GENERALIZED WARDOWSKI TYPE FIXED POINT THEOREMS VIA α-ADMISSIBLE FG-CONTRACTIONS IN b-METRIC SPACES∗

Vahid PARVANEHDepartment of Mathematics,Gilan-E-Gharb Branch,Islamic Azad University,Gilan-E-Gharb,Iran E-mail:zam.dalahoo@gmail.com

Nawab HUSSAIN Department of Mathematics,King Abdulaziz University P.O.Box 80203,Jeddah 21589,Saudi Arabia E-mail:nhusain@kau.edu.sa

Zoran KADELBURG Faculty of Mathematics,University of Belgrade,Studentski trg 16/IV,11000 Beograd,Serbia E-mail:kadelbur@matf.bg.ac.rs

GENERALIZED WARDOWSKI TYPE FIXED POINT THEOREMS VIA α-ADMISSIBLE FG-CONTRACTIONS IN b-METRIC SPACES∗

Vahid PARVANEH†Department of Mathematics,Gilan-E-Gharb Branch,Islamic Azad University,Gilan-E-Gharb,Iran E-mail:zam.dalahoo@gmail.com

Nawab HUSSAIN Department of Mathematics,King Abdulaziz University P.O.Box 80203,Jeddah 21589,Saudi Arabia E-mail:nhusain@kau.edu.sa

Zoran KADELBURG Faculty of Mathematics,University of Belgrade,Studentski trg 16/IV,11000 Beograd,Serbia E-mail:kadelbur@matf.bg.ac.rs

Recently,Wardowski[Fixed Point Theory Appl.,2012:94,2012]introduced and studied a new contraction called F-contraction to prove a fixed point result as a generalization of the Banach contraction principle.In this paper,we introduce an α-β-FG-contraction and generalize the Wardowski fixed point result in b-metric and ordered b-metric spaces.As an application of our results we deduce Suzuki type fixed point results for β-FG-contractions. Moreover,we discuss some illustrative examples to highlight the realized improvements.

α-admissible mapping;F-contraction;α-continuous function

2010 MR Subject Classification47H10;54H25

1　Introduction and Preliminaries

Banach contractive principle or Banach fixed point theorem is the most celebrated result in fixed point theory which illustrates that in a complete metric space,each contractive mapping has a unique fixed point.There were a great number of generalizations of this principle by using different forms of contractive conditions in various spaces.Recently,Wardowski[1]introduced a new contraction called F-contraction and proved a fixed point result as a generalization of the Banach contraction principle.Abbas et al.[2],Wardowski and Van Dung[3],as well as Piri and Kumam[4]generalized the concept of F-contraction and proved certain fixed and common fixed point results.

One of the interesting results which also generalizes the Banach contraction principle was given by Samet et al.[5]by defining α-ψ-contractive and α-admissible mappings.

Definition 1.1(see[5])Let T be a self-mapping on a set X and let α:X×X→[o,∞) be a function.We say that T is an α-admissible mapping if

Denote by Ψ the family of all nondecreasing functions ψ:[o,∞)→[o,∞)such that<∞for all t>o,where ψnis the n-th iterate of ψ.Samet et al.proved the following result.

Theorem 1.2(see[5])Let(X,d)be a complete metric space and let T be an α-admissible mapping.Assume that

for all x,y∈X,where ψ∈Ψ.Also,suppose that the following assertions hold:

(i)there exists xo∈X such that α(xo,Txo)≥1;

(ii)either T is continuous,or,for any sequence{xn}in X with α(xn,xn+1)≥1 for all n∈ℕ∪{o}such that xn→x as n→∞,we have α(xn,x)≥1 for all n∈ℕ∪{o}.

Then T has a fixed point.

Hussain et al.[6],as well as Karapinar et al.[7]extended this result as follows(see also [8,9]).

Definition 1.3(see[6])Let(X,d)be a metric space.Let α:X×X→[o,∞)and T:X→X be mappings.We say that T is an α-continuous mapping on(X,d),if,for given x∈X and sequence{xn},

Example 1.4(see[6])Let X=[o,∞)and d(x,y)=|x−y|be a metric on X.Assume that T:X→X and α:X×X→[o,+∞)be defined by

Clearly,T is not continuous,but T is α-continuous on(X,d).

Definition 1.5(see[7])Let f:X→X and α:X×X→[o,+∞).We say that f is a triangular α-admissible mapping if

Lemma 1.7(see[7])Let f be a triangular α-admissible mapping.Assume that there exists xo∈X such that α(xo,fxo)≥1.Define sequence{xn}by xn=fnxo.Then

Bahtin in[1o]and Czerwik in[11,12]introduced the concept of a b-metric space.Since then,several papers dealt with fixed point theory for single-valued and multi-valued operators in b-metric spaces(see,e.g.[13–19]).

Definition 1.8(see[11])Let X be a(nonempty)set and s≥1 be a given real number. A function d:X×X→ℝ+is a b-metric if,for all x,y,z∈X,the following conditions are satisfied:

(b1)d(x,y)=o iff x=y,

(b2)d(x,y)=d(y,x),

(b3)d(x,z)≤s[d(x,y)+d(y,z)].

In this case,the pair(X,d)is called a b-metric space.

Definition 1.9(see[15])Let(X,d)be a b-metric space.

(a)A sequence{xn}in X is called b-convergentif there exists x∈X such that d(xn,x)→o, as n→∞.In this case,we write

(b){xn}in X is said to be b-Cauchy if d(xn,xm)→o as n,m→∞.

(c)The b-metric space(X,d)is b-complete if every b-Cauchy sequence in X is b-convergent.

Note that a b-metric need not be a continuous function.The following example(see[16,17]) illustrates this fact.

Example 1.10Let X=ℕ∪{∞}and let d:X×X→ℝ be defined by

It can be checked that for all m,n,p∈X,we have

Thus,(X,d)is a b-metric space(with s=5/2).Let xn=2n for each n∈ℕ.Then

that is,xn→∞,but d(xn,1)=2/→5=d(∞,1)as n→∞.

Lemma 1.11(see[19])Let(X,d)be a b-metric space with parameter s≥1,and suppose that{xn}and{yn}are b-convergent to x and y,respectively.Then we have

In particular,if x=y,then we have=o.Moreover,for each z∈X,we have

In this paper,we introduce the concept of α-β-FG-contraction and obtain some fixed point results in b-metric and partially ordered b-metric spaces.Examples are provided in order to illustrate the results.Our results extend those of Wardowski and several other authors.

2　Fixed Point Results for α-Admissible β-FG-Contractions

Let s>1 be a fixed real number.We will consider the following classes of functions.

ΔFwill denote the set of all functions F:ℝ+→ℝ such that

(Δ1)F is continuous and strictly increasing;

(Δ2)for each sequence{tn}⊆R+,

Note that condition(F3)from[1,3]will not be used.

ΔG,βwill denote the set of pairs(G,β),where G:ℝ+→ℝ and β:[o,∞)→[o,1),such that

Example 2.1If F(t)=G(t)=lnt and β(t)=k∈(o,1)then F∈ΔFand(G,β)∈ΔG,β.

Definition 2.2Let(X,d)be a b-metric space with parameter s>1 and let T be a self-mapping on X.Also suppose that α:X×X→[o,∞)is a function.We say that T is an α-β-FG-contraction if for all x,y∈X with 1≤α(x,y)and d(Tx,Ty)>o,we have

where F∈ΔF,(G,β)∈ΔG,βand

Now we state and prove our main result of this section.

Theorem 2.3Let(X,d)be a complete b-metric space with parameter s>1.Let T: X→X be a self-mapping satisfying the following assertions:

(i)T is a triangular α-admissible mapping;

(ii)T is an α-β-FG-contraction;

(iii)there exists xo∈X such that α(xo,Txo)≥1;

(iv)T is α-continuous.

Then T has a fixed point.Moreover,T has a unique fixed point if α(x,y)≥1 for all x,y∈Fix(T).

ProofLet xo∈X be such that α(xo,Txo)≥1.For xo∈X,we define the sequence {xn}by xn=Tnxo=Txn−1.Now since,T is an α-admissible mapping then α(xo,x1)= α(xo,Txo)≥1.By continuing this process we have

for all n∈ℕ.Also,let there exists no∈ℕ such that,xno=xno+1.Then xnois a fixed point of T and we have nothing to prove.Hence,we assume that xn/=xn+1,i.e.,d(Txn−1,Txn)>o for all n∈ℕ∪{o}.Since,T is an α-β-FG-contraction,so we derive

which implies

Consequently,we deduce that

Continuing in this way,we get

By taking the limit as n→∞in(2.3),−∞,and since,F∈ΔFwe obtain

Next,we show that{xn}is a b-Cauchy sequence in X.

Suppose the contrary,that is,{xn}is not a b-Cauchy sequence.Then there exists ε>o for which we can find two subsequences{xmi}and{xni}of{xn}such that niis the smallest index for which

This means that

From(2.5)and using the triangular inequality,we get

Taking the upper limit as i→∞,we get

Also,from(2.6),

On the other hand,we have

Using(2.4)and(2.6)and taking the upper limit as i→∞,we get

Again,using the triangular inequality,we have

Taking the upper limit as i→∞in the above inequality and using(2.4)and(2.6),we get

Since T is a triangular α-admissible mapping,we have that α(xmi,xni)≥1,and so we can apply(2.1)to conclude that

Now,taking the upper limit as i→∞in(2.11)and using(2.7),(2.8),(2.1o)and(2.12),we have

which further implies that

Therefore,

which is impossible because of(2.5)and(2.12).

Thus,we prov that{xn}is a b-Cauchy sequence in the b-metric space(X,d).Since(X,d) is b-complete,the sequence{xn}b-converges to some z∈X,that is,

Suppose that z/=Tz.Then,from Lemma 1.11,as T is α-continuous,

Hence,we have d(Tz,z)=o and so Tz=z.Thus,z is a fixed point of T.

Let x,y∈Fix(T)where x/=y.Then froma contradiction.Hence,x=y.Therefore,T has a unique fixed point.

Taking G(t)=lnt,β(t)=k where k∈(o,1),and putting τ=−lnk in the above theorem, we obtain a generalization of the results from[1,3]in the setup of b-metric spaces.

Corollary 2.4Let(X,d)be a complete b-metric space and let F∈ΔF.Let T be a self-mapping on X and α:X×X→[o,∞)be a function such that the mapping T satisfies the following assertions:

(i)T is a triangular α-admissible mapping;

(ii)

for some τ>o and for all x,y∈X with 1≤α(x,y)and d(Tx,Ty)>o,where Msis defined by(2.2);

(iii)there exists xo∈X such that α(xo,Txo)≥1;

(iv)T is α-continuous.

Then T has a fixed point.Moreover,T has a unique fixed point when α(x,y)≥1 for all x,y∈Fix(T).

Taking F(t)=G(t)=lnt in the above theorem,we obtain the following result.

Corollary 2.5Let(X,d)be a complete b-metric space and let T be a self-mapping on X.Also suppose that α:X×X→[o,∞)is a function such that the mapping T satisfies the following assertions:

(i)T is a triangular α-admissible mapping;for all x,y∈X with 1≤α(x,y)and d(Tx,Ty)>o,where(lnt,β)∈ΔG,β,and Msis defined by(2.2);

(iii)there exists xo∈X such that α(xo,Txo)≥1;

(iv)T is α-continuous.

Then T has a fixed point.Moreover,T has a unique fixed point if α(x,y)≥1 for all x,y∈Fix(T).

Corollary 2.6Let(X,d)be a complete b-metric space.Let T:X→X be a selfmapping and suppose that α:X×X→[o,∞)is a function such that the mapping T satisfies the following assertions:

(i)T is a triangular α-admissible mapping;

for all x,y∈X with 1≤α(x,y)and d(Tx,Ty)>o,where(lnt,β)∈ΔG,β,and Msis defined by(2.2);

(iii)there exists xo∈X such that α(xo,Txo)≥1; (iv)T is α-continuous.

Then T has a fixed point.Moreover,T has a unique fixed point if α(x,y)≥1 for all x,y∈Fix(T).

Taking β(t)=k,where k∈(o,1)in the above corollary,and denoting k′=−k,we obtain the following result.

Corollary 2.7Let(X,d)be a complete b-metric space.Let T:X→X be a selfmapping and suppose that α:X×X→[o,∞)is a function such that the mapping T satisfies the following assertions:

(i)T is a triangular α-admissible mapping;

for all x,y∈X with 1≤α(x,y)and d(Tx,Ty)>o,where k′>o,and Msis defined by(2.2); (iii)there exists xo∈X such that α(xo,Txo)≥1; (iv)T is α-continuous.

Then T has a fixed point.Moreover,T has a unique fixed point if α(x,y)≥1 for all x,y∈Fix(T).

In the following theorem we replace the α-continuity of T by another condition(iv)′.

Theorem 2.8Let(X,d)be a complete b-metric space with parameter s>1.Let T: X→X be a self-mapping and suppose that α:X×X→[o,∞)is a function such that the mapping T satisfies the following assertions:

(i)T is a triangular α-admissible mapping;

(ii)T is an α-β-FG-contraction;

(iii)there exists xo∈X such that α(xo,Txo)≥1;

(iv)′if{xn}is a sequence in X such that α(xn,xn+1)≥1 with xn→x as n→∞,then 1≤α(Txn,x)holds for all n∈ℕ.

Then T has a fixed point.Moreover,T has a unique fixed point whenever α(x,y)≥1 for all x,y∈Fix(T).

ProofLet xo∈X be such that α(xo,Txo)≥1.As in the proof of Theorem 2.3 we can conclude that

where,xn+1=Txn.So,from(iv)′,

holds for all n∈ℕ.Hence,from(2.1)we deduce that

Suppose that z/=Tz.Then,from Lemma 1.11 we get

Now,using property(F1)of function F and(2.14),we obtain

Thus,z=Tz and z is a fixed point of T.The uniqueness follows similarly as in Theorem 2.3.

We provide now examples showing how our results can be used.

Example 2.9Let X=[o,+∞)be endowed with the b-metric d(x,y)=(x−y)2with s =2.Define T:X→X,α:X×X→[o,∞),β:[o,∞)→[o,1)and F,G:R+→R by

Let α(x,y)≥1,i.e.,x,y∈[o,1].Since Tw∈[o,1]for all w∈[o,1],it is α(Tx,Ty)≥1. That is,T is an α-admissible mapping.Let{xn}be a sequence in X such that α(xn,xn+1)≥1, with xn→x as n→∞.Then,Txn,x∈[o,1]for all n∈ℕ.That is,1≤α(Txn,x)holds for all n∈ℕ.Clearly,α(o,To)≥1.Let α(x,y)≥1.So,x,y∈[o,1]and hence we obtain

Thus,all the conditions of Theorem 2.8 hold and so T has a fixed point(which is z=o).

On the other hand,let x=o,y=2 and τ>o.Then,

where ρ(x,y)=|x−y|.That is,neither[1,Theorem 2.1]nor[3,Theorem 2.4]can be applied for this example(in standard metric,but also in the given b-metric),whatever function F is chosen.

Example 2.10Consider the set X={α,β,γ,δ,∊}and choose k,u>o such that u+1ok

d(x,x)=o and d(x,y)=d(y,x)for all x,y∈X,is a b-metric on X with parameter s=9/2. Define mappings T:X→X and α:X×X→ℝ by

Then,T is an α-continuous triangular α-admissible mapping.We will check that the contractive condition(2.13)of Corollary 2.5 is fulfilled with β given by β(t)=e−(u+t).Indeed,the only cases when α(x,y)≥1 and d(Tx,Ty)>o are when{x,y}={α,δ}or{x,y}={γ,δ}.In both cases,Ms(x,y)=1ok,d(Tx,Ty)=d(α,γ)=k and hence

That is,all the conditions of Corollary 2.5 are satisfied and T has a unique fixed point(z=c).

Note that Ms(β,∊)=d(Tβ,T∊)=d(β,∊)=4k and again the results of[1,3]cannot be used to obtain the conclusion.

3　Some Applications

3.1Application to Mappings in Ordered Spaces

Ran and Reurings initiated the study of fixed point results on partially ordered sets in [2o],where they gave many useful applications to matrix equations.Nieto and Rodr´ıguez-Lopez[21]followed this line of research and gave several applications to differential equations. Subsequently,many researchers have focused on different contractive conditions in complete metric(and other)spaces endowed with a partial order and obtained many fixed point results in such spaces,together with various applications.

From the above obtained theorems,we derive following new results in partially ordered metric spaces.

Theorem 3.1Let(X,d,≾)be a complete partially ordered b-metric space and let T: X→X be a self-mapping.Assume that the following assertions hold true:

(i)T is a nondecreasing ordered β-FG-contraction,i.e.,

for all x,y∈X with x≾y and d(Tx,Ty)>o,where(β,F,G)∈Δ,and Msis defined by(2.2);

(ii)there exists xo∈X such that xo≾Txo;

(iii)either for a given x∈X and a sequence{xn}

or,if{xn}is a sequence such that xn≾xn+1with xn→x as n→∞,then Txn≾x holds for all n∈ℕ.

Then T has a fixed point.

3.2Suzuki-Wardowski-Type Fixed Point Results

In this section,as an application of our results proved above,we deduce certain Suzuki-Wardowski type fixed point theorems(see[22–24]).

Theorem 3.2Let(X,d)be a complete b-metric space and let T be a continuous selfmapping on X.If for all x,y∈X with d(x,Tx)≤d(x,y)and d(Tx,Ty)>o we have

where(β,F,G)∈Δ,and Msis defined by(2.2),then T has a unique fixed point.

ProofDefine α:X×X→[o,∞)by α(x,y)=d(x,y)−d(x,Tx)+1 for all x,y∈X with d(x,y)−d(x,Tx)≥o and α(x,y)=1 for all x,y∈X with d(x,y)−d(x,Tx)o.Equivalently,let d(x,Tx)≤d(x,y)with d(Tx,Ty)>o. Then,we have that(3.1)holds true.This means that T is an α-β-FG-contraction mapping. Hence,all conditions of Theorem 2.3 hold and so T has a unique fixed point.

Corollary 3.3Let(X,d)be a complete b-metric space and let T be a continuous selfmapping on X.If for all x,y∈X with d(x,Tx)≤d(x,y)and d(Tx,Ty)>o we havewhere τ>o,F satisfies conditions(F1)and(F2),and Msis defined by(2.2),then T has a unique fixed point.

AcknowledgementsThis article was funded by the Deanship of Scientific Research (DSR),King Abdulaziz University,Jeddah.Therefore,the second author acknowledges with thanks DSR,KAU for financial support.The third author is thankful to the Ministry of Education,Science and Technological Development of Serbia,Grant No.174oo2.

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∗Jan 9,2015.The third author was supported by Ministry of Education,Science and Technological Development(174002).†

Vahid PARVANEH.