Fixed Point and Common Fixed Point Theorems for Generalized Weak Contraction Mappings of Integral Type in Modular Spaces

where 0 < k < 1. The Banach Contraction Mapping Principle appeared in explicit form in Banach’s thesis in 1922 1 . For 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 2–6 . In 1997Alber andGuerreDelabriere 7 introduced the concept of weak contraction in Hilbert spaces, and Rhoades 8 has showed that the result by Akber et al. is also valid in complete metric spaces A mapping T : X → X is said to be weakly contractive if


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 where 0 < k < 1.The Banach Contraction Mapping Principle appeared in explicit form in Banach's thesis in 1922 1 .For 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 2-6 .In 1997 Alber and Guerre-Delabriere 7 introduced the concept of weak contraction in Hilbert spaces, and Rhoades 8 has showed that the result by Akber et al. is also valid in complete metric spaces A mapping T : X → X is said to be weakly contractive if where φ : 0, ∞ → 0, ∞ is continuous and nondecreasing function such that φ t 0 if and only if t 0. If one takes φ t 1 − k t where 0 < k < 1, then 1.2 reduces to 1.1 .In 2002, Branciari 9 gave a fixed point result for a single mapping an analogue of Banach's contraction principle for an integral-type inequality, which is stated as follow.
Theorem 1.1.Let X, d be a complete metric space, α ∈ 0, 1 , f : X → X a mapping such that for each x, y ∈ X, where ϕ : Ê → Ê is a Lebesgue integrable which is summable, nonnegative, and for all ε > 0, Afterward, many authors extended this work to more general contractive conditions.The works noted in 10-12 are some examples from this line of research.
The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano 13 and redefined by Musielak and Orlicz 14 .A lot of mathematicians are interested, fixed points of Modular spaces, for example 15-22 .In 2009, Razani and Moradi 23 studied fixed point theorems for ρ-compatible maps of integral type in modular spaces.
Recently, Beygmohammadi and Razani 24 proved the existence for mapping defined on a complete modular space satisfying contractive inequality of integral type.
In this paper, we study the existence of fixed point and common fixed point theorems for ρ-compatible mapping satisfying a generalize weak contraction of integral type in modular spaces.
First, we start with a brief recollection of basic concepts and facts in modular spaces.
If ρ is modular in X, then the set defined by is called a modular space.X ρ is a vector subspace of X.
Definition 1.4.Let X ρ be a modular space.Then, 3 a subset C of X ρ is said to be ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belong to C, 4 a subset C of X ρ is said to be ρ-complete if any ρ-Cauchy sequence in C is ρconvergent sequence and its is in C, Definition 1.6.Let X ρ be a modular space, where ρ satisfies the Δ 2 -condition.Two self-mappings T and f of

A Common Fixed Point Theorem for ρ-Compatible Generalized Weak Contraction Maps of Integral Type
Theorem 2.1.Let X ρ be a ρ-complete modular space, where ρ satisfies the for all x, y ∈ X ρ , where ϕ : 0, ∞ → 0, ∞ is a Lebesgue integrable which is summable, nonnegative, and for all ε > 0, ε 0 ϕ t dt > 0 and φ : 0, ∞ → 0, ∞ is lower semicontinuous function with φ t > 0 for all t > 0 and φ t 0 if and only if t 0. If one of T or f is continuous, then there exists a unique common fixed point of T and f.Proof.Let x ∈ X ρ and generate inductively the sequence {Tx n } n∈AE as follow: Tx n fx n 1 .First, we prove that the sequence {ρ c Tx n − Tx n−1 } converges to 0. Since, } is decreasing and bounded below.Hence, there exists r ≥ 0 such that Taking n → ∞ in the inequality 2.2 which is a contradiction, thus r 0. This implies that Next, we prove that the sequence {Tx n } n∈AE is ρ-Cauchy.Suppose {cTx n } n∈AE is not ρ-Cauchy, then there exists ε > 0 and sequence of integers We can assume that Let m k be the smallest number exceeding n k for which 2.5 holds, and

2.7
Since θ k ⊂ AE and clearly θ k / ∅, by well ordering principle, the minimum element of θ k is denoted by m k and obviously 2.6 holds.Now, let α ∈ Ê be such that l/c 1/α 1, then we

2.9
Using the Δ 2 -condition and 2.4 , we obtain From 2.8 and 2.11 , we also have 12 which is a contradiction.Hence, {cTx n } n∈AE is ρ-Cauchy and by the Δ 2 -condition, {Tx n } n∈AE is ρ-Cauchy.Since X ρ is ρ-complete, there exists a point u ∈ X ρ such that ρ Tx n − u → 0 as n → ∞.If T is continuous, then T 2 x n → Tu and Tfx n → Tu as n → ∞.Since ρ c fTx n − Tfx n → 0 as n → ∞, by ρ-compatible, fTx n → Tu as n → ∞.Next, we prove that u is a unique fixed point of T. Indeed,

2.13
Taking n → ∞ in the inequality 2.13 , we have International Journal of Mathematics and Mathematical Sciences which implies that ρ c Tu − u 0 and Tu u.Since T X ρ ⊆ f X ρ , there exists u 1 such that u Tu fu 1 .The inequality, and, thus, 2.17 which implies that, u Tu 1 fu 1 and also fu fTu 1 Tfu 1 Tu u see 25 .Hence, fu Tu u.Suppose that there exists w ∈ X ρ such that w Tw fw and w / u, we have which is a contradiction.Hence, u w and the proof is complete.
In fact, if take φ t

A Fixed Point Theorem for Generalized Weak Contraction Mapping of Integral Type
Theorem 3.1.Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.Let c, l ∈ Ê , c > l and T : X ρ → X ρ be a mapping such that for each x, y ∈ X ρ , where ϕ : 0, ∞ → 0, ∞ is a Lebesgue integrable which is summable, nonnegative, and for all ε > 0, ε 0 ϕ t dt > 0 and φ : 0, ∞ → 0, ∞ is lower semicontinuous function with φ t > 0 for all t > 0 and φ t 0 if and only if t 0.Then, T has a unique fixed point.
Proof.First, we prove that the sequence {ρ c T n x − T n−1 x } converges to 0. Since, ϕ t dt r > 0, taking n → ∞ in the inequality 3.2 which is a contradiction, thus r 0. So, we have 3.4 Next, we prove that the sequence {T n x } n∈AE is ρ-Cauchy.Suppose {cT n x } n∈AE is not ρ-Cauchy, there exists ε > 0 and sequence of integers We can assume that Let m k be the smallest number exceeding n k for which 3.5 holds, and

3.7
Since θ k ⊂ AE and clearly θ k / ∅, by well ordering principle, the minimum element of θ k is denoted by m k and obviously 3.6 holds.Now, let α ∈ Ê be such that l/c 1/α 1, then we

3.9
Using the Δ 2 -condition and 3.4 , we obtain lim From 3.8 and 3.11 , we have which is a contradiction.Hence, {cT n x } n∈AE is ρ-Cauchy and again by the Δ 2 -condition, {T n x } n∈AE is ρ-Cauchy.Since X ρ is ρ-complete, there exists a point u ∈ X ρ such that ρ T n x − u → 0 as n → ∞.Next, we prove that u is a unique fixed point of T. Indeed, So, we have 3.17 which is a contradiction.Hence, u w and the proof is complete.
Corollary 3.2.Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.Let f : X ρ → X ρ be a mapping such that there exists an λ ∈ 0, 1 and c, l ∈ Ê where l < c and for each x, y ∈ X ρ , Assume that ψ : Ê → 0, ∞ is an increasing and upper semicontinuous function satisfying ψ t < t for all t > 0. Let ϕ : 0, ∞ → 0, ∞ be a Lebesgue integrable which is summable, nonnegative, and for all ε > 0, ε 0 ϕ t dt > 0 and let f : X ρ → X ρ be a mapping such that there are c, l ∈ Ê where for each x, y ∈ X ρ .Then, T has a unique fixed point in X ρ .

2 . 2 International
Journal of Mathematics and Mathematical SciencesThis means that the sequence { ρ c Tx n −Tx n−1 0

3 . 2 International
Journal of Mathematics and Mathematical Sciences it follows that the sequence { ρ c T n x−T n−1 x 0 } is decreasing and bounded below.Hence, there exists r ≥ 0 such that lim n → ∞ ρ c T n x−T n−1 x 0 ϕ t dt r. 3.3 If r > 0, then lim n → ∞ ρ c T n x−T n−1 x 0

Thus ρ c/ 2
u − Tu 0 and Tu u.Suppose that there exists w ∈ X ρ such that Tw w and w / u, we have
Corollary 2.2 see 23 .Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.Suppose c, l ∈ Ê , c > l and T, h : X ρ → X ρ are two ρ-compatible mappings such that T X ρ ⊆ If one of h or T is continuous, then there exists a unique common fixed point of h and T.Corollary 2.3 see 23 .Let X ρ be a ρ-complete modular space, where ρ satisfies the Δ 2 -condition.Suppose c, l ∈ Ê , c > l and T, h : X ρ → X ρ are two ρ-compatible mappings such that T X ρ ⊆ 1 − k t where 0 < k < 1 and take φ t t − ψ t , respectively, where ψ : Ê → Ê is a nondecreasing and right continuous function with ψ t < t for all t > 0, we obtain following corollaries.whereϕ : Ê → Ê is a Lebesgue integrable which is summable, nonnegative, and for all ε > 0, ε 0 ϕ t dt > 0 and ψ : Ê → Ê is a nondecreasing and right continuous function with ψ t < t for all t > 0. If one of h or T is continuous, then there exists a unique common fixed point of h and T.