Common fixed points of almost generalized contractive mappings in ordered metric spaces

Rezaei Roshan et al. Fixed Point Theory and Applications 2013, 2013:159http://www.fixedpointtheoryandapplications.com/content/2013/1/159

RESEARCH Open Access

Common fixed points of almost generalized(ψ ,ϕ)s-contractive mappings in orderedb-metric spacesJamal Rezaei Roshan1, Vahid Parvaneh2*, Shaban Sedghi1, Nabi Shobkolaei3 and Wasfi Shatanawi4

*Correspondence:[email protected] of Mathematics,Gilan-E-Gharb Branch, Islamic AzadUniversity, Gilan-E-Gharb, IranFull list of author information isavailable at the end of the article

AbstractIn this paper, we introduce the notion of almost generalized (ψ ,ϕ)s-contractivemappings and we establish some fixed and common fixed point results for this classof mappings in ordered complete b-metric spaces. Our results generalize severalwell-known comparable results in the literature. Finally, two examples support ourresults.MSC: 54H25; 47H10; 54E50

Keywords: fixed point; generalized contractions; complete ordered b-metric spaces

1 IntroductionA fundamental principle in computer science is iteration. Iterative techniques are usedto find roots of equations and solutions of linear and nonlinear systems of equations anddifferential equations. So, the attractiveness of the fixed point iteration is understandableto a large number of mathematicians.The Banach contraction principle [] is a very popular tool for solving problems in non-

linear analysis. Some authors generalized this interesting theorem in different ways (see,e.g., [–]).Berinde in [, ] initiated the concept of almost contractions and obtained many in-

teresting fixed point theorems for a Ćirić strong almost contraction.Now, let us recall the following definition.

Definition [] A single-valued mapping f : X → X is called a Ćirić strong almost con-traction if there exist a constant α ∈ [, ) and some L ≥ such that

d(fx, fy) ≤ αM(x, y) + Ld(y, fx)

for all x, y ∈ X, where

M(x, y) = max

{d(x, y),d(x, fx),d(y, fy), d(x, fy) + d(y, fx)

}.

Babu in [] introduced the class of mappings which satisfy condition (B).

© 2013 Rezaei Roshan et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

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Definition [] Let (X,d) be a metric space. A mapping f : X → X is said to satisfycondition (B) if there exist a constant δ ∈ (, ) and some L ≥ such that

d(fx, fy) ≤ δd(x, y) + Lmin{d(x, fx),d(x, fy),d(y, fx)

}

for all x, y ∈ X.

Moreover, Babu in [] proved the existence of a fixed point for such mappings on com-plete metric spaces.Ćirić et al. in [] introduced the concept of almost generalized contractive condition

and they proved some existing results.

Definition [] Let (X,�) be a partially ordered set. Two mappings f , g : X → X aresaid to be strictly weakly increasing if fx ≺ gfx and gx≺ fgx, for all x ∈ X.

Definition [] Let f and g be two self mappings on ametric space (X,d). Then they aresaid to satisfy almost generalized contractive condition, if there exist a constant δ ∈ (, )and some L ≥ such that

d(fx, fy) ≤ δ max

{d(x, y),d(x, fx),d(y, gy), d(x, gy) + d(y, fx)

}

+ Lmin{d(x, fx),d(x, gy),d(y, fx)

}(.)

for all x, y ∈ X.

Ćirić et al. in [] proved the following theorems.

Theorem Let (X,�) be a partially ordered set and suppose that there exists a metric don X such that the metric space (X,d) is complete. Let f : X → X be a strictly increasingcontinuous mapping with respect to �. Suppose that there exist a constant δ ∈ [, ) andsome L ≥ such that

d(fx, fy) ≤ δM(x, y) + Lmin{d(x, fx),d(x, fy),d(y, fx)

}

for all comparable elements x, y ∈ X, where

M(x, y) = max

{d(x, y),d(x, fx),d(y, fy),

d(x, fy) + d(y, fx)

}.

If there exists x ∈ X such that x � fx, then f has a fixed point in X.

Theorem Let (X,�) be a partially ordered set and suppose that there exists a metric don X such that themetric space (X,d) is complete. Let f , g : X → X be two strictly weakly in-creasingmappings which satisfy (.)with respect to�, for all comparable elements x, y ∈ X.If either f or g is continuous, then f and g have a common fixed point in X .

Khan et al. [] introduced the concept of an altering distance function as follows.

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Definition [] The function ϕ : [, +∞)→ [, +∞) is called an altering distance func-tion, if the following properties hold:. ϕ is continuous and non-decreasing.. ϕ(t) = if and only if t = .

So far, many authors have studied fixed point theorems which are based on alteringdistance functions (see, e.g., [, –]).The concept of a b-metric space was introduced by Czerwik in []. After that, several

interesting results about the existence of a fixed point for single-valued and multi-valuedoperators in b-metric spaces have been obtained (see [, –]). Pacurar [] provedsome results on sequences of almost contractions and fixed points in b-metric spaces. Re-cently, Hussain and Shah [] obtained some results on KKMmappings in cone b-metricspaces.Consistent with [] and [], the following definitions and results will be needed in the

sequel.

Definition [] Let X be a (nonempty) set and s ≥ be a given real number. A functiond : X ×X → R+ is a b-metric iff for all x, y, z ∈ X, the following conditions hold:

(b) d(x, y) = iff x = y,(b) d(x, y) = d(y,x),(b) d(x, z) ≤ s[d(x, y) + d(y, z)].

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

It should be noted that, the class of b-metric spaces is effectively larger than the class ofmetric spaces, since a b-metric is a metric, when s = .Here, we present an example to show that in general, a b-metric need not necessarily be

a metric (see also [, p.]):

Example Let (X,d) be ametric space and ρ(x, y) = (d(x, y))p, where p > is a real number.We show that ρ is a b-metric with s = p–.Obviously, conditions (b) and (b) of Definition are satisfied.If < p < ∞, then the convexity of the function f (x) = xp (x > ) implies

(a + b

)p≤

(ap + bp

),

and hence, (a + b)p ≤ p–(ap + bp) holds.Thus, for each x, y, z ∈ X,

ρ(x, y) =(d(x, y)

)p ≤ (d(x, z) + d(z, y)

)p ≤ p–((d(x, z)

)p + (d(z, y)

)p)= p–

(ρ(x, z) + ρ(z, y)

).

So, condition (b) of Definition is also satisfied and ρ is a b-metric.

Definition [] Let (X,d) be a b-metric space. Then a sequence {xn} in X is called:

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(a) b-convergent if and only if there exists x ∈ X such that d(xn,x) → , as n→ +∞. Inthis case, we write limn→∞ xn = x.

(b) b-Cauchy if and only if d(xn,xm) → as n,m → +∞.

Proposition (See Remark . in []) In a b-metric space (X,d) the following assertionshold:

(p) A b-convergent sequence has a unique limit.(p) Each b-convergent sequence is b-Cauchy.(p) In general, a b-metric is not continuous.

Definition [] The b-metric space (X,d) is b-complete if every b-Cauchy sequence inX b-converges.

It should be noted that, in general a b-metric function d(x, y) for s > is not jointly con-tinuous in all its variables. The following example is an example of a b-metric which is notcontinuous.

Example (see Example in []) Let X =N∪ {∞} and let D : X ×X →R be defined by

D(m,n) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

ifm = n,| m –

n | ifm,n are even ormn = ∞, ifm and n are odd andm = n, otherwise.

Then it is easy to see that for allm,n,p ∈ X, we have

D(m,p) ≤ (D(m,n) +D(n,p)

).

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

D(n,∞) =n

→ , as n→ ∞,

that is, xn → ∞, but D(xn, ) = �D(∞, ) as n→ ∞.

Aghajani et al. [] proved the following simple lemmaabout the b-convergent sequences.

Lemma Let (X,d) be a b-metric space with s ≥ , and suppose that {xn} and {yn}b-converge to x, y, respectively. Then, we have

sd(x, y)≤ lim inf

n→∞ d(xn, yn)≤ lim supn→∞

d(xn, yn) ≤ sd(x, y).

In particular, if x = y, then, limn→∞ d(xn, yn) = .Moreover, for each z ∈ X we have

sd(x, z) ≤ lim inf

n→∞ d(xn, z) ≤ lim supn→∞

d(xn, z) ≤ sd(x, z).

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In this paper, we introduce the notion of an almost generalized (ψ ,ϕ)s-contractive map-ping and we establish some results in complete ordered b-metric spaces, where ψ and ϕ

are altering distance functions. Our results generalize Theorems and and all results in[] and several comparable results in the literature.

2 Main resultsIn this section, we define the notion of almost generalized (ψ ,ϕ)s-contractive mappingand prove our new results. In particular, we generalize Theorems ., . and . of Ćirićet al. in [].Let (X,�,d) be an ordered b-metric space and let f : X → X be a mapping. Set

Ms(x, y) = max

{d(x, y),d(x, fx),d(y, fy), d(x, fy) + d(y, fx)

s

}

and

N(x, y) = min{d(x, fx),d(y, fx)

}.

Definition Let (X,d) be a b-metric space. We say that a mapping f : X → X is an al-most generalized (ψ ,ϕ)s-contractivemapping if there exist L ≥ and two altering distancefunctions ψ and ϕ such that

ψ(sd(fx, fy)

) ≤ ψ(Ms(x, y)

)– ϕ

(Ms(x, y)

)+ Lψ

(N(x, y)

)(.)

for all x, y ∈ X.

Now, let us prove our first result.

Theorem Let (X,�) be a partially ordered set and suppose that there exists a b-metricd on X such that (X,d) is a b-complete b-metric space. Let f : X → X be a non-decreasingcontinuous mapping with respect to �. Suppose that f satisfies condition (.), for all com-parable elements x, y ∈ X. If there exists x ∈ X such that x � fx, then f has a fixed point.

Proof Let x ∈ X. Then, we define a sequence (xn) in X such that xn+ = fxn, for all n ≥ .Since x � fx = x and f is non-decreasing, we have x = fx � x = fx. Again, as x � xand f is non-decreasing, we have x = fx � x = fx. By induction, we have

x � x � · · · � xn � xn+ � · · · .

If xn = xn+, for some n ∈ N, then xn = fxn and hence xn is a fixed point of f . So, we mayassume that xn = xn+, for all n ∈N. By (.), we have

ψ(d(xn,xn+)

) ≤ ψ(sd(xn,xn+)

)= ψ

(sd(fxn–, fxn)

)≤ ψ

(Ms(xn–,xn)

)– ϕ

(Ms(xn–,xn)

)+ Lψ

(N(xn–,xn)

), (.)

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where

Ms(xn–,xn) = max

{d(xn–,xn),d(xn–, fxn–),d(xn, fxn),

d(xn–, fxn) + d(xn, fxn–)s

}

= max

{d(xn–,xn),d(xn,xn+),

d(xn–,xn+)s

}

≤ max

{d(xn–,xn),d(xn,xn+),

sd(xn–,xn) + sd(xn,xn+)s

}

= max

{d(xn–,xn),d(xn,xn+),

d(xn–,xn) + d(xn,xn+)

}

= max{d(xn–,xn),d(xn,xn+)

}(.)

and

N(xn–,xn) = min{d(xn–, fxn–),d(xn, fxn–)

}= min

{d(xn–,xn),

}= . (.)

From (.)-(.) and the properties of ψ and ϕ, we get

ψ(d(xn,xn+)

) ≤ ψ(max

{d(xn–,xn),d(xn,xn+)

})– ϕ

(max

{d(xn–,xn),d(xn,xn+)

})< ψ

(max

{d(xn–,xn),d(xn,xn+)

}). (.)

If

max{d(xn–,xn),d(xn,xn+)

}= d(xn,xn+),

then by (.) we have

ψ(d(xn,xn+)

) ≤ ψ(d(xn,xn+)

)– ϕ

(d(xn,xn+)

)< ψ

(d(xn,xn+)

),

which gives a contradiction. Thus,

max{d(xn–,xn),d(xn,xn+)

}= d(xn–,xn).

Therefore (.) becomes

ψ(d(xn,xn+)

) ≤ ψ(d(xn,xn–)

)– ϕ

(d(xn–,xn)

)< ψ

(d(xn,xn–)

). (.)

Since ψ is a non-decreasing mapping, {d(xn,xn+) : n ∈ N ∪ {}} is a non-increasing se-quence of positive numbers. So, there exists r ≥ such that

limn→∞d(xn,xn+) = r.

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Letting n → ∞ in (.), we get

ψ(r)≤ ψ(r) – ϕ(r)≤ ψ(r).

Therefore, ϕ(r) = , and hence r = . Thus, we have

limn→∞d(xn,xn+) = . (.)

Next, we show that {xn} is a b-Cauchy sequence in X. Suppose the contrary, that is, {xn} isnot a b-Cauchy sequence. Then there exists ε > for which we can find two subsequences{xmi} and {xni} of {xn} such that ni is the smallest index for which

ni >mi > i, d(xmi ,xni ) ≥ ε. (.)

This means that

d(xmi ,xni–) < ε. (.)

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

ε ≤ d(xmi ,xni )

≤ sd(xmi ,xmi–) + sd(xmi–,xni )

≤ sd(xmi ,xmi–) + sd(xmi–,xni–) + sd(xni–,xni ).

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

ε

s≤ lim sup

i→∞d(xmi–,xni–).

On the other hand, we have

d(xmi–,xni–) ≤ sd(xmi–,xmi ) + sd(xmi ,xni–).

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

lim supi→∞

d(xmi–,xni–) ≤ εs.

So, we have

ε

s≤ lim sup

i→∞d(xmi–,xni–) ≤ εs. (.)

Again, using the triangular inequality, we have

d(xmi–,xni ) ≤ sd(xmi–,xni–) + sd(xni–,xni ),

ε ≤ d(xmi ,xni ) ≤ sd(xmi ,xmi–) + sd(xmi–,xni )

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and

ε ≤ d(xmi ,xni ) ≤ sd(xmi ,xni–) + sd(xni–,xni ).

Taking the upper limit as i→ ∞ in the first and second inequalities above, and using (.)and (.) we get

ε

s≤ lim sup

i→∞d(xmi–,xni ) ≤ εs. (.)

Similarly, taking the upper limit as i → ∞ in the third inequality above, and using (.)and (.), we get

ε

s≤ lim sup

i→∞d(xmi ,xni–) ≤ ε. (.)

From (.), we have

ψ(sd(xmi ,xni )

)= ψ

(sd(fxmi–, fxni–)

)≤ ψ

(Ms(xmi–,xni–)

)– ϕ

(Ms(xmi–,xni–)

)+ Lψ

(N(xmi–,xni–)

), (.)

where

Ms(xmi–,xni–) = max

{d(xmi–,xni–),d(xmi–, fxmi–),d(xni–, fxni–),

d(xmi–, fxni–) + d(fxmi–,xni–)s

}

= max

{d(xmi–,xni–),d(xmi–,xmi ),d(xni–,xni ),

d(xmi–,xni ) + d(xmi ,xni–)s

}(.)

and

N(xmi–,xni–) = min{d(xmi–, fxmi–),d(fxmi–,xni–)

}= min

{d(xmi–,xmi ),d(xmi ,xni–)

}. (.)

Taking the upper limit as i→ ∞ in (.) and (.) and using (.), (.), (.) and (.),we get

ε

s= min

{ε

s,

εs +

εs

s

}

≤ lim supi→∞

Ms(xmi–,xni–)

= max

{lim supi→∞

d(xmi–,xni–), , ,

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lim supi→∞ d(xmi–,xni ) + lim supi→∞ d(xmi ,xni–)s

}

≤ max

{εs, εs

+ ε

s

}= εs.

So, we have

ε

s≤ lim sup

i→∞Ms(xmi–,xni–) ≤ εs (.)

and

lim supi→∞

N(xmi–,xni–) = . (.)

Similarly, we can obtain

ε

s≤ lim inf

i→∞ Ms(xmi–,xni–) ≤ εs. (.)

Now, taking the upper limit as i→ ∞ in (.) and using (.), (.) and (.), we have

ψ(εs) ≤ ψ(s lim sup

i→∞d(xmi ,xni )

)

≤ ψ(

lim supi→∞

Ms(xmi–,xni–))– lim inf

i→∞ ϕ(Ms(xmi–,xni–)

)≤ ψ(εs) – ϕ

(lim infi→∞ Ms(xmi–,xni–)

),

which further implies that

ϕ(

lim infi→∞ Ms(xmi–,xni–)

)= ,

so lim infi→∞ Ms(xmi–,xni–) = , a contradiction to (.). Thus, {xn+ = fxn} is a b-Cauchysequence in X. As X is a b-complete space, there exists u ∈ X such that xn → u as n → ∞,and

limn→∞xn+ = lim

n→∞ fxn = u.

Now, suppose that f is continuous. Using the triangular inequality, we get

d(u, fu) ≤ sd(u, fxn) + sd(fxn, fu).

Letting n → ∞, we get

d(u, fu) ≤ s limn→∞d(u, fxn) + s lim

n→∞d(fxn, fu) = .

So, we have fu = u. Thus, u is a fixed point of f . �

Note that the continuity of f in Theorem is not necessary and can be dropped.

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Theorem Under the same hypotheses of Theorem , without the continuity assumptionof f , assume that whenever {xn} is a non-decreasing sequence in X such that xn → x ∈ X,xn � x, for all n ∈N, then f has a fixed point in X.

Proof Following similar arguments to those given in Theorem , we construct an increas-ing sequence {xn} in X such that xn → u, for some u ∈ X. Using the assumption on X, wehave xn � u, for all n ∈N. Now, we show that fu = u. By (.), we have

ψ(sd(xn+, fu)

)= ψ

(sd(fxn, fu)

)≤ ψ

(Ms(xn,u)

)– ϕ

(Ms(xn,u)

)+ Lψ

(N(xn,u)

), (.)

where

Ms(xn,u) = max

{d(xn,u),d(xn, fxn),d(u, fu),

d(xn, fu) + d(fxn,u)s

}

= max

{d(xn,u),d(xn,xn+),d(u, fu),

d(xn, fu) + d(xn+,u)s

}(.)

and

N(xn,u) = min{d(xn, fxn),d(u, fxn)

}= min

{d(xn,xn+),d(u,xn+)

}. (.)

Letting n → ∞ in (.) and (.) and using Lemma , we get

d(u, fu)s

= min

{d(u, fu)

s,d(u, fu)s

}

≤ lim supn→∞

Ms(xn,u)

≤ max

{d(u, fu),

sd(u, fu)s

}= d(u, fu) (.)

and

N(xn,u) → .

Similarly, we can obtain

d(u, fu)s

≤ lim infn→∞ Ms(xn,u) ≤ d(u, fu). (.)

Again, taking the upper limit as n → ∞ in (.) and using Lemma and (.) we get

ψ(d(u, fu)

)= ψ

(ssd(u, fu)

)≤ ψ

(s lim sup

n→∞d(xn+, fu)

)

≤ ψ(

lim supn→∞

Ms(xn,u))– lim inf

n→∞ ϕ(Ms(xn,u)

)≤ ψ

(d(u, fu)

)– ϕ

(lim infn→∞ Ms(xn,u)

).

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Therefore, ϕ(lim infn→∞ Ms(xn,u)) ≤ , equivalently, lim infn→∞ Ms(xn,u) = . Thus, from(.) we get u = fu and hence u is a fixed point of f . �

Corollary Let (X,�) be a partially ordered set and suppose that there exists a b-metricd on X such that (X,d) is a b-complete b-metric space. Let f : X → X be a non-decreasingcontinuous mapping with respect to �. Suppose that there exist k ∈ [, ) and L ≥ suchthat

d(fx, fy) ≤ ks

max

{d(x, y),d(x, fx),d(y, fy),

d(x, fy) + d(y, fx)s

}

+Ls

min{d(x, fx),d(y, fx)

}for all comparable elements x, y ∈ X. If there exists x ∈ X such that x � fx, then f has afixed point.

Proof Follows from Theorem by taking ψ(t) = t and ϕ(t) = ( – k)t, for all t ∈ [, +∞).�

Corollary Under the hypotheses of Corollary , without the continuity assumption of f ,for any non-decreasing sequence {xn} in X such that xn → x ∈ X, let us have xn � x for alln ∈N. Then, f has a fixed point in X.

Let (X,d) be an ordered b-metric space and let f , g : X → X be two mappings. Set

Ms(x, y) = max

{d(x, y),d(x, fx),d(y, gy), d(x, gy) + d(y, fx)

s

}

and

N(x, y) = min{d(x, fx),d(y, fx),d(x, gy)

}.

Now, we present the following definition.

Definition Let (X,d) be a partially ordered b-metric space and let ψ and ϕ be alteringdistance functions. We say that a mapping f : X → X is an almost generalized (ψ ,ϕ)s-contractive mapping with respect to a mapping g : X → X, if there exists L ≥ such that

ψ(sd(fx, gy)

) ≤ ψ(Ms(x, y)

)– ϕ

(Ms(x, y)

)+ Lψ

(N(x, y)

)(.)

for all x, y ∈ X.

Definition Let (X,�) be a partially ordered set. Then two mappings f , g : X → X aresaid to be weakly increasing if fx � gfx and gx� fgx, for all x ∈ X.

Theorem Let (X,�) be a partially ordered set and suppose that there exists a b-metricd on X such that (X,d) is a b-complete b-metric space, and let f , g : X → X be two weaklyincreasing mappings with respect to �. Suppose that f satisfies ., for all comparableelements x, y ∈ X. If either f or g is continuous, then f and g have a common fixed point.

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Proof Let us divide the proof into two parts as follows.First part:We prove that u is a fixed point of f if and only if u is a fixed point of g . Suppose

that u is a fixed point of f ; that is, fu = u. As u � u, by (.), we have

ψ(sd(u, gu)

)= ψ

(sd(fu, gu)

)≤ ψ

(max

{d(u, fu),d(u, gu),

s

(d(u, gu) + d(u, fu)

)})

– ϕ

(max

{d(u, fu),d(u, gu),

s(d(u, gu) + d(u, fu)

)})

+ Lmin{d(u, fu),d(u, gu)

}= ψ

(d(u, gu)

)– ϕ

(d(u, gu)

)≤ ψ

(sd(u, gu)

)– ϕ

(d(u, gu)

).

Thus, we have ϕ(d(u, gu)) = . Therefore, d(u, gu) = and hence gu = u. Similarly, we canshow that if u is a fixed point of g , then u is a fixed point of f .Second part (construction of a sequence by iterative technique):Let x ∈ X. We construct a sequence {xn} in X such that xn+ = fxn and xn+ = gxn+,

for all non-negative integers. As f and g are weakly increasing with respect to �, we have:

x = fx � gfx = x = gx � fgx = x � · · ·xn+ = fxn � gfxn = xn+ � · · · .

If xn = xn+, for some n ∈ N, then xn = fxn. Thus xn is a fixed point of f . By the firstpart, we conclude that xn is also a fixed point of g .If xn+ = xn+, for some n ∈ N, then xn+ = gxn+. Thus, xn+ is a fixed point of g . By

the first part, we conclude that xn+ is also a fixed point of f . Therefore, we assume thatxn = xn+, for all n ∈N. Now, we complete the proof in the following steps.Step : We will prove that

limn→∞d(xn,xn+) = .

As xn and xn+ are comparable, by (.), we have

ψ(d(xn+,xn+)

) ≤ ψ(sd(xn+,xn+)

)= ψ

(sd(fxn, gxn+)

)≤ ψ

(Ms(xn,xn+)

)– ϕ

(Ms(xn,xn+)

)+ Lψ

(N(xn,xn+)

),

where

Ms(xn,xn+) = max

{d(xn,xn+),d(xn, fxn),d(xn+, gxn+),

d(fxn,xn+) + d(xn, gxn+)s

}

= max

{d(xn,xn+),d(xn+,xn+),

d(xn,xn+)s

}

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≤ max

{d(xn,xn+),d(xn+,xn+),

sd(xn,xn+) + sd(xn+,xn+)s

}

= max{d(xn,xn+),d(xn+,xn+)

}and

N(xn,xn+) = min{d(xn, fxn),d(xn+, fxn),d(xn, gxn+)

}= min

{d(xn,xn+),d(xn+,xn+),d(xn,xn+)

}= .

So, we have

ψ(d(xn+,xn+)

) ≤ ψ(max

{d(xn,xn+),d(xn+,xn+)

})– ϕ

(max

{d(xn,xn+),d(xn+,xn+)

}). (.)

If

max{d(xn,xn+),d(xn+,xn+)

}= d(xn+,xn+),

then (.) becomes

ψ(d(xn+,xn+)

) ≤ ψ(d(xn+,xn+)

)– ϕ

(d(xn+,xn+)

)<ψ

(d(xn+,xn+)

),

which gives a contradiction. So,

max{d(xn,xn+),d(xn+,xn+)

}= d(xn,xn+)

and hence (.) becomes

ψ(d(xn+,xn+)

) ≤ ψ(d(xn,xn+)

)– ϕ

(d(xn,xn+)

) ≤ ψ(d(xn,xn+)

). (.)

Similarly, we can show that

ψ(d(xn+,xn)

) ≤ ψ(d(xn–,xn)

)– ϕ

(d(xn–,xn)

) ≤ ψ(d(xn–,xn)

). (.)

By (.) and (.), we get that {d(xn,xn+);n ∈ N} is a non-increasing sequence of positivenumbers. Hence there is r ≥ such that

limn→∞d(xn,xn+) = r.

Letting n → ∞ in (.), we get

ψ(r)≤ ψ(r) – ϕ(r)≤ ψ(r),

which implies that ϕ(r) = and hence r = . So, we have

limn→∞d(xn,xn+) = . (.)

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Step : We will prove that {xn} is a b-Cauchy sequence. It is sufficient to show that {xn}is a b-Cauchy sequence. Suppose the contrary, that is, {xn} is not a b-Cauchy sequence.Then there exists ε > , for which we can find two subsequences of positive integers {xmi}and {xni} such that ni is the smallest index for which

ni >mi > i, d(xmi ,xni ) ≥ ε. (.)

This means that

d(xmi ,xni–) < ε. (.)

From (.), (.) and the triangular inequality, we get

d(xni+,xmi )

≤ sd(xni+,xni ) + sd(xni ,xmi )

< sd(xni+,xni ) + sd(xni ,xni–) + sd(xni–,xmi )

≤ sd(xni+,xni ) + sd(xni ,xni–) + sd(xni–,xni–) + sd(xni–,xmi )

< sd(xni+,xni ) + sd(xni ,xni–) + sd(xni–,xni–) + εs.

Taking the upper limit in the above inequality and using (.), we have

lim supi→∞

d(xni+,xmi ) ≤ εs.

Again, from (.) and the triangular inequality, we get

ε ≤ d(xmi ,xni ) ≤ sd(xmi ,xni+) + sd(xni+,xni ).

Taking the upper limit in the above inequality and using (.), we have

ε

s≤ lim sup

i→∞d(xni+,xmi ).

So, we obtain

ε

s≤ lim sup

i→∞d(xni+,xmi ) ≤ εs. (.)

Similarly, we can obtain

ε

s≤ lim sup

i→∞d(xni ,xmi–) ≤ εs,

ε ≤ lim supi→∞

d(xni ,xmi ) ≤ εs,

ε

s≤ lim sup

i→∞d(xni+,xmi–) ≤ εs.

(.)

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Since xni and xmi– are comparable, using (.) we have

ψ(sd(xni+,xmi )

)= ψ

(sd(fxni , gxmi–)

)≤ ψ

(Ms(xni ,xmi–)

)– ϕ

(Ms(xni ,xmi–)

)+ Lψ

(N(xni ,xmi–)

), (.)

where

Ms(xni ,xmi–) = max

{d(xni ,xmi–),d(xni , fxni ),d(xmi–, gxmi–),

d(xni , gxmi–) + d(xmi–, fxni )s

}

= max

{d(xni ,xmi–),d(xni ,xni+),d(xmi–,xmi ),

d(xni ,xmi ) + d(xni+,xmi–)s

}(.)

and

N(xni ,xmi–) = min{d(xni , fxni ),d(xmi–, fxni ),d(xni , gxmi–)

}= min

{d(xni ,xni+),d(xmi–,xni+),d(xni ,xmi )

}. (.)

Taking the upper limit in (.) and (.) and using (.) and (.), we get

ε

s+

ε

s= min

{ε

s,ε + ε

s

s

}

≤ lim supi→∞

Ms(xni ,xmi–)

= max

{lim supi→∞

d(xni ,xmi–), , ,

lim supi→∞ d(xni ,xmi ) + lim supi→∞ d(xni+,xmi–)s

}

≤{εs,

εs + εs

s

}= εs.

So, we have

ε

s+

ε

s≤ lim sup

i→∞Ms(xni ,xmi–) ≤ εs (.)

and

lim supi→∞

N(xni ,xmi–) = . (.)

Similarly, we can obtain

ε

s+

ε

s≤ lim inf

i→∞ Ms(xni ,xmi–)≤ εs. (.)

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Now, taking the upper limit as i→ ∞ in (.) and using (.), (.) and (.), we have

ψ(εs

)= ψ

(s ε

s

)≤ ψ

(s lim sup

i→∞d(xni+,xmi )

)

≤ ψ(

lim supi→∞

Ms(xni ,xmi–))– lim inf

i→∞ ϕ(Ms(xni ,xmi–)

)≤ ψ

(εs

)– ϕ

(lim infi→∞ Ms(xni ,xmi–)

),

which implies that

ϕ(

lim infi→∞ Ms(xni ,xmi–)

)= ,

so lim infi→∞ Ms(xmi–,xni–) = , a contradiction to (.). Hence {xn} is a b-Cauchy se-quence in X.Step (Existence of a common fixed point):As {xn} is a b-Cauchy sequence in X which is a b-complete b-metric space, there exists

u ∈ X such that xn → u as n→ ∞, and

limn→∞xn+ = lim

n→∞ fxn = u.

Now, without any loss of generality, we may assume that f is continuous. Using the trian-gular inequality, we get

d(u, fu) ≤ sd(u, fxn) + sd(fxn, fu).

Letting n → ∞, we get

d(u, fu) ≤ s limn→∞d(u, fxn) + s lim

n→∞d(fxn, fu) = .

So, we have fu = u. Thus, u is a fixed point of f . By the first part, we conclude that u is alsoa fixed point of g . �

The continuity of one of the functions f or g in Theorem is not necessary.

Theorem Under the hypotheses of Theorem , without the continuity assumption of oneof the functions f or g , for any non-decreasing sequence {xn} in X such that xn → x ∈ X, letus have xn � x, for all n ∈N. Then, f and g have a common fixed point in X .

Proof Reviewing the proof of Theorem , we construct an increasing sequence {xn} in Xsuch that xn → u, for some u ∈ X. Using the assumption on X, we have xn � u, for alln ∈N. Now, we show that fu = gu = u. By (.), we have

ψ(sd(xn+, gu)

)= ψ

(sd(fxn, gu)

)≤ ψ

(Ms(xn,u)

)– ϕ

(Ms(xn,u)

)+ Lψ

(N(xn,u)

), (.)

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where

Ms(xn,u) = max

{d(xn,u),d(xn, fxn),d(u, gu),

d(xn, gu) + d(fxn,u)s

}

= max

{d(xn,u),d(xn,xn+),d(u, gu),

d(xn, gu) + d(xn+,u)s

}(.)

and

N(xn,u) = min{d(xn, fxn),d(u, fxn),d(xn, gu)

}= min

{d(xn,xn+),d(u,xn+),d(xn, gu)

}. (.)

Letting n → ∞ in (.) and (.) and using Lemma , we get

d(u, gu)s

= min

{d(u, gu)

s,d(u, gu)s

}

≤ lim supn→∞

Ms(xn,u) ≤ max

{d(u, gu), sd(u, gu)

s

}= d(u, gu) (.)

and

N(xn,u) → .

Similarly, we can obtain

d(u, gu)s

≤ lim infn→∞ Ms(xn,u) ≤ d(u, gu). (.)

Again, taking the upper limit as n→ ∞ in (.) and using Lemma and (.), we get

ψ(sd(u, gu)

)= ψ

(s sd(u, gu)

)≤ ψ

(s lim sup

n→∞d(xn+, gu)

)

≤ ψ(

lim supn→∞

Ms(xn,u))– lim inf

n→∞ ϕ(Ms(xn,u)

)≤ ψ

(d(u, gu)

)– ϕ

(lim infn→∞ Ms(xn,u)

)≤ ψ

(sd(u, gu)

)– ϕ

(lim infn→∞ Ms(xn,u)

).

Therefore, ϕ(lim infn→∞ Ms(xn,u)) ≤ , equivalently, lim infn→∞ Ms(xn,u) = . Thus,from (.) we get u = gu and hence u is a fixed point of g . On the other hand, similarto the first part of the proof of Theorem , we can show that fu = u. Hence, u is a commonfixed point of f and g . �

Also, we have the following results.

Corollary Let (X,�) be a partially ordered set and suppose that there exists a b-metricd on X such that (X,d) is a b-complete b-metric space. Let f , g : X → X be two weakly

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increasing mappings with respect to �. Suppose that there exist k ∈ [, ) and L ≥ suchthat

d(fx, gy) ≤ ks

max

{d(x, y),d(x, fx),d(y, gy), d(x, gy) + d(fx, y)

s

}

+Ls

min{d(x, fx),d(y, fx),d(x, gy)

}

for all comparable elements x, y ∈ X. If either f or g is continuous, then f and g have acommon fixed point.

Corollary Under the hypotheses of Corollary , without the continuity assumption ofone of the functions f or g , assume that whenever {xn} is a non-decreasing sequence in Xsuch that xn → x ∈ X, then xn � x, for all n ∈ N. Then f and g have a common fixed pointin X.

Now, in order to support the usability of our results, we present the following examples.

Example Let X = [,∞) be equipped with the b-metric d(x, y) = |x – y| for all x, y ∈ X,where s = – = .Define a relation � on X by x� y iff y≤ x, the functions f , g : X → X by

fx = ln

( +

x

)

and

gx = ln

( +

x

),

and the altering distance functions ψ ,ϕ : [, +∞) → [, +∞) by ψ(t) = bt and ϕ(t) = (b –)t, where ≤ b ≤

. Then, we have the following:() (X,�) is a partially ordered set having the b-metric d, where the b-metric space

(X,d) is b-complete.() f and g are weakly increasing mappings with respect to �.() f and g are continuous.() f is an almost generalized (ψ ,ϕ)s-contractive mapping with respect to g , that is,

ψ(sd(fx, gy)

) ≤ ψ(Ms(x, y)

)– ϕ

(Ms(x, y)

)+ Lψ

(N(x, y)

)

for all x, y ∈ X with x� y and L ≥ , where

Ms(x, y) = max

{d(x, y),d(x, fx),d(y, gy), d(x, gy) + d(y, fx)

s

}

and

N(x, y) = min{d(x, fx),d(y, fx),d(x, gy)

}.

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Proof The proof of () is clear. To prove (), for each x ∈ X, we know that + x ≤ e x

and + x

≤ e x . So, fx = ln( + x

) ≤ x and gx = ln( + x ) ≤ x. Hence, fgx = ln( + gx

) ≤ gx andgfx = ln( + fx

) ≤ fx, for each x ∈ X. Therefore, f and g are weakly increasing mappingswith respect to �. It is easy to see that f and g are continuous.To prove (), let x, y ∈ X with x � y. So, y≤ x. Thus, we have the following cases.Case : If y

≤ x , then we have

≤ + x

+ y

≤ + x

+ y

�⇒ ≤ ln

( + x

+ y

)≤ ln

( + x

+ y

).

Now, using the mean value theorem for function ln( + t), for t ∈ [ y ,x ], we have

ψ(sd(fx, gy)

)= bd(fx, gy)

= b(

ln

( +

x

)– ln

( +

y

))= b

(ln

( + x

+ y

))

≤ b(

ln

( + x

+ y

))= b

(ln

( +

x

)– ln

( +

y

))

≤ b(x–y

)≤

(x – y)

≤ d(x, y) ≤M(x, y) = ψ(M(x, y)

)– ϕ

(M(x, y)

),

that is, we have

ψ(sd(fx, fy)

) ≤ ψ(Ms(x, y)

)– ϕ

(Ms(x, y)

)+ Lψ

(N(x, y)

)for each L ≥ .Case : If x

<y ≤ x

, then we have

<y–

x

≤ y

�⇒(y–

x

)≤ y

.

Using the mean value theorem for function ln( + t), for t ∈ [ x ,

y ], we have

ψ(sd(fx, gy)

)= bd(fx, gy)

= b(

ln

( +

x

)– ln

( +

y

))

≤ b(y–

x

)≤

by ≤

y

=(y

)≤

(y – ln

( +

y

))= d(y, gy)

≤ M(x, y) = ψ(M(x, y)

)– ϕ

(M(x, y)

).

So, we have

ψ(sd(fx, fy)

) ≤ ψ(Ms(x, y)

)– ϕ

(Ms(x, y)

)+ Lψ

(N(x, y)

)

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for each L ≥ . Combining Cases and together, we conclude that f is an almost general-ized (ψ ,ϕ)s-contractive mapping with respect to g . Thus, all the hypotheses of Theorem are satisfied and hence f and g have a common fixed point. Indeed, is the unique com-mon fixed point of f and g . �

Remark A subset W of a partially ordered set X is said to be well ordered if every twoelements of W are comparable []. Note that in Theorems and , f has a unique fixedpoint provided that the fixed points of f are comparable. Also, in Theorems and , theset of common fixed points of f and g is well ordered if and only if f and g have one andonly one common fixed point.

Example Let X = {, , , , } be equipped with the following partial order �:

�:={(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )

}.

Define b-metric d : X ×X → R+ by

d(x, y) =

⎧⎨⎩ if x = y,(x + y) if x = y.

It is easy to see that (X,d) is a b-complete b-metric space.Define the self-maps f and g by

f =(

), g =

(

).

We see that f and g are weakly increasing mappings with respect to � and f and g arecontinuous.Define ψ ,ϕ : [,∞) → [,∞) by ψ(t) =

√t and ϕ(t) = t

. One can easily check that f isan almost generalized (ψ ,ϕ)s-contractive mapping with respect to g , with L ≥

.Thus, all the conditions of Theorem are satisfied and hence f and g have a common

fixed point. Indeed, and are two common fixed points of f and g . Note that the orderedset (X,�) is not well ordered.

3 ApplicationsLet denote the set of all functions φ : [, +∞) → [, +∞) satisfying the following hy-potheses:. Every φ ∈ is a Lebesgue integrable function on each compact subset of [, +∞).. For any φ ∈ and any ε > ,

∫ ε

φ(τ )dτ > .It is an easy matter to check that the mapping ψ : [, +∞)→ [, +∞) defined by

ψ(t) =∫ t

φ(τ )dτ

is an altering distance function. Therefore, we have the following results.

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Corollary Let (X,�) be a partially ordered set having a b-metric d such that the b-metric space (X,d) is b-complete. Let f : X → X be a non-decreasing continuous mappingwith respect to �. Suppose that there exist k ∈ [, ) and L ≥ such that

∫ d(fx,fy)

φ(τ )dτ ≤ k

s

∫ max{d(x,y),d(x,fx),d(y,fy), d(x,fy)+d(y,fx)s }

φ(τ )dτ

+Ls

∫ min{d(x,fx),d(y,fx)}

φ(τ )dτ

for all comparable elements x, y ∈ X. If there exists x ∈ X such that x � fx, then f has afixed point.

Proof Follows from Theorem by taking ψ(t) =∫ t φ(τ )dτ and ϕ(t) = ( – k)t, for all t ∈

[, +∞). �

Corollary Let (X,�) be a partially ordered set having a b-metric d such that the b-metricspace (X,d) is b-complete. Let f , g : X → X be two weakly increasing mappings with respectto �. Suppose that there exist k ∈ [, ) and L ≥ such that

∫ d(fx,gy)

φ(τ )dτ ≤ k

s

∫ max{d(x,y),d(x,fx),d(y,gy), d(x,gy)+d(y,fx)s }

φ(τ )dτ

+Ls

∫ min{d(x,fx),d(y,fx),d(x,gy)}

φ(τ )dτ

for all comparable elements x, y ∈ X. If either f or g is continuous, then f and g have acommon fixed point.

Proof Follows from Theorem by taking ψ(t) =∫ t φ(τ )dτ and ϕ(t) = ( – k)t, for all t ∈

[, +∞). �

Finally, let us finish this paper with the following remarks.

Remark Theorem . of [] is a special case of Corollary .

Remark Theorem . of [] is a special case of Corollary .

Remark Theorem ., Corollary . andCorollary . of [] are special cases of Corol-lary .

Remark Since a b-metric is a metric when s = , so our results can be viewed as a gen-eralization and extension of corresponding results in [] and several other comparableresults.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Rezaei Roshan et al. Fixed Point Theory and Applications 2013, 2013:159 Page 22 of 23http://www.fixedpointtheoryandapplications.com/content/2013/1/159

Author details1Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran. 2Department ofMathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran. 3Department of Mathematics, BabolBranch, Islamic Azad University, Babol, Iran. 4Department of Mathematics, Hashemite University, Zarqa, 13115, Jordan.

AcknowledgementsThe authors thank the referees for their valuable comments which helped them to correct the first version of themanuscript.

Received: 8 March 2013 Accepted: 2 May 2013 Published: 19 June 2013

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doi:10.1186/1687-1812-2013-159Cite this article as: Rezaei Roshan et al.: Common fixed points of almost generalized (ψ ,ϕ)s-contractive mappings inordered b-metric spaces. Fixed Point Theory and Applications 2013 2013:159.