Some applications of Caristi’s fixed point theorem in metric spaces
 Farshid Khojasteh^{1}Email author,
 Erdal Karapinar^{2} and
 Hassan Khandani^{3}
https://doi.org/10.1186/s136630160501z
© Khojasteh et al. 2016
Received: 27 July 2015
Accepted: 13 January 2016
Published: 27 January 2016
Abstract
In this work, partial answers to Reich, Mizoguchi and Takahashi’s and AminiHarandi’s conjectures are presented via a light version of Caristi’s fixed point theorem. Moreover, we introduce the idea that many of known fixed point theorems can easily be derived from the Caristi theorem. Finally, the existence of bounded solutions of a functional equation is studied.
Keywords
MSC
1 Introduction and preliminaries
In the literature, the Caristi fixed point theorem is known as one of the very interesting and useful generalizations of the Banach fixed point theorem for selfmappings on a complete metric space. In fact, the Caristi fixed point theorem is a modification of the εvariational principle of Ekeland ([1, 2]), which is a crucial tool in nonlinear analysis, in particular, optimization, variational inequalities, differential equations, and control theory. Furthermore, in 1977 Western [3] proved that the conclusion of Caristi’s theorem is equivalent to metric completeness. In the last decades, Caristi’s fixed point theorem has been generalized and extended in several directions (see e.g., [4, 5] and the related references therein).
The Caristi’s fixed point theorem asserts the following.
Theorem 1.1
([6])
In 1972, Reich [8] introduced the following open problem.
Problem 1.1

Daffer et al. assumed that \(\mu: \mathbb {R}^{+}\rightarrow \mathbb {R}^{+}\)

is upper right semicontinuous,

\(\mu(t)< t\) for all \(t>0\), and

\(\mu(t)\leq tat^{b}\), where \(a>0\), \(1< b<2\) on some interval \([0,s]\), \(s>0\).


Jachymski assumed that \(\mu: \mathbb {R}^{+}\rightarrow \mathbb {R}^{+}\)

is superadditive, i.e., \(\mu(x+y)>\mu(x)+\mu(y)\), for all \(x,y\in \mathbb {R}^{+}\), and

\(t\mapsto t\mu(t)\) is nondecreasing.

Problem 1.2
In 1989, Mizoguchi and Takahashi [12], gave a partial answer to Problem 1.2 as follows.
Theorem 1.2
Another analogous open problem was raised in 2010 by AminiHarandi [13], which we state after the following notations.
Problem 1.3
The answer is yes if Tx is compact for every x (AminiHarandi [13], Theorem 3.3).
In this work, we show that many of the known Banach contraction generalizations can be deduced and generalized by Caristi’s fixed point theorem and its consequences. Also, partial answers to the mentioned open problems are given via our main results. For more details as regards a fixed point generalization of multivalued mappings we refer to [14].
2 Main result
In this section, we show that many of the known fixed point results can be deduced from the following light version of Caristi’s theorem.
Corollary 2.1
Proof
Corollary 2.2
([15], Banach contraction principle)
Proof
Corollary 2.3
Proof
The following results are the main results of this paper and play a crucial role to find the partial answers for Problem 1.1, Problem 1.2, and Problem 1.3. Comparing the partial answers for Reich’s problems, our answers include simple conditions. Also, the compactness condition on Tx is not needed.
Theorem 2.1
Proof
The following theorem is a partial answer to Problem 1.1.
Theorem 2.2
Proof
The following theorem is a partial answer to Problem 1.2.
Corollary 2.4
([12], MizoguchiTakahashi’s type)
Proof
Let \(\theta(t)=\eta(t)t\), \(\theta(t)< t\) for all \(t\in R_{+}\), and \(\frac{\theta(t)}{t}=\eta(t)\) is a nondecreasing mapping. By the assumption \(d(Tx,Ty)\leq\eta(d(x,y))d(x,y)=\theta(d(x,y))\) for all \(x,y\in X\), therefore by Theorem 2.2 T has a fixed point. □
The following theorem is a partial answer to Problem 1.3.
Corollary 2.5
Proof
Let \(\eta(t)=t\theta(t)\), for each \(t>0\). Then \(\eta(t)< t\), for each \(t>0\), and \(\frac{\eta(t)}{t}=1\frac{\theta(t)}{t}\) is nondecreasing. Thus, the desired result is obtained by Theorem 2.2. □
3 Existence of bounded solutions of functional equations
We will prove the following theorem.
Theorem 3.1
 (i)
\(f:{\mathcal{Z}}\times {\mathcal{T}}\to \mathbb {R}\) and \(\Im: {\mathcal{Z}}\times {\mathcal{T}}\times \mathbb {R}\to \mathbb {R}\) are continuous and bounded;
 (ii)for all \(h,k\in {\mathcal{B}}({\mathcal{Z}})\), ifwhere \(x\in {\mathcal{Z}}\) and \(y\in {\mathcal{T}}\). Then the functional equation (20) has a bounded solution.$$ \begin{aligned} & 0< d(h,k)< 1 \quad \textit{implies}\quad \bigl\vert \Im\bigl(x,y,h(x)\bigr)\Im\bigl(x,y,k(x)\bigr)\bigr\vert \leq \frac {1}{2}d^{2}(h,k), \\ &d(h,k)\geq1 \quad \textit{implies}\quad \bigl\vert \Im\bigl(x,y,h(x)\bigr)\Im \bigl(x,y,k(x)\bigr)\bigr\vert \leq\frac{2}{3}d(h,k), \end{aligned} $$(23)
Proof
Declarations
Acknowledgements
The first author would like to thank Prof. S. Mansour Vaezpour for valuable suggestions.
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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