Some fixed point results for non-decreasing and mixed monotone mappings with auxiliary functions
- Shuang Wang^{1}Email author,
- Arslan Hojat Ansari^{2} and
- Sumit Chandok^{3}
https://doi.org/10.1186/s13663-015-0456-5
© Wang et al. 2015
Received: 20 March 2015
Accepted: 4 November 2015
Published: 17 November 2015
Abstract
The purpose of this paper is to provide sufficient conditions for the existence and uniqueness of fixed points for non-decreasing and mixed monotone mappings with auxiliary functions in the framework of metric space endowed with a partial order. As applications of our results we obtain several interesting corollaries and fixed point theorems in the underlying spaces. In order to illustrate our results, we provide two examples in which other theorems from the literature cannot be applied. In addition, the equivalence property between unidimensional and multidimensional fixed point theorems is investigated. We also present some applications to the existence of solutions of integral and differential equations.
Keywords
MSC
1 Introduction
One of the newest branches of fixed point theory is devoted to the study of coupled fixed points, introduced by Guo and Lakshmikantham [1] in 1987. Thereafter, Gnana Bhaskar and Lakshmikantham [2] introduced the concept of the mixed monotone property for contractive operators in two variables in the setting of a partially ordered metric space, and they established some coupled fixed point theorems. Their results were extended and generalized by several authors in the last few years; see [3–21] and the references cited therein. Recently, Roldán et al. [22] introduced the notion of coincidence point between mappings in any number of variables, and several special extended notions of so-called coupled, tripled, quadrupled, and multidimensional fixed/coincidence points appeared in the literature; see, for example, [2, 3, 13, 23], respectively.
Harjani and Sadarangani [24] investigated some unidimensional fixed point theorems for generalized contractions in complete partially ordered metric spaces and applications to ordinary differential equations. In [23] and [25] the authors obtained some multidimensional fixed point theorems for mixed monotone mappings, which extended the corresponding coupled, tripled, and quadrupled fixed point results appearing in the literature. In 2014, Wang [26] obtained some multidimensional fixed point theorems for isotone mappings and extended some of the results in coupled, tripled, quadrupled and multidimensional fixed/coincidence points for mixed monotone and non-decreasing mappings in the framework of complete partially ordered metric spaces. She also gave a simple and unified approach to coupled, tripled, quadrupled, and multidimensional fixed point theorems for mixed monotone mappings.
Motivated and inspired by the results of [22–26], we establish some fixed point theorems for non-decreasing and mixed monotone mappings with auxiliary functions in a complete partially ordered metric space. The auxiliary functions used in the paper are more general than the gauge functions appearing in the literature. Our results improve and generalize the well-known results of Harjani and Sadarangani [24] and Wang [26]. By using the theorems, we also obtain several interesting corollaries and fixed point theorems in the underlying spaces. In order to illustrate our results, we provide two examples in which the theorems of [24] cannot be applied. In addition, the equivalence property between unidimensional and multidimensional fixed point theorems is investigated. Also as applications of our results, we provide two examples for the solution of integral and differential equations.
2 Basic concepts
In order to fix the framework needed to state our main results, we recall the following notions.
For simplicity, we denote from now on \(\underbrace{X\times X\cdot \cdot\cdot X\times X}_{k}\) by \(X^{k}\), where \(k\in\mathbb{N}\) and X is a non-empty set. If elements x, y of a partially ordered set \((X,\leq)\) are comparable (i.e., \(x\leq y\) or \(y\leq x\) holds), we write \(x\asymp y\). Let \(\{A,B\}\) be a partition of the set \(\Lambda_{k}=\{1,2,\ldots,k\}\), that is, \(A\cup B=\Lambda_{k}\) and \(A\cap B=\emptyset\), \(\Omega_{A,B}=\{\sigma :\Lambda_{k}\rightarrow\Lambda_{k}:\sigma(A)\subseteq A \mbox{ and } \sigma (B)\subseteq B\} \) and \(\Omega^{{\prime }}_{A,B}=\{\sigma:\Lambda_{k}\rightarrow\Lambda_{k}:\sigma(A)\subseteq B \mbox{ and } \sigma(B)\subseteq A\}\). Henceforth, let \(\sigma_{1},\sigma _{2},\ldots,\sigma_{k}\) be k mappings from \(\Lambda_{k}\) into itself, and ϒ be the k-tuple \((\sigma_{1},\sigma_{2},\ldots,\sigma_{k})\). For brevity, \((y_{1},y_{2},\ldots,y_{k}),(v_{1},v_{2},\ldots,v_{k}), (y^{n}_{1},y^{n}_{2},\ldots,y^{n}_{k})\) and \((x^{1}_{0},x^{2}_{0},\ldots,x^{k}_{0})\) will be denoted by Y, V, \(Y_{n}\) and \(X_{0}\), respectively.
Definition 2.1
([7])
- (i)
if \(\{x_{n}\}\) is a non-decreasing sequence such that \(x_{n}\rightarrow x\), then \(x_{n}\leq x\) for all n,
- (ii)
if \(\{y_{n}\}\) is a non-increasing sequence such that \(y_{n}\rightarrow y\), then \(y_{n}\geq y\) for all n.
Definition 2.2
([2])
Definition 2.3
([3])
Definition 2.4
([22])
Definition 2.5
- (i)
a coupled fixed point [2] if \(k=2\), \(F(x_{1},x_{2})=x_{1}\), and \(F(x_{2},x_{1})=x_{2}\);
- (ii)
a tripled fixed point [3] if \(k=3\), \(F(x_{1},x_{2},x_{3})=x_{1}\), \(F(x_{2},x_{1},x_{2})=x_{2}\), and \(F(x_{3},x_{2},x_{1})=x_{3}\);
- (iii)
a ϒ-fixed point [25] of F if \(F(x_{\sigma_{i}(1)},x_{\sigma_{i}(2)},\ldots,x_{\sigma_{i}(k)})=x_{i}\) for \(i\in \Lambda_{k}\).
Definition 2.6
([24])
Let \((X,\leq)\) be a partially ordered set. A mapping \(f:X\rightarrow X\) is monotone non-decreasing if \(x,y\in X\), \(x\leq y\Rightarrow f(x)\leq f(y)\).
Definition 2.7
([26])
An element \(Y\in X^{k}\) is called a fixed point of the mapping \(T:X^{k}\rightarrow X^{k}\) if \(T(Y)=Y\).
Lemma 2.8
([27])
Let \((X,\leq)\) be a partially ordered set and d a metric on X. If \((X,\leq,d)\) is regular, then \((X^{k},\preceq,\rho_{k})\) is regular.
Definition 2.9
- (a)
\(f (s,t)\le s\);
- (b)
f is continuous;
- (c)
\(f (s,t)=s\) implies that either \(s=0\) or \(t=0\).
For brevity, we denote the C-class by \(\mathcal{C}\).
Example 2.10
([28])
- (1)
\(f(s,t)=s-t\), \(f (s,t)=s \Rightarrow t=0\);
- (2)
\(f(s,t)=\frac{s-t}{1+t}\), \(f (s,t)=s \Rightarrow t=0\);
- (3)
\(f(s,t)=\frac{s}{1+t}\), \(f (s,t)=s \Rightarrow s=0\) or \(t=0\);
- (4)
\(f(s,t)=\log_{a}\frac{t+a^{s}}{1+t}\), \(a>1\), \(f (s,t)=s \Rightarrow s=0\) or \(t=0\);
- (5)
\(f(s,t)=ks\), \(0< k<1\), \(f(s,t)=s \Rightarrow s=0\);
- (6)
\(f(s,t)=(s+l)^{\frac{1}{1+t}}-l\), \(l>1\), \(f(s,t)=s \Rightarrow t=0\);
- (7)
\(f(s,t)=s\log_{a+t}a\), \(a>1\), \(f(s,t)=s \Rightarrow s=0\) or \(t=0\);
- (8)
\(f(s,t)=s-\frac{t}{1+t}\), \(f(s,t)=s \Rightarrow t=0\);
- (9)
\(f(s,t)=s-(\frac{1+s}{2+s})t\), \(f(s,t)=s \Rightarrow t=0\);
- (10)
\(f(s,t)=\frac{s}{k+t}\), \(k>1\), \(f(s,t)=s \Rightarrow s=0\).
Remark 2.11
It is easy to verify that C-class is a natural generalization of classical Banach contraction principle (see (5) of Example 2.10).
Definition 2.12
([29])
- (a)
ψ is continuous and strictly increasing.
- (b)
\(\psi(t)=0\) if and only if \(t=0\).
Lemma 2.13
([30])
- (i)
\(\lim_{k\rightarrow\infty}d(x_{m(k)-1},x_{n(k)+1})=\epsilon\);
- (ii)
\(\lim_{k\rightarrow\infty}d(x_{m(k)},x_{n(k)})=\epsilon\);
- (iii)
\(\lim_{k\rightarrow\infty}d(x_{m(k)-1},x_{n(k)})=\epsilon\).
Remark 2.14
3 Existence of fixed points
In this section, we state and prove the existence of fixed points for non-decreasing and mixed monotone mappings with auxiliary functions in the setting of complete partially ordered metric spaces. In addition, the equivalence property between unidimensional and multidimensional fixed point theorems is investigated.
We denote Φ the set of all continuous and strictly increasing functions \(\varphi:[0,\infty)\rightarrow[0,\infty)\), and Ψ the set of all functions, such that \(\lim_{t\rightarrow r}\psi(t)>0\) for every \(r>0\) and \(\psi (t)=0\Longleftrightarrow t=0\).
Theorem 3.1
- (a)
T is continuous, or
- (b)
\((X, \leq, d)\) is regular.
Proof
Starting with \(z_{0}\) in X, define the sequence \(\{z_{n}\}\subset X\) by \(z_{n+1}=T(z_{n})\) for \(n\geq0\). Obviously, if \(z_{n_{0}+1}=z_{n_{0}}\) for some \(n_{0}\geq0\), then \(z_{n_{0}}\) is a fixed point of T. So, assume that \(z_{n+1}\neq z_{n}\) for every \(n\geq0\).
Now suppose that (a) holds. It follows from \(z_{n+1}=T(z_{n})\) that z̄ is a fixed point of T, that is, \(T(\bar{z})=\bar{z}\).
Remark 3.2
In Theorem 3.1, we use \(h\in\mathcal{C}\), and \(\mathcal{C}\) is a class of more general functions than the gauge function used in Theorems 2.1 and 2.2 of [24]. Indeed, the gauge function, \(h(s,t)=s-t\) in Theorems 2.1 and 2.2 of [24] is an element of \(\mathcal{C}\).
Taking \(h(s,t)=s-t\) or \(h(s,t)=\frac{s}{1+t}\) in Theorem 3.1, we obtain the following results immediately.
Corollary 3.3
- (a)
T is continuous, or
- (b)
\((X,\leq,d)\) is regular.
Remark 3.4
We note that, if ψ is an altering distance function, then \(\psi \in\Psi\). But the reverse is not true in general (see Example 3.5). Therefore, Corollary 3.3 generalizes the well-known results of Harjani and Sadarangani’s [24] in the framework of partially ordered metric spaces (see Theorems 2.1 and 2.2 in [24]).
Example 3.5
- (1)
\((X,d)\) is a complete metric space and \((X,\leq,d)\) is regular.
- (2)
T is a continuous mapping.
- (3)
\(\varphi\in\Phi\) and \(\psi\in\Psi\).
- (4)
Take \(z_{0}=1\). Then \(1=T(1)\).
- (5)
T is a non-decreasing mapping. Indeed, let \(x,y\in X\) such that \(x\leq y\), (a) if \(x=y\), then \(T(x)=T(y)\); (b) if \((x,y)=(0,2)\), then \(T(x)=T(y)\). Therefore, T is a non-decreasing mapping.
- (6)
This shows that all the conditions of Corollary 3.3 are satisfied. Therefore by using Corollary 3.3, T has a fixed point. Indeed, 0 and 1 are two fixed points of T. However, Theorems 2.1 and 2.2 of Harjani and Sadarangani’s [24] cannot be applied to this example because ψ is not an altering distance function.
Corollary 3.6
- (a)
T is continuous, or
- (b)
\((X,\leq,d)\) is regular.
Example 3.7
Let \((X,\leq)\) be the partially ordered set with \(X=[0,\infty)\) and the natural ordering of the real number as the partial ordering ≤. Consider the metric on X: \(d(x,y)=|x-y|\) for all \(x,y\in X\). Then \((X,d)\) is complete and \((X,\leq,d)\) is regular.
Let \(T:X\rightarrow X\) be defined by \(T(x)=\frac{x}{1+x}\) for all \(x\in X\). Then T is a non-decreasing and continuous mapping. Consider \(\psi,\varphi:[0,\infty)\rightarrow[0,\infty)\) defined by \(\psi (t)=\varphi(t)=t\). It is obvious that \(\varphi\in\Phi\) and \(\psi \in\Psi\).
Now we give the following multidimensional fixed point theorem for mixed monotone mappings.
Theorem 3.8
Proof
Remark 3.9
Note that the multidimensional fixed point theorem (Theorem 3.8) is equivalent to the unidimensional fixed point theorem (Theorem 3.1). Also, we find that Theorem 3.8 is a consequence of Theorem 3.1. Conversely, taking \(k=1\), \(A=\{1\}\), \(B=\emptyset\), and \(F=T\) in Theorem 3.8, we obtain Theorem 3.1 immediately.
Using similar arguments to the proof of Theorem 3.8, the following results are immediate consequences of the unidimensional fixed point theorem (Theorem 3.1).
Corollary 3.10
- (a)
F is continuous, or
- (b)
\((X,d,\leq)\) is regular.
Corollary 3.11
- (a)
F is continuous, or
- (b)
\((X,\leq,d)\) is regular.
Using Theorem 3.1, we obtain the following result on multidimensional fixed points, which generalizes Theorem 3.1 of Wang [26].
Theorem 3.12
- (a)
T is continuous, or
- (b)
\((X, \leq, d)\) is regular.
Proof
Using Lemma 2.8, we find that \((X^{k},\preceq,\rho_{k})\) is regular. By our assumptions, all the conditions of Theorem 3.1 are satisfied in the setting of a complete partially ordered metric space \((X^{k},\preceq,\rho_{k})\). Therefore, by using Theorem 3.1, T has a fixed point. □
Remark 3.13
(b) We also find that the multidimensional fixed point theorem (Theorem 3.12) is equivalent to the unidimensional fixed point theorem (Theorem 3.1). In fact, Theorem 3.12 is a consequence of Theorem 3.1. Conversely, taking \(k=1\) in Theorem 3.12, we obtain Theorem 3.1 immediately.
4 Uniqueness of fixed points
Now, we state and prove the uniqueness of fixed points in the setting of a complete partially ordered metric space.
Theorem 4.1
In addition to the hypotheses of Theorem 3.1, suppose that, for all fixed points \(\bar{y},y^{*}\in X\) of T, there exists \(z\in X\) such that z is comparable to ȳ and to \(y^{*}\). Then T has a unique fixed point.
Proof
Using similar arguments to the proof of Theorem 3.8, we deduce the following corollaries from Theorem 4.1.
Corollary 4.2
In addition to the hypotheses of Theorem 3.8, suppose that, for all ϒ-fixed points \(\bar{Y},Y^{*}\in X^{k}\) of F, there exists \(Z\in X^{k}\) such that Z is comparable to Ȳ and to \(Y^{*}\). Then F has a unique ϒ-fixed point.
Corollary 4.3
In addition to the hypotheses of Corollary 3.10, suppose that, for all coupled fixed points \(\bar{Y},Y^{*}\in X^{2}\) of F, there exists \(Z\in X^{2}\) such that Z is comparable to Ȳ and to \(Y^{*}\). Then F has a unique coupled fixed point.
Corollary 4.4
In addition to the hypotheses of Corollary 3.11, suppose that, for all tripled fixed points \(\bar{Y},Y^{*}\in X^{3}\) of F, there exists \(Z\in X^{3}\) such that Z is comparable to Ȳ and to \(Y^{*}\). Then F has a unique tripled fixed point.
5 Application to integral equations
Theorem 5.1
- (i)
\(K:[0,T]\times[0,T]\times\mathbb{R}\rightarrow[0,\infty )\) and \(g:[0,T]\rightarrow[0,\infty)\) are continuous,
- (ii)there exist \(h \in\mathcal{C}\), \(\psi\in\Psi\), and a continuous function \(\tilde{G}:[0,T]\times[ 0,T]\rightarrow[0,\infty)\) such thatfor all \(s,t\in[0,T]\) and \(x,y\in C[0,T]\) with \(x\leq y\);$$ 0\leq K(t,s,y)-K(t,s,x)\leq\tilde{G}(t,s)h\bigl(d(x,y),\psi\bigl(d(x,y)\bigr) \bigr) $$
- (iii)
\(\sup_{t\in[0,T]}\int_{0}^{T}\tilde {G}(t,s)\,ds\leq1\) or \(\sup_{t\in[0,T]}(\int_{0}^{T}\tilde{G}(t,s)^{2}\,ds)^{\frac{1}{2}}\leq\frac{1}{\sqrt{T}}\).
Proof
Corollary 5.2
([24])
Proof
Take \(K(t,s,x(s))=G(t,s)f(s,x(s))\) and \(g(t)=0\) for all \(s,t\in[0,1]\). It is clear that condition (i) of Theorem 5.1 holds since \(f(t,x)\) and \(G(t,s)\) are non-negative continuous functions.
Declarations
Acknowledgements
The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper. This work was supported by the Natural Science Foundation of Jiangsu Province under grant (13KJB110028).
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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