Fixed Point Theorems for Modular Contraction Mappings on Modulared Spaces

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1 Int. Journal of Math. Analysis, Vol. 7, 2013, no. 20, HIKARI Ltd, Fixed Point Theorems for Modular Contraction Mappings on Modulared Spaces Mariatul Kiftiah Dept. of Math., Tanjungpura University Pontianak, Indonesia Supama Dept. of Math., Gadjah Mada University Yogyakarta 55281, Indonesia Copyright c 2013 Mariatul Kiftiah and Supama. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we present some conditions that guarantee the existence of the fixed point for a certain type of modular contraction mapping on a modulared space. Mathematics Subject Classification: 46A40, 46B40, 47H10 Keywords: modular, Ciric ρ-contraction, Meir-Keeler ρ-contraction, Jachymski ρ-contraction 1 Introduction One knows that fixed point theorems have important roles and applications in many areas, particularly in differential and integral equations [1]. One most popular fixed point theorem is those so-called Banach Fixed Point Theorem.

2 966 M. Kiftiah and Supama In its development, some authors have generalized the concept of Banach fixed point into modulared spaces. The theory of modulared spaces was firstly initiated by Nakano. Some properties and characterizations of the spaces are discussed in [6]. Geometrically, many concepts in normed spaces closed relate to those in modulared spaces. That is why some authors can successfully generalize the concept of Banach fixed point theorem into some classes of modulared spaces. One of the most important conditions that plays a crucial role so that a certain mapping has a unique fixed point is a contraction condition [1,3,7]. By using a certain type of gauge functions, Farajzadeh et.al [2011] generalized the contraction condition to present some fixed point theorems [3]. Some equivalent conditions between each type of gauge functions and their related contraction conditions are discussed in [4]. Ciric, L. [1981] developed a new fixed point theorem for contractive mappings [2]. Some characterizations for contractive mappings delivered by Meir and Keeler in [5]. In this paper, we will present some conditions that guarantee the existence of the fixed point for a certain type of modular contraction mapping on a modulared space. To this end, we will firstly introduce some type of modular contraction mappings in the beginning of Section 3. 2 Some Basic Notions and Properties As usual, N and R denote the set of all positive integers and real numbers system, respectively. The extended real numbers system will be denoted by R. Let X be a linear space over R. A non-negative function ρ : X R is called a modular if for every f,g X the following conditions hold, (i) ρ(f) =0ifff =0. (ii) ρ( f) =ρ(f). (iii) ρ(αf + βg) ρ(f)+ρ(g) for every α, β 0 such that α + β =1. A linear space X equipped with a modular ρ, written by (X, ρ), is called a modulared space. We shall also denote a modulared space by the single character X, when the modular ρ is explicitly understood. By considering the definition of the modular, then we can easily prove the following theorems.

3 Fixed point theorems for modular contraction mappings 967 Theorem 2.1 Let (X, ρ) be a modulared space. (i) If α, β R, 0 α β then ρ(αf) ρ(βf) for every f X. (ii) If ρ(f) <ɛfor every ɛ>0 then f =0. Theorem 2.2 Let (X, ρ) be a modulared space. If f 1,f 2,f 3,...,f n X, and α 1,α 2,α 3,...,α n are non-negative real numbers such that n i=1 α i =1, then ρ( n i=1 α if i ) n i=1 ρ((f i). Let (X, ρ) be a modulared space. Then, the set X ρ = {f X : ρ(αf) <, for some α>0}. is a linear space modulared by ρ. Trough out this paper, we always mean that X is a space modulared by ρ and X ρ is as mentioned above. A sequence {f n } X ρ is said to be ρ-convergent (modular convergent) to f X ρ if for every real number ɛ>0 there exists a positive integer N such that for every n N we have: ρ(f n f) <ɛ. Further, f is called a ρ-limit (modular limit) of{f n }, and we write ρ lim n f n = f If the sequence {f n } X ρ is ρ-convergent, then we can prove that its ρ-limit is unique. A sequence {f n } X ρ is called a ρ-cauchy (modular Cauchy) sequence if for every real number ɛ>0 there exists a positive integer N such that for every m, n N we have: ρ(f n f m ) <ɛ. It is clear that in every modulared space X ρ, every ρ-convergent sequence is ρ-cauchy. However, the converse may be failed. The modulared space X ρ is said to be ρ-complete (modular complete) if every ρ-cauchy sequence in X ρ is ρ-convergent. It is easy to prove the following lemmas. Lemma 2.3 The sequence {f n } X ρ is ρ-convergent to f X ρ iff for every α, the sequence {αf n } X ρ is ρ-convergent to αf X ρ.

4 968 M. Kiftiah and Supama Lemma 2.4 The sequence {f n } X ρ is ρ-cauchy iff for every α, the sequence {αf n } X ρ is ρ-cauchy. Any set F X ρ is said to be modular closed (ρ-closed) if for any sequence {f n } F which is ρ-convergent to f X ρ implies f F. Any set B X ρ is said to be modular bounded (ρ-bounded) if sup{ρ(f g) : f, g B} <. Definition 2.5 (i) A gauge ϕ :[0, ) [0, ) is said to be of Φ 1 -class, denoted by ϕ Φ 1, if for every ɛ>0 there exists a number δ>ɛsuch that for every t (ɛ, δ), ϕ(t) ɛ. (ii) A gauge ϕ :[0, ) [0, ) is said to be of Φ 2 -class, denoted by ϕ Φ 2, if for every ɛ>0 there exists a number δ>ɛsuch that for every t [ɛ, δ), ϕ(t) ɛ. (iii) A gauge ϕ :[0, ) [0, ) is said to be of Φ-class, denoted by ϕ Φ, if it is non-decreasing and for every ɛ>0 there exists a number δ>ɛ such that ϕ(δ) ɛ. Some correlation between the gauges are given in the following theorems. Their proofs can be found in [7]. Theorem 2.6 Let X, f : X X, and E,F : X [0, ). IfE(f(x)) F (x) for every x X, then the following statements are equivalent. (i) There exists a gauge ψ Φ 1 such that E(f(x)) ψ(f (x)) for every x X. (ii) For every ɛ > 0 there exists δ > ɛ such that E(f(x)) ɛ whenever F (x) (ɛ, δ). The following theorems are needed to prove the main theorem which will be presented in the next section. Their proof can be read in [3]. Note that Theorem 2.7 below is a special case of the Theorem 2.6. Theorem 2.7 Let X be a modulared space, f : X ρ X ρ, and E,F : X ρ X ρ [0, ). If ρ(f(x) f(y)) F (x, y) for every x, y X ρ, then the following statements are equivalent. (i) There exists a gauge ψ Φ 1 such that ρ(f(x) f(y)) ψ(f (x, y)) for every x, y X ρ.

5 Fixed point theorems for modular contraction mappings 969 (ii) For every ɛ>0 there exists δ>ɛsuch that ρ(f(x) f(y)) ɛ whenever F (x, y) (ɛ, δ). Theorem 2.8 Let f : X ρ X ρ and f : X ρ X ρ [0, ). If for any x, y X ρ with F (x, y) =0implies ρ(f(x) f(y)) = 0, then the following statements are equivalent. (i) There exists a gauge ψ Φ 2 such that for every x, y X ρ with F (x, y) = 0 implies ρ(f(x) f(y)) ψ(f (x, y)). (ii) For every ɛ>0 there exists δ>ɛsuch that ρ(f(x) f(y)) ɛ whenever F (x, y) (ɛ, δ). 3 Fixed Point Theorems on Modulared Spaces We begin this section by introducing the definition of some type of ρ-contraction mappings on modular spaces, which are generalizations of those on metric spaces. Definition 3.1 Let X be a modulared space. (i) A function f : X ρ X ρ is said to be Ciric modular contraction (Ciric ρ-contraction) if for every ɛ>0 there exists δ>ɛsuch that for any x, y X ρ with ρ(x y) (ɛ, δ), we have ρ(f(x) f(y)) <ɛ (ii) A function f : X ρ X ρ is said to be Meir-Keeler modular contraction (Meir-Keeler ρ-contraction) if for every ɛ>0 there exists δ>ɛsuch that for any x, y X ρ with ρ(x y) [ɛ, δ), we have ρ(f(x) f(y)) <ɛ (iii) A function f : X ρ X ρ is said to be Jachymski modular contraction (Jachymski ρ-contraction) if there exists a gauge ϕ Φ 1 such that for any x, y X ρ, ρ(f(x) f(y)) <ϕ(ρ(x y))

6 970 M. Kiftiah and Supama It is easy to see that every Meir-Keeler ρ-contraction mapping is Ciric ρ- contraction. Followings are examples of each type of ρ-contraction mappings on modular spaces. Example 3.2 The linear space X = M[0, 1], i.e. the collection of all real valued measurable functions on [0, 1], is a modulared space with respect to: ρ(f) = 1 0 f, f X Further, if the function T : X X is defined by T (f)(x) = f(x) 2 whenever f(x) < 1 and T (f)(x) = f(x) for any else, then T is a Ciric ρ-contraction mapping. Beside that, we can also prove that T is Jachymski ρ-operator too. Example 3.3 The real number system R is a space modulared by ρ(x) = x If the function f : R Ris defined by f(x) = x if x < x 1, then f is a Meir-Keeler ρ contraction mapping. 2 and f(x) =0if Further, by using each type of the ρ-contraction mappings mentioned in the Definition 3.1, we construct some fixed point theorems. Theorem 3.4 Let (X, ρ) be a modulared space such that X ρ is ρ-complete, and B X ρ ρ-closed and ρ-bounded. If f : B B is a Ciric ρ-contraction mapping, then f has a unique fixed point. Proof: For any x B and n N {0}, we define f 0 (x) =x, f n+1 (x) =f(f n (x)), and x n = f n (x) Since the function f is Ciric ρ-contraction, then for every ɛ > 0, there exists δ>ɛsuch that the sequence {ρ(x n x n+1 )} n=0 is decreasing and bounded in (ɛ, δ). This implies that lim ρ(x n x n+1 ) = inf n ρ(x n x n+1 )=r exists. We are going to prove that r = 0. If it is not true, then there exists δ>rsuch that {ρ(x n x n+1 ) n=0 } is decreasing in (r, δ). This implies ρ(x n x n+1 ) n=0 r

7 Fixed point theorems for modular contraction mappings 971 for every n 1. This lead to a contradiction, so r =0. Following the fact lim ρ(x n x n+1 ) = 0, then {x n } is a ρ-cauchy sequence. So, it is ρ-convergent to a unique x X ρ, since X ρ is ρ-complete. The ρ- closedness of B implies x B. Further, since x = ρ lim x n and {ρ(x n x n+1 )} n=0 is decreasing in (ɛ, δ), then for some N N, we have: ρ( x f(x) ) 2 ρ(x f(x N+1 )) + ρ(f(x N+1 ) f(x)) ρ(x f(x N+1 )) + ρ(f(x N ) x) < 2ɛ This implies x = f(x), i.e. x is a fixed point of f. Since every Meir-Keeler ρ-contraction mapping is Ciric ρ-contraction, then we have the following corollary. Corollary 3.5 Let (X, ρ) be a modulared space such that X ρ is ρ-complete, and B X ρ ρ-closed and ρ-bounded. If f : B B is a Meier-Keeler ρ- contraction mapping, then f has a unique fixed point. As a consequence of Theorem 2.8 and Corollary 3.5, we have the following corollary. Corollary 3.6 Let (X, ρ) be a modulared space such that X ρ is ρ-complete, B X ρ ρ-closed and ρ-bounded, and f : B B. If there exists a gauge ψ Φ 2 such that for every x, y B with x y, ρ(f(x) f(y)) ψ(ρ(x y)), then the function f has a unique fixed point. Theorem 3.7 Let X be a modulared space such that X ρ is ρ-complete, and B X ρ ρ-closed and ρ-bounded. If the function f : B B is Jachymski ρ-contraction, then f has a unique fixed point. Proof: Following Theorem 2.7 and Theorem 3.4, the assertion follows. References [1] Agarwal, R.P., Meehan,M., and O Regan,D., 2001, Fixed Point Theory and Application, Cambridge University Press. [2] Ciric, L., 1981, A New Fixed-Point Theorem For Contractive Mappings, Publications De L Institut Mathematique (NS), 30 (44),

8 972 M. Kiftiah and Supama [3] Farajzadeh,A.P., Mohammad, M.B., and Noor, M.A., 2011, Fixed Pont Theorems in Modular Spaces, Mathematical Communications, 16, [4] Jachymski, J., 1995, Equivalent Conditions and The Meir-Keeler Type Theorems, Journal of mathematical Analysis and Applications, 194, [5] Meir, A. and Keeler, E., 1969, A Theorem on Contractive Mappings, Journal of mathematical Analysis and Applications, 28, [6] Musielak, J., 1983, Orlicz Spaces and Modular Spaces, Springer-Verlag, New York. [7] Proinov, P.D., 2006, Fixed Point Theorems in Metric Spaces, Nonlinear Analysis, 64, Received: January, 2013