This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Fixed point theorems for contraction mappings in modular metric spaces Fixed Point Theory and Applications 2011, 2011:93 doi:10.1186/1687-1812-2011-93 Chirasak Mongkolkeha (cm.mongkol@hotmail.com) Wultiphol Sintunavarat (poom_teun@hotmail.com) Poom Kumam (poom.kum@kmutt.ac.th) ISSN 1687-1812 Article type Research Submission date 20 June 2011 Acceptance date 2 December 2011 Publication date 2 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/93 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Fixed point theorems for contraction mappings in modular metric spaces Chirasak Mongkolkeha, Wutiphol Sintunavarat and Poom Kumam ∗ Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand ∗ Corresponding author: poom.kum@kmutt.ac.th Email addresses: CM: cm.mongkol@hotmail.com WS: poom teun@hotmail.com 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 spaces; modular spaces; contraction map- pings; fixed points. 1 2 1 Introduction Let (X, d) be a metric space. A mapping T : X → X is a contraction if d(T (x), T (y)) ≤ kd(x, y), (1.1) for all x, y ∈ X, where 0 ≤ k < 1. The Banach Contraction Mapping Princi- ple 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 [2–10]. The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [11] and was intensively developed by Koshi, Shimogaki, Yamamuro [11–13] and others. Fur- ther and the most complete development of these theories are due to Luxemburg, Musielak, Orlicz, Mazur, Turpin [14–18] and their collaborators. A lot of math- ematicians are interested fixed points of Modular spaces, for example [4, 19–26]. In 2008, Chistyakov [27] intro duced 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 the- ory 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. 3 2 Preliminaries We will start with a brief recollection of basic concepts and facts in modular spaces and modular metric spaces (see [14,15, 27–29] for more details). Definition 2.1. Let X be a vector space over R(or C). 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(λx) → 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) 4 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(λx) < ∞} (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 w λ (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) w λ+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, ∞). 5 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 →+0 w λ+ (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) for all λ, µ > 0 and 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 x 0 ∈ 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, x 0 ) < ∞} 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) 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) = w 1 (x, 0) with (2.6) holds, where x ∈ X, then (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 ; (ii) if w is convex, X ∗ ρ ≡ X ∗ w (0) = X ρ is a linear subspace of X and the func- tional  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∈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∈N in X w is said to be Cauchy if w λ (x m , x n ) → 0, as m, n → ∞ for al l 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. 7 (4) A subset C of X w is said to be complete if any Cauchy sequence in C is convergent sequence and its 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 con- traction 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) ≤ kw 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 x 0 ba an arbitrary point in X w and we write x 1 = T x 0 , x 2 = T x 1 = T 2 x 0 , and in general, x n = T x n−1 = T n x 0 for all n ∈ N. Then, 8 w λ (x n+1 , x n ) = w λ (T x n , T x n−1 ) ≤ kw λ (x n , x n−1 ) = kw 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 ∈ N. If m > n, by Definition 2.2(iii), implies w λ (x n , x m ) = w λ·(m−n) m−n (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 ) ≤ k n w λ m−n (x 0 , x 1 ) + k n+1 w λ m−n (x 0 , x 1 ) + · · · + k m−1 w λ m−n (x 0 , x 1 ) = (k n + k n+1 + · · · + k m−1 )w λ m−n (x 0 , x 1 ) ≤ ( k n 1−k )w λ m−n (x 0 , x 1 ). (3.2) Taking n, m → ∞ in (3.2), we have w λ (x n , x m ) → 0. Thus, {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 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) ≤ kw λ 2 (x, x n ) + w λ 2 (x n+1 , x) (3.3) for all λ > 0 and for each n ∈ N. Taking n → ∞ in (3.3) implies that w λ (T x, x) = 0 for all λ > 0 and thus T x = 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 9 see that w λ (x, z) = w λ (T x, T z) ≤ kw 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 0 ∈ 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 0 . Proof. For each m ∈ N, 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 ) ≤ kw λ·(m−1) m (T m x, y m ) + ε m ≤ kw λ·(m−2) m (T m x, T y m−1 ) + kw 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 ) + kε m−1 + ε m . . . ≤ m  i=0 k m−i ε i (3.4) [...]... contractions in modular spaces Appl Math Lett 24, 1795–1798 (2011) 20 [24] Kumam, P: On uniform opial condition, uniform Kadec–Klee property in modular spaces and application to fixed point theory J Interdisciplinary Math 8, 377–385 (2005) [25] Kumam, P: Fixed point theorems for nonexpansive mapping in modular spaces Arch Math 40, 345–353 (2004) [26] Mongkolkeha, C, Kumam, P: Fixed point and common fixed point theorems. .. 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X and Xw be a modular metric space induced by w If Xw is a complete modular metric space and T : Xw → Xw is a mapping, which T N is a contraction mapping for some positive integer N Then, T has a unique fixed point in Xw 11 Proof By Theorem 3.2, T N has a unique fixed point u ∈ Xw From T N (T u) = T N +1 u = T (T N u) = T u, so T u is a fixed point of T N By the uniqueness of fixed point of T N , we... common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces Int J Math Math Sci 2011, Article ID 705943, 12 (2011) [27] Chistyakov, VV: Modular metric spaces generated by F -modulars Folia Math 14, 3–25 (2008) [28] Chistyakov, VV: Modular metric spaces I basic concepts Nonlinear Anal 72, 1–14 (2010) [29] Chistyakov, VV: Metric modulars and their application... 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Fixed point theorems for contraction mappings in modular metric spaces Fixed Point Theory and Applications. notice below). For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about. 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. 3 2