Cclass functions with new approach on coincidence point results for generalized \((\psi ,\varphi )\)weakly contractions in ordered bmetric spaces
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Abstract
In this paper, by using the Cclass functions and a new approach we present some coincidence point results for four mappings satisfying generalized \((\psi ,\phi )\)weakly contractive condition in the setting of ordered bmetric spaces. Also, an application and example are given to support our results.
Keywords
bMetric space Partially ordered set Fixed point CClass functionsMathematics Subject Classification
Primary 47H10 Secondary 54H25Background
Metric fixed point theorem is playing a major role in mathematics and the applied sciences. Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to many applications in different areas such as variational, linear inequalities and optimization problems.
Banach contraction principle states that every contractive mapping defined on a complete metric space has a unique fixed point. This principle has been generalized by many researchers in different ways Abbas and Dorić (2010), Abbas et al. (2011), Abbas et al. (2012), Abbas and Rhoades (2009), Agarwal et al. (2008) and Shatanawi and Postolache (2013), Shatanawi et al. (2011), Shatanawi and Mustafa (2012), Choudhury et al. (2013), Aydi et al. (2013), Aydi et al. (2012), Shatanawi et al. (2014), Radenović and Kadelburg (2012).
In 1997, Alber and GuerreDelabriere (1997) introduced the concept of weak contraction in the setup of Hilbert spaces as follows: A self mapping f on X is a weak contraction, if \(d(fx,fy)\le d(x,y)\varphi (d(x,y))\) for all \(x,y\in X\), where \(\varphi\) is an altering distance function. Thereafter, in Rhoades (2001), generalized the Banach contraction principle by considering the class of weak contraction in the setup of metric spaces and proved that every weakly contractive mapping defined on a complete metric space has a unique fixed point.
Later on, in Zhang and Song (2009) introduced the concept of a generalized \(\varphi\)weak contractive mappings and proved the following common fixed point result: Let (X, d) be a complete metric space. If \(f,\,g:X\rightarrow X\) are generalized \(\varphi\)weak contractive mappings, then there exists a unique point \(u\in X\) such that \(u=fu=gu.\)
We refer the reader to Abbas and Dorić (2010), Dorić (2009), Moradi et al. (2011) and Razani et al. (2012) for more works in this area.
The concept of bmetric space was introduced by Czerwik in Czerwik (1998). Since then, several papers have been published on the fixed point theory of various classes of singlevalued and multivalued operators in bmetric spaces (see also Akkouchi 2011; Aydi et al. 2012; Boriceanu 2009a, b; Boriceanu et al. 2010; Bota et al. 2011; Hussain et al. 2012; Hussain and Shah 2011; Olatinwo 2008; Mustafa 2014; Pacurar 2010; Mustafa 2013; Ansari et al. 2014).
Mathematical preliminaries
Definition 1
(Altun and Simsek 2010) Let f and g be two selfmaps on partially ordered set X. A pair (f, g) is said to be weakly increasing if \(fx\preceq gfx\) and \(gx\preceq fgx\) for all \(x\in X\).
Definition 2
(Abbas et al. 2011) Let f and g be two selfmaps on partially ordered set X. A pair (f, g) is said to be partially weakly increasing if \(fx\preceq gfx\) for all \(x\in X\).
Let X be a nonempty set and \(T:X\rightarrow X\) be a given mapping. For every \(x\in X\), let \(T^{1}(x)=\{u\in X:Tu=x\}\).
Definition 3
(Nashine and Samet 2011) Let \((X,\preceq )\) be a partially ordered set and \(f,g,T:X\rightarrow X\) are mappings such that \(fX\subseteq TX\) and \(gX\subseteq TX\). The ordered pair (f, g) is said to be weakly increasing with respect to T if and only if for all \(x\in X\), \(fx\preceq gy\) for all \(y\in T^{1}(fx)\) and \(gx\preceq fy\) for all \(y\in T^{1}(gx)\).
Definition 4
(Esmaily et al. 2012) Let \((X,\preceq )\) be a partially ordered set and \(f,g,T:X\rightarrow X\) are mappings such that \(fX\subseteq TX\) and \(gX\subseteq TX\). The ordered pair (f, g) is said to be partially weakly increasing with respect to T if \(fx\preceq gy\) for all \(y\in T^{1}(fx)\).
Remark 5
 (1)
If \(f=g\), we say that f is weakly increasing (partially weakly increasing) with respect to T.
 (2)
If \(T=I_X\) (the identity mapping on X), then the above definitions reduces to the weakly increasing (partially weakly increasing) mapping (See, Nashine and Samet 2011; Shatanawi and Samet 2011).
Jungck in Jungck (1986) introduced the following definition.
Definition 6
(Jungck 1986) Let (X, d) be a metric space and \(f, g:X\rightarrow X\). The pair (f, g) is said to be compatible if \(\lim \nolimits _{n\rightarrow \infty }d(fgx_n,gfx_n)=0\), whenever \(\{x_n\}\) is a sequence in X such that \(\lim \nolimits _{n\rightarrow \infty }fx_n=\lim \nolimits _{n\rightarrow \infty }gx_n=t\) for some \(t\in X\).
Definition 7
Let f and g be two self mappings on a nonempty set X. If \(x=fx=gx\) for some x in X, then x is called a common fixed point of f and g.
Definition 8
(Jungck 1996) Let \(f,g:X\rightarrow X\) be given selfmappings on X. The pair (f, g) is said to be weakly compatible if f and g commute at their coincidence points (i.e., \(fgx=gfx\), whenever \(fx=gx\)).
Definition 9
 (1)
If a nondecreasing sequence \(x_n\rightarrow x\), then \(x_n\preceq x\) for all n.
 (2)
If a nonincreasing sequence \(y_n\rightarrow y\), then \(y_n\succeq y\) for all n.
Definition 10
 (1)
\(\psi\) is monotone increasing and continuous,
 (2)
\(\psi (t) =0\) if and only if \(t = 0\).
In Nashine and Samet (2011), established some coincidence point and common fixed point theorems for mappings satisfying a generalized weakly contractive condition in an ordered complete metric space by considering a pair of altering distance functions \((\psi ,\varphi )\). In fact, they proved the following theorem.
Theorem 11
 (i)
T and R are continuous,
 (ii)
\(TX\subseteq RX\),
 (iii)
T is weakly increasing with respect to R,
 (iv)
the pair (T, R) is compatible.
Then, T and R have a coincidence point, that is, there exists \(u\in X\) such that \(Ru=Tu.\)
Further, they showed that by replacing the continuity hypotheses on T and R with the regularity of \((X,d,\preceq )\) and omitting the compatibility of the pair (T, R), the above theorem is still valid (see, Theorem 2.6 of Nashine and Samet 2011).
Also, in Shatanawi and Samet (2011), Shatanawi and Samet studied common fixed point and coincidence point for three self mappings T, S and R satisfying \((\psi , \varphi )\)weakly contractive condition in an ordered metric space (X, d), where S and T are weakly increasing with respect to R and \(\psi ,\varphi\) are altering distance functions. Their result generalize Theorem 11.
Analogous to the work in Nashine and Samet (2011), Shatanawi and Samet proved the above result by replacing the continuity hypotheses of T, S and R with the regularity of X and omitting the compatibility of the pair (T, R) and (S, R) (See, Theorem 2.2 of Shatanawi and Samet 2011).
Consistent with Czerwik (1998), Jovanović et al. (2010) and Singh and Prasad (2008), the following definitions and results will be needed in the sequel.
Definition 12
 \((b_1)\)

\(d(x,y)=0\) iff \(x=y,\)
 \((b_2)\)

\(d(x,y)=d(y,x),\)
 \((b_3)\)

\(d(x,z)\le s[d(x,y)+d(y,z)].\)
The pair (X, d) is called a bmetric space.
Note that, the class of bmetric spaces is effectively larger than the class of metric spaces, since a bmetric is a metric, when \(s=1.\)
The following example shows that in general a bmetric need not necessarily be a metric (see, also, Singh and Prasad 2008, p. 264).
Example 13
(Aghajani et al. 2014) Let (X, d) be a metric space, and \(\rho (x,y)=(d(x,y))^{p},\) where \(p>1\) is a real number. Then, \(\rho\) is a b metric with \(s=2^{p1}.\)
However, if (X, d) is a metric space, then \((X,\rho )\) is not necessarily a metric space.
For example, if \(X= {\mathbb {R}}\) is the set of real numbers and \(d(x,y)=\left xy\right\) is the usual Euclidean metric, then \(\rho (x,y)=(xy)^{2}\) is a bmetric on \({\mathbb {R}}\) with \(s=2,\) but not a metric on \({\mathbb {R}}\).
Definition 14
Let X be a nonempty set. Then \(({X},d,\preceq )\) is called a partially ordered bmetric space if and only if d is a bmetric on a partially ordered set \((X,\preceq ).\)
Definition 15
(Boriceanu et al. 2010) Let (X, d) be a bmetric space. Then a sequence \(\{x_{n}\}\) in X is called bconvergent if and only if there exists \(x\in X\) such that \(d(x_n,x)\rightarrow 0\), as \(n\rightarrow +\infty\). In this case, we write \(\lim \nolimits _{n\rightarrow \infty }x_n=x.\)
Definition 16
(Boriceanu et al. 2010) Let (X, d) be a bmetric space. Then a sequence \(\{x_{n}\}\) in X is called bCauchy if and only if \(d(x_n,x_m)\rightarrow 0,\) as \(n,m\rightarrow +\infty.\)
Proposition 17
 (i)

A bconvergent sequence has a unique limit.
 (ii)

Each bconvergent sequence is bCauchy.
 (iii)

In general, a bmetric need not be continuous.
Definition 18
(Boriceanu et al. 2010) The bmetric space (X, d) is bcomplete if every b Cauchy sequence in X bconverges.
Definition 19
Let (X, d) and \((X^{\prime },d^{\prime })\) be two bmetric spaces. Then a function \(f:X\rightarrow X^{\prime }\) is bcontinuous at a point \(x\in X\) if and only if it is bsequentially continuous at x, that is, whenever \(\{x_n\}\) is bconvergent to x, \(\{f(x_n)\}\) is bconvergent to f(x).
Definition 20
 (1)
\(\varphi\) is continuous
 (2)
\(\varphi (0)\ge 0\), and \(\varphi (t)\ne 0,t\ne 0\).
In 2014 Ansari (2014) introduced the concept of Cclass functions which cover a large class of contractive conditions.
Definition 21
 (1)
\(F(r,t)\le r\);
 (2)
\(F(r,t)=r\) implies that either \(r=0\) or \(t=0\); for all \(r,t\in [0,\infty )\).
We denote a Cclass functions as \({\mathcal {C}}\).
Example 22
 (1)
\(F(r,t)=mr\), \(0{<}m{<}1\), \(F(r,t)=r\Rightarrow r=0\);
 (2)
\(F(r,t)=rt\), \(F(r,t)=r\Rightarrow t=0\);
 (3)
\(F(r,t)=\frac{r}{(1+t)^{\alpha }}\); \(\alpha \in (0,\infty )\), \(F(r,t)=r\) \(\Rightarrow\) \(r=0\) or \(t=0\).
Lemma 23
Motivated by the works in Nashine and Samet (2011), Shatanawi and Samet (2011) and Jamal (2015), In this paper, by using the Cclass functions and a new approach, we present some coincidence point results for four mappings satisfying generalized \((\psi ,\phi )\)weakly contractive condition in the setting of ordered bmetric spaces where \(\psi\) is altering distance function and \(\varphi\) is Ultraaltering distance function. Also, an application and example are given to support our results.
Main results
Theorem 24
Proof
Let \(x_{0} \in X\) be an arbitrary point. Since \(f(X)\subseteq T(X)\) and \(g(X)\subseteq h(X)\), one can find \(x_{1}, x_{2} \in X\) such that \(fx_{0}= Tx_{1}\) and \(gx_{1}=hx_{2}\).
Repeating this process, we obtain \(w_{2n+1}\preceq w_{2n+2}\) for all \(n\ge 0.\)
The proof will be done in three steps.
Step I We will show that \(\lim \nolimits _{k\rightarrow \infty }d(w_{k},w_{k+1})=0.\)
On the other hand, the pairs (f, h) and (g, T) are compatible. So, they are weakly compatible. Hence, \(fh(x_{2n})=hf(x_{2n})\) and \(gT(x_{2n+1})=Tg(x_{2n+1})\), or, equivalently, \(fw_{2n}=hw_{2n+1}\) and \(gw_{2n+1}=Tw_{2n+2}\). Now, since, \(w_{2n} =w_{2n+1} =w_{2n+2},\) we have, \(fw_{2n}=hw_{2n}\) and \(gw_{2n}=Tw_{2n}\).
In the other case, when \(k_{0}=2n+1\), similarly, one can show that \(w_{2n+1}\) is a coincidence point of the pairs (f, h) and (g, T). Also for \({\mathcal {N}}(x_{2n},x_{2n+1})=0\) or \({\mathcal {N}}(x_{2n},x_{2n+1})=d(w_{2n},w_{2n+2})\), one can obtain the desired result.
Using argument similar to the above, one can show the inequality (6) is true for \(k=2n+1\). Therefore, (6) is true for all \(k=1,2,3,\cdots\).
Step II Using 10 and Lemma (23) we get \(\{w_{n}\}\) is a bCauchy sequence in X.
Step III In this step we prove that f, g, T and h have a coincidence point.
which implies \(\left[F(\psi \left( {d(fw,gw)} \right),\varphi \left( {d(fw,gw)) = \psi (d(fw,gw)} \right)\right]\), hence, either
\(\psi \big (d(fw,gw)\big ) = 0\hbox { or }\varphi \big (d(fw,gw)\big )=0\), then in both cases we get \(fw=gw\).
which implies \(\psi \big (s^{a}d(fw,gw)\big ) =0\), and so \(fw=gw\). So, in all cases we get that, \(fw=gw=hw=Tw\). \(\square\)
By taking \(\psi (t)=\varphi (t)=t\) and \(F(r,t)=\lambda r\), \(\lambda > 1,\) we get the following corollary.
Corollary 25
In the following theorem, we replace the compatibility of the pairs (f, h) and (g, T) by weak compatibility of the pairs and we omit the continuity assumption of f, g, T and h and
Theorem 26
Proof
Since \(Tx_{2n+1}\rightarrow w=hv\), as \({n\rightarrow \infty }\) and the regularity of X, \(Tx_{2n+1}\preceq hv\). But from triangle inequality of b metric space we have \(d(fv,w) \le sd(w,gx_{2n+1}) + sd(fv,gx_{2n+1})\)
so, \(fv=w=hv\).
so, \(fv=w=hv\).
As f and h are weakly compatible, we have \(fw=fhv=hfv=hw.\) Thus, w is a coincidence point of f and h.
Similarly it can be shown that w is a coincidence point of the pair (g, T).
The rest of the proof can be done using similar arguments as in Theorem 24. \(\square\)
Taking \(h=T\) in Theorem 24, we obtain the following result.
Corollary 27
 a.

the pair (f, T) is compatible and f is continuous, or,
 b.

the pair (g, T) is compatible and g is continuous.
Taking \(T=h\) and \(f=g\) in Theorem 24, we obtain the following coincidence point result.
Corollary 28
Example 29
Corollary 30
Corollary 31
Taking \(T=h=I_X\) (the identity mapping on X) in Theorems 24 and 26, we obtain the following common fixed point result.
Corollary 32
 a.

f or g is continuous, or,
 b.

X is regular.
Application
Let \(X = (C[a,b],{\mathbf {R}})\) denote the set of all continuous functions from [a, b] to \({\mathbf {R}}\). Consider the partial order on X to be define as: \(x, y \in X, \,\,\,\, x \preceq y \text{ iff } x(t) \le y(t), \,\,\, \forall t \in [a,b]\).
Theorem 33
 (1)
\(G:[a,b]\times [a,b] \rightarrow [0,\infty )\) is a continuous function,
 (2)
\(H_{1}, H_{2}:[a,b]\times {\mathbf {R}} \rightarrow {\mathbf {R}}\) are continuous functions,
 (3)
\(\sup _{t\in [a,b]}\int _{a}^{b}G(t,r)dr<\frac{1}{\sqrt{2^{m}}} ,m>1\)
 (4)for all \(r \in [a,b]\) and \(x \in X\) we have$$\begin{aligned} H_{1}(r,x(r))&\le H_{2}\Big (r,\int _{a}^{b}G(t,r)H_{1}(r,x(r)) dr\Big )\\ H_{2}(r,x(r))& \le H_{1}\Big (r,\int _{a}^{b}G(t,r)H_{2}(r,x(r)) dr\Big ) \end{aligned}$$
 (5)For all \(x(r),y(r)\in X\) with \(x(r)\le y(r)\); \(r\in [a,b]\) we have$$\begin{aligned} H_{1}(r,x(r))H_{2}(r,y(r)^{2}\le \sqrt{\ln (1+x(r)y(r)^{2})}. \end{aligned}$$
Then, the integral Eq. (30) have a solution \(x \in X\).
Proof
Therefore, all conditions of corollary 32 are satisfied with \({\mathcal {N}}(x,y) = d(x,y)\) and \(a = m\). As a result of corollary 32 the mappings f and g has a common fixed point in X which is a solution of the Eq. 30. \(\square\)
Conclusions
By using the Cclass function F such that F is increasing with respect to first variable and decreasing with respect to second variable, we proved some coincidence point results for four continuous mappings f, g, T and h, where the pairs (f, h) and (g, T) are compatible satisfying generalized \((\psi ,\phi )\)weakly contractive condition in the setting of ordered bmetric spaces, \(\psi\) is altering distance function and \(\varphi\) is Ultraaltering distance function. Also, we can replace the compatibility of the pairs (f, h) and (g, T) by weak compatibility of the pairs and we omit the continuity assumption of f, g, T and h. This approach can be extended to other spaces.
Notes
Authors' contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors are highly appreciated the referees efforts of this paper who helped us to improve it in several places.
Competing interests
The authors declare that they have no competing interests.
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