (3.16), we obtained that
w λ ( T x , x ) k w λ ( T x , x ) .
(3.17)
Since k [ 0 , 1 2 ) , we have Tx = x. Thus, x is a fixed point of T. Next, we prove that x is a unique fixed point. Suppose that z be another fixed point of T. We note that
w λ ( x , z ) = w λ ( T x , T z ) k ( w λ 2 ( T x , x ) + w λ 2 ( T z , z ) ) = 0

for all λ > 0, which implies that x = z. Therefore, x is a unique fixed point of T.   □

Now, we shall give a validate example of Theorem 3.2 .

Example 3.7. Let X = {(a, 0) 2|0 ≤ a ≤ 1} {(0, b) 2|0 ≤ b ≤ 1}.

Defined the mapping w : (0, ∞) × X × X → [0, ∞] by
w λ ( ( a 1 , 0 ) , ( a 2 , 0 ) ) = 4 | a 1 - a 2 | 3 λ ,
w λ ( ( 0 , b 1 ) , ( 0 , b 2 ) ) = | b 1 - b 2 | λ ,
and
w λ ( ( a , 0 ) , ( 0 , b ) ) = 4 a 3 λ + b λ = w λ ( ( 0 , b ) , ( a , 0 ) ) .
We note that if we take λ → ∞, then we see that X = X w and also X w is a complete modular metric space. We let a mapping T : X w X w is define by
T ( ( a , 0 ) ) = ( 0 , a )
and
T ( ( 0 , b ) ) = b 2 , 0 .
Simple computations show that
w λ ( T ( ( a 1 , b 1 ) ) , T ( ( a 2 , b 2 ) ) ) 3 4 w λ ( ( a 1 , b 1 ) , ( a 2 , b 2 ) )

for all (a1, b1), (a2, b2) X w . Thus, T is a contraction mapping with constant k = 3 4 . Therefore, T has a unique fixed point that is (0, 0) X w .

On the Euclidean metric d on X w , we see that
d ( T ( ( 0 , 0 ) ) , T ( ( 1 , 0 ) ) ) = d ( ( 0 , 0 ) , ( 0 , 1 ) ) = 1 > k = k d ( ( 0 , 0 ) , ( 1 , 0 ) )

for all k [0, 1). Thus, T is not a contraction mapping and then the Banach contraction mapping cannot be applied to this example.

Notes

Declarations

Acknowledgements

The authors thank the referee for comments and suggestions on this manuscript. The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0029/2553). The second author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) and the Faculty of Science, KMUTT for financial support during the preparation of this manuscript for the Ph.D. Program. The third author was supported by the Commission on Higher Education and the Thailand Research Fund (Grant No.MRG5380044). Moreover, this study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC Project No. 54000267).

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT)

References

  1. Banach S: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fund Math 1922, 3: 133–181.Google Scholar
  2. Jungck G: Compatible mappings and common fixed points. Int J Math Math Sci 1986, 9: 771–779. 10.1155/S0161171286000935MathSciNetView ArticleGoogle Scholar
  3. Jungck G, Rhoades BE: Fixed points for set valued functions without continuity. Indian J Pure Appl Math 1998, 29: 227–238.MathSciNetGoogle Scholar
  4. Razani A, Nabizadeh E, Beyg Mohamadi M, Homaeipour S: Fixed point of nonlinear and asymptotic contractions in the modular space. Abstr Appl Anal 2007., 2007: Article ID 40575, 10Google Scholar
  5. Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleGoogle Scholar
  6. Sintunavarat W, Kumam P: Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition. Appl Math Lett 2009, 22: 1877–1881. 10.1016/j.aml.2009.07.015MathSciNetView ArticleGoogle Scholar
  7. Sintunavarat W, Kumam P: Weak condition for generalized multi-valued ( f , α , β )-weak contraction mappings. Appl Math Lett 2011, 24: 460–465. 10.1016/j.aml.2010.10.042MathSciNetView ArticleGoogle Scholar
  8. Sintunavarat W, Kumam P: Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J Inequal Appl 2011, 2011: 3. 10.1186/1029-242X-2011-3View ArticleGoogle Scholar
  9. Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput Math Appl 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040MathSciNetView ArticleGoogle Scholar
  10. Sintunavarat W, Kumam P:Common fixed point theorems for generalized J H -operator classes and invariant approximations. J Inequal Appl 2011, 2011: 67. 10.1186/1029-242X-2011-67View ArticleGoogle Scholar
  11. Nakano H: Modulared Semi-Ordered Linear Spaces. In Tokyo Math Book Ser. Volume 1. Maruzen Co., Tokyo; 1950.Google Scholar
  12. Koshi S, Shimogaki T: On F-norms of quasi-modular spaces. J Fac Sci Hokkaido Univ Ser I 1961,15(3–4):202–218.MathSciNetGoogle Scholar
  13. Yamamuro S: On conjugate spaces of Nakano spaces. Trans Amer Math Soc 1959, 90: 291–311. 10.1090/S0002-9947-1959-0132378-1MathSciNetView ArticleGoogle Scholar
  14. Luxemburg WAJ: Banach function spaces. Thesis, Delft, Inst of Techn Assen, The Netherlands 1955.Google Scholar
  15. Mosielak J, Orlicz W: On modular spaces. Studia Math 1959, 18: 49–65.MathSciNetGoogle Scholar
  16. Musielak J, Orlicz W: Some remarks on modular spaces. Bull Acad Polon Sci Sr Sci Math Astron Phys 1959, 7: 661–668.MathSciNetGoogle Scholar
  17. Mazur S, Orlicz W: On some classes of linear spaces. Studia Math 1958, 17: 97–119. Reprinted in [21]: 981–1003MathSciNetGoogle Scholar
  18. Turpin Ph: Fubini inequalities and bounded multiplier property in generalized modular spaces. Comment. Math Tomus specialis in honorem Ladislai Orlicz I 1978, 331–353.Google Scholar
  19. Beygmohammadi M, Razani A: Two fixed point theorems for mappings satisfying a general contractive condition of integral type in the modular spaces. Int J Math Math Sci 2010, 2010: Article ID 317107, 10.MathSciNetView ArticleGoogle Scholar
  20. Dominguez-Benavides T, Khamsi MA, Samadi S: Uniformly Lipschitzian mappings in modular function spaces. Nonlinear Anal Theory Methods Appl 2001, 46: 267–278. 10.1016/S0362-546X(00)00117-6MathSciNetView ArticleGoogle Scholar
  21. Khamsi MA: Quasicontraction mappings in modular spaces without Δ 2 -condition. Fixed Point Theory Appl 2008., 2008: Article ID 916187, 6Google Scholar
  22. Khamsi MA, Kozlowski WM, Reich S: Fixed point theory in modular function spaces. Nonlinear Anal. Theory Methods Appl 1990, 14: 935–953. 10.1016/0362-546X(90)90111-SMathSciNetView ArticleGoogle Scholar
  23. Kuaket K, Kumam P: Fixed point for asymptotic pointwise contractions in modular spaces. Appl Math Lett 2011, 24: 1795–1798. 10.1016/j.aml.2011.04.035MathSciNetView ArticleGoogle Scholar
  24. Kumam P: On uniform opial condition, uniform Kadec-Klee property in modular spaces and application to fixed point theory. J Interdisciplinary Math 2005, 8: 377–385.MathSciNetView ArticleGoogle Scholar
  25. Kumam P: Fixed point theorems for nonexpansive mapping in modular spaces. Arch Math 2004, 40: 345–353.MathSciNetGoogle Scholar
  26. Mongkolkeha C, Kumam P: Fixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces. Int J Math Math Sci 2011., 2011: Article ID 705943, 12Google Scholar
  27. Chistyakov VV: Modular metric spaces generated by F -modulars. Folia Math 2008, 14: 3–25.MathSciNetGoogle Scholar
  28. Chistyakov VV: Modular metric spaces I basic concepts. Nonlinear Anal 2010, 72: 1–14. 10.1016/j.na.2009.04.057MathSciNetView ArticleGoogle Scholar
  29. Chistyakov VV: Metric modulars and their application. Dokl Math 2006,73(1):32–35. 10.1134/S106456240601008XView ArticleGoogle Scholar

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