# A Best Proximity Point Result in Modular Spaces with the Fatou Property

Authors: Mohamed Jleli; Erdal Karapınar; Bessem Samet
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Consider a nonself-mappingT:A→B, where (A,B) is a pair of nonempty subsets of a modular space Xρ. A best proximity point of T is a point z∈A satisfying the condition: ρ(z−Tz)=inf{ρ(x−y):(x,y)∈A×B}. In this paper, we introduce the class of proximal quasicontraction nonself-mappings in modular spaces with the Fatou property. For such mappings, we provide sufficient conditions assuring the existence and uniqueness of best proximity points.


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## Body

## 1. Introduction and Preliminaries

Through this paper, we denote byℕ the set of positive integers including zero. Let X be a vector space over ℝ. We denote by 0X its zero vector. According to Orlicz [1], a functional ρ:X→[0,∞] is said to be modular, if, for any pair (x,y)∈X2, the following conditions are satisfied:(i)
ρ(x)=0 if and only if x=0X;(ii)
ρ(-x)=ρ(x);(iii)
ρ(αx+βy)≤ρ(x)+ρ(y) whenever α,β≥0 and α+β=1.Ifρ is a modular in X, then the set
(1)Xρ:={x∈X:ρ(λx)⟶0asλ⟶0},
called a modular space, is a vector space.As a classical example of modulars, we may give the Orlicz modular defined for every measurable real functionf by
(2)ρφ(f)=∫ℝ‍φ(|f(t)|)dλ(t),
where λ is the Lebesgue measure in ℝ and φ:ℝ→[0,∞) is a function satisfying some conditions. The modular space induced by the Orlicz modular ρφ is called the Orlicz space. For more examples of modular spaces, we refer the reader to [2–4].Definition 1.
LetXρ be a modular space.(1)
The sequence{xn}⊂Xρ is said to be ρ-convergent to x∈Xρ if ρ(xn-x)→0, as n→∞.(2)
The sequence{xn}⊂Xρ is said to be ρ-Cauchy if ρ(xn-xm)→0, as n,m→∞.(3)
A subsetC of Xρ is called ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belongs to C.(4)
A subsetC of Xρ is called ρ-complete if any ρ-Cauchy sequence in C is ρ-convergent and its ρ-limit belongs to C.Definition 2.
The modularρ has the Fatou property if ρ(x)≤liminfn→∞ρ(xn) whenever {xn}ρ-converges to x.Recently, the existence and uniqueness of best proximity points in metric spaces were investigated by many authors; see [2, 5–14] and references therein. In this paper, we introduce the family of proximal quasicontraction nonself-mappings on modular spaces with the Fatou property. Our main result is a best proximity point theorem providing sufficient conditions assuring the existence and uniqueness of best proximity points for such mappings.Let(A,B) be a pair of nonempty closed subsets of a modular space Xρ. Through this paper, we will use the following notations:
(3)γ(A,B):=inf{ρ(x-y):(x,y)∈A×B},A0:={a∈A:ρ(a-b)=γ(A,B)forsomeb∈B},B0:={b∈B:ρ(a-b)=γ(A,B)forsomea∈A}.Definition 3.
LetT:A→B be a given nonself-mapping. We say that z∈A0 is a best proximity point of T if
(4)ρ(z-Tz)=γ(A,B).Clearly, from condition (i), ifA=B, a best proximity point of T will be a fixed point of T.Definition 4.
A nonself-mappingT:A→B is said to be a proximal quasicontraction if there exists a number q∈(0,1) such that
(5)ρ(u-Tx)=γ(A,B)ρ(v-Ty)=γ(A,B)}⟹ρ(u-v)≤qmax{ρ(x-y),ρ(x-u),ppppppppρ(y-v),ρ(x-v),ppppppppρ(y-u)},
where x,y,u,v∈A.Lemma 5.
LetT:A→B be a nonself-mapping. Suppose that  (i)
A0≠∅;(ii)
T(A0)⊆B0.Then, for any a∈A0, there exists a sequence {xn}⊂A0 such that
(6)x0=a,ρ(xn+1-Txn)=γ(A,B),∀n∈ℕ.Proof.
Leta∈A0. From (ii), we have Ta∈B0. By definition of the set B0, there exists x1∈A0 such that ρ(x1-Ta)=γ(A,B). Again, we have Tx1∈B0, which implies that there exists x2∈A0 such that ρ(x2-Tx1)=γ(A,B). Continuing this process, by induction, we obtain a sequence {xn}⊂A0 satisfying (6).Definition 6.
Under the assumptions of Lemma5, any sequence {xn}⊂A0 satisfying (6) is called a proximal Picard sequence associated to a∈A0. We denote by PP(a) the set of all proximal sequences associated to a∈A0.Definition 7.
Under the assumptions of Lemma5, we say that A0 is proximal T-orbitally ρ-complete if every ρ-Cauchy sequence {xn}∈PP(a) for some a∈A0ρ-converges to an element in A0.Leta∈A0 and {xn}∈PP(a). For all n∈ℕ, We denote
(7)δp(xn):=sup{ρ(xn+s-xn+r):r,s∈ℕ}.
Since x0=a, we have
(8)δp(a)=sup{ρ(xs-xr):r,s∈ℕ}.


## 2. A Best Proximity Point Theorem

The following lemmas will be useful later.Lemma 8.
LetXρ be a modular space. Suppose that a nonself-mapping T:A→B, where (A,B) is a pair of subsets of X, satisfies the following conditions:(i)
∃a∈A0|δp(a)<∞;(ii)
T(A0)⊆B0;(iii)
T is proximal quasi-contraction.Then, for any {xn}∈PP(a), one has
(9)ρ(xn-xn+m)≤δp(xn)≤qnδp(a),
for any n≥1 and m∈ℕ.Proof.
Let{xn}∈PP(a) and (s,r)∈ℕ2. From the definition of PP(a), for all n≥1, we have
(10)ρ(xn+s-Txn-1+s)=ρ(xn+r-Txn-1+r)=γ(A,B),
which implies, since T is a proximal quasi-contraction, that
(11)ρ(xn+s-xn+r)≤qmax{ρ(xn-1+s-xn-1+r),ppppppppρ(xn-1+s-xn+s),ppppppppρ(xn-1+r-xn+r),ppppppppρ(xn-1+s-xn+r),ppppppppρ(xn-1+r-xn+s)}≤qδp(xn-1).
This implies immediately that
(12)δp(xn)≤qδp(xn-1),
for all n≥1. Hence, for any n∈ℕ, we have
(13)δp(xn)≤qnδp(a).
Using the above inequality, for all n≥1 and m∈ℕ, we have
(14)ρ(xn-xn+m)≤δp(xn)≤qnδp(a).Lemma 9.
Let(A,B) be a pair of subsets of a modular space Xρ. Let T:A→B be a given nonself-mapping. Suppose that  (i)
A0 is proximal T-orbitally ρ-complete;(ii)
T(A0)⊆B0;(iii)
∃a∈A0 such that δp(a)<∞;(iv)
T is proximal quasi-contraction;(v)
ρ satisfies the Fatou property.Then, any sequence {xn}∈PP(a)ρ-converges to some z∈A0 such that
(15)ρ(xn-z)≤qnδp(a),
for all n≥1. Moreover, there exists w∈A0 such that
(16)ρ(w-Tz)=γ(A,B).Proof.
Let{xn}∈PP(a). From Lemma 8, we know that {xn} is ρ-Cauchy. Since A0 is proximal T-orbitally ρ-complete, then there exists z∈A0 such that {xn}ρ-converges to z. Again, by Lemma 8, we have
(17)ρ(xn-xn+m)≤qnδp(a),
for any n≥1 and m∈ℕ. Letting m→∞ in the above inequality and using the Fatou property, we obtain
(18)ρ(xn-z)≤qnδp(a),
for all n≥1. Now, since Tz∈B0, by the definition of B0, there exists some w∈A0 such that ρ(w-Tz)=γ(A,B).Now, we are ready to state and prove our main result.Theorem 10.
Suppose that the assumptions of the previous lemma are satisfied. Assumeρ(z-w)<∞ and ρ(a-w)<∞. Then, the ρ-limit z∈A0 of {xn}∈PP(a) is a best proximity point of T. Moreover, if u∈A0 is any best proximity point of T such that ρ(z-u)<∞, then one has z=u.Proof.
By Lemma9, we have
(19)ρ(w-Tz)=γ(A,B).
On the other hand, from the definition of {xn}, we have
(20)ρ(x1-Ta)=γ(A,B).
Since T is proximal quasi-contraction, we get that
(21)ρ(w-x1)≤qmax{ρ(a-z),ρ(z-w),ppppppρρ(a-x1),ρ(z-x1),ρ(a-w)}.
Using Lemmas 8 and 9, we obtain that
(22)ρ(w-x1)≤max{qδp(a),qρ(z-w),qρ(a-w)}.
Again, from the definition of {xn}, we have
(23)ρ(x2-Tx1)=γ(A,B).
Since T is proximal quasi-contraction, we get that
(24)ppρ(w-x2)≤qmax{ρ(z-x1),ρ(z-w),ρ(x1-x2),ppppppppppppppppppρ(z-x2),ρ(x1-w)}pppppppppp≤qmax{q2qδp(a),ρ(z-w),δp(x1),ppppppppppppppppppq2δp(a),ρ(x1-w)}pppppppppp≤qmax{q2qδp(a),ρ(z-w),qδp(a),ppppppppppppppppppq2δp(a),ρ(x1-w)}pppppppppp=qmax{qδp(a),ρ(z-w),ρ(x1-w)}(from(22))≤max{q2δp(a),qρ(z-w),q2ρ(a-w)}.
Thus, we proved that
(25)ρ(w-x2)≤max{q2δp(a),qρ(z-w),q2ρ(a-w)}.
Continuing this process, by induction, we get that
(26)ρ(w-xn)≤max{qnδp(a),qρ(z-w),qnρ(a-w)},
for all n≥1. Therefore, we have
(27)limsupn→∞ρ(xn-w)≤qρ(z-w).
Using the Fatou property, we get
(28)ρ(z-w)≤qρ(z-w),
which implies, since q<1, that ρ(z-w)=0; that is, z=w. Thus, from (19), we get that
(29)ρ(z-Tz)=γ(A,B).
Hence, z is a best proximity point of T.
Suppose now thatu∈A0 is a best proximity point of T such that ρ(z-u)<∞. Since T is proximal quasi-contraction, we obtain that
(30)ρ(z-u)≤qmax{ρ(z-u),ρ(z-z),ρ(u-u),ppppppppρ(z-u),ρ(u-z)}=qρ(z-u).
Since q<1, we have ρ(z-u)=0, which implies that u=z.Consider now the caseA=B. In this case, a best proximity point of T:A→B will be a fixed point of the self-mapping T.Definition 11.
We say thatA is T-orbitally ρ-complete if {Tna} is a ρ-Cauchy for every a∈A, then it is ρ-convergent to an element of A.Similarly to Ćirić [15] definition, Khamsi [16] introduced the concept of quasicontraction self-mappings in modular spaces.Definition 12.
The self-mappingT:A→A is said to be a quasicontraction if there exists a constant q∈(0,1) such that
(31)ρ(Tx-Ty)≤qmax{ρ(x-y),ρ(x-Tx),ρ(y-Ty),ppppppppρ(x-Ty),ρ(y-Tx)},
for all x,y∈A.From Theorem10, we can deduce the following result, that is, a slight extension of the fixed point theorem established by Khamsi in [16].Corollary 13.
Consider a self-mappingT:A→A, where A is a nonempty subset of Xρ. Suppose that the following conditions hold:(i)
A is T-orbitally ρ-complete;(ii)
∃a∈A such that sup{ρ(Tsa-Tra):s,r∈ℕ}<∞;(iii)
ρ satisfies the Fatou property;(iv)
T is quasi-contraction.Then, the sequence {Tna}ρ-converges to some z∈A. Moreover, if ρ(z-Tz)<∞ and ρ(a-Tz)<∞, then z is a fixed point of  T. If u∈A is a fixed point of  T with ρ(z-u)<∞, then u=z.



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