Sharpening some core theorems of Nieto and RodríguezLópez with application to boundary value problem
 Marwan Amin Kutbi^{1},
 Aftab Alam^{2} and
 Mohammad Imdad^{2}Email author
https://doi.org/10.1186/s1366301504467
© Kutbi et al. 2015
Received: 23 June 2015
Accepted: 20 October 2015
Published: 2 November 2015
Abstract
In this paper, we prove sharpened versions of some classical ordertheoretic metrical fixed point theorems due to Nieto and RodríguezLópez (Order 22(3):223239, 2005) using ordertheoretic variants of completeness and continuity besides some another notions such as: the ICC property, the DCC property, and the MCC property. In this continuation, we further extend our results for BoydWong type nonlinear contractions. Finally, as an application of our certain newly proved results, we establish the existence and uniqueness of solution of a first order periodic boundary value problem.
Keywords
MSC
1 Introduction
Throughout this paper, the pair \((X,\preceq)\), stands for a nonempty set X equipped with a partial order ⪯ often called an ordered set wherein \(x\succeq y\) means \(y\preceq x\). Two elements x and y in an ordered set \((X,\preceq)\) are said to be comparable if either \(x\preceq y\) or \(x\succeq y\) and denote it by \(x\prec\succ y\). A subset E of an ordered set \((X,\preceq)\) is called totally ordered if \(x\prec\succ y\) for all \(x,y\in E\). A selfmapping f defined on an ordered set \((X,\preceq)\) is called increasing (or isotone or orderpreserving) if for any \(x,y\in X\), \(x\preceq y\) implies \(f(x)\preceq f(y)\). As per standard practice, we can define the notions of increasing, decreasing, monotone, bounded above and bounded below sequences besides bounds (upper as well as lower) of a sequence in an ordered set \((X,\preceq)\), which on the set of real numbers under natural ordering coincide with their usual senses. Following O’Regan and Petruşel [1], the triple \((X,d,\preceq)\) is called ordered metric space wherein X denotes a nonempty set endowed with a metric d and a partial order ⪯. If in addition, d is a complete metric on X, then we say that \((X,d,\preceq)\) is an ordered complete metric space.
In the recent years, a multitude of ordertheoretic metrical fixed point theorems have been proved for orderpreserving contractions. In such results, the involved contraction condition is considerably weakened as one is merely required to hold only on those elements which are comparable in the underlying partial ordering. The techniques involved in the proofs of such results is the combination of ideas used in the contraction principle together with the one employed in monotone iterative technique. This trend is essentially initiated by Turinici [2, 3]. Later, Ran and Reurings [4] proved a slightly more natural version of the corresponding fixed point theorems of Turinici (cf. [2, 3]) for continuous monotone mappings with some applications to matrix equations. In the same lieu, Nieto and RodríguezLópez [5] proved some variants of Ran and Reuring fixed point theorem for increasing mappings. Nieto and RodríguezLópez fixed point theorems were also extended by several authors (see [1, 6–10]).
Before discussing such results, we summarize some relevant basic terminologies needed in our subsequent discussion. Throughout this manuscript, \(\mathbb{N}\) stands for the set of natural numbers and \(\mathbb{N}_{0}\) for the set of whole numbers (i.e. \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\)).
 (i)
If \(\{x_{n}\}\) is increasing and \(x_{n}\stackrel {d}{\longrightarrow} x\), then we denote it symbolically by \(x_{n}\uparrow x\).
 (ii)
If \(\{x_{n}\}\) is decreasing and \(x_{n}\stackrel {d}{\longrightarrow} x\), then we denote it symbolically by \(x_{n}\downarrow x\).
 (iii)
If \(\{x_{n}\}\) is monotone and \(x_{n}\stackrel {d}{\longrightarrow} x\), then we denote it symbolically by \(x_{n}\uparrow\downarrow x\).
Alam et al. [9] formulated the following notions by using certain properties on ordered metric space (in order to avoid the necessity of continuity requirement on underlying mapping) utilized by earlier authors especially from [5, 11] besides some other ones.
Definition 1
[9]
 (i)\((X,d,\preceq)\) has the ICU (increasingconvergenceupper bound) property if every increasing convergent sequence \(\{x_{n}\}\) in X is bounded above by its limit (as an upper bound), i.e.,$$x_{n}\uparrow x \quad\Rightarrow\quad x_{n}\preceq x\quad \forall n\in \mathbb{N}_{0}, $$
 (ii)\((X,d,\preceq)\) has the DCL (decreasingconvergencelower bound) property if every decreasing convergent sequence \(\{x_{n}\}\) in X is bounded below by its limit (as a lower bound), i.e.,$$x_{n}\downarrow x \quad\Rightarrow\quad x_{n}\succeq x\quad \forall n\in \mathbb {N}_{0}, \mbox{and} $$
 (iii)
\((X,d,\preceq)\) has the MCB (monotoneconvergenceboundedness) property if X has both the ICU and the DCL property.
Alam et al. [10] further weakened the notions embodied in Definition 1 as follows.
Definition 2
[10]
 (i)\((X,d,\preceq)\) has the ICC (increasingconvergencecomparable) property if every increasing convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{x_{n_{k}}\}\) is comparable with the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\uparrow x \quad\Rightarrow\quad\exists\mbox{ a subsequence } \{x_{n_{k}}\}\mbox{ of } \{x_{n}\} \mbox{ with } x_{n_{k}} \prec\succ x \quad\forall k\in\mathbb{N}_{0}, $$
 (ii)\((X,d,\preceq)\) has the DCC (decreasingconvergencecomparable) property if every decreasing convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{x_{n_{k}}\}\) is comparable with the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\downarrow x \quad\Rightarrow\quad\exists\mbox{ a subsequence } \{x_{n_{k}}\}\mbox{ of } \{x_{n}\} \mbox{ with } x_{n_{k}} \prec\succ x \quad\forall k\in\mathbb{N}_{0}, \mbox{and} $$
 (iii)\((X,d,\preceq)\) has the MCC (monotoneconvergencecomparable) property if every monotone convergent sequence \(\{x_{n}\}\) in X has a subsequence \(\{x_{n_{k}}\}\) such that every term of \(\{x_{n_{k}}\}\) is comparable with the limit of \(\{x_{n}\}\), i.e.,$$x_{n}\uparrow\downarrow x \quad\Rightarrow\quad\exists\mbox{ a subsequence } \{x_{n_{k}}\}\mbox{ of } \{x_{n}\} \mbox{ with } x_{n_{k}} \prec\succ x \quad\forall k\in\mathbb{N}_{0}. $$
Remark 1

ICU property ⇒ ICC property.

DCL property ⇒ DCC property.

MCB property ⇒ MCC property ⇒ ICC property as well as DCC property.
Jleli et al. [12] formulated the following notion by using a certain property on an ordered set (in order to prove the uniqueness of fixed points) utilized by Nieto and RodríguezLópez [5].
Definition 3
[12]
Let \((X,\preceq)\) be an ordered set and f a selfmappings on X. We say that \((X,\preceq)\) is directed if for each pair \(x,y\in X\), \(\exists z\in X\) such that \(x\prec\succ z\) and \(y\prec\succ z\).
Notice that \((X,\preceq)\) is directed if and only if every pair of elements of X has a lower bound or an upper bound (cf. [5]).
Definition 4
[13]
 (i)
\(k\geq2\),
 (ii)
\(z_{1}=x\) and \(z_{k}=y\),
 (iii)
\(z_{1}\prec\succ z_{2}\prec\succ\cdots\prec\succ z_{k1}\prec\succ z_{k}\).
We denote by \(\mathrm{C}(x,y,\prec\succ)\) the class of all ≺≻chains between x and y. If \((X,\preceq)\) is directed then \(\mathrm{C}(x,y,\prec\succ)\) is nonempty, for each \(x,y\in X\) (cf. [13]).
For the sake of completeness, we record the following wellknown core results.
Theorem 1
(Nieto and RodríguezLópez [5])
 (a):

\((X,d)\) is complete,
 (b):

f is increasing,
 (c):

either f is continuous or \((X,d,\preceq)\) has the ICU property,
 (d):

there exists \(x_{0}\in X\) such that \(x_{0} \preceq f(x_{0})\),
 (e):

there exists \(\alpha\in[0,1)\) such that$$d(fx,fy)\leq\alpha d(x,y) \quad\forall x,y\in X \textit{ with } x\prec\succ y. $$
Theorem 2
(Nieto and RodríguezLópez [5])
 \((\mathrm{c})^{\prime}\) :

either f is continuous or \((X,d,\preceq)\) has the DCL property,
 \((\mathrm{d})^{\prime}\) :

there exists \(x_{0}\in X\) such that \(x_{0} \succeq f(x_{0})\).
Theorem 3
(Nieto and RodríguezLópez [5])
Theorem 4
(Turinici [13])
 \((\mathrm{f})^{\prime}\) :

\(\mathrm{C}(x,y,\prec\succ)\) is nonempty for each \(x,y\in X\).
Recently, Alam et al. [10] adopted the notions of completeness and continuity with respect to ordertheoretic metrical structure, which run as follows.
Definition 5
[10]
 (i)
O̅complete if every increasing Cauchy sequence in X converges,
 (ii)
\(\underline{\mathrm{O}}\)complete if every decreasing Cauchy sequence in X converges, and
 (iii)
Ocomplete if every monotone Cauchy sequence in X converges.
Remark 2
In an ordered metric space, completeness ⇒ Ocompleteness ⇒ O̅completeness as well as \(\underline{\mathrm{O}}\)completeness.
Definition 6
[10]
 (i)O̅continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\subset X\),$$x_{n}\uparrow x\quad\Rightarrow\quad f(x_{n})\stackrel{d}{ \longrightarrow} f(x), $$
 (ii)\(\underline{\mathrm{O}}\)continuous at \(x\in X\) if for any sequence \(\{x_{n}\}\subset X\),$$x_{n}\downarrow x\quad\Rightarrow\quad f(x_{n})\stackrel{d}{ \longrightarrow} f(x), \mbox{and} $$
 (iii)Ocontinuous at \(x\in X\) if for any sequence \(\{x_{n}\} \subset X\),$$x_{n} \uparrow\downarrow x\quad\Rightarrow \quad f(x_{n})\stackrel{d}{ \longrightarrow} f(x). $$
Here it can be pointed out that the notion of O̅continuity is previously defined by Turinici [14] by recalling that f is \((d,\preceq)\)continuous.
Remark 3
In an ordered metric space, continuity ⇒ Ocontinuity ⇒ O̅continuity as well as \(\underline{\mathrm{O}}\)continuity.
2 Fixed point theorems for linear contractions
In this section, we present sharpened forms of Theorems 14, which are new results on their own and are proved without completeness (of the metric space), without continuity (of the underlying mapping) and without the ICU property/DCL property/MCB property (of an ordered metric space).
Theorem 5
 (a):

\((X,d,\preceq)\) is O̅complete,
 (b):

f is increasing,
 (c):

either f is O̅continuous or \((X,d,\preceq)\) has the ICC property,
 (d):

there exists \(x_{0}\in X\) such that \(x_{0} \preceq f(x_{0})\),
 (e):

there exists \(\alpha\in[0,1)\) such that$$d(fx,fy)\leq\alpha d(x,y) \quad\forall x,y\in X \textit{ with } x\prec\succ y. $$
Proof
Next, we present a dual result to Theorem 5.
Theorem 6
 \((\mathrm{d})^{\prime}\) :

there exists \(x_{0}\in X\) such that \(x_{0}\succeq f(x_{0})\).
Proof
Remark 4
Notice that Theorems 5 and 6 sharpen Theorems 1 and 2, respectively. Here we observe that in Nieto and RodríguezLópez theorems the completeness, continuity, the ICU property, and the DCL property are not necessary as the same can alternately be replaced by their respective relatively weaker notions.
On combining Theorem 5 and Theorem 6, we obtain the following result.
Theorem 7
 \((\mathrm{d})^{\prime\prime}\) :

there exists \(x_{0}\in X\) such that \(x_{0}\prec\succ f(x_{0})\).
Finally, we prove certain unique fixed point results corresponding to Theorems 57.
Theorem 8
Proof
3 Fixed point theorems for nonlinear contractions
We need the following known results in the proof of our main results of this section.
Lemma 1
[9]
Let \(\varphi\in\Omega\). If \(\{a_{n}\}\subset(0,\infty)\) is a sequence such that \(a_{n+1}\leq \varphi(a_{n})\) \(\forall n\in\mathbb{N}_{0}\), then \(\lim_{n\to\infty}a_{n}=0\).
Lemma 2
[12]
 (i)
\(n_{k}>m_{k}\geq k\),
 (ii)
\(d(x_{m_{k}},x_{n_{k}})\geq\epsilon\),
 (iii)
\(d(x_{m_{k}},x_{n_{k}1})<\epsilon\),
 (iv)the following four sequences tend to ϵ when \(k\rightarrow\infty\):$$d(x_{m_{k}},x_{n_{k}}), d(x_{m_{k}+1},x_{n_{k}}), d(x_{m_{k}},x_{n_{k}+1}), d(x_{m_{k}+1},x_{n_{k}+1}). $$
Now, we extend Theorems 5 and 6 for nonlinear contractions as follows.
Theorem 9
 \((\mathrm{e})^{\prime}\) :

there exists \(\varphi\in\Omega\) such that$$d(fx,fy)\leq\varphi\bigl(d(x,y)\bigr) \quad\forall x,y\in X \textit{ with } x\prec \succ y. $$
Proof
We start the proof proceeding the lines similar to the proof of Theorem 5 (resp. Theorem 6 or Theorem 7). Following its lines, we define the sequence \(\{x_{n}\}\) of Picard iterates and then we can prove that the sequence \(\{x_{n}\}\) is increasing (resp. decreasing or monotone).
 (i)
\(d(x_{n_{k}},x)=0 \) \(\forall k\in\mathbb{N}^{0}\),
 (ii)
\(d(x_{n_{k}},x)>0 \) \(\forall k\in\mathbb{N}^{+}\).
In case (i), we have \(d(fx_{n_{k}},fx)=0\) \(\forall k\in\mathbb{N}^{0}\), which implies that \(d(x_{n_{k}+1},fx)=0\) \(\forall k\in\mathbb{N}^{0}\) and hence (15) holds for all \(k\in\mathbb{N}^{0}\). If case (ii) holds, by the definition of Ω, we have \(d(x_{n_{k}+1},fx)\leq \varphi(d(x_{n_{k}},x))< d(x_{n_{k}},x)\) \(\forall k\in\mathbb{N}^{+}\) and hence (15) holds for all \(k\in\mathbb{N}^{+}\). Thus (15) holds for all \(k\in\mathbb{N}\).
Taking the limit of (15) as \(n\rightarrow\infty\) and using \(x_{n_{k}}\stackrel{d}{\longrightarrow} x\), we obtain \(x_{n_{k}+1}\stackrel{d}{\longrightarrow} f(x)\). Owing to the uniqueness of limit, we obtain \(f(x)=x\) so that x is a fixed point of f. □
Remark 5
Notice that for \(\varphi(t)=\alpha\cdot t\) with \(\alpha\in [0,1)\), Theorem 9 reduces to Theorems 57.
Theorem 10
In addition to the hypotheses of Theorem 9, suppose that the following assumption (f) (of Theorem 8) holds, then f has a unique fixed point.
Proof
We can prove this result easily by using the similar technique as utilized in Theorem 8. □
4 Application to boundary value problem
Let \(\mathcal{C}(I)\) denote the space of all continuous functions defined on I. Now, we recall the following definitions.
Definition 7
[5]
Definition 8
[5]
 (i)
ϕ is continuous and increasing,
 (ii)
\(\phi(t)< t\) for each \(t>0\).
Now, we prove the following result regarding the existence and uniqueness of a solution of problem (16) in the presence of a lower solution or an upper solution.
Theorem 11
Proof
(a) Clearly, \((\mathcal{C}(I),d,\preceq)\) is an Ocomplete ordered metric space.
Hence, all the conditions of Theorem 9 are satisfied consequently \(\mathcal{A}\) has a fixed point.
Finally choose arbitrary \(u,v\in\mathcal{C}(I)\), then \(w:=\max\{\mathcal{A}u,\mathcal{A}v\}\in\mathcal{C}(I)\). As \(\mathcal{A}(u)\preceq w\) and \(\mathcal{A}(v)\preceq w\), \(\{\mathcal{A}u,w,\mathcal{A}v\}\) is a ≺≻chain between \(\mathcal{A}(u)\) and \(\mathcal{A}(v)\). Thus, by Theorem 10, \(\mathcal{A}\) has a unique fixed point, which is, indeed, a unique solution of problem (16). □
Declarations
Acknowledgements
The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) in Jeddah, Kingdom of Saudi Arabia during this research. All the authors are thankful to two anonymous learned referees for their encouraging comments.
Open Access This 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.
Authors’ Affiliations
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