On Generalized Pata Type Contractions

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Article

On Generalized Pata Type Contractions

Geno Kadwin Jacob¹, M. S. Khan², Choonkil Park^3,* and Sungsik Yun^4,*

1 Department of Mathematics, Bharathidasan University, Tiruchirappalli-620 024, Tamil Nadu, India;

2 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, Muscat 123, Sultanate of Oman; [email protected]

3 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

4 Department of Financial Mathematics, Hanshin University, Gyeonggi-do 18101, Korea

* Correspondence: [email protected] (C.P.); [email protected] (S.Y.)

Received: 4 January 2018; Accepted: 30 January 2018; Published: 13 February 2018

Abstract:In this paper, the existence of fixed point for Pata type Zamfirescu mapping in a complete metric space is proved. Our result give existence of fixed point for a wider class of functions and also prove the existence of best proximity point to the result on “A fixed point theorem in metric spaces”, given by vittorino Pata.

Keywords:Zamfirescu mapping; Pata mapping; fixed point; proximity point MSC:primary 47H10; 54H25

1. Introduction

In 1922, Banach proved the existence of fixed point on a complete metric space (X,d). The mapping f has been considered to be a contraction andf takes points ofXto itself. Later, several interpretations for the existence of fixed point with weaker conditions to contraction mappings were given. Later, Kannan type and Chatterjea type mappings were introduced. These were significant type of mappings since they provided the existence of fixed point for non-continuous mappings for the first time in literature. In 1972, Zamfirescu [1] introduced and gave the existence of fixed point for a generalized contraction mapping. This class of functions generalized the results of [2–4]. All of these mappings were compared by [5].

Throughout the paper,Θdenotes the class of all increasing functionsΨ:[0, 1]→[0,∞)such that Ψis continuous at 0 withΨ(₀) =_0.

Definition 1. Let(X,d)be a metric space. A mapping f :X→X is said to be a Zamfirescu mapping if, for all x,y∈ X and a,b,c∈[0, 1), it satisfies the condition

d(f(x),f(y))≤maxn

ad(x,y),b 2

d(x,f(x)) +d(y,f(y)),c 2

d(x,f(y)) +d(y,f(x))^o. In a recent paper, Pata [6] obtained the following refinement of the classical Banach Contraction Principle. LetΛ≥0,α≥1,β∈[0,α]be any constants. For eache∈[0, 1],

d(f(x),f(y))≤(1−e)d(x,y) +_Λe^α_Ψ(e)^h1+kxk+kykⁱ^β, (1) wherekxk=d(x,x0)for arbitraryx0∈XandΨ∈Θ.

In this paper, we define Pata type Zamfirescu mappings and prove the existence of fixed point in metric spaces, which generalizes the result of [1,6]. We also prove a best proximity point result that generalizes the result of [6].

Mathematics2018,6, 25; doi:10.3390/math6020025 www.mdpi.com/journal/mathematics

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The following lemma is used to prove our results.

Lemma 1. Suppose(X,d) is a metric space. Let{xn}be a sequence in X such that d(xn,x_n+1) → 0as n→∞. If{xn}is not a Cauchy sequence, then there exist aδ>₀and sequences of positive integers{m_k}and {nk}with mk>nk>k such that d(xm_k,xn_k)≥δ, d(xm_k−1,xn_k)<δand

1. limk→∞d(xm_k−1,xn_k+1) =δ;

2. limk→∞d(xm_k,xn_k) =δ;

3. limk→∞d(xm_k−1,xn_k) =δ.

Using the above lemma, we get

limk→∞d(x_m_k₊₁,x_n_k₊₁) =δ, limk→∞d(xm_k,xn_k−1) =δ.

2. Existence of Fixed Point for Pata Type Zamfirescu Mappings

In this section, we prove the existence of unique fixed point for Pata type Zamfirescu mappings. Let (X,d) be a metric space. In the sequel, we writekxk=d(x,x0), wherex0is an arbitrary element inX.

Definition 2. Let(X,d)be a complete metric space. A mapping f :X→X is said to be a Pata type Zamfirescu mapping if for all x,y∈ X,Ψ∈_Θand for everye∈[0, 1], f satisfies the inequality

d(f(x),f(y))≤(1−e)M(x,y) +_Λe^α_Ψ(e)^h1+kxk+kyk+kf(x)k+kf(y)kⁱ^β, where M(x,y) = maxn

d(x,y),^d(x,f^(x))+d(y,₂ ^f^(y)),^d(x,f^(y))+d(y,f₂ ^(x))o

and Λ ≥ 0, α ≥ 1, β ∈ [0,α] are constants.

Now, we show that all Zamfirescu mappings fall under a particular case of Pata type Zamfirescu mappings. Letd=max{a,b,c}in Definition1and consider the Bernoulli’s inequality(1+rt)≤(1+t)^r, r≥1 andt∈[−1,∞). Then,

d(f(x),f(y))≤d max

d(x,y),d(x,f(x)) +d(y,f(y))

2 ,d(x,f(y)) +d(y,f(x)) 2

≤(1−e)max

2 ,d(x,f(y)) +d(y,f(x)) 2

+ (d+e−1)

1+max

kxk+kyk,kxk+kyk+kf(x)k+kf(y)k 2

≤(1−e)max

2 ,d(x,f(y)) +d(y,f(x)) 2

+d

1+^e−1 d

[1+kxk+kyk+kf(x)k+kf(y)k]

≤(1−e)max

2 ,d(x,f(y)) +d(y,f(x)) 2

+de¹^d [1+kxk+kyk+kf(x)k+kf(y)k]

≤(1−e)max

2 ,d(x,f(y)) +d(y,f(x)) 2

+dee¹⁻^d^d [1+kxk+kyk+kf(x)k+kf(y)k].

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Comparing this with Pata type Zamfirescu mappings, we have that Zamfirescu mapping is actually a special case of Pata type Zamfirescu mappings withΛ=d,Ψ(e) =e

1−d

d andβ=1. It is also clear that mappings given by [6–8] were also Pata type Zamfirescu mappings.

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Now, we prove the main result of this paper.

Theorem 1. Let(X,d)be a complete metric space and let f : X →X be a Pata type Zamfirescu mapping.

Then, f has a unique fixed point in X.

Proof. Letx0be an arbitrary element inX. Definexn+1 = f(xn)andcn = d(xn,x0). To prove that d(xn+1,xn)is a nonincreasing sequence, takee=0. Therefore,

d(x_n+1,xn)≤maxn

d(xn,xn−1),d(xn,f(xn)) +d(xn−1,f(xn−1))

2 ,

d(xn,f(xn−1)) +d(xn−1,f(xn)) 2

o

≤maxn

d(xn,xn−1),d(xn,x_n+1) +d(xn−1,xn)

2 ,d(xn−1,x_n+1) 2

o

≤maxn

d(xn,xn−1),d(xn,xn+1) +d(xn−1,xn)

2 ,d(xn,xn−1) +d(xn+1,xn) 2

o

≤maxn

d(xn,xn−1),d(xn,xn+1) +d(xn−1,xn) 2

o . Therefore,d(x_n+1,xn)≤d(xn,xn−1)≤ · · · ≤d(x₁,x0) =c₁. Claim (1):{cn}is bounded.

cn =d(xn,x0)

≤d(xn,xn+1) +d(xn+1,x1) +d(x1,x0)

≤(1−e)_maxⁿd(xn,x₀)_,^d(xn,x_n+1) +d(x₀,x₁)

2 ,d(xn,x₁) +d(x₀,x_n+1) 2

o

+2c₁+_Λe^α_Ψ(e)^h1+kxnk+0+kx_n+1k+kx₁kⁱ^β

≤(1−e)maxn

d(xn,x₀),d(xn,x_n+1) +d(x0,x₁)

2 ,d(xn,x₁) +d(x0,x_n+1) 2

o

+2c₁+_Λe^α_Ψ(e)^h1+kxnk+d(xn+1,xn) +d(xn,x0) +kx₁kⁱ^β

≤(1−e)maxn

d(xn,x0),d(xn,xn+1) +d(x0,x1)

2 ,d(xn,x1) +d(x0,xn+1) 2

o

+2c1+_Λe^α_Ψ(e)^h1+kxnk+kx1k+kxnk+kx1kⁱ^β

≤(1−e)maxn

cn,c1,d(xn,x0) +d(x1,x0) +d(xn+1,xn) +d(xn,x0) 2

o

+2c1+Λe^αΨ(e)^h1+kxnk+kxnk+kx1k+kx1kⁱ^β

≤(1−e)_max{cn,c1,cn+_c₁}+_2c₁+_ΛeΨ(e)^h₁+_2c_n+_2c₁ⁱ^β

≤(1−e)[cn+c1] +2c1+Λe^αΨ(e)^h1+2cn+2c1

iα

.

By the same reasoning as in [8], it follows that the sequence{cn}is bounded.

Let lim

n→∞d(xn,xn−1) =d. Sinced(xn,xn−1)is nonincreasing,

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d(xn+1,xn) =d(f(xn),f(xn−1))

≤(1−e)maxn

d(xn,xn−1),d(xn,x_n+1) +d(xn−1,xn)

2 ,

d(xn,xn) +d(xn−1,x_n+1) 2

o

+_Λe^α_Ψ(e)^h1+kxnk+kxn−1k+kx_n+1k+kxnkⁱ^β

≤(1−e)maxn

d(xn,xn−1),d(xn,x_n+1) +d(xn−1,xn) 2

o+_KeΨ(e). Now, asn→∞, we getd≤KΨ(e)and henced=0.

Claim (2): The sequence{xn}is Cauchy. Suppose that{xn}is not a Cauchy sequence. Then, by Lemma1, there exist subsequences{xn_k}and{xm_k}of{xn}withn_k>m_k>ksuch that

δ≤d(xm_k,xn_k) =d(f(xm_k−1),f(xn_k−1))

≤(1−e)maxn

d(xm_k−1,xn_k−1),d(xm_k−1,xm_k) +d(xn_k−1,xn_k)

2 ,

d(xn_k−1,xm_k) +d(xm_k−1,xn_k) 2

o+_KeΨ(e)

≤(1−e)maxn

d(xm_k−1,xn_k) +d(xn_k,xn_k−1),d(x_m_k−1,xm_k) +d(x_n_k−1,xn_k)

2 ,

d(xn_k−1,xm_k) +d(xm_k−1,xn_k) 2

o

+KeΨ(e).

Now, ask→_{∞, we get}δ≤_KΨ(e), which is a contradiction. Therefore,{xn}is Cauchy. SinceX is complete, there existsx ∈Xsuch thatxn →x. Now, for alln∈Nand fore=0, we obtain

d(f(x),x)≤d(f(x),x_n+1) +d(x_n+1,x)

≤maxn

d(x,xn),d(x,f(x)) +d(xn,xn+1)

2 ,

d(x,x_n+1) +d(xn,f(x)) 2

o+d(x_n+1,x).

Asn→∞, the above inequality concludes thatd(f(x),x)≤ ¹₂d(f(x),x). Hence,xis a fixed point of f. For the uniqueness of fixed point, suppose thatxandyare fixed points ofF. Then,

d(_f(_x)_,_f(_y))≤(1−e)_maxⁿ_d(_x,_y)_,^d(x,f(x)) +d(y,f(y))

2 ,

d(_x,_f(_y)) +_d(_y,_f(_x)) 2

o

+KeΨ(e).

Therefore, we getd(x,y)≤_KΨ(e)and hencex=y. Therefore,f has a unique fixed point inX.

Corollary 1. Let(X,d)be a complete metric space and f₁:X→X be a Zamfirescu mapping satisfying, for all x,y∈ X and a,b,c∈[_{0, 1}), the inequality

d(f1(x),f1(y))≤maxn

ad(x,y),b 2

d(x,f1(x)) +d(y,f1(y)),c 2

d(x,f1(y)) +d(y,f1(x))^o. Then, f1has a unique fixed point in X.

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Proof. Using inequality (2), we obtain that d(f₁(x),f₁(y))≤(1−e)maxn

d(x,y),d(x,f₁(x)) +d(y,f₁(y))

2 ,d(x,f₁(y)) +d(y,f₁(x)) 2

o

+dee¹⁻^d^dh

1+kxk+kyk+kf₁(x)k+kf₁(y)kⁱ. Therefore, by Theorem1, f1has a unique fixed point inX.

Corollary 2. Let(X,d)be a complete metric space and f : X→X be a mapping which satisfies(1). Then, f has a unique fixed point in X.

3. Existence of Best Proximity Point for Pata Type Proximal Contraction

In this section, we define Pata type proximal mappings and prove the existence of best proximity points. Our work generalizes the result of [6]. LetAandBbe two closed subsets of a complete metric space (X,d). We denote byA0the subset ofAdefined by

A0={x∈ A: d(x,y) =d(A,B), for somey∈B}. Similarly, we denote byB₀the subset ofBdefined by

B0={y∈B: d(_x,_y) =_d(_A,_B)_{, for some}_x ∈A}.

Throughout this section, we assume thatA0andB0are closed subsets ofAandB.

Definition 3. A mapping f :A→B is said to be a Pata type proximal contraction if for all x,y∈ A,Ψ∈Θ and for everye∈[0, 1], f satisfies the inequality

d(u,v)≤(1−e)d(x,y) +_Λe^α_Ψ(e)^h1+kxk+kykⁱ^β, where d(f(x)_,u) =d(f(y)_,v) =d(A,B)andΛ≥0,α≥1,β∈[_0,α]are any constants.

Theorem 2. Let A and B be two closed subsets of a complete metric space(X,d). Let f :A→B be a Pata type proximal contraction such that f(A0)⊂B0. Then, f has a best proximity point in A.

Proof. Letx0be an element inA0. Then, f(x0) ∈ B0and so there exists an element x₁ ∈ A0such that d(f(x₀),x₁) = d(A,B). Similiarly, define x_n+1 ∈ A₀such thatd(x_n+1,f(xn)) = d(A,B)and cn =d(xn,x0). Then, we get

d(xn,xn+1))≤(1−e)d(xn−1,xn) +Λe^αΨ(e)^h1+kxnk+kxn−1kⁱ^β (3) for alle∈[0, 1].

In particular, letting e = 0 in the inequality (3), we obtain that d(x_n+1,xn) is a nonincreasing sequence.

Therefore,d(xn+1,xn)≤d(xn,xn−1)≤...≤d(x1,x0) =c1.

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Next, we prove that{cn}is bounded:

cn =d(xn,x0)

≤d(_x_n_,_x_n+1) +_d(_x_n+1_,_x₁) +_d(_x₁_,_x₀)

≤(1−e)d(xn,x0) +2c1+_Λe^α_Ψ(e)^h1+kxnk+kx0kⁱ^β

≤(1−e)cn+2c1+ΛeΨ(e)^h1+cn+c1

iβ

. Now, as in [6], we see that{cn}is bounded.

Let lim

n→∞d(xn,xn−1) =d. Sinced(xn,xn−1)is nonincreasing,

d(xn+1,xn)≤(1−e)d(xn,xn−1) +Λe^αΨ(e)^h1+kxnk+kxn−1kⁱ^β

≤(1−e)d(xn,xn−1) +_KeΨ(e).

Asn→_{∞, we get}d≤_KΨ(e)and henced=0. Now, we claim that{xn}is a Cauchy sequence.

Suppose that{xn}is not a Cauchy sequence. Then, by Lemma1, there exist subsequences{xn_k}and {xm_k}of{xn}withnk>mk>ksuch that

δ≤d(xm_k,xn_k)≤(1−e)d(x_m_k−1,x_n_k−1) +_KeΨ(e)

≤(1−e)^hd(xm_k−1,xn_k) +d(xn_k,xn_k−1)ⁱ+_KeΨ(e).

Now, ask→_{∞, we get}δ≤_KΨ(e), which is a contradiction. Therefore,{xn}is a Cauchy sequence.

SinceXis complete, there existsx∈Xsuch that{xn} →x. Letd(u,f(x)) =d(A,B)ande=0. Then, for alln∈N, we get

d(u,x)≤d(u,xn+1) +d(xn+1,x)

≤d(x,xn) +d(x_n+1,x)_, which concludes thatd(u,x) =0. Hence,xis a proximity point of f.

Suppose thatxandyare proximity points of f. Then,

d(x,y)≤(1−e)d(x,y) +KeΨ(e).

Therefore, we getd(x,y)≤KΨ(e)and hencex=y. Thus,xis the only proximity point offinA.

Corollary 3. Let(X,d)be a complete metric space and f :X→X be a mapping which satisfies condition(1). Then, f has a unique fixed point in X.

Proof. The proof follows directly from the previous theorem, whenA=B.

In complete metric space setting, the following example shows the existence of best proximity of Pata type proximal contraction.

Example 1. Consider A = {(0,a)|a∈[0, 1]} and B = {(1,b)|b∈[0, 1]} on R² ^under ¹−norm with d(A,B) =1. For x∈ A, define f :A→B as f(0,x) = (1,^x₄²),Λ=α=β=1, and forδ∈(0,¹₂)

ψ(x) = (_x

2 if x∈[0,δ), 1if x∈[δ, 1].

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It has to be shown that, for alle∈[0, 1],f satisfies the inequality of Pata type proximal contraction (i.e.,)...







f or e=0 =⇒d(u,v)≤d(x,y),

f or e∈(0, 1] =⇒d(u,v)≤(1−e)d(x,y) +_eΨ(e)^h1+kxk+kykⁱ. (4) The next inequality shows that f satisfies the first case ife=0. Fore=0and for all(0,x),(0,y)∈A,

d(u,v) =d 0,x²

4

, 0,y² 4

=0 +^x

2

4 −^y

2

4

= ^x 2 +^y

2 x

2 −^y 2

≤d(x,y). The following inequalities shows that f satisfies second case ife∈(0, 1]. Fore∈(0,δ)and for all(0,x),(0,y)∈ A,

d(u,v) =d 0,x²

4

, 0,y² 4

=0 +^x

2

4 −^y

2

4

≤ ¹ 2 x−y

= ¹ 2d(x,y)

= (1−e)d(x,y) +¹

2+e−1 d(x,y)

= (1−e)d(x,y) + ¹ 2

1+^e−1

1 2

d(x,x0) +d(y,x0)

≤(₁−e)_d(_x,_y) + ¹ 2e²

d(_x,_x₀) +_d(_y,_x₀)

= (1−e)d(x,y) +eΨ(e)^h1+kxk+kykⁱ. Fore∈[δ, 1]and for all(0,x),(0,y)∈ A,

d(u,v) =d 0,x²

4

, 0,y² 4

=0 +^x

2

4 −^y

2

4

≤d(x,y)

= (1−e)d(x,y) +ed(x,y)

≤(1−e)d(x,y) +e

d(x,x0) +d(y,x0)

≤(1−e)d(x,y) +eΨ(e)^h1+kxk+kykⁱ.

Therefore, f is Pata type proximal contraction and hence there exists a best proximity point(0, 0)in A.

Author Contributions:Geno Kadwin Jacob , M. S. Khan, Choonkil Park and Sungsik Yun conceived and wrote the paper.

Conflicts of Interest:The authors declare no conflict of interest.

References

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c

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