Vol. 25, No. 1 (2019), pp. 16–23.
ON BEST PROXIMITY POINTS OF GENERALIZED ALMOST-F -CONTRACTION MAPPINGS
Mahdi Salamatbakhsh
1, Robab Hamlbarani Haghi
2∗1,2Payame Noor University, Department of Mathematics, P.O. Box 19395-3697, Tehran, Iran,
mahdi.salamatbakhsh@gmail.com,∗robab.haghi@gmail.com
Abstract. We provide some results about best proximity points of generalized almost-F-contraction mappings in metric spaces which generalize and extend recent fixed point theorems. Also, we give an example to illustrate our main result.
Key words and Phrases: Best proximity point, generalized almost-F-contraction, propertyP.
Abstrak. Pada papaer ini kami menunjukkan beberapa hasil tentang titik-titik kedekatan yang terbaik dari pemetaan konraksi-hampir-F pada ruang metrik yang memperluas dan memperumum teorema titik tetap. Kami juga memberikan sebuah contoh untuk mengilustrasikan hasil utama kami.
Kata kunci: Titik kedekatan terbaik, kontraksi-hampir-F yang diperumum, sifatP.
1. Introduction
Fixed point theory focusses on the strategies for solving non-linear equations of the kind T x =xin which T is a self mapping defined on a subset of a metric space. But when T is a non-self mapping, it is probable that T x = x has no solution. So, for non self mapping, we try to find an approximate solution for the equation. On the other hand, best proximity point theorems offer an approximation solution that is optimal. It should be noted that best proximity point theorems furnish an approximate solution to the mentioned equation when T has no fixed point. Thus the best proximity point plays a crucial role in fixed point theory and consequently this research subject has attracted attention of many authors, as confirmed referring to [1, 2, 3, 4, 5]. In [1], the authors introduced the F−Suzuki contraction mappings and proved an existence and uniqueness theorem of fixed point. Following this direction of research and motivated by the works of [1, 3],
2000 Mathematics Subject Classification: 47H10, 54H25, 37C25.
Received: 22-06-2017, revised: 28-04-2018, accepted: 02-06-2018.
16
we introduce the new class of generalizedF−Suzuki contractions and prove a best proximity point theorem concerning such contractions. Moreover, an example is given to illustrate the usability of the new theory.
2. Main Results
To introduce our new results, it is fundamental to recall the definition of a best proximity point of a non-self mapping T and the notion of P−property.
Let Aand B be two nonempty subsets of a metric space (X, d). To facilitate the arguments, let
A0={a∈A:d(a, b) =d(A, B),for some b∈B}
B0={b∈B:d(a, b) =d(A, B),for somea∈A}
and
d(A, B) = inf{d(a, b) :a∈A, b∈B}.
Definition 2.1. Let A andB be two nonempty subsets of a metric space (X, d).
An element u ∈ A is said to be a best proximity point of the non-self mapping T:A→B if it satisfies the condition
d(u, T u) =d(A, B).
Definition 2.2. [4]LetAandB be two nonempty subsets of a metric space(X, d) withA0is nonempty. Then pair(A, B)is said to have the(P)-property if for each x1, x2∈Aandy1, y2∈B, the following implication holds:
d(x1, y1) =d(x2, y2) =d(A, B) ⇒ d(x1, x2) =d(y1, y2).
Throughout the article N, R+ and R denote the set of natural numbers, positive real numbers and real numbers, respectively.
Definition 2.3. [5]Let F:R+→Rbe a mapping satisfying:
(F1): F is strictly increasing, that is, α < β⇒F(α)< F(β)for all α, β∈R+,
(F2): for every sequence{αn} in R+ we have lim
n→∞αn = 0iff
n→∞lim F(αn) =−∞,
(F3): there exists a number k∈(0,1) such that lim
α→0αkF(α) = 0.
We denote with zthe family of all functionsF that satisfy the conditions (F1)− (F3).
Example 2.4. [1]The following function F:R+ →Rbelongs toz. F(α) = ln(α), F(α) = ln(α) +α, F(α) =−1
α12.
Definition 2.5. [5]Let (X, d)be a metric space. A mappingT:X →X is called an F-contraction on X if there exist F ∈zand τ >0 such that for all x, y∈X with d(T x, T y)>0, we have
τ+F(d(T x, T y))≤F(d(x, y)).
Definition 2.6. [2] Let (X, d) be a metric space and T:X → X be a mapping.
Then the mappingT is said to be an almost-F-contraction if there existF ∈zand τ >0, L≥0 such that
d(T x, T y)>0 ⇒ τ+F(d(T x, T y))≤F(d(x, y) +Ld(y, T x)), d(T x, T y)>0 ⇒ τ+F(d(T x, T y))≤F(d(x, y) +Ld(x, T y)).
Definition 2.7. Let (X, d)be a complete metric space and T:X →X be a map- ping. Mapping T is said to be a generalized almost-F-contraction if there exist F ∈z, τ >0 andL≥0 such that the following implication holds:
T x6=T y and 1
2d(x, T x)≤d(x, y) ⇒ τ+F(d(T x, T y))≤F(M(x, y) +Ld(y, T x)) whereM(x, y) = max{d(x, y), d(x, T x), d(y, T y)}.
In the sequel, letd∗(x, y) =d(x, y)−d(A, B) for allx, y∈X.
Definition 2.8. Let(A, B)be a pair of nonempty subsets of a metric space(X, d) and T:A → B be a mapping. Mapping T is said to be a non-self generalized almost-F-contraction if there existF ∈z, τ >0 andL≥0 such that the following implication holds:
T x6=T y and 0<1
2d∗(x, T x)≤d(x, y) ⇒
τ+F(d(T x, T y))≤F(M(x, y)−d(A, B) +Ld∗(y, T x)) (1) whereM(x, y) = max{d(x, y), d(x, T x), d(y, T y)}.
Note that the domain of functionF in Definition 2.8 is (0,+∞). Since M(x, y)−d(A, B)> d∗(x, T x)>0 for allx, y∈A,
so Definition 2.8 is well defined.
Now we are ready to state and prove our main result.
Theorem 2.9. Let (A, B) be a pair of nonempty closed subsets of a complete metric space X.Also assume that T: A→B is a generalized almost-F-contraction satisfying:
(i): the pair(A, B)has propertyP, (ii): T(A0)⊆B0,
(iii): F(α+)−F(α)< τ for anyα >0.
ThenT has a best proximity point.
Proof. Let x0 ∈ A0. Since T(A0) ⊆ B0, T x0 ∈ B0. So there exists x1 ∈ A0
such thatd(x1, T x0) =d(A, B). Since T x1 ∈B0, again for some x2 ∈A0, we get d(x2, T x1) =d(A, B).By continuing this process, we can find the sequence{xn}in A0 as{T xn}in B0 andd(xn+1, T xn) =d(A, B) for anyn∈N∪ {0}.Thus for any n∈N,we have d(xn, T xn−1) =d(xn+1, T xn) =d(A, B).The propertyP implies
d(xn, xn+1) =d(T xn, T xn−1) for alln∈N. (2) Note
d(xn−1, T xn−1)≤d(xn−1, xn) +d(xn, T xn−1)≤d(xn−1, xn) +d(A, B). (3) Similarly, d(xn, T xn)≤d(xn, xn+1) +d(A, B).Also,
M(xn−1, xn) = max{d(xn−1, xn), d(xn−1, T xn−1), d(xn, T xn)}
≤ max{d(xn−1, xn), d(xn, xn+1)}+d(A, B). (4) If for somen0, d(xn0, xn0+1) = 0,consequentlyd(T xn0−1, T xn0) = 0.SoT xn0−1= T xn0, henced(A, B) =d(xn0, T xn0). Thus the conclusion is immediate. So let for anyn≥0, d(xn, xn+1)>0.
From 3,
d∗(xn−1, T xn−1)≤d(xn−1, xn) for alln∈N. (5) Suppose that d∗(xn−1, T xn−1) > 0 for all n ≥ 1. Otherwise there is nothing to prove.
By 4 and 5, We get
τ+F(d(T xn−1, T xn)) ≤ F(M(xn−1, xn)−d(A, B) +Ld∗(xn, T xn−1))
= F(M(xn−1, xn))−d(A, B))
≤ F(max{d(xn−1, xn), d(xn, xn+1)}). (6) By 2 and 6, We derive
τ+F(d(xn, xn+1)) = τ+F(d(T xn−1, T xn))
≤ F(max{d(xn−1, xn), d(xn, xn+1)}.
Ifd(xn, xn+1)≥d(xn−1, xn),then
τ+F(d(xn, xn+1))≤F(d(xn, xn+1)) which is a contradiction and hence for anyn∈N,
d(xn, xn+1)< d(xn−1, xn). (7) Consequently the sequence{d(xn, xn+1)}is decreasing and bounded below. We are going to show that lim
n→∞d(xn, xn+1) = 0.
From 7, we deduce that
F(d(xn, xn+1))< F(d(xn, xn−1))−τ, for all n∈N
By repeating this process for anyn,we get
F(d(xn, xn+1))< F(d(x0, x1))−nτ (8) by tendingnto infinity, it follows that lim
n→∞F(d(xn, xn+1)) =−∞and so
n→∞lim d(xn, xn+1) = 0. (9)
We claim that{xn}is cauchy. By assumptionF3 in Definition 2.3 and 9, for some k∈(0,1),
n→∞lim d(xn, xn+1)kF(d(xn, xn+1)) = 0.
Note that by 8,
d(xn, xn+1)kF(d(xn, xn+1))≤d(xn, xn+1)kF(d(x0, x1)−nτ).
If n → ∞, we get nd(xn, xn+1)k = 0. Therefore, there exists n0 ∈ N such that d(xn+1, xn)< 1
n1k
for anyn > n0.Ifm > n > n0,then
d(xn, xm)≤
m−1
X
i=n
d(xi, xi+1)≤ X
n≥n0
1 n1k. So lim
n→∞d(xn, xm) = 0. From the completeness of X and closedness of A, there existsx∗∈A such thatxn→x∗.Consider
d(x∗, T x∗) ≤ d(x∗, T xn) +d(T xn, T x∗)
≤ d(x∗, xn+1) +d(xn+1, T xn) +d(T xn, T x∗)
≤ d(x∗, xn+1) +d(A, B) +d(T xn, T x∗).
By takingn→ ∞,
d∗(x∗, T x∗)≤lim sup
n→∞
d(T xn, T x∗). (10)
On the other hand,d(xn, T xn)≤d(xn, xn+1) +d(A, B),then
n→∞lim d(xn, T xn) =d(A, B).
SinceM(xn, x∗) = max{d(x∗, T x∗), d(xn, T xn), d(xn, x∗)},we deduce easily
n→∞lim M(xn, x∗) =d(x∗, T x∗). (11) By using the triangular inequality
d∗(xn, T xn) =d(xn, T xn)−d(A, B)
≤d(xn, xn+1) +d(xn+1, T xn)−d(A, B)
=d(xn, xn+1),
d∗(xn+1, T xn+1) =d(xn+1, T xn+1)−d(A, B)
≤d(xn+1, T xn) +d(T xn+1, T xn)−d(A, B)
=d(T xn, T xn+1) =d(xn+1, xn+2)
< d(xn, xn+1).
Above inequalities imply that 1
2(d∗(xn, T xn) +d∗(xn+1, T xn+1))≤d(xn, xn+1).
Suppose that the following inequalities hold for somen∈N: 1
2d∗(xn, T xn)> d(xn, x∗), 1
2d∗(xn+1, T xn+1)> d(xn+1, x∗).
Thend(xn, xn+1)≤d(xn, x∗) +d(xn+1, x∗)< d(xn, xn+1) which is a contradiction.
Then for anyn∈N∪ {0},either 1
2d∗(xn, T xn) ≤ d(xn, x∗) or (12) 1
2d∗(xn+1, T xn+1) ≤ d(xn+1, x∗). (13) If 12 holds, then
τ+F(d(T xn, T x∗))≤F(M(xn, x∗)−d(A, B) +Ld∗(T xn+1, x∗)). (14) In addition
d∗(T xn+1, x∗) = d(T xn+1, x∗)−d(A, B)
≤ d(T xn+1, xn+2) +d(xn+2, x∗)−d(A, B).
We conclude that lim
n→∞d∗(T xn+1, x∗) = 0.Then by 10, 11 and relation 14 and that F is increasing, we deduce that
τ+F(d∗(x∗, T x∗))≤τ+ lim sup
n→∞
F(d(T xn, T x∗))
≤lim sup
n→∞
F(M(xn, x∗)−d(A, B) +Ld∗(T xn+1, x∗)). (15) Putd∗(x∗, T x∗) =α.Note that the sequenceM(xn, x∗)−d(A, B)+Ld∗(T xn+1, x∗) tends toαfrom right. Ifα >0,then by takingn→ ∞in 14, we obtainτ+F(α)≤ F(α+), that is,F(α+)−F(α)≥τ which is a contradiction with the assumption.
Soα= 0, that isd(x∗, T x∗) =d(A, B). If 13 holds, the proof is similar.
Theorem 2.10. In addition to the hypothesis of Theorem 2.9, assume thatT also satisfies the following condition: there existG∈zand somel≥0, τ >0such that for allx, y∈Awith T x6=T y,
τ+G(d(T x, T y))≤G(d(x, y) +ld∗(x, T x)), then best proximity point of T is unique.
Proof. Letx0, y0 be disjoint best proximity points ofT.Then by propertyP,we obtaind(x0, y0) =d(T x0, T y0).SoT x06=T y0,
τ+G(d(x0, y0)) =τ+G(d(T x0, T y0))≤G(d(x0, y0)), which is a contradiction. Hence best proximity point ofT is unique.
Example 2.11. Let X= [0,4], A= [0,1]∪[3,4]andB = [32,2]. Define T x=
3
2, x∈[0,1]
2, x∈[3,4]
One can consider d(A, B) = 12, A0={1} andB0={32}. We show thatT satisfies in Theorem 2.9. First let x∈[0,1]andy∈[3,4]. we can observe that
1
2d∗(x, T x)≤d(x, y)
for any x∈[0,1] andy ∈[3,4] and also for anyx∈[3,4]and y∈[0,1]. Now, we are going to prove thatT satisfies in contraction condition 1 For this, takex∈[0,1]
andy ∈[3,4]. Let F ∈zis continuous. Replacing the values of x, y in relation 1, we have
τ+F(1
2) ≤ F(3
2)≤(F(max{3
2 −x, y−2, y−x} −1 2 + L(y−3
2))
(16) Note that when x∈[3,4]andy∈[0,1],then we obtain
τ+F(1
2)≤F(max{x−2,3
2 −y, x−y} − 1
2+L(2−y)).
Put L= 0andτ =F(1)−F(12). Then all condition of Theorem 2.9 are satisfied.
It is easy to see that T satisfies in Theorem 2.10 with F =G. Note thatx= 1 is the unique best proximity point of T.
Theorem 2.12. Let (A, B) be a pair of nonempty closed subsets of a complete metric space X.Also assume that T: A→B is a generalized almost-F-contraction satisfying:
(i): the pair(A, B)has propertyP, (ii): T(A0)⊆B0,
(iii): F is continuous.
ThenT has a best proximity point.
Proof. SinceF is continuous, thenF satisfies in assumption (iii) of Theorem 2.9.
So by Theorem 2.9,T has a best proximity point.
Theorem 2.13. Let (X, d) be a complete metric space andT:X →X be a gen- eralized almost-F-contraction such thatF(α+)−F(α)< τ for anyα >0. ThenT has a fixed point.
Proof. Put A =B =X in Theorem 2.9. Then by Theorem 2.9, T has a fixed point.
Corollary 2.14. Let(X, d)be a complete metric space andT:X →X be a gener- alized almost-F-contraction. Also assume thatF is continuous, ThenT has a fixed point.
Corollary 2.15. Let (X, d) be a complete metric space and let T: X → X be a F-contraction such thatF(α+)−F(α)< τ for anyα >0.ThenT has a fixed point.
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