# Fixed Point and Common Fixed Point Theorems for Generalized Weak

Authors: Chirasak Mongkolkeha; Poom Kumam
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We prove new fixed point and common fixed point theorems for generalized weak contractive mappings of integral type in modular
spaces. Our results extend and generalize the results of A. Razani and R. Moradi (2009) and M. Beygmohammadi and A. Razani (2010).


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## Body

## 1. Introduction

Let(X,d) be a metric space. A mapping T:X→X is a contraction if (1.1)d(T(x),T(y))≤kd(x,y),
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(1.2)d(T(x),T(y))≤d(x,y)-ϕ(d(x,y)),
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,
(1.3)∫0d(f(x),f(y))φ(t)dt≤α∫0d(x,y)φ(t)dt,
where φ:ℝ+→ℝ+ is a Lebesgue integrable which is summable, nonnegative, and for all ε>0, ∫0εφ(t)dt>0. Then, f has a unique fixed point z∈X such that for each x∈X, limn→∞fnx=z.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 forρ-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.Definition 1.2.
LetX be a vector space over ℝ(or ℂ). A functional ρ:X→[0,∞] is called a modular if for arbitrary f and g, elements of X satisfy the following conditions:  (1)
ρ(f)=0 if and only if f=0;(2)
ρ(αf)=ρ(f) for all scalar α with |α|=1;(3)
ρ(αf+βg)≤ρ(f)+ρ(g), whenever α,β≥0 and α+β=1. If we replace (3) by(4)
ρ(αf+βg)≤αsρ(f)+βsρ(g), for α,β≥0,  αs+βs=1 with an s∈(0,1], then the modular ρ is called s-convex modular, and if s=1,  ρ is called convex modular.Ifρ is modular in X, then the set defined by (1.4)Xρ={x∈X:ρ(λx)⟶0asλ⟶0}
is called a modular space. Xρ is a vector subspace of X.Definition 1.3.
A modularρ is said to satisfy the Δ2-condition if ρ(2fn)→0 as n→∞, whenever ρ(fn)→0 as n→∞.Definition 1.4.
LetXρ be a modular space. Then,(1)
the sequence(fn)n∈ℕ in Xρ is said to be ρ-convergent to f∈Xρ if ρ(fn-f)→0, as n→∞,(2)
the sequence(fn)n∈ℕ in Xρ is said to be ρ-Cauchyif ρ(fn-fm)→0, as n,m→∞,(3)
a subsetC of Xρ is said to be ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belong to C,(4)
a subsetC of Xρ is said to be ρ-complete if any ρ-Cauchy sequence in C is ρ-convergent sequence and its is in C,(5)
a subsetC of Xρ is said to be ρ-bounded if δρ(C) = sup {ρ(f-g);f,g∈C}<∞.Definition 1.5.
LetC be a subset of Xρ and T:C→C an arbitrary mapping. T is called a ρ-contraction if for each f,g∈Xρ there exists k<1 such that
(1.5)ρ(T(f)-T(g))≤kρ(f-g).Definition 1.6.
LetXρ be a modular space, where ρ satisfies the Δ2-condition. Two self-mappings T and f of Xρ are called ρ-compatible if ρ(Tfxn-fTxn)→0 as n→∞, whenever {xn}n∈ℕ is a sequence in Xρ such that fxn→z and Txn→z for some point z∈Xρ.


## 2. A Common Fixed Point Theorem forρ-Compatible Generalized Weak Contraction Maps of Integral Type

Theorem 2.1.
LetXρ be a  ρ-complete modular space, where  ρ satisfies the  Δ2-condition. Let  c,l∈ℝ+,  c>l and  T,f:Xρ→Xρ are two  ρ-compatible mappings such that  T(Xρ)⊆f(Xρ) and
(2.1)∫0ρ(c(Tx-Ty))φ(t)dt≤∫0ρ(l(fx-fy))φ(t)dt-ϕ(∫0ρ(l(fx-fy))φ(t)dt),
for all x,y∈Xρ, where φ:[0,∞)→[0,∞) is a Lebesgue integrable which is summable, nonnegative, and for all ε>0, ∫0εφ(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.
Letx∈Xρ and generate inductively the sequence {Txn}n∈ℕ as follow: Txn=fxn+1. First, we prove that the sequence {ρ(c(Txn-Txn-1))} converges to 0. Since,
(2.2)∫0ρ(c(Txn-Txn-1))φ(t)dt≤∫0ρ(l(fxn-fxn-1))φ(t)dt-ϕ(∫0ρ(l(fxn-fxn-1))φ(t)dt)≤∫0ρ(l(fxn-fxn-1))φ(t)dt≤∫0ρ(l(Txn-1-Txn-2))φ(t)dt<∫0ρ(c(Txn-1-Txn-2))φ(t)dt.
This means that the sequence {∫0ρ(c(Txn-Txn-1))} is decreasing and bounded below. Hence, there exists r≥0 such that
(2.3)lim⁡n→∞∫0ρ(c(Txn-Txn-1))φ(t)dt=r.
If r>0, then limn→∞∫0ρ(c(Txn-Txn-1))φ(t)dt=r>0. Taking n→∞ in the inequality (2.2) which is a contradiction, thus r=0. This implies that
(2.4)ρ(c(Txn-Txn-1))⟶0asn⟶∞.
Next, we prove that the sequence{Txn}n∈ℕ is ρ-Cauchy. Suppose {cTxn}n∈ℕ is not ρ-Cauchy, then there exists ε>0 and sequence of integers {mk},{nk} with mk>nk≥k such that
(2.5)ρ(c(Txmk-Txnk))≥εfork=1,2,3,…
We can assume that
(2.6)ρ(c(Txmk-1-Txnk))<ε.
Letmk be the smallest number exceeding nk for which (2.5) holds, and
(2.7)θk={m∈N∣∃nk∈N;ρ(c(Txm-Txnk))≥ε,m>nk≥k}.
Since θk⊂ℕ and clearly θk≠∅, by well ordering principle, the minimum element of θk is denoted by mk and obviously (2.6) holds. Now, let α∈ℝ+ be such that l/c+1/α=1, then we get
(2.8)∫0εφ(t)dt≤∫0ρ(c(Txmk-Txnk))φ(t)dt≤∫0ρ(l(fxmk-fxnk))φ(t)dt-ϕ(∫0ρ(l(fxmk-fxnk))φ(t)dt)≤∫0ρ(l(fxmk-fxnk))φ(t)dt≤∫0ρ(l(Txmk-1-Txnk-1))φ(t)dt,(2.9)ρ(l(Txmk-1-Txnk-1))=ρ(l(Txmk-1-Txnk+Txnk-Txnk-1))=ρ(lcc(Txmk-1-Txnk)+1ααl(Txnk-Txnk-1))≤ρ(c(Txmk-1-Txnk))+ρ(αl(Txnk-Txnk-1))<ε+ρ(αl(Txnk-Txnk-1)).
Using theΔ2-condition and (2.4), we obtain
(2.10)lim⁡n→∞ρ(αl(Txnk-Txnk-1))=0.
It follows that
(2.11)lim⁡k→∞∫0ρ(l(Txmk-1-Txnk-1))φ(t)dt<∫0εφ(t)dt.
From (2.8) and (2.11), we also have
(2.12)∫0εφ(t)dt≤∫0ρ(l(Txmk-1-Txnk-1))φ(t)dt<∫0εφ(t)dt,
which is a contradiction. Hence, {cTxn}n∈ℕ is ρ-Cauchy and by the Δ2-condition, {Txn}n∈ℕ is ρ-Cauchy. Since Xρ is ρ-complete, there exists a point u∈Xρ such that ρ(Txn-u)→0 as n→∞. If T is continuous, then T2xn→Tu and Tfxn→Tu as n→∞. Since ρ(c(fTxn-Tfxn))→0 as n→∞, by ρ-compatible, fTxn→Tu as n→∞. Next, we prove that u is a unique fixed point of T. Indeed,
(2.13)∫0ρ(c(T2xn-Txn))φ(t)dt=∫0ρ(c(T(Txn)-Txn))φ(t)dt≤∫0ρ(l(fTxn-fxn))φ(t)dt-ϕ(∫0ρ(l(fTxn-fxn))φ(t)dt)≤∫0ρ(l(fTxn-fxn))φ(t)dt.
Taking n→∞ in the inequality (2.13), we have
(2.14)∫0ρ(c(Tu-u))φ(t)dt≤∫0ρ(l(Tu-u))φ(t)dt,
which implies that ρ(c(Tu-u))=0 and Tu=u. Since T(Xρ)⊆f(Xρ), there exists u1 such that u=Tu=fu1. The inequality,
(2.15)∫0ρ(c(T2xn-Tu1))φ(t)dt≤∫0ρ(l(fTxn-fu1))φ(t)dt-ϕ(∫0ρ(l(fTxn-fu1))φ(t)dt)≤∫0ρ(l(fTxn-fu1))φ(t)dt
as n→∞, yields
(2.16)∫0ρ(c(Tu-Tu1))φ(t)dt≤∫0ρ(l(Tu-fu1))φ(t)dt
and, thus,
(2.17)∫0ρ(c(u-Tu1))φ(t)dt≤∫0ρ(l(u-fu1))φ(t)dt≤∫0ρ(l(u-u))φ(t)dt=0,
which implies that, u=Tu1=fu1 and also fu=fTu1=Tfu1=Tu=u (see [25]). Hence, fu=Tu=u. Suppose that there exists w∈Xρ such that w=Tw=fw and w≠u, we have ∫0ρ(c(w-u))φ(t)dt>0 and
(2.18)∫0ρ(c(w-u))φ(t)dt=∫0ρ(c(Tw-Tu))φ(t)dt≤∫0ρ(l(fw-fu))φ(t)dt-ϕ(∫0ρ(l(fw-fu))φ(t)dt)<∫0ρ(l(fw-fu))φ(t)dt<∫0ρ(c(w-u))φ(t)dt,
which is a contradiction. Hence, u=w and the proof is complete.In fact, if takeϕ(t)=(1-k)t where 0<k<1 and take ϕ(t)=t-ψ(t), respectively, where ψ:ℝ+→ℝ+ is a nondecreasing and right continuous function with ψ(t)<t for all t>0, we obtain following corollaries.Corollary 2.2 (see [23]).
LetXρ be a  ρ-complete modular space, where  ρ satisfies the  Δ2-condition. Suppose  c,l∈ℝ+,  c>l and  T,h:Xρ→Xρ are two  ρ-compatible mappings such that  T(Xρ)⊆h(Xρ) and
(2.19)∫0ρ(c(Tx-Ty))φ(t)dt≤k∫0ρ(l(hx-hy))φ(t)dt,
for some k∈(0,1), where φ:ℝ+→ℝ+ is a Lebesgue integrable which is summable, nonnegative, and for all ε>0, ∫0εφ(t)dt>0. 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]).
LetXρ be a  ρ-complete modular space, where  ρ satisfies the  Δ2-condition. Suppose  c,l∈ℝ+,  c>l and  T,h:Xρ→Xρ are two  ρ-compatible mappings such that  T(Xρ)⊆h(Xρ) and
(2.20)∫0ρ(c(Tx-Ty))φ(t)dt≤ψ(∫0ρ(l(hx-hy))φ(t)dt),
where φ:ℝ+→ℝ+ is a Lebesgue integrable which is summable, nonnegative, and for all ε>0, ∫0εφ(t)dt>0 and ψ:ℝ+→ℝ+ 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.


## 3. A Fixed Point Theorem for Generalized Weak Contraction Mapping of Integral Type

Theorem 3.1.
LetXρ be a  ρ-complete modular space, where  ρ satisfies the  Δ2-condition. Let  c,l∈ℝ+,  c>l and  T:Xρ→Xρ be a mapping such that for each  x,y∈Xρ,
(3.1)∫0ρ(c(Tx-Ty))φ(t)dt≤∫0ρ(l(x-y))φ(t)dt-ϕ(∫0ρ(l(x-y))φ(t)dt),
where φ:[0,∞)→[0,∞) is a Lebesgue integrable which is summable, nonnegative, and for all ε>0, ∫0εφ(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(Tnx-Tn-1x))} converges to 0. Since,
(3.2)∫0ρ(c(Tnx-Tn-1x))φ(t)dt≤∫0ρ(l(Tn-1x-Tn-2x))φ(t)dt-ϕ(∫0ρ(l(Tn-1x-Tn-2x))φ(t)dt)≤∫0ρ(l(Tn-1x-Tn-2x))φ(t)dt<∫0ρ(c(Tn-1x-Tn-2x))φ(t)dt,
it follows that the sequence {∫0ρ(c(Tnx-Tn-1x))} is decreasing and bounded below. Hence, there exists r≥0 such that
(3.3)lim⁡n→∞∫0ρ(c(Tnx-Tn-1x))φ(t)dt=r.
If r>0, then limn→∞∫0ρ(c(Tnx-Tn-1x))φ(t)dt=r>0, taking n→∞ in the inequality (3.2) which is a contradiction, thus r=0. So, we have
(3.4)ρ(c(Tnx-Tn-1x))⟶0asn⟶∞.
Next, we prove that the sequence {Tn(x)}n∈ℕ is ρ-Cauchy. Suppose {cTn(x)}n∈ℕ is not ρ-Cauchy, there exists ε>0 and sequence of integers {mk},{nk} with mk>nk≥k such that
(3.5)ρ(c(Tmkx-Tnkx))≥εfork=1,2,3,….
We can assume that
(3.6)ρ(c(Tmk-1x-Tnkx))<ε.
Let mk be the smallest number exceeding nk for which (3.5) holds, and
(3.7)θk={m∈N∣∃nk∈N;ρ(c(Tmx-Tnkx))≥ε,m>nk≥k}.
Since θk⊂ℕ and clearly θk≠∅, by well ordering principle, the minimum element of θk is denoted by mk and obviously (3.6) holds. Now, let α∈ℝ+ be such that l/c+1/α=1, then we get
(3.8)∫0εφ(t)dt≤∫0ρ(c(Tmkx-Tnkx))φ(t)dt≤∫0ρ(l(Tmk-1x-Tnk-1x))φ(t)dt-ϕ(∫0ρ(l(Tmk-1x-Tnk-1x))φ(t)dt)≤∫0ρ(l(Tmk-1x-Tnk-1x))φ(t)dt,,(3.9)ρ(l(Tmk-1x-Tnk-1x))=ρ(l(Tmk-1x-Tnkx+Tnkx-Tnk-1x))=ρ(lcc(Tmk-1x-Tnkx)+1ααl(Tnkx-Tnk-1x))≤ρ(c(Tmk-1x-Tnkx))+ρ(αl(Tnkx-Tnk-1x))<ε+ρ(αl(Tnkx-Tnk-1x)).
Using theΔ2-condition and (3.4), we obtain
(3.10)lim⁡k→∞ρ(αl(Tnkx-Tnk-1x))=0,(3.11)lim⁡k→∞∫0ρ(l(Tmk-1x-Tnk-1x))φ(t)dt<∫0εφ(t)dt.
From (3.8) and (3.11), we have
(3.12)∫0εφ(t)dt≤∫0ρ(l(Tmk-1x-Tnk-1x))φ(t)dt<∫0εφ(t)dt,
which is a contradiction. Hence, {cTn(x)}n∈ℕ is ρ-Cauchy and again by the Δ2-condition, {Tn(x)}n∈ℕ is ρ-Cauchy. Since Xρ is ρ-complete, there exists a point u∈Xρ such that ρ(Tnx-u)→0 as n→∞. Next, we prove that u is a unique fixed point of T. Indeed,
(3.13)ρ(c2(u-Tu))=ρ(c2(u-Tn+1x+Tn+1x-Tu))≤ρ(c(u-Tn+1x))+ρ(c(Tn+1x-Tu)),(3.14)∫0ρ(c(Tn+1x-Tu))φ(t)dt≤∫0ρ(l(Tnx-u))φ(t)dt-ϕ(∫0ρ(l(Tnx-u))φ(t)dt)≤∫0ρ(l(Tnx-u))φ(t)dt.
Since ρ(Tnx-u)→0 as n→∞, we obtain
(3.15)lim⁡n→∞∫0ρ(c(Tn+1x-Tu))φ(t)dt≤0,
which implies that
(3.16)ρ(c(Tn+1x-Tu))⟶0asn⟶∞.
So, we have
(3.17)ρ(c(u-Tn+1x))+ρ(c(Tn+1x-Tu))⟶0asn⟶∞.
Thus ρ(c/2(u-Tu))=0 and Tu=u. Suppose that there exists w∈Xρ such that Tw=w and w≠u, we have ∫0ρ(c(w-u))φ(t)dt>0 and
(3.18)∫0ρ(c(w-u))φ(t)dt=∫0ρ(c(Tw-Tu))φ(t)dt≤∫0ρ(l(w-u))φ(t)dt-ϕ(∫0ρ(l(w-u))φ(t)dt)<∫0ρ(l(w-u))φ(t)dt<∫0ρ(c(w-u))φ(t)dt,
which is a contradiction. Hence, u=w and the proof is complete.Corollary 3.2.
LetXρ be a  ρ-complete modular space, where  ρ satisfies the  Δ2-condition. Let  f:Xρ→Xρ be a mapping such that there exists an  λ∈(0,1) and  c,l∈ℝ+ where  l<c and for each  x,y∈Xρ,
(3.19)∫0ρ(c(fx-fy))φ(t)dt≤λ∫0ρ(l(x-y))φ(t)dt
where φ:[0,∞)→[0,∞) is a Lebesgue integrable which is summable, nonnegative, and for all ε>0, ∫oεφ(t)dt>o. Then, T has a unique fixed point in Xρ.Corollary 3.3 (see [24]).
LetXρ be a  ρ-complete modular space where  ρ satisfies the  Δ2-condition. Assume that  ψ:ℝ+→[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,  ∫0εφ(t)dt>0 and let  f:Xρ→Xρ be a mapping such that there are  c,l∈ℝ+ where  l<c,
(3.20)∫0ρ(c(Tx-Ty))φ(t)dt≤ψ(∫0ρ(l(x-y))φ(t)dt),
for each x,y∈Xρ. Then, T has a unique fixed point in Xρ.



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