Cclass functions with new approach on coincidence point results for generalized \((\psi ,\varphi )\)weakly contractions in ordered bmetric spaces
 562 Downloads
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.
References
 Abbas M, Dorić D (2010) Common fixed point theorem for four mappings satisfying generalized weak contractive condition. Filomat 24(2):1–10CrossRefGoogle Scholar
 Abbas M, Nazir T, Radenović S (2011) Common fixed points of four maps in partially ordered metric spaces. Appl Math Lett 24:1520–1526CrossRefGoogle Scholar
 Abbas M, Nazir T, Radenović S (2012) Common fixed point of generalized weakly contractive maps in partially ordered \(G\)metric spaces. Appl Math Comput 218:9383–9395Google Scholar
 Abbas M, Rhoades BE (2009) Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl Math Comput 215:262–269Google Scholar
 Agarwal RP, ElGebeily MA, O’Regan D (2008) Generalized contractions in partially ordered metric spaces. J Appl Anal 87(1):109–116CrossRefGoogle Scholar
 Aghajani A, Abbas M, Roshan JR (2014) Common fixed point of generalized weak contractive mappings in partially ordered bmetric spaces. Math Slov 4:941–960Google Scholar
 Akkouchi M (2011) Common fixed point theorems for two selfmappings of a \(b\)metric space under an implicit relation. Hacet J Math Statist 40(6):805–810Google Scholar
 Alber Ya I, GuerreDelabriere S (1997) Principle of weakly contractive maps in Hilbert spaces. In: Gohberg I, Lyubich Y (eds) New results in operator theory and its advances and applications. Birkhä, Basel, pp 7–22CrossRefGoogle Scholar
 Altun I, Simsek H (2010) Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl. Article ID 621492Google Scholar
 Ansari AH, Chandok S, Ionescu C (2014) Fixed point theorems on \(b\)metric spaces for weak contractions with auxiliary functions. J Inequal Appl 2014:429CrossRefGoogle Scholar
 Ansari AH (2014) Note on \(\varphi\)–\(\psi\)contractive type mappings and related fixed point. In: The 2nd regional conference on mathematics and applications, Payame Noor University, p 377–380Google Scholar
 Aydi H, Bota M, Karapınar E, Mitrović S (2012) A fixed point theorem for setvalued quasicontractions in \(b\)metric spaces. Fixed Point Theory Appl 2012:88CrossRefGoogle Scholar
 Aydi H, Karapinar E, Mustafa Z (2013) Some tripled fixed point theorems in partially ordered metric spaces. Tamkang J Math 44:3. doi: 10.5556/j.tkjm.44.2013.990
 Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N (2012) Theorems for BoydWong type contractions in ordered metric spaces. Abstr Appl Anal. Article ID: 359054Google Scholar
 Boriceanu M (2009) Strict fixed point theorems for multivalued operators in \(b\)metric spaces. Int J Mod Math 4(3):285–301Google Scholar
 Boriceanu M, Bota M, Petrusel A (2010) Multivalued fractals in \(b\)metric spaces. Cent Eur J Math 8(2):367–377CrossRefGoogle Scholar
 Boriceanu M (2009) Fixed point theory for multivalued generalized contraction on a set with two \(b\)metrics. Studia Universitatis, “Babes–Bolyai”, Mathematica, Vol. LIV, Number 3Google Scholar
 Bota M, Molnar A, Varga C (2011) On Ekeland’s variational principle in \(b\)metric spaces. Fixed Point Theory 12(2):21–28Google Scholar
 Choudhury BS, Metiya N, Postolache M (2013) A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. Art. No. 152Google Scholar
 Czerwik S (1998) Nonlinear setvalued contraction mappings in \(b\)metric spaces. Atti Sem Mat Fis Univ Modena 46(2):263–276Google Scholar
 Dorić D (2009) Common fixed point for generalized \((\psi,\varphi )\)weak contractions. Appl Math Lett 22:1896–1900CrossRefGoogle Scholar
 Esmaily J, Vaezpour SM, Rhoades BE (2012) Coincidence point theorem for generalized weakly contractions in ordered metric spaces. Appl Math Comput 219:1536–1548Google Scholar
 Hussain N, Dorić N, Kadelburg Z, Radenović S (2012) Suzukitype fixed point results in metric type spaces. Fixed Point Theory Appl 2012:126CrossRefGoogle Scholar
 Hussain N, Shah MH (2011) KKM mappings in cone \(b\)metric spaces. Comput Math Appl 62:1677–1684CrossRefGoogle Scholar
 Jamal R (2015) Roshan, Vahid Parvaneh, Stojan Radenović, Miloje Rajović, Some coincidence point results for generalized (\(\psi,\varphi\))weakly contractions in ordered \(b\)metric spaces. Fixed Point Theory and Applications 2015:68. doi: 10.1186/s1366301503136 CrossRefGoogle Scholar
 Jovanović M, Kadelburg Z, Radenović S (2010) Common Fixed Point Results in MetricType Spaces. Abstr. Applied Anal. Article ID 978121. doi: 10.1155/2010/978121
 Jovanović M, Kadelburg Z, Radenović S (2010) Common fixed point results in metrictype spaces. Fixed Point Theory Appl. Article ID 978121Google Scholar
 Jungck G (1986) Compatible mappings and common fixed points. Int J Math Math Sci 9:771–779CrossRefGoogle Scholar
 Jungck G (1996) Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East J Math Sci 4:199–215Google Scholar
 Khan MS, Swalesh M, Sessa S (1984) Fixed points theorems by altering distances between the points. Bull Aust Math Soc 30:1–9CrossRefGoogle Scholar
 Moradi S, Fathi Z, Analouee E (2011) Common fixed point of single valued generalized \(\varphi _{f}\)weak contractive mappings. Appl Math Lett 24(5):771–776CrossRefGoogle Scholar
 Mustafa Z, Roshan JR, Parvaneh V, Kadelburg Z (2013) Some common fixed point results in ordered partial bmetric space. J Inequal Appl 2013:562CrossRefGoogle Scholar
 Mustafa Z, Roshan JR, Parvaneh V, Kadelburg Z (2014) Fixed point theorems for Weakly TChatterjea and weakly \(T\)Kannan contractions in \(b\)metric spaces. J Inequal Appl 2014:46CrossRefGoogle Scholar
 Nashine HK, Samet B (2011) Fixed point results for mappings satisfying \((\psi,\varphi )\)weakly contractive condition in partially ordered metric spaces. Nonlinear Anal 74:2201–2209CrossRefGoogle Scholar
 Nieto JJ, RodaiguezLoez R (2007) Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sinica 23(12) 2205â€“ 2212Google Scholar
 Olatinwo MO (2008) Some results on multivalued weakly jungck mappings in \(b\)metric space. Cent Eur J Math 6(4):610–621CrossRefGoogle Scholar
 Pacurar M (2010) Sequences of almost contractions and fixed points in \(b\)metric spaces. Analele Universitatii de Vest, Timisoara Seria Matematica Informatica XLVIII 3:125–137Google Scholar
 Radenović S, Kadelburg Z, Jandrlić D, Jandrlić A (2012) Some results on weakly contractive maps. Bull Iranian Math Soc 38(3):625–645Google Scholar
 Razani A, Parvaneh V, Abbas M (2012) A common fixed point for generalized \((\psi,\varphi )_{f, g}\)weak contractions. Ukrainian Math J 63(11):1756–1769CrossRefGoogle Scholar
 Rhoades BE (2001) Some theorems on weakly contractive maps. Nonlinear Anal 47:2683–2693CrossRefGoogle Scholar
 Shatanawi W, Mustafa Z (2012) On coupled random fixed point results in partially ordered metric spaces. Matematicki Vesnik 64:139–146Google Scholar
 Shatanawi W, Samet B (2011) On \((\psi,\phi )\)weakly contractive condition in partially ordered metric spaces. Comput Math Appl 62:3204–3214CrossRefGoogle Scholar
 Shatanawi W (2011) Mustafa Z, Tahat N (2011) Some coincidence point theorems for nonlinear contraction in ordered metric spaces. Fixed Point Theory Appl. doi: 10.1186/16871812201168
 Shatanawi W, Pitea A, Lazovic R (2014) Contraction conditions using comparison functions on \(b\)metric spaces. Fixed Point Theory Appl. Art. No. 135Google Scholar
 Shatanawi W, Postolache, M (2013) Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. Fixed Point Theory Appl.. Art. No. 271Google Scholar
 Singh SL, Prasad B (2008) Some coincidence theorems and stability of iterative proceders. Comput Math Appl 55:2512–2520CrossRefGoogle Scholar
 Zhang Q, Song Y (2009) Fixed point theory for generalized \(\varphi\)weak contractions. Appl Math Lett 22:75–78CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.