 Research
 Open Access
 Published:
A generalization on weak contractions in partially ordered bmetric spaces and its application to quadratic integral equations
Journal of Inequalities and Applications volume 2014, Article number: 355 (2014)
Abstract
We introduce the notion of almost generalized (\psi ,\phi ,L)contractive mappings, and establish the coincidence and common fixed point results for this class of mappings in partially ordered complete bmetric spaces. Our results extend and improve several known results from the context of ordered metric spaces to the setting of ordered bmetric spaces. As an application, we prove the existence of a unique solution to a class of nonlinear quadratic integral equations.
1 Introduction
Fixed points theorems in partially ordered metric spaces were firstly obtained in 2004 by Ran and Reurings [1], and then by Nieto and RodríguezLópez [2]. In this direction several authors obtained further results under weak contractive conditions (see, e.g., [3–8]). Berinde initiated in [9] the concept of almost contractions and obtained several interesting fixed point theorems. This has been a subject of intense study since then; see, e.g., [10–20]. Some authors used related notions as ‘condition (B)’ (Babu et al. [21]) and ‘almost generalized contractive condition’ for two maps (Ćirić et al. [22]), and for four maps (Aghajani et al. [23]). See also a note by Pacurar [15]. On the other hand, the concept of bmetric space was introduced by Czerwik in [24]. After that, several interesting results of the existence of fixed point for singlevalued and multivalued operators in bmetric spaces have been obtained (see [25–40]). Pacurar [41] proved some results on sequences of almost contractions and fixed points in bmetric spaces. Recently, Hussain and Shah [42] obtained results on KKM mappings in cone bmetric spaces. Using the concepts of partially ordered metric spaces, almost generalized contractive condition, and bmetric spaces, we define a new concept of almost generalized (\psi ,\phi ,L)contractive condition. In this paper, some coincidence and common fixed point theorems for mappings satisfying almost generalized (\psi ,\phi ,L)contractive condition in the setup of partially ordered complete bmetric spaces are proved. Consistent with [43] and [[40], p.264], the following definitions and results will be needed in the sequel.
Definition 1.1 [43]
Let X be a (nonempty) set and s\ge 1 be a given real number. A function d:X\times X\to {\mathbb{R}}^{+} is said to be a bmetric space iff for all x,y,z\in X, the following conditions are satisfied:

(i)
d(x,y)=0 iff x=y,

(ii)
d(x,y)=d(y,x),

(iii)
d(x,y)\le s[d(x,z)+d(z,y)].
The pair (X,d) is called a bmetric space with the parameter s.
It should be noted that the class of bmetric spaces is effectively larger than that of metric spaces, since a bmetric is a metric, when s=1.
The following example shows that in general a bmetric does not necessarily need to be a metric (see, also, [40]).
Example 1.1 [44]
Let (X,d) be a metric space and \rho (x,y)={(d(x,y))}^{p}, where p>1 is a real number. Then ρ is a bmetric 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)=xy is the usual Euclidean metric, then \rho (x,y)={(xy)}^{s} is a bmetric on ℝ with s=2, but it is not a metric on ℝ.
Also, the following example of a bmetric space is given in [45].
Example 1.2 [45]
Let X be the set of Lebesgue measurable functions on [0,1] such that {\int}_{0}^{1}{f(x)}^{2}\phantom{\rule{0.2em}{0ex}}dx<\mathrm{\infty}. Define D:X\times X\to [0,\mathrm{\infty}) by D(f,g)={\int}_{0}^{1}{f(x)g(x)}^{2}\phantom{\rule{0.2em}{0ex}}dx. As {({\int}_{0}^{1}{f(x)g(x)}^{2}\phantom{\rule{0.2em}{0ex}}dx)}^{\frac{1}{2}} is a metric on X, then, from the previous example, D is a bmetric on X, with s=2, where the bmetric D is defined with D(x,y)=\parallel d(x,y)\parallel, d is a cone metric (also see [46–49]).
Khamsi [50] also showed that each cone metric space over a normal cone has a bmetric structure.
Definition 1.2 [6]
We shall say that the mapping T is gnondecreasing if
2 Main results
Throughout the paper, let Ψ be the family of all functions \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying the following conditions:

(a)
ψ is continuous,

(b)
ψ is nondecreasing,

(c)
\psi (0)=0<\psi (t) for every t>0.
We denote by Φ the set of all functions \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying the following conditions:

(i)
φ is right continuous,

(ii)
φ is nondecreasing,

(iii)
\phi (t)<t for every t>0.
Let (X,d,\le ) be a partially ordered bmetric space and T:X\to X and g:X\to X be two mappings. Set
and
Now, we introduce the following definition.
Definition 2.1 Let (X,d,\le ) be a partially ordered bmetric space. We say that T:X\to X is an almost generalized (\psi ,\phi ,L)contractive mapping with respect to g:X\to X for some \psi \in \mathrm{\Psi}, \phi \in \mathrm{\Phi}, and L\ge 0 if
for all x,y\in X with gx\le gy.
Now, we establish some results for the existence of coincidence point and common fixed point of mappings satisfying almost generalized (\psi ,\phi ,L)contractive condition in the setup of partially ordered bmetric spaces. The first result in this paper is the following coincidence point theorem.
Theorem 2.1 Suppose that (X,d,\le ) is a partially ordered complete bmetric space. Let T:X\to X be an almost generalized (\psi ,\phi ,L)contractive mapping with respect to g:X\to X, and T and g are continuous such that T is a monotone gnondecreasing mapping, commutative with g and T(X)\subseteq g(X). If there exists {x}_{0}\in X such that g{x}_{0}\le T{x}_{0}, then T and g have a coincidence point in X.
Proof By the given assumptions, there exists {x}_{0}\in X such that g{x}_{0}\le T{x}_{0}. Since T(X)\subseteq g(X), we can define {x}_{1}\in X such that g{x}_{1}=T{x}_{0}, then g{x}_{0}\le T{x}_{0}=g{x}_{1}. Also there exists {x}_{2}\in X such that g{x}_{2}=T{x}_{1}. Since T is a monotone gnondecreasing mapping, we have
Continuing in this way, we construct a sequence \{{x}_{n}\} in X such that for all n=0,1,2,\dots ,
for which
If there exists {k}_{0}\in \mathbb{N} such that g{x}_{{k}_{0}+1}=g{x}_{{k}_{0}}, then g{x}_{{k}_{0}}=T{x}_{{k}_{0}}. This means that {x}_{{k}_{0}} is a coincidence point of T, g, and the proof is finished. Thus, g{x}_{n+1}\ne g{x}_{n} for all n\in \mathbb{N}. From (2.2) and (2.3) and the inequality (2.1) with (x,y)=({x}_{n},{x}_{n+1}), we have
where
and
Since
then we get
By (2.4) and (2.5), we have
Suppose that max\{d(g{x}_{n},g{x}_{n+1}),d(g{x}_{n+1},g{x}_{n+2})\}=d(g{x}_{n+1},g{x}_{n+2})>0 for some n\in \mathbb{N}, then by (2.6)
a contradiction. Hence,
and thus
Thus, we get
for all n\in \mathbb{N}. Now, from
and the property of φ, we obtain {lim}_{n\to \mathrm{\infty}}\psi (d(g{x}_{n},g{x}_{n+1}))=0, and consequently
Now, we shall prove that \{g{x}_{n}\} is a Cauchy sequence in (X,d). Suppose, on the contrary, that \{g{x}_{n}\} is not a Cauchy sequence. Then there exist \u03f5>0 and subsequences \{g{x}_{m(k)}\}, \{g{x}_{n(k)}\} of \{g{x}_{n}\} with m(k)>n(k)\ge k such that
Additionally, corresponding to n(k), we may choose m(k) such that it is the smallest integer satisfying (2.8) and m(k)>n(k)\ge k. Thus,
Using the triangle inequality in bmetric space and (2.8) and (2.9) we obtain
Taking the upper limit as k\to \mathrm{\infty} and using (2.7) we obtain
Also
So from (2.7) and (2.10), we have
Also
So from (2.7) and (2.10), we have
Also
so from (2.7) and (2.12), we have
Linking (2.7), (2.10), (2.11) together with (2.12) we get
So,
Similarly, we have
Since m(k)>n(k) from (2.2), we have
Thus,
Passing to the upper limit as k\to \mathrm{\infty}, and using (2.13), (2.14), and (2.15), we get
which is a contradiction. Thus, we proved that \{g{x}_{n}\} is a Cauchy sequence in (X,d). Since X is a complete bmetric space, there exists x\in X such that
From the commutativity of T and g, we have
Letting n\to \mathrm{\infty} in (2.17) and from the continuity of T and g, we get
This implies that x is a coincidence point of T and g. This completes the proof. □
Now, we will prove the following result.
Theorem 2.2 Suppose that (X,d,\le ) is a partially ordered complete bmetric space. Let T:X\to X be an almost generalized (\psi ,\phi ,L)contractive mapping with respect to g:X\to X, T is a gnondecreasing mapping and T(X)\subseteq g(X). Also suppose
Also suppose gX is closed. If there exists {x}_{0}\in X such that g{x}_{0}\le T{x}_{0}, then T and g have a coincidence. Further, if T and g commute at their coincidence points, then T and g have a common fixed point.
Proof As in the proof of Theorem 2.1, we can show that \{g{x}_{n}\} is a Cauchy sequence. Since gX is a closed, there exists x\in X such that
Now we show that x is a coincidence point of T and g. Since from (2.18) and (2.19) we have g{x}_{n}\le gx for all n, then by the triangle inequality in a bmetric space and (2.1), we get
Indeed,
and
Hence d(gx,Tx)=0, that is, Tx=gx. Thus we proved that T and g have a coincidence. Suppose now that T and g commute at x. Set y=Tx=gx. Then
Since from (2.18) we have gx\le g(gx)=gy and as gx=Tx and gy=Ty, from (2.1) we obtain
Hence d(Tx,Ty)=0, that is, y=Tx=Ty. Therefore, Ty=gy=y. Thus we proved that T and g have a common fixed point. □
In the following, we deduce some fixed point theorems from our main results given by Theorems 2.1 and 2.2.
Corollary 2.3 Let (X,d,\le ) be a partially ordered complete bmetric space and T:X\to X is a nondecreasing mapping. Suppose there exist \psi \in \mathrm{\Psi}, \phi \in \mathrm{\Phi}, and L\ge 0 such that
where
and
for all x,y\in X with x\le y. Also suppose either

(a)
if \{{x}_{n}\}\subset X is a nondecreasing sequence with {x}_{n}\to z in X, then {x}_{n}\le z, for all n, holds, or

(b)
T is continuous.
If there exists {x}_{0}\in X such that {x}_{0}\le T{x}_{0}, then T has a fixed point in X.
Example 2.1 Let X be the set of Lebesgue measurable functions on [0,1] such that {\int}_{0}^{1}x(t)\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty}. Define D:X\times X\to [0,\mathrm{\infty}) by
Then D is a bmetric on X, with s=2. Also, this space can also be equipped with a partial order given by
The operator T:X\to X defined by
Now, we prove that T has a fixed point. For all x,y\in X with x\le y, we have
Now, if we define \phi (t)=ln(1+t), \psi (t)=\sqrt{t}, and {x}_{0}=0. Thus, by Corollary 2.3 we see that T has a fixed point.
Remark 2.1 Corollary 2.3 extends and generalizes many existing fixed point theorems in the literature [2, 3, 51, 52].
The following result is the immediate consequence of Corollary 2.3.
Corollary 2.4 Let (X,d,\le ) be a partially ordered complete bmetric space and T:X\to X is a nondecreasing mapping. Suppose there exists \phi \in \mathrm{\Phi} such that
for all x,y\in X with x\le y. Also suppose either

(a)
if \{{x}_{n}\}\subset X is a nondecreasing sequence with {x}_{n}\to z in X, then {x}_{n}\le z, for all n, holds, or

(b)
T is continuous.
If there exists {x}_{0}\in X such that {x}_{0}\le T{x}_{0}, then T has a fixed point in X.
Remark 2.2 Corollary 2.4 is a generalization to [[3], Theorem 1.3].
Taking \phi (t)=\lambda t, 0<\lambda <1, in Corollary 2.4 we obtain the following generalization of the results in [1, 53].
Corollary 2.5 Let (X,d,\le ) be a partially ordered complete bmetric space and T:X\to X is a nondecreasing mapping. Suppose there exists \phi \in \mathrm{\Phi} such that
for all x,y\in X with x\le y. Also suppose either

(a)
if \{{x}_{n}\}\subset X is a nondecreasing sequence with {x}_{n}\to z in X, then {x}_{n}\le z, for all n, holds, or

(b)
T is continuous.
If there exists {x}_{0}\in X such that {x}_{0}\le T{x}_{0}, then T has a fixed point in X.
Corollary 2.6 Let (X,d,\le ) be a partially ordered complete bmetric space and T:X\to X is a nondecreasing mapping. Suppose there exist \psi \in \mathrm{\Psi} and 0\le \lambda <1 such that
for all x,y\in X with x\le y. Also suppose either

(a)
if \{{x}_{n}\}\subset X is a nondecreasing sequence with {x}_{n}\to z in X, then {x}_{n}\le z, for all n, holds, or

(b)
T is continuous.
If there exists {x}_{0}\in X such that {x}_{0}\le T{x}_{0}, then T has a fixed point in X.
3 Application to integral equations
Here, in this section, we wish to study the existence of a unique solution to a nonlinear quadratic integral equation, as an application to the our fixed point theorem. Consider the integral equation
Let Γ denote the class of those functions \gamma :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) for which \gamma \in \mathrm{\Phi} and {(\gamma (t))}^{p}\le \gamma ({t}^{p}), for all p\ge 1.
For example, {\gamma}_{1}(t)=kt, where 0\le k<1 and {\gamma}_{2}(t)=\frac{t}{t+1} are in Γ.
We will analyze (3.1) under the following assumptions:
(a_{1}) f:I\times \mathbb{R}\to \mathbb{R} is continuous monotone nondecreasing in x, f(t,x)\ge 0 and there exist constant 0\le L<1 and \gamma \in \mathrm{\Gamma} such that for all x,y\in \mathbb{R} and x\ge y
(a_{2}) h:I\to \mathbb{R} is a continuous function.
(a_{3}) k:I\times I\to \mathbb{R} is continuous in t\in I for every s\in I and measurable in s\in I for all t\in I such that
and k(t,s)\ge 0.
(a_{4}) There exists \alpha \in C(I) such that
(a_{5}) {L}^{p}{\lambda}^{p}{K}^{p}\le \frac{1}{{2}^{3p3}}.
We consider the space X=C(I) of continuous functions defined on I=[0,1] with the standard metric given by
This space can also be equipped with a partial order given by
Now for p\ge 1, we define
It is easy to see that (X,d) is a complete bmetric space with s={2}^{p1} [44].
For any x,y\in X and each t\in I, max\{x(t),y(t)\} and min\{x(t),y(t)\} belong to X and are upper and lower bounds of x, y, respectively. Therefore, for every x,y\in X, one can take max\{x,y\},min\{x,y\}\in X which are comparable to x, y. Now, we formulate the main result of this section.
Theorem 3.1 Under assumptions (a_{1})(a_{5}), (3.1) has a unique solution in C(I).
Proof We consider the operator T:X\to X defined by
By virtue of our assumptions, T is well defined (this means that if x\in X then T(x)\in X). For x\le y, and t\in I we have
Therefore, T has the monotone nondecreasing property. Also, for x\le y, we have
Since the function γ is nondecreasing and x\le y, we have
hence
Then we obtain
This proves that the operator T satisfies the contractive condition (2.21) appearing in Corollary 2.4. Also, let α, β be the functions appearing in assumption (a_{4}); then, by (a_{4}), we get \alpha \le T(\alpha ). So, (3.1) has a solution and the proof is complete. □
Example 3.1 Consider the following functional integral equation:
for t\in [0,1]. Observe that this equation is a special case of (3.1) with
Indeed, by using \gamma (t)=\frac{1}{3}t we see that \gamma \in \mathrm{\Phi} and {(\gamma (t))}^{p}={(\frac{1}{3}t)}^{p}=\frac{1}{{3}^{p}}{t}^{p}\le \frac{1}{3}{t}^{p}=\gamma ({t}^{p}), for all p\ge 1. Further, for arbitrarily fixed x,y\in \mathbb{R} such that x\ge y and for t\in [0,1] we obtain
Thus, the function f satisfies assumption (a_{1}) with L=\frac{1}{6}. It is also easily seen that h is a continuous function. Further, notice that the function k is continuous in t\in I for every s\in I and measurable in s\in I for all t\in I and k(t,s)\ge 0. Moreover, we have
If we put \alpha (t)=\frac{3{t}^{2}}{4(1+{t}^{4})}, we have
This shows that assumption (a_{4}) holds. Taking L=\frac{1}{6}, K=\frac{2}{3} and \lambda =\frac{1}{27}, then inequality {L}^{p}{\lambda}^{p}{K}^{p}\le \frac{1}{{2}^{3p3}} appearing in assumption (a_{5}) has the following form:
It is easily seen that each number p\ge 1 satisfies the above inequality. Consequently, all the conditions of Theorem 3.1 are satisfied. Hence the integral equation (3.2) has a unique solution in C(I).
References
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Soc. 2004, 132: 14351443. 10.1090/S0002993903072204
Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223239. 10.1007/s1108300590185
Agarwal RP, ElGebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008,87(1):109116. 10.1080/00036810701556151
Altun I, Durmaz G: Some fixed point theorems on ordered cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 319325. 10.1007/s122150090026y
Beg I, Rashid Butt A: Fixed point for setvalued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal. 2009, 71: 36993704. 10.1016/j.na.2009.02.027
Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008: Article ID 131294, (2008)
Grana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 13791393. 10.1016/j.na.2005.10.017
Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 34023410.
Berinde V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 2004, 9: 4353.
Berinde V: Some remarks on a fixed point theorem for Ćirićtype almost contractions. Carpath. J. Math. 2009, 25: 157162.
Berinde V: Common fixed points of noncommuting almost contractions in cone metric spaces. Math. Commun. 2010, 15: 229241.
Berinde V: Approximating common fixed points of noncommuting almost contractions in metric spaces. Fixed Point Theory 2010, 11: 179188.
Hussain N, Ðorić D, Kadelburg Z, Radenović S: Suzukitype fixed point results in metric type spaces. Fixed Point Theory Appl. 2012: Article ID 126, (2012)
Jovanović M, Kadelburg Z, Radenović S: Common fixed point results in metrictype spaces. Fixed Point Theory Appl. 2010: Article ID 978121, (2010)
Pacurar M: Fixed point theory for cyclic Berinde operators. Fixed Point Theory 2012, 11: 419428.
Radenović S, Kadelburg Z: Quasicontractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 2011,5(1):3850. 10.15352/bjma/1313362978
Radenović S, Kadelburg Z, Jandrlić D, Jandrlić A: Some results on weakly contractive maps. Bull. Iran. Math. Soc. 2012,38(3):625645.
Shah MH, Simić S, Hussain N, Sretenović A, Radenović S: Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces. J. Comput. Anal. Appl. 2012,14(2):290297.
Shukla S: Partial rectangular metric spaces and fixed point theorems. Sci. World J. 2014: Article ID 756298, (2014)
Suzuki T: Fixed point theorems for Berinde mappings. Bull. Kyushu Inst. Technol., Pure Appl. Math. 2011, 58: 1319.
Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contractions. Carpath. J. Math. 2008, 24: 812.
Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217: 57845789. 10.1016/j.amc.2010.12.060
Aghajani A, Radenović S, Roshan JR: Common fixed point results for four mappings satisfying almost generalized (S,T)contractive condition in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 56655670. 10.1016/j.amc.2011.11.061
Czerwik S: Contraction mappings in b metric spaces. Acta Math. Inform. Univ. Ostrav. 1993, 1: 511.
Aghajani A, Arab R: Fixed points of(\psi ,\varphi ,\theta )contractive mappings in partially ordered b metric spaces and application to quadratic integral equation. Fixed Point Theory Appl. 2013: Article ID 245, (2013)
Azam A, Mehmood N, Ahmad J, Radenović S: Multivalued fixed point theorems in cone b metric spaces. J. Inequal. Appl. 2013.2013: Article ID 582, (2013)
Boriceanu M: Fixed point theory for multivalued generalized contraction on a set with two b metrics. Stud. Univ. BabeşBolyai, Math. 2009,LIV(3):314.
Boriceanu M: Strict fixed point theorems for multivalued operators in b metric spaces. Int. J. Mod. Math. 2009,4(3):285301.
George, R, Radenović, S, Reshma, KP, Shukla, S: Rectangular bmetric spaces and contraction principle. J. Nonlinear Sci. Appl. (2014, in press)
Hussain N, Parvaneh V, Roshan JR, Kadelburg Z: Fixed points of cyclic(\psi ,\phi ,L,A,B)contractive mappings in ordered b metric spaces with applications. Fixed Point Theory Appl. 2013: Article ID 256, (2013)
Mustafa Z, Roshan JR, Parvaneh V:Coupled coincidence point results for (\psi ,\phi )weakly contractive mappings in partially ordered {G}_{b}metric spaces. Fixed Point Theory Appl. 2013: Article ID 206, (2013)
Mustafa Z, Roshan JR, Parvaneh V, Kadelburg Z: Some common fixed point results in ordered partial b metric spaces. J. Inequal. Appl. 2013: Article ID 562, (2013)
Popović B, Radenović S, Shukla S: Fixed point results to TVScone b metric spaces. Gulf J. Math. 2013, 1: 5164.
Roshan JR, Shobkolaei N, Sedhi S, Parvaneh V, Radenović S:Common fixed point theorems for three maps in discontinuous {G}_{b}metric spaces. Acta Math. Sci. Ser. B 2014,34(5):112.
Shukla S: Partial b metric spaces and fixed point theorems. Mediterr. J. Math. 2014, 11: 703711. 10.1007/s0000901303274
Zabihi F, Razani A: Fixed point theorems for hybrid rational Geraghty contractive mappings in ordered b metric spaces. J. Appl. Math. 2014: Article ID 929821, 2014
Roshan JR, Parvaneh V, Altun I: Some coincidence point results in ordered b metric spaces and applications in a system of integral equations. Appl. Math. Comput. 2014, 226: 725737.
Roshan, JR, Parvaneh, V, Kadelburg, Z: Common fixed point theorems for weakly isotone increasing mappings in ordered bmetric spaces. J. Nonlinear Sci. Appl. (to appear). www.tjnsa.com
Roshan JR, Parvaneh V, Sedghi S, Shobkolaei N, Shatanawi W: Common fixed points of almost generalized {(\psi ,\phi )}_{s}contractive mappings in ordered b metric spaces. Fixed Point Theory Appl. 2013: Article ID 159, (2013)
Singh SL, Prasad B: Some coincidence theorems and stability of iterative procedures. Comput. Math. Appl. 2008, 55: 25122520. 10.1016/j.camwa.2007.10.026
Pacurar M: Sequences of almost contractions and fixed points in b metric spaces. An. Univ. Vest. Timiş., Ser. Mat.Inform. 2010, 48: 125137.
Hussain N, Shah MH: KKM mappings in cone b metric spaces. Comput. Math. Appl. 2011, 62: 16771684. 10.1016/j.camwa.2011.06.004
Czerwik S: Nonlinear setvalued contraction mappings in b metric spaces. Atti Semin. Mat. Fis. Univ. Modena 1998,46(2):263276.
Aghajani, A, Abbas, M, Roshan, JR: Common fixed point of generalized weak contractive mappings in partially ordered bmetric spaces. Math. Slovaca (in press)
Khamsi MA, Hussain N: KKM mappings in metric type spaces. Nonlinear Anal. 2010,73(9):31233129. 10.1016/j.na.2010.06.084
Ðukić D, Kadelburg Z, Radenović S: Fixed points of Geraghtytype mappings in various generalized metric spaces. Abstr. Appl. Anal. 2011: Article ID 561245, (2011)
Parvaneh V, Roshan JR, Radenović S: Existence of tripled coincidence point in ordered b metric spaces and application to a system of integral equations. Fixed Point Theory Appl. 2013: Article ID 130, (2013)
Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces. Comput. Math. Appl. 2010, 60: 17761783. 10.1016/j.camwa.2010.07.008
An TV, Dung NV, Kadelburg Z, Radenović S: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2014. 10.1007/s1339801401737
Khamsi MA: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010: Article ID 315398 (2010). 10.1155/2010/315398
Altman M: A fixed point theorem in compact metric spaces. Am. Math. Mon. 1975, 82: 827829. 10.2307/2319801
Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30: 19. 10.1017/S0004972700001659
Ray BK: On Ciric’s fixed point theorem. Fundam. Math. 1977,94(3):221229.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Allahyari, R., Arab, R. & Shole Haghighi, A. A generalization on weak contractions in partially ordered bmetric spaces and its application to quadratic integral equations. J Inequal Appl 2014, 355 (2014). https://doi.org/10.1186/1029242X2014355
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2014355
Keywords
 fixed point
 common fixed point
 coincidence point
 integral equations
 bmetric space
 partially ordered set