On new evolution of Ri’s result via wdistances and the study on the solution for nonlinear integral equations and fractional differential equations
 Hassen Aydi^{1, 2},
 Teerawat Wongyat^{3} and
 Wutiphol Sintunavarat^{3}Email author
https://doi.org/10.1186/s1366201815902
© The Author(s) 2018
Received: 18 December 2017
Accepted: 5 April 2018
Published: 12 April 2018
Abstract
The aim of this work is to establish a new fixed point theorem for generalized contraction mappings with respect to wdistances in complete metric spaces. An illustrative example is provided to advocate the usability of our results. Also, we give a numerical experiment for approximating a fixed point in these examples. As an application, the received results are used to summarize the existence and uniqueness of the solution for nonlinear integral equations and nonlinear fractional differential equations of Caputo type.
Keywords
MSC
1 Introduction

integral equations;

ordinary differential equations;

partial differential equations;

matrix equations;

functional equations.
Theorem 1.1
([3])
This result generalized Boyd and Wong’s fixed point theorem in [1] and Matkowski’s fixed point theorem in [2] and hence it also contains Banach’s contraction mapping principle.
On the another hand, the concept of a wdistance on a metric space was introduced and investigated by Kada et al. [4], and this concept was applied to several famous fixed point theorems. Meanwhile, Kada et al. [4] gave the important tool related to wdistances which will be discussed in the next section. Many generalizations of fixed point results with the idea of wdistances have been investigated heavily by many authors (see in [5–8] and references therein).
To the best of our knowledge, there has been no discussion so far concerning Ri’s fixed point result in [3] in the sense of wdistances. Based on the above mentioned fact, we present new fixed point theorems for generalized contraction mappings with respect to wdistances in complete metric spaces, which is an extension of Ri’s fixed point result, and give an example for showing the usability of our results while Theorem 1.1 is not applicable. We also give numerical experiments for finding a fixed point in this example. As an application, the acquired results are used to aggregate the existence and uniqueness of the solution for nonlinear integral equations and nonlinear fractional differential equations.
2 Preliminaries
In this section, we recall some important notations, needed definitions, and primary results joint with the literature.
Definition 2.1
([4])
 (W1)
\(q(x,y)\leq q(x,z)+q(z,y)\);
 (W2)
\(q(x,\cdot):X \rightarrow[0,\infty)\) is lower semicontinuous;
 (W3)
for each \(\varepsilon>0\), there exists \(\delta>0\) such that \(q(x,y)\leq\delta\) and \(q(x,z)\leq\delta\) imply \(d(y,z)\leq \varepsilon\).
It is well known that each metric on a nonempty set X is a wdistance on X. Here, we give some other examples of wdistances.
Example 2.2
Let \((X, d)\) be a metric space. A function \(q: X\times X\rightarrow[0, \infty)\) defined by \(q(x, y) = c\) for every \(x, y \in X\) is a wdistance on X, where c is a positive real number. But q is not a metric since \(q(x, x) = c \neq0\) for any \(x \in X\).
Example 2.3
The following lemma will be used in the next section.
Lemma 2.4
([4])
 (i)
If \(\lim_{n\rightarrow\infty}q(x_{n},x)= \lim_{n\rightarrow\infty}q(x_{n},y)=0\), then \(x=y\). In particular, if \(q(z,x)=q(z,y)=0\), then \(x=y\).
 (ii)
If \(q(x_{n},y_{n})\leq\alpha_{n}\) and \(q(x_{n},y)\leq\beta_{n}\) for any \(n\in\mathbb{N}\), where \(\{\alpha_{n}\}\) and \(\{\beta_{n}\}\) are sequences in \([0,\infty)\) converging to 0, then \(\{y_{n}\}\) converges to y.
 (iii)
If, for each \(\varepsilon>0\), there exists \(N_{\varepsilon}\in\mathbb{N}\) such that \(m>n>N_{\varepsilon}\) implies \(q(x_{n},x_{m})<\varepsilon\) (or \(\lim_{m,n\rightarrow\infty}q(x_{n},x_{m})=0\)), then \(\{x_{n}\}\) is a Cauchy sequence.
3 Main results
First, we will create two lemmas to prove the main result.
Lemma 3.1
Proof
Lemma 3.2
Proof
Next, we exhibit the main result in this paper.
Theorem 3.3
Proof
Now, we give an example which is possible to apply by the contractive condition (3.3) but not the contractive condition in Theorem 1.1.
Example 3.4
 Case 1.If \(x \in[0,\infty)\) and \(y\geq1\), then$$\begin{aligned} q(fx,fy) =&fy \\ =&\frac{y}{4} \\ \leq&\frac{y}{3} \\ =&\varphi(y) \\ =&\varphi \bigl(q(x,y) \bigr). \end{aligned}$$
 Case 2.If \(x \in[0,\infty)\) and \(y< 1\), then$$ q(fx,fy) = fy = 0 \leq\varphi \bigl(q(x,y) \bigr). $$
The iterates of Picard iterations in Example 3.4
\(x_{0}=50\)  \(x_{0}=100\)  \(x_{0}=150\)  \(x_{0}=200\)  

\(x_{1}\)  12.500000  25.000000  37.500000  50.000000 
\(x_{2}\)  3.125000  6.250000  9.375000  12.500000 
\(x_{3}\)  0.781250  1.562500  2.343750  3.125000 
\(x_{4}\)  0.000000  0.390625  0.585938  0.781250 
\(x_{5}\)  0.000000  0.000000  0.000000  0.000000 
\(x_{6}\)  0.000000  0.000000  0.000000  0.000000 
\(x_{7}\)  0.000000  0.000000  0.000000  0.000000 
\(x_{8}\)  0.000000  0.000000  0.000000  0.000000 
\(x_{9}\)  0.000000  0.000000  0.000000  0.000000 
\(x_{10}\)  0.000000  0.000000  0.000000  0.000000 
⋮  ⋮  ⋮  ⋮  ⋮ 
Here, we give the wellknown lemma about the relation between some conditions of the control function without the proof.
Lemma 3.5
Let \(\varphi:[0,\infty)\rightarrow[0,\infty)\) be a function.
 (\(\spadesuit_{1}\)):

If φ is right continuous such that \(\varphi(t)< t \) for all \(t>0\), then \(\varphi(0)=0\).
 (\(\spadesuit_{2}\)):

If φ is increasing and right continuous, then φ is upper semicontinuous.
 (\(\spadesuit_{3}\)):

If φ is upper semicontinuous from the right such that \(\varphi(t)< t \) for all \(t>0\), then \(\limsup_{s\rightarrow t^{+}}\varphi(s)< t\) for all \(t>0\).
By using Theorem 3.3 and Lemma 3.5, we get the following results.
Corollary 3.6
Corollary 3.7
Taking \(q=d\) in Theorem 3.3 and Corollaries 3.6, 3.7, we obtain the following results.
Corollary 3.8
([3])
Corollary 3.9
Corollary 3.10
([2])
4 Applications
The theory of nonlinear integral equations nowadays is a large topic which is found in many applications of various branches in mathematics and other fields such as biology, engineering, economics, etc. Meanwhile, the fractional order models and the theory of nonlinear fractional differential equations are very important to study natural problems because the manner of the trajectory of the fractional order derivatives is nonlocal, which describes that the fractional order derivative has memory effect features. So the theory of nonlinear fractional differential equations can be widely applied in many branches such as the optimal control, finance, chaos, physics, etc. For more details, we refer the reader to [9–16] and the references therein. Nowadays, many mathematicians proved the existence and uniqueness of a solution of nonlinear fractional differential equations by using the fixed point results (see in [7, 17] and the references therein).

nonlinear Fredholm integral equations;

nonlinear Volterra integral equations;

fractional differential equations of Caputo type.
Throughout this section, let us denote by \(C[a, b]\), where \(a,b\in \mathbb{R}\) with \(a< b\), the set of all continuous functions from \([a,b]\) into \(\mathbb{R}\).
4.1 The nonlinear integral equations
In this subsection, we prove the existence and uniqueness results of a solution for the nonlinear Fredholm integral equation and nonlinear Volterra integral equation by using our main results in the previous section.
Theorem 4.1
Proof
Using the identical method in the proof of the above theorem, we get the following result.
Theorem 4.2
4.2 The nonlinear fractional differential equations
The aim of this subsection is to prove the existence and uniqueness result of solutions for the nonlinear fractional differential equations of Caputo type by using Theorem 3.3.
Now, we prove the following existence theorem.
Theorem 4.3
Proof
5 Conclusions
Motivated by the great impact of the models in the form of an integral equation and fractional differential equations of Caputo type, we introduced a new contractive condition by using the idea of a wdistance in metric spaces and established fixed point results for a mapping satisfying the purposed contractive condition. Then we used the received analysis theoretical results for investigating the existence and uniqueness of the solution for nonlinear integral equations and nonlinear fractional differential equations of Caputo type.
Declarations
Acknowledgements
The second author would like to thank the Thailand Research Fund and Office of the Higher Education Commission under grant No. MRG5980242 for financial support during the preparation of this manuscript.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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