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Fixed points of cyclic weakly $(\psi ,\phi ,L,A,B)$contractive mappings in ordered bmetric spaces with applications
Fixed Point Theory and Applications volume 2013, Article number: 256 (2013)
Abstract
We introduce the notion of ordered cyclic weakly $(\psi ,\phi ,L,A,B)$contractive mappings, and we establish some fixed and common fixed point results for this class of mappings in complete ordered bmetric spaces. Our results extend several known results from the context of ordered metric spaces to the setting of ordered bmetric spaces. They are also cyclic variants of some very recent results in ordered bmetric spaces with even weaker contractive conditions. Some examples support our results and show that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.
MSC:47H10, 54H25.
1 Introduction and preliminaries
The Banach contraction principle is a very popular tool for solving problems in nonlinear analysis. One of the interesting generalizations of this basic principle was given by Kirk et al. [1] in 2003 by introducing the following notion of cyclic representation.
Definition 1 [1]
Let A and B be nonempty subsets of a metric space $(X,d)$ and $T:A\cup B\to A\cup B$. Then T is called a cyclic map if $T(A)\subseteq B$ and $T(B)\subseteq A$.
The following interesting theorem for a cyclic map was given in [1].
Theorem 1 Let A and B be nonempty closed subsets of a complete metric space $(X,d)$. Suppose that $T:A\cup B\to A\cup B$ is a cyclic map such that
for all $x\in A$ and $y\in B$, where $k\in [0,1)$ is a constant. Then T has a unique fixed point u and $u\in A\cap B$.
It should be noted that cyclic contractions (unlike Banachtype contractions) need not be continuous, which is an important gain of this approach. Following the work of Kirk et al., several authors proved many fixed point results for cyclic mappings, satisfying various (nonlinear) contractive conditions.
Berinde initiated in [2] 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., [3–7]. Some authors used related notions as ‘condition (B)’ (Babu et al. [8]) and ‘almost generalized contractive condition’ for two maps (Ćirić et al. [9]), and for four maps (Aghajani et al. [10]). See also a note by Pacurar [11]. Here, we recall one of the respective definitions.
Definition 2 [9]
Let f and g be two selfmappings on a metric space $(X,d)$. They are said to satisfy almost generalized contractive condition, if there exist a constant $\delta \in (0,1)$ and some $L\ge 0$ such that
for all $x,y\in X$.
Khan et al. [12] introduced the concept of an altering distance function as follows.
Definition 3 [12]
A function $\phi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is called an altering distance function if the following properties hold:

1.
φ is continuous and nondecreasing.

2.
$\phi (t)=0$ if and only if $t=0$.
So far, many authors have studied fixed point theorems, which are based on altering distance functions.
The concept of a bmetric space was introduced by Bakhtin in [13], and later used by Czerwik in [14, 15]. After that, several interesting results about the existence of fixed points for singlevalued and multivalued operators in bmetric spaces have been obtained (see, e.g., [16–28]). Recently, Hussain and Shah [29] obtained some results on KKM mappings in cone bmetric spaces.
Consistent with [15] and [28], the following definitions and results will be needed in the sequel.
Definition 4 [15]
Let X be a (nonempty) set, and let $s\ge 1$ be a given real number. A function $d:X\times X\to {R}^{+}$ is a bmetric if for all $x,y,z\in X$, the following conditions hold:
(b_{1}) $d(x,y)=0$ iff $x=y$,
(b_{2}) $d(x,y)=d(y,x)$,
(b_{3}) $d(x,z)\le s[d(x,y)+d(y,z)]$.
In this case, the pair $(X,d)$ is called a bmetric space.
It should be noted that the class of bmetric spaces is effectively larger than the class of metric spaces, since a bmetric is a metric if (and only if) $s=1$. Here, we present an easy example to show that in general, a bmetric need not necessarily be a metric (see also [[28], p.264]).
Example 1 Let $(X,\rho )$ be a metric space and $d(x,y)={(\rho (x,y))}^{p}$, where $p>1$ is a real number. Then d is a bmetric with $s={2}^{p1}$. Condition (b_{3}) follows easily from the convexity of the function $f(x)={x}^{p}$ ($x>0$).
The notions of bconvergent and bCauchy sequences, as well as of bcomplete bmetric spaces are introduced in an obvious way (see, e.g., [18]).
It should be noted that in general, a bmetric function $d(x,y)$ for $s>1$ need not be jointly continuous in both variables. The following example (corrected from [22]) illustrates this fact.
Example 2 Let $X=\mathbb{N}\cup \{\mathrm{\infty}\}$, and let $d:X\times X\to \mathbb{R}$ be defined by
Then considering all possible cases, it can be checked that for all $m,n,p\in X$, we have
Thus, $(X,d)$ is a bmetric space (with $s=5/2$). Let ${x}_{n}=2n$ for each $n\in \mathbb{N}$. Then
that is, ${x}_{n}\to \mathrm{\infty}$, but $d({x}_{n},1)=2\nrightarrow 5=d(\mathrm{\infty},1)$ as $n\to \mathrm{\infty}$.
Aghajani et al. [16] proved the following simple lemma about the bconvergent sequences.
Lemma 1 Let $(X,d)$ be a bmetric space with $s\ge 1$, and suppose that $\{{x}_{n}\}$ and $\{{y}_{n}\}$ bconverge to x, y, respectively. Then we have
In particular, if $x=y$, then ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{y}_{n})=0$. Moreover, for each $z\in X$, we have
The existence of fixed points for mappings in partially ordered metric spaces was first investigated in 2004 by Ran and Reurings [30], and then by Nieto and Lopez [31]. Afterwards, this area was a field of intensive study of many authors.
Shatanawi and Postolache proved in [32] the following common fixed point results for cyclic contractions in the framework of ordered metric spaces.
Theorem 2 [32]
Let $(X,\u2aaf,d)$ be a complete ordered metric space, and let A, B be closed nonempty subsets of X with $X=A\cup B$. Let $f,g:X\to X$ be two mappings, which are $(A,B)$weakly increasing (see further Definition 6). Assume that

(a)
$A\cup B$ is a cyclic representation of X w.r.t. the pair $(f,g)$, i.e., $f(A)\subset B$ and $g(B)\subset A$;

(b)
there exist $0<\delta <1$ and an altering distance function ψ such that for any two comparable elements $x,y\in X$ with $x\in A$ and $y\in B$, we have
$$\psi (d(fx,gy))\le \delta \psi (max\{d(x,y),d(x,fx),d(y,gy),\frac{1}{2}(d(x,gy)+d(y,fx))\});$$ 
(c)
f or g is continuous, or
(c′) the space $(X,\u2aaf,d)$ is regular.
Then f and g have a common fixed point.
Here, the ordered metric space $(X,\u2aaf,d)$ is called regular if for any nondecreasing sequence $\{{x}_{n}\}$ in X such that ${x}_{n}\to x\in X$, as $n\to \mathrm{\infty}$, one has ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
By an ordered bmetric space, we mean a triple $(X,\u2aaf,d)$, where $(X,\u2aaf)$ is a partially ordered set, and $(X,d)$ is a bmetric space. Fixed points in such spaces were studied, e.g., by Aghajani et al. [16] and Roshan et al. [27]. In the last mentioned paper, the following common fixed point results for contractions in ordered bmetric spaces were proved.
Theorem 3 [27]
Let $(X,\u2aaf,d)$ be a complete ordered bmetric space, and let $f,g:X\to X$ be two weakly increasing mappings. Suppose that there exist two altering distance functions ψ, φ and a constant $L\ge 0$ such that the inequality
holds for all comparable $x,y\in X$, where
and
If either [f or g is continuous], or the space $(X,\u2aaf,d)$ is regular, then f and g have a common fixed point.
In this paper, we introduce the notion of ordered cyclic weakly $(\psi ,\phi ,L,A,B)$contractions and then derive fixed point and common fixed point theorems for these cyclic contractions in the setup of complete ordered bmetric spaces. Our results extend some fixed point theorems from the framework of ordered metric spaces, in particular Theorem 2. On the other hand, they are cyclic variants of Theorem 3 with even weaker contractive conditions.
We show by examples that the obtained extensions are proper. Moreover, an application to integral equations is given here to illustrate the usability of the obtained results.
2 Common fixed point results
In this section, we introduce the notion of ordered cyclic weakly $(\psi ,\phi ,L,A,B)$contractive pair of selfmappings and prove our main results.
Definition 5 Let $(X,\u2aaf,d)$ be an ordered bmetric space, let $f,g:X\to X$ be two mappings, and let A and B be nonempty closed subsets of X. The pair $(f,g)$ is called an ordered cyclic weakly $(\psi ,\phi ,L,A,B)$contraction if

(1)
$X=A\cup B$ is a cyclic representation of X w.r.t. the pair $(f,g)$; that is, $fA\subseteq B$ and $gB\subseteq A$;

(2)
there exist two altering distance functions ψ, φ and a constant $L\ge 0$, such that for arbitrary comparable elements $x,y\in X$ with $x\in A$ and $y\in B$, we have
$$\psi ({s}^{2}d(fx,gy))\le \psi ({M}_{s}(x,y))\phi ({M}_{s}(x,y))+L\psi (N(x,y)),$$(2.1)
where
and
Definition 6 [32]
Let $(X,\u2aaf)$ be a partially ordered set, and let A and B be closed subsets of X with $X=A\cup B$. Let $f,g:X\to X$ be two mappings. The pair $(f,g)$ is said to be $(A,B)$weakly increasing if $fx\u2aafgfx$ for all $x\in A$ and $gy\u2aaffgy$ for all $y\in B$.
Theorem 4 Let $(X,\u2aaf,d)$ be a complete ordered bmetric space, and let A and B be closed subsets of X. Let $f,g:X\to X$ be two $(A,B)$weakly increasing mappings with respect to ⪯. Suppose that

(a)
the pair $(f,g)$ is an ordered cyclic weakly $(\psi ,\phi ,L,A,B)$contraction;

(b)
f or g is continuous.
Then f and g have a common fixed point $u\in A\cap B$.
Proof Let us divide the proof into two parts.
First part. We prove that $u\in A\cap B$ 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. As $u\u2aafu$ and $u\in A\cap B$, by (2.1), we have
It follows that $\phi (d(u,gu))=0$. Therefore, $d(u,gu)=0$, and hence $gu=u$. Similarly, we can show 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 ${x}_{0}\in A$, and let ${x}_{1}=f{x}_{0}$. Since $fA\subseteq B$, we have ${x}_{1}\in B$. Also, let ${x}_{2}=g{x}_{1}$. Since $gB\subseteq A$, we have ${x}_{2}\in A$. Continuing this process, we can construct a sequence $\{{x}_{n}\}$ in X such that ${x}_{2n+1}=f{x}_{2n}$, ${x}_{2n+2}=g{x}_{2n+1}$, ${x}_{2n}\in A$ and ${x}_{2n+1}\in B$. Since f and g are $(A,B)$weakly increasing, we have
If ${x}_{2n}={x}_{2n+1}$, for some $n\in \mathbb{N}$, then ${x}_{2n}=f{x}_{2n}$. Thus, ${x}_{2n}$ is a fixed point of f. By the first part of proof, we conclude that ${x}_{2n}$ is also a fixed point of g. Similarly, if ${x}_{2n+1}={x}_{2n+2}$, for some $n\in \mathbb{N}$, then ${x}_{2n+1}=g{x}_{2n+1}$. Thus, ${x}_{2n+1}$ is a fixed point of g. By the first part of proof, we conclude that ${x}_{2n+1}$ is also a fixed point of f. Therefore, we assume that ${x}_{n}\ne {x}_{n+1}$ for all $n\in \mathbb{N}$. Now, we complete the proof in the following steps.
Step 1. We will prove that
As ${x}_{2n}$ and ${x}_{2n+1}$ are comparable and ${x}_{2n}\in A$ and ${x}_{2n+1}\in B$, by (2.1), we have
where
and
Hence, we have
If
then (2.4) becomes
which gives a contradiction. So,
and hence, (2.4) becomes
Similarly, we can show that
By (2.5) and (2.6), we get that $\{d({x}_{n},{x}_{n+1}):n\in \mathbb{N}\}$ is a nonincreasing sequence of positive numbers. Hence, there is $r\ge 0$ such that
Letting $n\to \mathrm{\infty}$ in (2.5), we get
which implies that $\phi (r)=0$, and hence $r=0$. So, we have
Step 2. We will prove that $\{{x}_{n}\}$ is a bCauchy sequence. Because of (2.7), it is sufficient to show that $\{{x}_{2n}\}$ is a bCauchy sequence. Suppose on the contrary, i.e., that $\{{x}_{2n}\}$ is not a bCauchy sequence. Then there exists $\epsilon >0$, for which we can find two subsequences $\{{x}_{2{m}_{i}}\}$ and $\{{x}_{2{n}_{i}}\}$ of $\{{x}_{2n}\}$ such that ${n}_{i}$ is the smallest index, for which
This means that
From (2.8) and using the triangular inequality, we get
Using (2.7) and taking the upper limit as $i\to \mathrm{\infty}$, we get
On the other hand, we have
Using (2.7), (2.9) and taking the upper limit as $i\to \mathrm{\infty}$, we get
Again, using the triangular inequality, we have
and
Taking the upper limit as $i\to \mathrm{\infty}$ in the above inequalities, and using (2.7), (2.9) and (2.11), we get
and
Since ${x}_{2{m}_{i}}$ and ${x}_{2{n}_{i}1}$ are comparable and ${x}_{2{m}_{i}}\in A$ and ${x}_{2{n}_{i}1}\in B$, using (2.1) we have
where
and
Taking the upper limit in (2.15) and using (2.7) and (2.11)(2.13), we get
Hence, we have
and, from (2.16),
Now, taking the upper limit as $i\to \mathrm{\infty}$ in (2.14) and using (2.10), (2.17) and (2.18), we have
which implies that $\phi ({lim\hspace{0.17em}inf}_{i\to \mathrm{\infty}}{M}_{s}({x}_{2{m}_{i}},{x}_{2{n}_{i}1}))=0$. By (2.15), it follows that ${lim\hspace{0.17em}inf}_{i\to \mathrm{\infty}}d({x}_{2{m}_{i}},{x}_{2{n}_{i}})=0$, which is in contradiction with (2.8). Hence $\{{x}_{n}\}$ is a bCauchy sequence in X.
Step 3 (existence of a common fixed point).
As $\{{x}_{n}\}$ is a bCauchy sequence in X which is a bcomplete bmetric space, there exists $u\in X$ such that ${x}_{n}\to u$ as $n\to \mathrm{\infty}$, and
Now, without loss of generality, we may assume that f is continuous. Using the triangular inequality, we get
Letting $n\to \mathrm{\infty}$, we get
Hence, we have $fu=u$. Thus, u is a fixed point of f and, since A and B are closed subsets of X, $u\in A\cap B$. By the first part of proof, we conclude that u is also a fixed point of g. □
The assumption of continuity of one of the mappings f or g in Theorem 4 can be replaced by another condition, which is often used in similar situations. Namely, we shall use the notion of a regular ordered bmetric space, which is defined analogously to the case of the standard metric (see the paragraph following Theorem 2).
Theorem 5 Let the hypotheses of Theorem 4 be satisfied, except that condition (b) is replaced by the assumption
(b′) the space $(X,\u2aaf,d)$ is regular.
Then f and g have a common fixed point in X.
Proof Repeating the proof of Theorem 4, we construct an increasing sequence $\{{x}_{n}\}$ in X such that ${x}_{n}\to u$ for some $u\in X$. As A and B are closed subsets of X, we have $u\in A\cap B$. Using the assumption (b′) on X, we have ${x}_{n}\u2aafu$ for all $n\in \mathbb{N}$. Now, we show that $fu=gu=u$. By (2.1), we have
where
and
Letting $n\to \mathrm{\infty}$ in (2.20) and (2.21) and using Lemma 1, we get
and $N({x}_{2n},u)\to 0$. Now, taking the upper limit as $n\to \mathrm{\infty}$ in (2.19) and using Lemma 1 and (2.22), we get
It follows that $\phi ({lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{M}_{s}({x}_{2n},u))=0$, and hence, by (2.20), that $d(u,gu)=0$. Thus, u is a fixed point of g. On the other hand, similar to the first part of the proof of Theorem 4, we can show that $fu=u$. Hence, u is a common fixed point of f and g. □
3 Consequences and examples
As consequences, we have the following results.
By putting $A=B=X$ in Theorems 4 and 5, we obtain improvements of the main results (Theorems 5 and 6) of Roshan et al. [27], i.e., of Theorem 3 of the present paper (note that we have ${s}^{2}$ instead of ${s}^{4}$ in the contractive condition).
Taking $\phi =(1\delta )\psi $, $0<\delta <1$ in Theorems 4 and 5, we get the following.
Corollary 1 Let $(X,\u2aaf,d)$ be a complete ordered bmetric space, and let A and B be closed subsets of X. Let $f,g:X\to X$ be two $(A,B)$weakly increasing mappings with respect to ⪯. Suppose that

(a)
$X=A\cup B$ is a cyclic representation of X w.r.t. the pair $(f,g)$;

(b)
there exist $0<\delta <1$, $L\ge 0$ and an altering distance function ψ such that for any comparable elements $x,y\in X$ with $x\in A$ and $y\in B$, we have
$$\psi ({s}^{2}d(fx,gy))\le \delta \psi ({M}_{s}(x,y))+L\psi (N(x,y)),$$(3.1)
where ${M}_{s}(x,y)$ and $N(x,y)$ are given by (2.2) and (2.3), respectively;

(c)
f or g is continuous, or
(c′) the space $(X,\u2aaf,d)$ is regular.
Then f and g have a common fixed point $u\in A\cap B$.
Taking $s=1$ and $L=0$ in Corollary 1, we obtain Theorems 2.1 and 2.2 of Shatanawi and Postolache [32] (Theorem 2 in this paper).
Taking $\psi (t)=t$ for $t\in [0,+\mathrm{\infty})$ in Corollary 1, we get the following.
Corollary 2 Let $(X,\u2aaf,d)$ be a complete ordered bmetric space. Let A and B be nonempty closed subsets of X, and let $f,g:X\to X$ be two $(A,B)$weakly increasing mappings with respect to ⪯ such that $f(A)\subseteq B$ and $g(B)\subseteq A$. Suppose that there exist $\delta \in (0,1)$ and $L\ge 0$ such that
for all comparable elements $x,y\in X$ with $x\in A$ and $y\in B$. If either f or g is continuous, or the space $(X,\u2aaf,d)$ is regular, then f and g have a common fixed point.
Putting $f=g$ in Theorems 4 and 5, the following corollary is obtained which extends and improves Theorems 3 and 4 in [27].
Corollary 3 Let $(X,\u2aaf,d)$ be a complete ordered bmetric space, and let A and B be closed subsets of X. Let $f:X\to X$ be a mapping such that f is nondecreasing with respect to ⪯. Assume the following:

(a)
$A\cup B$ is a cyclic representation of X w.r.t. f, that is, $fA\subseteq B$, $fB\subseteq A$;

(b)
there exist two altering distance functions ψ, φ, and $L\ge 0$ such that
$$\psi ({s}^{2}d(fx,fy))\le \psi ({M}_{s}(x,y))\phi ({M}_{s}(x,y))+L\psi (N(x,y))$$(3.2)
for all comparable $x,y\in X$ with $x\in A$ and $y\in B$, where
and

(c)
f is continuous, or
(c′) the space $(X,\u2aaf,d)$ is regular.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\u2aaff{x}_{0}$, then f has a fixed point.
Again, taking $\phi =(1\delta )\psi $, $0<\delta <1$ in Corollary 3, we get the following.
Corollary 4 Let $(X,\u2aaf,d)$ be a complete ordered bmetric space, let and A and B be closed subsets of X. Let $f:X\to X$ be a nondecreasing map with respect to ⪯. Suppose that

(a)
$X=A\cup B$ is a cyclic representation of X w.r.t. f;

(b)
there exist $0<\delta <1$, $L\ge 0$ and an altering distance function ψ such that for any comparable elements $x,y\in X$ with $x\in A$ and $y\in B$, we have
$$\psi ({s}^{2}d(fx,fy))\le \delta \psi ({M}_{s}(x,y))+L\psi (N(x,y)),$$(3.3)
where ${M}_{s}(x,y)$ and $N(x,y)$ are given in Corollary 3;

(c)
f is continuous, or
(c′) the space $(X,\u2aaf,d)$ is regular.
Then f has a fixed point $u\in A\cap B$.
Remark 1 (Common) fixed points of the given mappings in Theorems 4 and 5 and Corollaries 3 and 4 need not be unique (see further Example 4). However, it is easy to show that they must be unique in the case that the respective sets of (common) fixed points are well ordered (recall that a subset W of a partially ordered set is said to be well ordered if every two elements of W are comparable).
We illustrate our results with the following two examples.
Example 3 Consider the bmetric space $(X,d)$ given in Example 2, ordered by natural ordering and a mapping $f:X\to X$ given as
If $A=\{n:n\in \mathbb{N}\}\cup \{\mathrm{\infty}\}$ and $B=\{8n:n\in \mathbb{N}\}\cup \{\mathrm{\infty}\}$, then $A\cup B$ is a cyclic representation of X with respect to f. Take $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ given as $\psi (t)=\sqrt{t}$, $\delta =5/4\sqrt{2}$ (<1) and $L\ge 0$ arbitrary. In order to check the contractive condition (3.3), consider the following cases.
If $x,y\in \mathbb{N}$, then
and (3.3) holds. If $x=\mathrm{\infty}$ and y is an even integer, then
Finally, if $x=\mathrm{\infty}$ and y is an odd integer, then $d(x,y)=5$ and (3.3) trivially holds.
Hence, all the conditions of Corollary 4 are satisfied. Obviously, f has a (unique) fixed point ∞, belonging to $A\cap B$.
We now present an example showing that there are situations where our results can be used to conclude about the existence of (common) fixed points, while some other known results cannot be applied.
Example 4 Let $X=\{0,1,2,3,4\}$ be equipped with the following partial order:
Define a bmetric $d:X\times X\to {\mathbb{R}}^{+}$ by
It is easy to see that $(X,d)$ is a bcomplete bmetric space with $s=49/25$. Set $A=\{0,1,2,3,4\}$ and $B=\{0,2\}$, and define selfmaps f and g by
It is easy to see that f and g are $(A,B)$weakly increasing mappings with respect to ⪯, and that f and g are continuous. Also, $A\cup B=X$, $f(A)\subseteq B$ and $g(B)\subseteq A$.
Define $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ by $\psi (t)=\sqrt{t}$. One can easily check that the pair $(f,g)$ satisfies the requirements of Corollary 1, with any δ and $L\ge 0$, as the lefthand side of the contractive condition (3.1) is equal to 0 for all comparable x, y such that $x\in A$ and $y\in B$. Hence, f and g have a common fixed point. Indeed, 0 and 2 are two common fixed points of f and g. (Note that the ordered set $(\{0,2\},\u2aaf)$ is not well ordered).
However, take $x=1\in A$ and $y=0\in B$ (which are not comparable). Then
where $0<\delta <1$ and $L\ge 0$ are arbitrary, since
and
Hence, this result cannot be applied in the context of bmetric spaces without order.
4 Application to existence of solutions of integral equations
Integral equations like (4.1) have been studied in many papers (see, e.g., [22, 33] and the references therein). In this section, we look for a nonnegative solution to (4.1) in $X=C([0,T],\mathbb{R})$.
Consider the integral equation
where $T>0$, $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ and $G:[0,T]\times [0,T]\to [0,\mathrm{\infty})$ are continuous functions.
Let $X=C([0,T])$ be the set of real continuous functions on $[0,T]$. We endow X with the bmetric
Clearly, $(X,\mathcal{D})$ is a complete bmetric space (with the parameter ${s}^{\prime}=2$). We endow X with the partial order ⪯ given by
Clearly, the space $(X,\u2aaf,\mathcal{D})$ is regular.
Let $\alpha ,\beta \in X$ and ${\alpha}_{0},{\beta}_{0}\in \mathbb{R}$ such that
Assume that for all $t\in [0,T]$, we have
and
Let for all $s\in [0,T]$, $f(s,\cdot )$ be a decreasing function, that is,
Assume that $\gamma >0$ is such that
Define a mapping $\mathcal{T}:X\to X$ by
Suppose that for all $s\in [0,T]$ and for all comparable $x,y\in X$ with ($x(s)\le {\beta}_{0}$ and $y(s)\ge {\alpha}_{0}$) or ($x(s)\ge {\alpha}_{0}$ and $y(s)\le {\beta}_{0}$),
Theorem 6 Under the assumptions (4.2)(4.7), the integral equation (4.1) has a solution in the set $\{u\in C([0,T]):\alpha \u2aafu\le \beta \}$.
Proof Define closed subsets of X, ${A}_{1}$ and ${A}_{2}$ by
Consider the mapping $\mathcal{T}:X\to X$ defined above. We will prove that
Suppose that $u\in {A}_{1}$, that is,
Applying, condition (4.5), since $G(t,s)\ge 0$ for all $t,s\in [0,T]$, we obtain that
The above inequality with condition (4.3) implies that
for all $t\in [0,T]$. Thus, we have $\mathcal{T}u\in {A}_{2}$.
Similarly, let $u\in {A}_{2}$, that is,
Using condition (4.5), since $G(t,s)\ge 0$ for all $t,s\in [0,T]$, we obtain that
The above inequality with condition (4.4) implies that
for all $t\in [0,T]$. Hence, we have $\mathcal{T}u\in {A}_{1}$. Thus, (4.8) holds.
Now, let $(u,v)\in {A}_{1}\times {A}_{2}$, that is, for all $t\in [0,T]$,
This implies from condition (4.2) that for all $t\in [0,T]$,
Also, if $x\u2aafy$, then by (4.7), we have
for all $t\in [0,T]$. That is, $\mathcal{T}x\u2aaf\mathcal{T}y$. Hence, $\mathcal{T}$ is increasing.
Now, by the conditions (4.6) and (4.7), we have for all $t\in [0,T]$ and for all
comparable $x\in {A}_{1}$ and $y\in {A}_{2}$,
which implies that
with $\delta =4\gamma {({max}_{t\in [0,T]}{\int}_{0}^{T}G(t,s)\phantom{\rule{0.2em}{0ex}}ds)}^{2}<1$.
Now, all the conditions of Corollary 2 (with $\mathcal{T}=g=f$ and $L=0$) hold, and $\mathcal{T}$ has a fixed point z in
That is, $z\in {A}_{1}\cap {A}_{2}$ is the solution to (4.1). □
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Acknowledgements
The authors are highly indebted to the referees of this paper who helped us to improve it in several places. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support. The fourth author is thankful to the Ministry of Education, Science and Technological Development of Serbia.
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Hussain, N., Parvaneh, V., Roshan, J.R. et al. Fixed points of cyclic weakly $(\psi ,\phi ,L,A,B)$contractive mappings in ordered bmetric spaces with applications. Fixed Point Theory Appl 2013, 256 (2013). https://doi.org/10.1186/168718122013256
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Keywords
 common fixed point
 cyclic contraction
 almost contraction
 ordered bmetric space
 altering distance function