Common Fixed Point Theorems for Novel Admissible Contraction with Applications in Fractional and Ordinary Differential Equations

Abstract

In our work, we offer a novel idea of contractions, namely an ${(\alpha ,\beta ,\gamma )}_P-$contraction, to prove the existence of a coincidence point and a common fixed point in complete metric spaces. This leads us to an extension of previous results in the literature. Furthermore, we applied our acquired results to prove the existence of a solution for ordinary and fractional differential equations with integral-type boundary conditions.

1. Introduction

Fixed point theory and its applications have become one of the most fascinating research subjects in nonlinear analysis. Researchers have put a lot of effort into this branch of mathematics in the last few decades because of its powerful application strengths. In many disciplines, fixed point theory has been utilized in various approaches. For instance, the existence of a solution to nonlinear integral equations (see [1] for an example), the existence and uniqueness topics for a solution of differential equations, and integral equations involving Caputo-type fractional operators (see, for instance, [2,3]), problems in signal recovery under some obscured filters (see [4,5,6,7]), image restoration problems (see [8,9,10,11,12] for more details), the study of the convergence of iterative methods (see [13,14,15,16,17] for some decent work in this field), and many other branches of mathematics.

Recently, there have been various alternative ways for researchers to approach fixed point theory (see for example [18,19,20]). One of the standard schemes is to extend the notion of contractions. In the celebrated paper [21], Geraghty introduced an important class $\Theta$ of Geraghty functions $\theta :[0,\infty )\to [0,1)$, satisfying

$$\theta \left(t_n\right)\to 1⟹t_n\to 0,$$

where $\left\{t_n\right\}$ is a sequence in $[0,\infty )$. He also proved remarkable outcomes that extended Banach’s contraction principle. Generally, the concept of Geraghty functions itself can be further generalized to admissible Geraghty functions, which is one of the most outstanding subjects in fixed point theory. In particular, in 2014, Karapınar [22,23] extended the result in [21] by employing admissible mapping for the contractive conditions on maps.

In 2017, Fulga and Proca [24] introduced a new and significant generalization of contraction, known as the ${\varphi }_E$-Geraghty contraction, which also extended the work presented in [21]. In a similar vein, Martínez-Moreno et al. demonstrated common fixed point theorems for Geraghty type contraction mappings endowed with the monotone property with two metrics in their work [25]. These contributions have undoubtedly provided valuable inspiration to the research community.

Building on these inspirations, Charoensawan and Atiponrat (refer to [18]) and Charoensawan (refer to [19]) presented some interesting existence and uniqueness results for common fixed point theorems concerning generalized contraction mappings with two metrics in metric spaces and JS-metric spaces endowed with a directed graph. This work further extended the scope of previous studies [21,25] in the field.

As a consequence of the above motivation, one of the main purposes of this paper was to amplify the previous concepts by enlarging the codomains of the renowned Geraghty functions and creating a new type of contraction equipped with the modified functions, which is called ${(\alpha ,\beta ,\gamma )}_P-$contraction. It is worth pointing out here that this new construction, in fact, beautifully extends several of the previous results in the literature.

To be more precise, in this work, we chose to utilize admissibility instead of requiring our spaces to be endowed with a directed graph. This decision has enabled us to extend our previous results, as mentioned in [18,19]. Additionally, it is worth noting that our current work slightly differs from our previous research (refer to [2,3,26]) in that we are now considering generalized Geraghty functions rather than auxiliary functions. As generalized Geraghty functions and auxiliary functions are indeed quite different, we believe this modification has enhanced our approach and contributes to the novelty of the present study.

Our construction offers the existence results of coincidence points and common fixed points for any pair of mappings under additional conditions. Indeed, we organized the following work into four consecutive sections. In Section 2, important background knowledge of the current work is given. To be more precise, we recall the key definitions and concepts necessary in our work. Additionally, the definition of ${(\alpha ,\beta ,\gamma )}_P-$contraction for two mappings is introduced. Moreover, we show that the coincidence points and common fixed points of these mappings exist under some specific requirements. Also, an explicit example is illustrated to support the main results. In Section 3, application in nonlinear fractional differential equations having nonlocal boundary conditions is explicitly demonstrated. Next, application in ordinary differential equations is thoroughly explained in Section 4. Lastly, we finish our work by providing inclusions in Section 5.

2. Main Results

For this part, we present a new contraction, which is called an ${(\alpha ,\beta ,\gamma )}_P$-contraction, to prove the existence of a coincidence point and a common fixed point in complete metric spaces.

To begin with, we prove the following lemma, which will play a significant role in our main theorem. It is worth mentioning here that this lemma is a standard result in the literature (see, for instance, [2,3,26]).

Lemma 1

([2,3,26]). Let $\left\{x_n\right\}$ be a sequence in a metric space $(X,d)$ with a function $g:X\to X$ such that

$$\underset{n\to \infty }{lim}d(g{x}_n,g{x}_{n+1})=0.$$

Suppose that $\left\{x_n\right\}$ contains two subsequences $\left\{x_{m\left(k\right)}\right\}$ and $\left\{x_{n\left(k\right)}\right\}$ together with $ϵ>0$ such that for each $k\in \mathbb{N}$, we have $n\left(k\right)>m\left(k\right)\ge k$, where $n\left(k\right)$ is the smallest number possible, satisfying

$$d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})\ge ϵ\mathit{and}d(g{x}_{n\left(k\right)-1},g{x}_{m\left(k\right)})<ϵ.$$

Then,

$${\displaystyle ϵ=\underset{k\to \infty }{lim}d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})=\underset{k\to \infty }{lim}d(g{x}_{m\left(k\right)+1},g{x}_{n\left(k\right)+1}).}$$

Moreover, if $\alpha :X\times X\to [0,\infty )$ is triangular, i.e., $\alpha (a,b)\ge 1$ and $\alpha (b,c)\ge 1$ imply $\alpha (a,c)\ge 1$ for any $a,b,c\in X$, such that

$$\alpha (g{x}_{n-1},g{x}_n)\ge 1\mathit{for}\mathit{any}n\in \mathbb{N},$$

then

$$\alpha (g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\ge 1\mathit{for}\mathit{any}k\in \mathbb{N}.$$

Throughout this work, let $(X,d,f,g,\alpha )$ denote a structure such that the following properties hold:

(1)

X is a non-empty set;

(2)

$(X,d)$ is a metric space;

(3)

$f,g$ are self-mappings on X;

(4)

$\alpha :X\times X\to [0,\infty )$.

Now, we introduce important classes of functions that will be related to our main results.

Let $\gamma :[0,\infty )\to [0,\infty )$ be a continuous non-decreasing self-mapping on $[0,\infty )$ such that, for all $r\in [0,\infty )$,

$$\gamma \left(r\right)=0⟺r=0.$$

Let us denote by $\Gamma$ the set of all such functions $\gamma$ above.

Next, we recall another class of functions introduced by Ayari, see [27], as follows.

Let $\mathcal{B}$ be a class of the mappings $\beta :[0,\infty )\to [0,1]$ such that

$$\underset{n\to \infty }{lim}\beta \left(t_n\right)=1⟹\underset{n\to \infty }{lim}{t}_n=0,$$

where $\left\{t_n\right\}$ is a sequence in $[0,\infty )$. In the following definition, a special kind of contraction is considered.

Definition 1.

On $(X,d,f,g,\alpha )$, the pair of functions $(f,g)$ will be called an${(\alpha ,\beta ,\gamma )}_P$-contraction if there are $\beta \in \mathcal{B}$ and $\gamma \in \Gamma$ such that, for any $x,y\in X$ with $\alpha (gx,gy)\ge 1$, we have

$$\alpha (gx,gy)\gamma \left(d\right(fx,fy\left)\right)\le \beta \left(P\right(gx,gy\left)\right)\gamma \left(P\right(gx,gy\left)\right),$$

where $P:X\times X\to [0,\infty )$ satisfies for all $x,y\in X$,

$$\begin{array}{cc}\hfill P(gx,gy)=max\{& \left|d\right(gy,fy)-d(gx,fx\left)\right|+d(gy,gx),\hfill \\ \hfill & d(fx,gx)+\left|d\right(gx,gy)-d(gy,fy\left)\right|,\hfill \\ \hfill & d(gy,fy)+\left|d\right(fx,gx)-d(gx,gy\left)\right|,\hfill \\ \hfill & {\displaystyle \frac{d(gx,fx)[1+d(fy,gy\left)\right]}{1+d(gx,gy)}}+|d(gx,gy)-d(gx,fx)|,\hfill \\ \hfill & {\displaystyle \frac{d(gy,fy)[1+d(gx,fx\left)\right]}{1+d(gx,gy)}}+|d(gx,fx)-d(gx,gy)|\}.\hfill \end{array}$$

Now, we prove a proposition that will be useful in proving our main results as follows.

Proposition 1.

On $(X,d,f,g,\alpha )$, suppose further that all the following conditions hold:

(1) 

There is a sequence $\left\{x_n\right\}$ in X such that, for every $n\in \mathbb{N}$, we have

(i) 

$g{x}_n\ne g{x}_{n-1},$

(ii) 

$g{x}_n=f{x}_{n-1},$

(iii) 

$\alpha (g{x}_{n-1},g{x}_n)\ge 1;$

(2) 

$(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction.

Then,

$$\underset{n\to \infty }{lim}d(g{x}_n,g{x}_{n+1})=0\mathit{for}\mathit{all}n\ge 0.$$

Proof. 

By assumption, $\left\{x_n\right\}$ is a sequence in X such that, for every $n\in \mathbb{N}$,

$$g{x}_n\ne g{x}_{n-1}\mathrm{and}g{x}_n=f{x}_{n-1},$$

whilst

$$\alpha (g{x}_{n-1},g{x}_n)\ge 1.$$

Since $(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction, for any $n\ge 0$,

$$\begin{array}{ccc}\hfill \gamma \left(d(g{x}_{n+1},g{x}_{n+2})\right)& =& \gamma \left(d(f{x}_n,f{x}_{n+1})\right)\hfill \\ & \le & \alpha (g{x}_n,g{x}_{n+1})\gamma \left(d(f{x}_n,f{x}_{n+1})\right)\hfill \\ & \le & \beta \left(P(g{x}_n,g{x}_{n+1})\right)\gamma \left(P(g{x}_n,g{x}_{n+1})\right)\hfill \\ & \le & \gamma \left(P(g{x}_n,g{x}_{n+1})\right).\hfill \end{array}$$

(1)

Also, a direct calculation shows that

$$\begin{array}{cc}\hfill P(g{x}_n,g{x}_{n+1})=max\{& d(g{x}_n,g{x}_{n+1})+|d(g{x}_n,f{x}_n)-d(g{x}_{n+1},f{x}_{n+1})|,\hfill \\ \hfill & d(g{x}_n,f{x}_n)+|d(g{x}_n,g{x}_{n+1})-d(g{x}_{n+1},f{x}_{n+1})|,\hfill \\ \hfill & d(g{x}_{n+1},f{x}_{n+1})+|d(g{x}_{n+1},g{x}_n)-d(g{x}_n,f{x}_n)|,\hfill \\ & {\displaystyle \frac{d(g{x}_n,f{x}_n)[1+d(g{x}_{n+1},f{x}_{n+1})]}{1+d(g{x}_n,g{x}_{n+1})}}+|d(g{x}_{n+1},g{x}_n)-d(g{x}_n,f{x}_n)|,\hfill \\ \hfill & {\displaystyle \frac{d(g{x}_{n+1},f{x}_{n+1})[1+d(g{x}_n,f{x}_n)]}{1+d(g{x}_n,g{x}_{n+1})}}+|d(g{x}_n,g{x}_{n+1})-d(g{x}_n,f{x}_n)|\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =max\{& d(g{x}_n,g{x}_{n+1})+|d(g{x}_n,g{x}_{n+1})-d(g{x}_{n+1},g{x}_{n+2})|,\hfill \\ \hfill & d(g{x}_n,g{x}_{n+1})+|d(g{x}_{n+1},g{x}_n)-d(g{x}_{n+1},g{x}_{n+2})|,\hfill \\ \hfill & |d(g{x}_{n+1},g{x}_n)-d(g{x}_{n+1},g{x}_n)|+d(g{x}_{n+1},g{x}_{n+2}),\hfill \\ & {\displaystyle \frac{d(g{x}_n,g{x}_{n+1})[1+d(g{x}_{n+1},g{x}_{n+2})]}{1+d(g{x}_n,g{x}_{n+1})}}+|d(g{x}_n,g{x}_{n+1})-d(g{x}_n,g{x}_{n+1})|,\hfill \\ \hfill & {\displaystyle \frac{d(g{x}_{n+1},g{x}_{n+2})[1+d(g{x}_{n+1},g{x}_n)]}{1+d(g{x}_n,g{x}_{n+1})}}+|d(g{x}_n,g{x}_{n+1})-d(g{x}_n,g{x}_{n+1})|\}\hfill \\ \hfill =max\{& d(g{x}_n,g{x}_{n+1})+|d(g{x}_n,g{x}_{n+1})-d(g{x}_{n+1},g{x}_{n+2})|,\hfill \\ \hfill & d(g{x}_n,g{x}_{n+1})+|d(g{x}_n,g{x}_{n+1})-d(g{x}_{n+1},g{x}_{n+2})|,\hfill \\ \hfill & d(g{x}_{n+1},g{x}_{n+2}),\hfill \\ \hfill & {\displaystyle \frac{d(g{x}_n,g{x}_{n+1})[1+d(g{x}_{n+1},g{x}_{n+2})]}{1+d(g{x}_n,g{x}_{n+1})}},\hfill \\ \hfill & d(g{x}_{n+1},g{x}_{n+2})\}.\hfill \end{array}$$

Here and subsequently, for each $n\in \mathbb{N}$, let

$$G_n=d(g{x}_n,g{x}_{n+1}).$$

We have that

$$\begin{array}{c}\hfill P(g{x}_n,g{x}_{n+1})=max\left\{G_n+|G_n-G_{n+1}|,G_{n+1},{\displaystyle \frac{G_n(1+G_{n+1})}{1+G_n}}\right\}.\end{array}$$

Now, suppose that $\left\{G_n\right\}$ is not decreasing. Then, there exists $C\in \mathbb{N}$ such that $G_C\le G_{C+1}$. Thus, we have

$$\begin{array}{c}\hfill P(g{x}_C,g{x}_{C+1})=G_{C+1}.\end{array}$$

Then, by inequality (1), we have

$$\begin{array}{ccc}\hfill \gamma \left(G_{C+1}\right)& =& \gamma \left(d(g{x}_{C+1},g{x}_{C+2})\right)\hfill \\ & \le & \beta \left(P(g{x}_C,g{x}_{C+1})\right)\gamma \left(P(g{x}_C,g{x}_{C+1})\right)\hfill \\ & =& \beta \left(G_{C+1}\right)\gamma \left(G_{C+1}\right)\hfill \\ & \le & \gamma \left(G_{C+1}\right).\hfill \end{array}$$

(2)

Since for every $n\in \mathbb{N}$, $g{x}_n\ne g{x}_{n-1}$,

$$G_{C+1}=d(g{x}_{C+1},g{x}_{C+2})>0.$$

By the property of $\gamma$, it follows that $\gamma \left(G_{C+1}\right)>0.$ Dividing inequality (2) by $\gamma \left(G_{C+1}\right)$ yields $\beta \left(G_{C+1}\right)=1$. By the fact that $\beta \in \mathcal{B}$, we have

$$G_{C+1}=d(g{x}_C,g{x}_{C+1})=0.$$

It is a contradiction. Therefore, $\left\{G_n\right\}$ is decreasing. That is, $G_n>G_{n+1}$ for all $n\ge 0$.

Since $\left\{G_n\right\}$ is a sequence that is bounded below, it must be a convergent sequence. So, there exists $K\in \mathbb{R}$ such that

$$\underset{n\to \infty }{lim}{G}_n=K\ge 0.$$

Suppose, on the contrary, that $K>0$. Thus, by the property of $\gamma$, we have

$$\underset{n\to \infty }{lim}\gamma \left(G_n\right)=\underset{n\to \infty }{lim}\gamma \left(P(g{x}_n,g{x}_{n+1})\right)=\gamma \left(K\right)>0.$$

As a result,

$$\begin{array}{c}\hfill P(g{x}_n,g{x}_{n+1})=max\{2G_n-G_{n+1},G_n\}\end{array}$$

and

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}P(g{x}_n,g{x}_{n+1})=K.\end{array}$$

By (1), it is true that

$$1=\underset{n\to \infty }{lim}{\displaystyle \frac{\gamma \left(G_{n+1}\right)}{\gamma \left(P(g{x}_n,g{x}_{n+1})\right)}}\le \underset{n\to \infty }{lim}\beta \left(G_n\right)\le 1.$$

Therefore, ${\displaystyle \underset{n\to \infty }{lim}\beta \left(G_n\right)=1}$. Note that the definition of $\beta$ implies

$$\underset{n\to \infty }{lim}{G}_n=0=K,$$

which contradicts our assumption. Thus, we obtain

$$\underset{n\to \infty }{lim}d(g{x}_n,g{x}_{n+1})=0\mathrm{for}\mathrm{any}n\ge 0.$$

This completes the proof. □

Next, we recall definitions of common fixed points and coincidence points as follows.

Definition 2.

On $(X,d,f,g,\alpha )$, we will employ the following notations throughout this work.

(1) 

$A(f,g):=\{x\in X:\alpha (gx,fx)\ge 1\};$

(2) 

$C(f,g)$ denotes the set of all coincidence points of f and g, i.e.,

$$C(f,g):=\{x\in X:fx=gx\};$$

(3) 

$Cm(f,g)$ denotes the set of all common fixed points of f and g, i.e.,

$$Cm(f,g):=\{x\in X:fx=gx=x\}.$$

Before we continue to our main results, let us recall important definitions which are necessary for the proof of the main theorems below.

Definition 3

([28]). Given $\alpha :X\times X\to [0,\infty )$ and $f,g:X\to X$, the function f is said to be triangular α-admissible with respect to g if, for all $x,y,z\in X$,

(1) 

$\alpha (gx,gy)\ge 1$ implies $\alpha (fx,fy)\ge 1$;

(2) 

$\alpha (x,z)\ge 1$ and $\alpha (z,y)\ge 1$ imply $\alpha (x,y)\ge 1$.

Definition 4

([29]). Given a metric space $(X,d)$ with $f,g:X\to X$, the maps f and g are called d-compatible if

$$\underset{n\to \infty }{lim}d(gf{x}_n,fg{x}_n)=0,$$

for all sequences $\left\{x_n\right\}$ in X with ${lim}_{n\to \infty }g{x}_n={lim}_{n\to \infty }f{x}_n$.

At this moment, we are ready to prove our most important theorem as follows.

Theorem 1.

Given a structure $(X,d,f,g,\alpha )$, such that $(X,d)$ is a complete metric space, assume further that all the following four conditions hold:

(1) 

$f\left(X\right)\subseteq g\left(X\right)$ such that $g\left(X\right)$ is closed;

(2) 

$(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction;

(3) 

f is triangular α-admissible with respect to g;

(4) 

At least one of the below requirements is fulfilled:

(a) 

f and g are continuous, and d-compatible; or

(b) 

for each sequence $\left\{x_n\right\}$ in X with $\alpha (x_n,x_{n+1})\ge 1$ for any $n\in \mathbb{N}$, if $x_n\to x\in X$, then $\alpha (x_n,x)\ge 1$ for each natural number n.

Also, if $A(f,g)\ne ⌀$, then $C(f,g)\ne ⌀.$

Proof. 

Suppose that $A(f,g)\ne ⌀$, we have $x_0\in X$ with $\alpha (g{x}_0,f{x}_0)\ge 1$. By our assumption that $f\left(X\right)\subseteq g\left(X\right)$, there is $x_1\in X$ with $g{x}_1=f{x}_0\in X$. Now, we may construct a sequence $\left\{x_n\right\}$ in X satisfying

$$g{x}_n=f{x}_{n-1}\mathrm{for}\mathrm{each}\mathrm{natural}\mathrm{number}n\in \mathbb{N}.$$

(3)

If it is satisfied that, for some $n_0\in \mathbb{N}$, $g{x}_{n_0}=g{x}_{n_0-1}$, then $x_{n_0-1}$ is clearly a coincidence point of g and f. As a consequence, we may assume from now on that

$$g{x}_n\ne g{x}_{n-1}\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(4)

Next, because

$$\alpha (g{x}_0,f{x}_0)=\alpha (g{x}_0,g{x}_1)\ge 1$$

and f is triangular $\alpha$-admissible with respect to g, we obtain that

$$\alpha (f{x}_0,f{x}_1)=\alpha (g{x}_1,g{x}_2)\ge 1.$$

Obviously, mathematical induction provides

$$\alpha (g{x}_{n-1},g{x}_n)\ge 1\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

(5)

Due to the fact that $(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction, from (3)–(5) and Proposition 1, we get

$$\underset{n\to \infty }{lim}d(g{x}_n,g{x}_{n+1})=0\mathrm{for}\mathrm{any}n\ge 0.$$

(6)

Now, we prove that the sequence $\left\{g{x}_n\right\}$ needs to be Cauchy. Assume that $\left\{g{x}_n\right\}$ is not a Cauchy sequence. Hence, there will be $ϵ>0$ such that for all $k\in \mathbb{N}$, there are $n\left(k\right),m\left(k\right)\in \mathbb{N}$ such that $\left\{g{x}_{m\left(k\right)}\right\}$ and $\left\{g{x}_{n\left(k\right)}\right\}$ are subsequences of $\left\{g{x}_n\right\}$ and $n\left(k\right)>m\left(k\right)\ge k$ with $n\left(k\right)$ being the tiniest number satisfying

$$d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})\ge ϵ\mathrm{and}d(g{x}_{n\left(k\right)-1},g{x}_{m\left(k\right)})<ϵ.$$

By the above inequalities, (6), and assumption (3), Lemma 1 implies

$${\displaystyle \underset{k\to \infty }{lim}d(g{x}_{m\left(k\right)+1},g{x}_{n\left(k\right)+1})=\underset{k\to \infty }{lim}d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})=\epsilon ,}$$

(7)

whilst

$$\alpha (g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\ge 1\mathrm{for}\mathrm{every}k\in \mathbb{N}.$$

(8)

Using the fact that $(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction yields

$$\begin{array}{cc}\hfill \gamma \left(d(g{x}_{m\left(k\right)+1},g{x}_{n\left(k\right)+1})\right)& =\gamma \left(d(f{x}_{m\left(k\right)},f{x}_{n\left(k\right)})\right)\hfill \\ \hfill & \le \alpha (g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\gamma \left(d(f{x}_{m\left(k\right)},f{x}_{n\left(k\right)})\right)\hfill \\ \hfill & \le \beta \left(P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right)\gamma \left(P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right)\hfill \\ \hfill & \le \gamma \left(P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right),\hfill \end{array}$$

(9)

where

$P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})$

$$\begin{array}{cc}\hfill =max\{& d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})+|d(g{x}_{m\left(k\right)},f{x}_{m\left(k\right)})-d(f{x}_{n\left(k\right)},g{x}_{n\left(k\right)})|,\hfill \\ \hfill & d(f{x}_{m\left(k\right)},g{x}_{m\left(k\right)})+|d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})-d(f{x}_{n\left(k\right)},g{x}_{n\left(k\right)})|,\hfill \\ \hfill & d(g{x}_{n\left(k\right)},f{x}_{n\left(k\right)})+|d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})-d(f{x}_{m\left(k\right)},g{x}_{m\left(k\right)})|,\hfill \\ & {\displaystyle \frac{d(g{x}_{m\left(k\right)},f{x}_{m\left(k\right)})[1+d(g{x}_{n\left(k\right)},f{x}_{n\left(k\right)})]}{1+d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})}}+|d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})-d(g{x}_{m\left(k\right)},f{x}_{m\left(k\right)})|,\hfill \\ \hfill & {\displaystyle \frac{d(g{x}_{n\left(k\right)},f{x}_{n\left(k\right)})[1+d(g{x}_{m\left(k\right)},f{x}_{m\left(k\right)})]}{1+d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})}}+|d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})-d(g{x}_{m\left(k\right)},f{x}_{m\left(k\right)})|\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =max\{& d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})+|d(g{x}_{n\left(k\right)},g{x}_{n\left(k\right)+1})-d(g{x}_{m\left(k\right)},g{x}_{m\left(k\right)+1})|,\hfill \\ \hfill & d(g{x}_{m\left(k\right)+1},g{x}_{m\left(k\right)})+|d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})-d(g{x}_{n\left(k\right)+1},g{x}_{n\left(k\right)})|,\hfill \\ \hfill & d(g{x}_{n\left(k\right)},g{x}_{n\left(k\right)+1})+|d(g{x}_{n\left(k\right)},g{x}_{m\left(k\right)})-d(g{x}_{m\left(k\right)},g{x}_{m\left(k\right)+1})|,\hfill \\ & {\displaystyle \frac{d(g{x}_{m\left(k\right)},g{x}_{m\left(k\right)+1})[1+d(g{x}_{n\left(k\right)},g{x}_{n\left(k\right)+1})]}{1+d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})}}+|d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})-d(g{x}_{m\left(k\right)},g{x}_{m\left(k\right)+1})|,\hfill \\ \hfill & {\displaystyle \frac{d(g{x}_{n\left(k\right)},g{x}_{n\left(k\right)+1})[1+d(g{x}_{m\left(k\right)},g{x}_{m\left(k\right)+1})]}{1+d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})}}+|d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})-d(g{x}_{m\left(k\right)},g{x}_{m\left(k\right)+1})|\}.\hfill \end{array}$$

Letting $k\to \infty$ in the above inequality and using (6), we obtain that

$$\underset{k\to \infty }{lim}P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})=\underset{k\to \infty }{lim}d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})=ϵ>0.$$

By the property of $\gamma$, we have

$$\underset{k\to \infty }{lim}\gamma \left(P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right)=\underset{k\to \infty }{lim}\gamma \left(d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right)=\gamma \left(ϵ\right)>0.$$

Also, inequality (9) and the above fact imply

$$1=\underset{k\to \infty }{lim}{\displaystyle \frac{\gamma \left(d(g{x}_{m\left(k\right)+1},g{x}_{n\left(k\right)+1})\right)}{\gamma \left(P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right)}}\le \underset{k\to \infty }{lim}\beta \left(P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right)\le 1.$$

As a result, $\underset{k\to \infty }{lim}\beta \left(P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})\right)=1.$ Since $\beta \in \mathcal{B}$,

$$0=\underset{k\to \infty }{lim}P(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})=\underset{k\to \infty }{lim}d(g{x}_{m\left(k\right)},g{x}_{n\left(k\right)})=ϵ>0,$$

which is impossible. Thus, we get that $\left\{g{x}_n\right\}$ is Cauchy in the complete metric space $(X,d)$.

Using the assumption that $g\left(X\right)$ is a closed subspace of the complete metric space $(X,d)$, there exist $x,c_0\in X$ with $c_0=gx\in g\left(X\right)$, which satisfy

$$\underset{n\to \infty }{lim}g{x}_n=\underset{n\to \infty }{lim}f{x}_n=c_0=gx.$$

(10)

Now, if assumption (a) is true, then f and g are d-compatible. So, it is true that

$$\underset{n\to \infty }{lim}d(gf{x}_n,fg{x}_n)=0.$$

(11)

Notice that

$$d(g{c}_0,f{c}_0)\le d(g{c}_0,gf{x}_n)+d(gf{x}_n,fg{x}_n)+d(fg{x}_n,f{c}_0).$$

Taking $n\to \infty$ yields $d(g{c}_0,f{c}_0)=0$ due to (11) and the continuity of f and g. Therefore, $g{c}_0=f{c}_0$. That is, $c_0$ is a coincidence point of f and g.

Next, if assumption (b) is true, according to (5) and (10), we have

$$\alpha (g{x}_n,gx)\ge 1\mathrm{for}\mathrm{every}n\in \mathbb{N}.$$

Now, we will show that a coincidence point of f and g is x. Conversely, assume that x fails to be a coincidence point of f and g. Consequently, $fx\ne gx$. Thence, $d(gx,fx)>0$. Also, for all $n\ge 0$,

$$d(gx,fx)\le d(gx,f{x}_n)+d(f{x}_n,fx).$$

As a result,

$$d(gx,fx)-d(gx,f{x}_n)\le d(f{x}_n,fx).$$

In fact, the definitions of $\gamma$ and ${(\alpha ,\beta ,\gamma )}_P$-contraction imply that

$$\begin{array}{ccc}\hfill \gamma (d(gx,fx)-d(gx,f{x}_n))& \le & \gamma \left(d(f{x}_n,fx)\right)\hfill \\ & \le & \alpha (g{x}_n,gx)\gamma \left(d(f{x}_n,fx)\right)\hfill \\ & \le & \beta \left(P(g{x}_n,gx)\right)\gamma \left(P(g{x}_n,gx)\right)\hfill \\ & \le & \gamma \left(P(g{x}_n,gx)\right),\hfill \end{array}$$

(12)

where

$$\begin{array}{cc}\hfill P(g{x}_n,gx)=max\{& d(g{x}_n,gx)+|d(g{x}_n,f{x}_n)-d(gx,fx)|,\hfill \\ \hfill & d(g{x}_n,f{x}_n)+|d(g{x}_n,gx)-d(gx,fx)|,\hfill \\ \hfill & d(gx,fx)+|d(g{x}_n,gx)-d(g{x}_n,f{x}_n)|,\hfill \\ \hfill & {\displaystyle \frac{d(g{x}_n,f{x}_n)[1+d(gx,fx)]}{1+d(g{x}_n,gx)}}+|d(g{x}_n,gx)-d(g{x}_n,f{x}_n)|,\hfill \\ \hfill & {\displaystyle \frac{d(gx,fx)[1+d(g{x}_n,f{x}_n)]}{1+d(g{x}_n,gx)}}+|d(g{x}_n,gx)-d(g{x}_n,f{x}_n)|\}.\hfill \end{array}$$

From the previous observation, using (10) yields

$$\underset{n\to \infty }{lim}P(g{x}_n,gx)=d(gx,fx)>0.$$

Also, by the property of $\gamma$, it follows that

$$\underset{n\to \infty }{lim}\gamma \left(P(g{x}_n,gx)\right)=\gamma \left(d(gx,fx)\right)=\gamma \left(0\right)>0.$$

Then, taking $n\to \infty$ in (12) gives us that

$$\begin{array}{c}\hfill 1=\underset{n\to \infty }{lim}\frac{\gamma (d(gx,fx)-d(gx,f{x}_n))}{\gamma \left(P(g{x}_n,gx)\right)}\le \underset{n\to \infty }{lim}\beta \left(P(g{x}_n,gx)\right)\le 1.\end{array}$$

We get that ${lim}_{k\to \infty }\beta \left(P(g{x}_n,gx)\right)=1$. Since $\beta \in \mathcal{B}$,

$$\underset{n\to \infty }{lim}P(g{x}_n,gx)=d(gx,fx)=0,$$

which is impossible. Therefore, $fx=gx$. That is, f and g have x as a coincidence point. □

It will be investigated in the succeeding theorem that including an additional condition in the previous result provides a pleasant outcome related to common fixed points.

Theorem 2.

Under all the hypotheses of Theorem 1, assume further that

(5) 

For all $x,y\in C(f,g)$, if $gx\ne gy$, then $\alpha (gx,gy)\ge 1$.

If $A(f,g)\ne ⌀$, then $Cm(f,g)\ne ⌀.$

Proof. 

To begin with, according to Theorem 1, we obtain $x\in X$ with $gx=fx$. Observe that, if $y\in X$ is also a coincidence point, then $P(gx,gy)=d(fx,fy)$ because $gx=fx$ and $gy=fy$, whilst

$$\begin{array}{cc}\hfill P(gx,gy)=max\{& \left|d\right(gy,fy)-d(gx,fx\left)\right|+d(gy,gx),\hfill \\ \hfill & d(gx,fx)+\left|d\right(gx,gy)-d(gy,fy\left)\right|,\hfill \\ \hfill & d(gy,fy)+\left|d\right(gx,fx)-d(gx,gy\left)\right|,\hfill \\ \hfill & {\displaystyle \frac{d(gx,fx)[1+d(fy,gy\left)\right]}{1+d(gx,gy)}}+|d(gx,gy)-d(gx,fx)|,\hfill \\ \hfill & {\displaystyle \frac{d(gy,fy)[1+d(gx,fx\left)\right]}{1+d(gx,gy)}}+|d(gx,gy)-d(gx,fx)|\}.\hfill \end{array}$$

We will show further that $gx=gy$. Contrarily, assume that $gx\ne gy$. Hence, $d(gx,gy)>0.$ By the additional condition (5) in our theorem, $\alpha (gx,gy)\ge 1$. Since $(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction,

$$\begin{array}{cc}\hfill \gamma \left(d\right(fx,fy\left)\right)& \le \alpha (gx,gy)\gamma \left(d\right(fx,fy\left)\right)\hfill \\ \hfill & \le \beta \left(P\right(gx,gy\left)\right)\gamma \left(P\right(gx,gy\left)\right)\hfill \\ \hfill & \le \gamma \left(P\right(gx,gy\left)\right)=\gamma \left(d\right(fx,fy\left)\right).\hfill \end{array}$$

Thus, $\beta \left(P\right(gx,gy\left)\right)=\beta \left(d\right(fx,fy\left)\right)=1$. By the property of $\beta$, we have $d(fx,fy)=0.$ It is a contradiction. As a result, $gx=fx=fy=gy$. This establishes the uniqueness of a coincidence point.

Next, set $x_0=x$. By applying assumption (1) in Theorem 1 repeatedly, we get a sequence $\left\{x_n\right\}$ with $g{x}_n=f{x}_{n-1}$ for all $n\in \mathbb{N}$. Now, because x is a coincidence point, we can assume that, for each $n\in \mathbb{N}$, $x_n=x$ so that $g{x}_n=fx$.

Now, letting $z=gx$ implies $gz=ggx=gfx.$ By the proof of Theorem 2.8 in [25], we obtain that $z\in Cm(f,g)$. □

In the next example, we illustrate the case that supports our theorems above.

Example 1.

Suppose that $X=[0,1]$, and $d:X\times X\to [0,\infty )$ is defined by

$$d(x,y)=|x-y|.$$

Moreover, let $f:X\to X$ and $g:X\to X$ be defined by

$$gx=x^2\mathit{and}fx=ln\left(1+{\displaystyle \frac{2x^4}{3}}\right)$$

for all $x\in X$. Next, suppose ${\displaystyle \gamma \left(t\right)=8t}$ for each real number $t\ge 0$, and let $\alpha :X\times X\to [0,\infty )$ such that

$${\displaystyle \alpha (x,y)=\left\{\begin{array}{cc}1\hfill & \mathit{if}x,y\in [0,\frac{1}{2}] with x\le y,\hfill \\ 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.}$$

It will be proved that all the below requirements are satisfied:

(1) 

$(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction;

(2) 

$f\left(X\right)\subseteq g\left(X\right)$ such that $g\left(X\right)$ is closed;

(3) 

f is triangular α-admissible with respect to g;

(4) 

f and g are continuous, and d-compatible.

To start with, let us prove that $(f,g)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction.

(1) Let

$$\beta \left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{arctan\left(t\right)}{t}}\hfill & \mathit{if}t>0,\hfill \\ 1\hfill & \mathit{if}t=0.\hfill \end{array}\right.$$

Note that $ln(1+x)\le arctan\left(x\right)$ for all $x\in [0,2]$.

If $gx=x^2$, $gy=y^2\in [0,\frac{1}{2}]$ and $x^2=gx<gy=y^2$, then it follows that

$$\begin{array}{cc}\hfill {\displaystyle \alpha (gx,gy)\gamma \left(d\right(fx,fy\left)\right)}& =8|ln\left(1+\frac{2x^4}{3}\right)-ln\left(1+\frac{2y^4}{3}\right)|\hfill \\ \hfill & \le 8ln\left(1+\frac{2}{3}|x^4-y^4|\right)\hfill \\ \hfill & \le {\displaystyle \frac{ln\left(1+P(gx,gy)\right)}{P(gx,gy)}}8P(gx,gy)\hfill \\ \hfill & \le {\displaystyle \frac{arctan\left(P(gx,gy)\right)}{P(gx,gy)}}8P(gx,gy)\hfill \\ \hfill & =\beta \left(P\right(gx,gy\left)\right)\gamma \left(P\right(gx,gy\left)\right).\hfill \end{array}$$

Thus, requirement (1) is satisfied.

(2) Clearly, $f\left(X\right)\subseteq g\left(X\right)$ and $g\left(X\right)$ is closed.

(3) We will show that f is triangular α-admissible with respect to g.

First, let $\alpha (gx,gy)\ge 1$. Observe that $gx=x^2$, $gy=y^2\in [0,\frac{1}{2}]$ and $x^2=gx\le gy=y^2$. As a result, $x^4\le y^4$ and

$$fx=ln\left(1+\frac{2x^4}{3}\right)\le ln\left(1+\frac{2y^4}{3}\right)=fy$$

with $fx,fy$ being elements in $[0,\frac{1}{2}]$. This implies $\alpha (fx,fy)\ge 1$.

Second, let $\alpha (gx,gy)\ge 1$ and $\alpha (gy,gz)\ge 1$; we have $gx=x^2$, $gy=y^2$, $gz=z^2\in [0,\frac{1}{2}]$ and

$$gx=x^2\le gy=y^2\le gz=z^2.$$

Therefore, $\alpha (gx,gz)\ge 1$. Thus, f is triangular α-admissible with respect to g.

(4) It is not difficult to see that f and g are continuous. So, let us show that f and g must be d-compatible. Assume that $\left\{x_n\right\}$ is a sequence in X satisfying

$$\underset{n\to \infty }{lim}g{x}_n=\underset{n\to \infty }{lim}f{x}_n=x.$$

It follows that $ln\left(1+\frac{2x^4}{3}\right)=x$, which yields $x=0$. Letting $n\to \infty$, we also get

$$d(gf{x}_n,fg{x}_n)=\left|{\left[ln\left(1+\frac{2{\left(x_n\right)}^4}{3}\right)\right]}^2-ln\left(1+\frac{2{\left(x_n\right)}^8}{3}\right)\right|\to 0.$$

Finally, observe that $\alpha (g0,f0)=\alpha (0,0)\ge 1$. So, $A(f,g)$ contains an element. Theorem 1 suggests that $C(f,g)$ must be non-empty. It is also not hard to see that 0 is, in fact, a coincidence point.

3. Application in Fractional Differential Equations

Our main goal in this part was to impose the existence of a solution for some fractional differential equations, namely Caputo fractional boundary value problems of order $\xi$ with the integral-type boundary conditions, where $\xi \in (n-1,n]$ for an integer $n\ge 2$.

To begin with, for each real number $\xi \ge 0$, the Caputo derivative of fractional order $\xi$ for any continuous function $u\left(t\right)$ is given by

$${}^c{D}^{\xi }u=I^{\lceil \xi \rceil -\xi }{D}^{\lceil \xi \rceil }u,$$

where $\lceil ·\rceil$ is the ceiling function. In addition, the Riemann–Liouville integral operator of order $\xi$, denoted by $I^{\xi }$, is so that

$$I^{\xi }u\left(t\right)=\frac{1}{\Gamma \left(\xi \right)}{\int }_0^t{(t-s)}^{\xi -1}u\left(s\right)ds.$$

It is worth mentioning that, when $\xi =0$, the notation $I^0$ denotes the identity operator. Additionally, let the mapping $\Gamma$ be such that

$$\Gamma \left(\xi \right)={\int }_0^{\infty }{t}^{\xi -1}{e}^{-t}dt.$$

Our main interest in this part is a Caputo fractional differential equation such that

$${}^c{D}^{\xi }u\left(t\right)=h(t,u\left(t\right))\mathrm{for}\mathrm{each}t\in [0,1],$$

(13)

where $n-1<\xi \le n$ and $n\ge 2$. Moreover, our boundary conditions will be

$$u\left(0\right)=u^{\prime }\left(0\right)=\cdots =u^{(n-2)}\left(0\right)=0\mathrm{and}u\left(1\right)={\int }_0^{\delta }u\left(s\right)ds,$$

(14)

with $\delta \in [0,1]$ and $h:[0,1]\times \mathbb{R}\to \mathbb{R}$.

According to [3], B. Wongsaijai et al. obtained a solution for the BVP (13) and (14) by considering a solution for the Volterra integral equation such that

$$\begin{array}{cc}\hfill u\left(t\right)=& \frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^{\delta }{\int }_0^s{(s-\tau )}^{\xi -1}h(\tau ,u\left(\tau \right))d\tau ds\hfill \\ \hfill & -\frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^1{(1-s)}^{\xi -1}h(s,u\left(s\right))ds\hfill \\ \hfill & +\frac{1}{\Gamma \left(\xi \right)}{\int }_0^t{(t-s)}^{\xi -1}h(s,u\left(s\right))ds.\hfill \end{array}$$

Next, it is a standard scheme in the literature that we construct an integral operator as a means to achieve an appropriate fixed point problem. To this end, we define $T:C[0,1]\to C[0,1]$ by the equation

$$\begin{array}{cc}\hfill Tu\left(t\right)=& \frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^{\delta }{\int }_0^s{(s-\tau )}^{\xi -1}h(\tau ,u\left(\tau \right))d\tau ds\hfill \\ \hfill & -\frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^1{(1-s)}^{\xi -1}h(s,u\left(s\right))ds\hfill \\ \hfill & +\frac{1}{\Gamma \left(\xi \right)}{\int }_0^t{(t-s)}^{\xi -1}h(s,u\left(s\right))ds.\hfill \end{array}$$

Here, we note that a solution of BVP (13)–(14) is given by $Tu=u$.

Next, we consider the following result.

Theorem 3.

Given a complete normed space $\left(C[0,1],{∥·∥}_{\infty }\right)$ with $\varpi :{\mathbb{R}}^2\to \mathbb{R}$, assume further that the following conditions (H${}_1$)–(H${}_5$) hold:

(H${}_1$) For any $t\in [0,1]$ and any $v,u\in C[0,1]$ with $\varpi \left(u\right(a),v(a\left)\right)\ge 0$ for each $a\in [0,1]$, it is satisfied that

$$|h(t,v\left(t\right))-h(t,u\left(t\right))|\le K^{*}P(u,v),$$

where $K^{*}$ is a constant such that

$$K^{*}<\frac{(n-{\delta }^n)\Gamma (\xi +2)}{n{\delta }^{\xi +1}+(\xi +1)(2n-{\delta }^n)}$$

and

$$\begin{array}{cc}\hfill P(u,v)=max\{& \parallel u-v\parallel +|\parallel u-Tu\parallel -\parallel v-Tv\parallel |,\hfill \\ \hfill & \parallel u-Tu\parallel +|\parallel u-v\parallel -\parallel v-Tv\parallel |,\hfill \\ \hfill & \parallel v-Tv\parallel +|\parallel u-v\parallel -\parallel u-Tu\parallel |,\hfill \\ \hfill & {\displaystyle \frac{\parallel u-Tu\parallel [1+\parallel v-Tv\parallel ]}{1+\parallel u-v\parallel }}+|\parallel u-v\parallel -\parallel u-Tu\parallel |,\hfill \\ \hfill & {\displaystyle \frac{\parallel v-Tv\parallel [1+\parallel u-Tu\parallel ]}{1+\parallel u-v\parallel }}+|\parallel u-v\parallel -\parallel u-Tu\parallel |\};\hfill \end{array}$$

(H${}_2$) There exists $u_0\in C[0,1]$ with $\varpi (u_0\left(t\right),T{u}_0\left(t\right))\ge 0$ for each $t\in [0,1];$

(H${}_3$) For all $v,u\in C[0,1]$ and $t\in [0,1]$,

$$\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathit{implies}\varpi \left(Tu\right(t),Tv(t\left)\right)\ge 0;$$

(H${}_4$) For any $v,u,w\in C[0,1]$ and $t\in [0,1]$,

$$\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathit{and}\varpi \left(v\right(t),w(t\left)\right)\ge 0\mathit{imply}\varpi \left(u\right(t),w(t\left)\right)\ge 0;$$

(H${}_5$) For any sequence $\left\{u_n\right\}$ in $C[0,1]$ such that $\varpi (u_n\left(a\right),u_{n+1}\left(a\right))\ge 0$ for all $a\in [0,1]$ and $n\in \mathbb{N}$,

$$u_n\to u\in C[0,1]\mathit{implies}\varpi (u_n\left(t\right),u\left(t\right))\ge 0\mathit{for}\mathit{all}t\in [0,1]\mathit{and}n\in \mathbb{N}.$$

Then, there is a fixed point $u^{*}\in C[0,1]$ of T. As a result, the BVP (13)–(14) has a solution $u^{*}\in C[0,1]$.

Proof. 

To start with, we note that the real-valued constant $K^{*}$ above is well defined because each term involved is positive. Next, we define $\alpha :C[0,1]\times C[0,1]\to [0,\infty )$ such that

$$\alpha (u,v)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathrm{for}\mathrm{any}t\in [0,1],\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

First, note that (H${}_2$) provides $A(f,g)\ne ⌀.$ Then, it is not hard to see that (H${}_3$) and (H${}_4$) imply that f is triangular $\alpha$-admissible with respect to $g=I$, where I is the identity function on $C[0,1]$. Furthermore, (H${}_5$) provides assumption $\left(b\right)$ in Theorem 1. Next, notice that the condition (H${}_1$) implies for any $u,v\in C[0,1]$ and $t\in [0,1]$ such that $\alpha (u,v)\ge 1$, we get

$$\begin{array}{cc}\hfill \left|Tu\right(t)-Tv(t\left)\right|& =|\frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^{\delta }{\int }_0^s{(s-\tau )}^{\xi -1}\left(h(\tau ,u(\tau \left)\right)-h(\tau ,v(\tau \left)\right)\right)d\tau ds\hfill \\ \hfill & -\frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^1{(1-s)}^{n-\xi -1}\left(h(\tau ,u(s\left)\right)-h(\tau ,v(s\left)\right)\right)ds\hfill \\ \hfill & +\frac{1}{\Gamma \left(\xi \right)}{\int }_0^t{(t-s)}^{\xi -1}\left(h(s,u(s\left)\right)-h(s,v(s\left)\right)\right)ds|\hfill \\ \hfill & \le \frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^{\delta }{\int }_0^s{|s-\tau |}^{\xi -1}\left|h(\tau ,u(\tau \left)\right)-h(\tau ,v(\tau \left)\right)\right|d\tau ds\hfill \\ \hfill & +\frac{n{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^1{|1-s|}^{\xi -1}\left|h(s,u(s\left)\right)-h(s,v(s\left)\right)\right|ds\hfill \\ \hfill & +\frac{1}{\Gamma \left(\xi \right)}{\int }_0^t{|t-s|}^{\xi -1}\left|h(s,u(s\left)\right)-h(s,v(s\left)\right)\right|ds\hfill \\ \hfill & \le \frac{n{K}^{*}{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^{\delta }{\int }_0^s{|s-\tau |}^{\xi -1}P(u,v)d\tau ds\hfill \\ \hfill & +\frac{n{K}^{*}{t}^{n-1}}{(n-{\delta }^n)\Gamma \left(\xi \right)}{\int }_0^1{|1-s|}^{\xi -1}P(u,v)ds\hfill \\ \hfill & +\frac{K^{*}}{\Gamma \left(\xi \right)}{\int }_0^t{|t-s|}^{\xi -1}P(u,v)ds.\hfill \end{array}$$

That is,

$$\begin{array}{c}\hfill |Tu\left(t\right)-Tv\left(t\right)|\le c_0{K}^{*}P(u,v),\end{array}$$

where

$$\begin{array}{cc}\hfill c_0& =\frac{n}{(n-{\delta }^n)\Gamma \left(\xi \right)}\underset{t\in (0,1)}{sup}\left({\int }_0^{\delta }{\int }_0^s{(s-\tau )}^{\xi -1}d\tau ds+{\int }_0^1{(1-s)}^{\xi -1}ds+\frac{n-{\delta }^n}{n}{\int }_0^t{(t-s)}^{\xi -1}ds\right).\hfill \end{array}$$

Hence,

$$d(Tu,Tv):={∥Tu-Tv∥}_{\infty }\le c_0{K}^{*}P(u,v).$$

(15)

Simple calculations give

$$\begin{array}{c}\hfill c_0=\frac{n}{(n-{\delta }^n)\Gamma \left(\xi \right)}\left(\frac{{\delta }^{\xi +1}}{\xi (\xi +1)}+\frac{2n-{\delta }^n}{\xi n}\right)=\frac{1}{\Gamma (\xi +2)}\left[\frac{n{\delta }^{\xi +1}+(\xi +1)(2n-{\delta }^n)}{(n-{\delta }^n)}\right].\end{array}$$

By (H${}_1$), we have ${\displaystyle K^{*}<\frac{1}{c_0}}$. Next, we define $\beta :[0,\infty )\to [0,1]$ and $\gamma :[0,\infty )\to [0,\infty )$ such that

$$\beta \left(t\right)=\left\{\begin{array}{cc}{c}_0{K}^{*}\hfill & \mathrm{if}t\ne 0,\hfill \\ 1\hfill & \mathrm{if}t=0,\hfill \end{array}\right.$$

and $\gamma \left(t\right)=2t$ for all $t\in [0,\infty )$. It can be checked that $\gamma \in \Gamma$ and $\beta \in \mathcal{B}$. Now, for any $v,u\in C[0,1]$ with $\alpha (u,v)\ge 0$, the inequality (15) yields

$$\begin{array}{cc}\hfill \gamma \left(d\right(Tu,Tv\left)\right)=2d(Tu,Tv)& \le 2c_0{K}^{*}P(u,v)\hfill \\ \hfill & =c_0{K}^{*}2P(u,v)\hfill \\ \hfill & =\beta \left(P\right(u,v\left)\right)\gamma \left(P\right(u,v\left)\right).\hfill \end{array}$$

As we can see from the above proof, all those requirements of Theorem 1 are fulfilled. Consequently, there is a function $u^{*}\in C[0,1]$ satisfying $T{u}^{*}=u^{*}$. □

Let us end this section by pointing out that one of the natural directions in which to extend this work is to consider applications in other fractional order derivatives, which can be carried out in future studies.

4. Application in Ordinary Differential Equations

Our main results also have a beautiful impact on ordinary differential equations. To start with, suppose that $u\in C[0,1]$ and consider a second order differential equation such that

$$u^{″}\left(t\right)=-g(t,u\left(t\right))\mathrm{for}\mathrm{all}t\in [0,1],$$

(16)

and nonlocal conditions are

$$u\left(0\right)=0\mathrm{and}u\left(1\right)=\xi u\left(\eta \right),$$

(17)

where $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function, $\eta \in (0,1)$, and $\xi$ is a given positive constant satisfying the nonresonance condition $\xi \eta \ne 1$. It is worth noting that the local conditions $u\left(0\right)=0$ and $u^{\prime }\left(1\right)=0$ can be considered as the limit case of (17) when $\eta \to 1^{-}$. Additionally, the problem (16) together with the boundary conditions (17) become two-point boundary conditions $u\left(0\right)=u\left(1\right)=0$ when $\xi =0$.

It is worth mentioning that the function $u\in C[0,1]$ becomes a solution for (16) if and only if it is a solution for the integral equation,

$$u\left(t\right)=-{\int }_0^1{k}_{\xi ,\eta }(s,t)g(t,u\left(t\right))ds,$$

(18)

where $k_{\xi ,\eta }(t,s)$ is defined by

$$k_{\xi ,\eta }(t,s)=\left\{\begin{array}{cc}{\displaystyle t-s+\frac{\xi t}{1-\xi \eta }(\eta -s)-\frac{(1-s)t}{1-\xi \eta }}\hfill & \mathrm{if}0\le s<min\{\eta ,t\},\hfill \\ {\displaystyle \frac{\xi t}{1-\xi \eta }(\eta -s)-\frac{(1-s)t}{1-\xi \eta }}\hfill & \mathrm{if}0\le t<s<\eta ,\hfill \\ {\displaystyle t-s-\frac{(1-s)t}{1-\xi \eta }}\hfill & \mathrm{if}0\le \eta <s<t\le 1,\hfill \\ {\displaystyle -\frac{(1-s)t}{1-\xi \eta }}\hfill & \mathrm{if}max\{\eta ,t\}<s\le 1.\hfill \end{array}\right.$$

Lemma 2.

Let ξ be a positive constant real number, and let $\eta \in (0,1)$ such that $\xi \eta \ne 1$. Then,

$${\int }_0^1{k}_{\xi ,\eta }(t,s)ds=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}{t}^2+\frac{\xi {\eta }^2-1}{2(1-\xi \eta )}t}& \mathit{if}t\le \eta ,\\ {\displaystyle -\frac{\xi }{2(1-\xi \eta )}{t}^3+\frac{\xi \eta }{1-\xi \eta }{t}^2+(\eta -\frac{1}{2(1-\xi \eta )})t-\frac{{\eta }^2}{2}}& \mathit{if}\eta <t.\end{array}\right.$$

Proof. 

In the case $t\le \eta$, we observe that

$$\begin{array}{cc}\hfill {\int }_0^1{k}_{\xi ,\eta }(t,s)ds& ={\int }_0^t{k}_{\xi ,\eta }(t,s)ds+{\int }_t^{\eta }{k}_{\xi ,\eta }(t,s)ds+{\int }_{\eta }^1{k}_{\xi ,\eta }(t,s)ds\hfill \\ \hfill & ={\int }_0^t\left(t-s+\frac{\xi t}{1-\xi \eta }(\eta -s)-\frac{(1-s)t}{1-\xi \eta }\right)ds\hfill \\ \hfill & +{\int }_t^{\eta }\left(\frac{\xi t}{1-\xi \eta }(\eta -s)-\frac{(1-s)t}{1-\xi \eta }\right)ds-{\int }_{\eta }^1\frac{(1-s)t}{1-\xi \eta }ds\hfill \\ \hfill & ={\int }_0^t(t-s)ds+\frac{\xi t}{1-\xi \eta }{\int }_0^{\eta }(\eta -s)ds-\frac{t}{1-\xi \eta }{\int }_0^1(1-s)ds\hfill \\ \hfill & =\frac{1}{2}{t}^2+\frac{\xi {\eta }^2-1}{2(1-\xi \eta )}t.\hfill \end{array}$$

Similarly, when $\eta \le t$, we have

$$\begin{array}{cc}\hfill {\int }_0^1{k}_{\xi ,\eta }(t,s)ds& ={\int }_0^{\eta }{k}_{\xi ,\eta }(t,s)ds+{\int }_{\eta }^t{k}_{\xi ,\eta }(t,s)ds+{\int }_t^1{k}_{\xi ,\eta }(t,s)ds\hfill \\ \hfill & ={\int }_0^{\eta }\left(t-s+\frac{\xi t}{1-\xi \eta }(\eta -s)-\frac{(1-s)t}{1-\xi \eta }\right)ds\hfill \\ \hfill & +{\int }_{\eta }^t\left(t-s-\frac{(1-s)t}{1-\xi \eta }\right)ds-{\int }_t^1\frac{(1-s)t}{1-\xi \eta }ds\hfill \\ \hfill & ={\int }_0^{\eta }(t-s)ds+\frac{\xi t}{1-\xi \eta }{\int }_0^t(\eta -s)ds-\frac{t}{1-\xi \eta }{\int }_0^1(1-s)ds\hfill \\ \hfill & =-\frac{\xi }{2(1-\xi \eta )}{t}^3+\frac{\xi \eta }{1-\xi \eta }{t}^2+(\eta -\frac{1}{2(1-\xi \eta )})t-\frac{{\eta }^2}{2}\hfill \end{array}$$

as required. □

As a consequence of Lemma 2 above, it can be observed that

$$\underset{t\in [0,1]}{sup}{\int }_0^1{k}_{\xi ,\eta }(t,s)ds\le \frac{1}{2}{\eta }^2+max\left\{\frac{|1-\xi {\eta }^2|}{2|1-\xi \eta |},\eta +\frac{\xi (1+2\eta )+1}{2|1-\xi \eta |}\right\}.$$

(19)

In addition, we recall that $(C[0,1],{∥·∥}_{\infty })$ is a complete normed space. So, $\left(C\right[0,1],d)$ is a complete metric space, where for any $u,v\in C[0,1]$,

$$d(u,v):={∥u-v∥}_{\infty }=sup\left\{\right|u\left(t\right)-v\left(t\right)|:t\in [0,1]\}.$$

Next, we define $F:C[0,1]\to C[0,1]$ by the following equation:

$$Fu\left(t\right)={\int }_0^1{k}_{\xi ,\eta }(t,s)g(s,u\left(s\right))ds.$$

Thus, the existence of a solution of (16) can be considered as the existence of a fixed point of the operator F above.

Theorem 4.

Given $\varpi :{\mathbb{R}}^2\to \mathbb{R}$, assume further that the following conditions (H${}_1$)–(H${}_4$) hold:

(H${}_1$) For any $t\in [0,1]$ and any $v,u\in C[0,1]$ with $\varpi \left(u\right(a),v(a\left)\right)\ge 0$ for each $a\in [0,1]$, it is satisfied that

$$|g(t,v\left(t\right))-g(t,u\left(t\right))|\le \frac{1}{K^{*}}ln(1+P(u,v)),$$

where

$$K^{*}=\frac{1}{2}{\eta }^2+max\left\{\frac{|1-\xi {\eta }^2|}{2|1-\xi \eta |},\eta +\frac{\xi (1+2\eta )+1}{2|1-\xi \eta |}\right\}$$

and

$$\begin{array}{cc}\hfill P(u,v)=max\{& \parallel u-v\parallel +|\parallel u-Fu\parallel -\parallel v-Fv\parallel |,\hfill \\ \hfill & \parallel u-Fu\parallel +|\parallel u-v\parallel -\parallel v-Fv\parallel |,\hfill \\ \hfill & \parallel v-Fv\parallel +|\parallel u-v\parallel -\parallel u-Fu\parallel |,\hfill \\ \hfill & {\displaystyle \frac{\parallel u-Fu\parallel [1+\parallel v-Fv\parallel ]}{1+\parallel u-v\parallel }}+|\parallel u-v\parallel -\parallel u-Fu\parallel |,\hfill \\ \hfill & {\displaystyle \frac{\parallel v-Fv\parallel [1+\parallel u-Fu\parallel ]}{1+\parallel u-v\parallel }}+|\parallel u-v\parallel -\parallel u-Fu\parallel |\};\hfill \end{array}$$

(H${}_2$) There is $u_0\in C[0,1]$ with $\varpi (u_0\left(t\right),F{u}_0\left(t\right))\ge 0$ for each $t\in [0,1];$

(H${}_3$) For all $v,u\in C[0,1]$ and $t\in [0,1]$,

$$\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathit{implies}\varpi \left(Fu\right(t),Fv(t\left)\right)\ge 0;$$

(H${}_4$) For all $v,u,w\in C[0,1]$ and $t\in [0,1]$,

$$\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathit{and}\varpi \left(v\right(t),w(t\left)\right)\ge 0\mathit{imply}\varpi \left(u\right(t),w(t\left)\right)\ge 0.$$

Then, the BVP (16) has a solution.

Proof. 

Consider $F:C[0,1]\to C[0,1]$, defined by

$$Fu\left(t\right)={\int }_0^1{k}_{\xi ,\eta }(t,s)g(s,u\left(s\right))ds.$$

Next, let $\alpha :C[0,1]\times C[0,1]\to [0,\infty )$ such that

$$\alpha (u,v)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathrm{for}\mathrm{any}t\in [0,1],\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Now, we will show that the pair $(F,I)$, where I is the identity map on $C[0,1]$, is an ${(\alpha ,\beta ,\gamma )}_P$-contraction.

Notice that the condition (H${}_1$) implies for any $u,v\in C[0,1]$ and $t\in [0,1]$ such that $\alpha (u,v)\ge 1$, we get

$$\begin{array}{cc}\hfill \left|Fu\right(t)-Fv(t\left)\right|& =\left|{\int }_0^1{k}_{\xi ,\eta }(t,s)(g(s,u\left(s\right))-g(s,v\left(s\right)))ds\right|\hfill \\ \hfill & \le {\int }_0^1{k}_{\xi ,\eta }(t,s)\left|(g(s,u\left(s\right))-g(s,v\left(s\right)))\right|ds\hfill \\ \hfill & \le {\int }_0^1{k}_{\xi ,\eta }(t,s)\frac{1}{K^{*}}ln(1+P(u,v))ds\hfill \\ \hfill & \le \frac{1}{K^{*}}ln(1+P(u,v))\underset{t\in [0,1]}{sup}{\int }_0^1{k}_{\xi ,\eta }(t,s)ds\hfill \\ \hfill & \le ln(1+P(u,v\left)\right).\hfill \end{array}$$

Next, we define $\beta :[0,\infty )\to [0,1]$ and $\gamma :[0,\infty )\to [0,\infty )$ such that

$$\beta \left(t\right)=\left\{\begin{array}{cc}\frac{ln(1+t)}{t}\hfill & \mathrm{if}t\ne 0,\hfill \\ 0\hfill & \mathrm{if}t=0,\hfill \end{array}\right.$$

and $\gamma \left(t\right)=t$ for each $t\in [0,\infty )$. It can be checked that $\gamma \in \Gamma$ and $\beta \in \mathcal{B}$. Now, for any $v,u\in C[0,1]$ with $\alpha (u,v)\ge 1$, we have

$$\begin{array}{cc}\hfill \gamma \left(d\right(Fu,Fv\left)\right)=d(Fu,Fv)& \le ln(1+P(u,v\left)\right)\hfill \\ \hfill & =\frac{ln(1+P(u,v\left)\right)}{P(u,v)}P(u,v)\hfill \\ \hfill & =\beta \left(P\right(u,v\left)\right)P(u,v)\hfill \\ \hfill & =\beta \left(P\right(u,v\left)\right)\gamma \left(P\right(u,v\left)\right).\hfill \end{array}$$

Thus, $(F,I)$ is an ${(\alpha ,\beta ,\gamma )}_P$-contraction. By assumptions (H${}_2$)–(H${}_4$), all those requirements of Theorem 1 are fulfilled. Consequently, there is a function $u^{*}\in C[0,1]$ satisfying $F{u}^{*}=u^{*}$. □

Remark 1.

When $\xi =0$, by taking $\eta \to 1^{-}$, we see that

$$\underset{t\in [0,1]}{sup}{\int }_0^1{k}_{\xi ,\eta }(t,s)ds=\frac{1}{8}.$$

In this case, let us define Green’s function ${\mathcal{K}}_{0,\eta }(t,s)$ such that

$${\mathcal{K}}_{0,\eta }(t,s)=\left\{\begin{array}{cc}t(1-s)\hfill & \mathrm{if}0\le t\le s\le 1,\hfill \\ s(1-t)\hfill & \mathrm{if}0\le s\le t\le 1.\hfill \end{array}\right.$$

In addition, define $F:C[0,1]\to C[0,1]$ by

$$Fu\left(t\right)={\int }_0^1{\mathcal{K}}_{0,\eta }(t,s)g(s,u\left(s\right))ds.$$

Thus, the existence of a solution of (16) when $\xi =0$ can be considered as the existence of a fixed point of the operator F above.

To finish this section, we indicate that our next corollary is a consequence of the preceding remark.

Corollary 1.

Given $\varpi :{\mathbb{R}}^2\to \mathbb{R}$, assume further that the following conditions (H${}_1$)–(H${}_4$) hold:

(H${}_1$) For any $t\in [0,1]$ and any $v,u\in C[0,1]$ with $\varpi \left(u\right(a),v(a\left)\right)\ge 0$ for each $a\in [0,1]$, it is satisfied that

$$\left|g\right(t,v\left(t\right))-g(t,u\left(t\right)\left)\right|\le 8ln(1+P(u,v\left)\right),$$

where $P(u,v)$ is defined as in Theorem 4.

(H${}_2$) There is $u_0\in C[0,1]$ with $\varpi (u_0\left(t\right),F{u}_0\left(t\right))\ge 0$ for each $t\in [0,1];$

(H${}_3$) For all $v,u\in C[0,1]$ and $t\in [0,1]$,

$$\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathit{implies}\varpi \left(Fu\right(t),Fv(t\left)\right)\ge 0;$$

(H${}_4$) For all $v,u,w\in C[0,1]$ and $t\in [0,1]$,

$$\varpi \left(u\right(t),v(t\left)\right)\ge 0\mathit{and}\varpi \left(v\right(t),w(t\left)\right)\ge 0\mathit{imply}\varpi \left(u\right(t),w(t\left)\right)\ge 0.$$

Then, the BVP (16) when $\xi =0$ has a solution.

5. Conclusions

In this work, we presented an idea of an ${(\alpha ,\beta ,\gamma )}_P$-contraction for a pair of functions $(f,g)$ to obtain the existence theorems for a common fixed point and a coincidence point of $(f,g)$ in a complete metric space equipped with additional requirements. The theorems and their consequences allowed us to extend previous results in the literature. Indeed, we applied our main conclusions to show the existence of a solution for ordinary differential equations with nonlocal conditions and a solution for fractional differential equations with integral-type boundary conditions. In the future, our findings can be extended to the case of generalized metric spaces such as JS-metric spaces (see, for instance, [19]), b-metric spaces (see, for example, [1]), partial metric spaces, etc. Additionally, exploring auxiliary functions instead of Geraghty functions could be a valuable direction because this will lead to an extension of the papers [2,3,26]. These two approaches hold the potential for various applications in different types of differential equations.