On a new fixed point theorem with an application on a coupled system of fractional differential equations

Abstract

In this work, new theorems and results related to fixed point theory are presented. The results obtained are used for the sake of proving the existence and uniqueness of a positive solution of a coupled system of equations that involves fractional derivatives in the Riemann–Liouville settings and is subject to boundary conditions in the form of integrals.

1 Introduction

For the last multiple of decades, the fractional calculus has drawn great interest of many sciences working on multifarious sciences on account of the good results obtained when this research applied the fractional tools to modeling of some real life problems [22, 26, 30, 36, 46–48].

On the other side, even though the fixed point theory is a branch of pure mathematics, it is showed that this theory is one of the main tools to use in order to tackle the qualitative properties of differential and integral equations in general and the existence and uniqueness of solutions to these equations in particular. A significant number of mathematicians utilized the classical results in the fixed point theory to discuss solutions of initial and boundary value problems (see [2–21, 23, 24, 27–29, 31–34, 37, 39–43]). Meanwhile, others established new fixed point theorems and applied them to proving the existence and oneness of solutions to a variety of differential equations [13, 25, 38, 44, 45].

In this work, we will propose new theorems related to the fixed points of operators. We discuss the admissibility of two multi-valued mappings in the category of complete b-metric spaces to obtain the existence of a common fixed point. Using the triangular admissibility, we prove the uniqueness of the common fixed point. Then we utilize the findings of this problem to discuss one of the most considerable qualitative aspects for differential equation with fractional order; that is, the existence and uniqueness of the positive solution of Riemann–Liouville fractional coupled system governed by boundary integral conditions.

2 Preliminaries

Let χ be a set of all increasing and continuous functions $\varphi: [0,+\infty ) \to \mathbb{[}0,+\infty )$ with the property $\varphi (\varsigma )=0\ \text{if and only if}\ \varsigma =0$ and $\varphi (c\varsigma )\leq c\varphi (\varsigma )$ for $c>1$.

Let Ω be the family of all functions $\vartheta: [0,+\infty ) \to [0,\frac{1}{s^{2}})$ such that, for any bounded sequence $\{\varsigma _{n}\}$ of positive real numbers, $\vartheta (\varsigma _{n})\rightarrow 1$ implies $\varsigma _{n}\rightarrow 0$.

Let $(X,d)$ be a b-metric space. Take $CB(X)$ the set of bounded and closed sets in X. For $x\in X$ and $A, B\in CB(X)$, we define

$$\begin{aligned} &D(x,A)=\inf_{a\in A} \,d(x,a), \\ &D(A,B)=\sup_{a\in A} D(a,B). \end{aligned}$$

Define a mapping $H: CB(X) \times CB(X) \to [0,\infty )$ such that

$$ H(A,B)=\max \Bigl\lbrace \sup_{x\in A} \,d(x,B), \sup _{y\in B}\,d(y,B) \Bigr\rbrace , $$

for every $A,B \in CB(X)$. Then the mapping H forms a b-metric.

Throughout the article I will denote the interval $[0,1]$. The definition for admissibility of a pair of single-valued mappings was first introduced in [1] and then generalized and used in [35] as well. Below we present the definition for the multi-valued case within the triangular admissibility.

Definition 2.1

Let $T_{1},T_{2}:X \rightarrow CB (X)$ be two multi-valued mappings and $\rho:X\times X\rightarrow [0,+\infty )$ be a function. Then the pair $(T_{1},T_{2})$ is said to be triangular $\rho _{*}$-admissible if the following conditions hold:

• (i) $(T_{1},T_{2})$ is $\rho _{*}$-admissible; that is, $\rho (\varsigma,\eta )\geq 1$ implies $\rho _{*}(T_{1}\varsigma,T_{2}\eta )\geq 1$ and $\rho _{*}(T_{2}\varsigma,T_{1}\eta )\geq 1$, where

  $$ \rho _{*}(A,B) =\inf \bigl\{ \rho (\varsigma,\eta ):\varsigma \in A, \eta \in B\bigr\} , $$

• (ii) $\rho (\varsigma,u)\geq 1$ and $\rho (u,\eta )\geq 1$ imply $\rho (\varsigma,\eta )\geq 1$.

Definition 2.2

([26, 36])

The Riemann–Liouville fractional integral of order $\rho >0$ of a continuous function $f:(0,+\infty )\rightarrow \mathbf{(-\infty, +\infty )}$ is given by

$$ I_{0^{+}}^{\rho }f(\varsigma )=\frac{1}{\varGamma (\rho )} \int _{0}^{ \varsigma }(\varsigma -\eta )^{\rho -1}f(\eta )\,d\eta, $$

provided the right-hand side is pointwise defined on $(0, +\infty )$.

Definition 2.3

([26, 36])

The Riemann–Liouville fractional derivative of order $\rho >0$ of a continuous function $f:(0,+\infty )\rightarrow \mathbf{(-\infty, +\infty )}$ is given by

$$ D_{0^{+}}^{\rho }f(\varsigma )=\frac{1}{\varGamma (n-\rho )} \biggl( \frac{d}{dt} \biggr)^{n} \int _{0}^{\varsigma } (\varsigma -\eta )^{n- \rho -1} f( \eta )\,d\eta, $$

where $n=[\rho ]+1$, $[\rho ]$ denotes the integer part of the number ρ, provided that the right-hand side is pointwise defined on $(0, +\infty )$.

In this paper, we discuss the local existence and uniqueness of positive solutions for the following coupled system of fractional boundary value problem subject to integral boundary conditions:

$$ \textstyle\begin{cases} D_{0^{+}}^{\rho } u(\varsigma ) + f(\varsigma,v(\varsigma ))=0, \qquad D_{0^{+}}^{ \varrho } v(\varsigma ) + g(\varsigma,u(\varsigma ))=0, \quad 0< \varsigma < 1, \\ u(0)=0, \qquad u(1)=\int _{0}^{1}\phi (\varsigma )u(\varsigma )\,d \varsigma, \qquad v(0)=0,\qquad v(1)=\int _{0}^{1}\varphi (\varsigma )v( \varsigma )\,d\varsigma \end{cases} $$

(1)

where $1<\rho,\varrho \leq 2, \phi,\varphi \in L^{1}I$ are nonnegative and $f,g\in C(I\times [0,+\infty ),[0,+\infty ))$ and D is the standard Riemann–Liouville fractional derivative. The functions $\phi (\varsigma )$, $\varphi (\varsigma )$ satisfy the following conditions:

$$\begin{aligned} (Q) \phi, \varphi:I \rightarrow [0, + \infty ) \quad\text{with } \phi, \varphi \in L^{1}I \end{aligned}$$

and

$$\begin{aligned} &\sigma _{1}:= \int _{0}^{1}\phi (\varsigma ) \varsigma ^{\rho -1} \,d \varsigma,\qquad \sigma _{2}:= \int _{0}^{1}\varphi (\varsigma ) \varsigma^{\varrho -1}\,d\varsigma,\quad \varsigma \in (0,1); \\ &\sigma _{3}:= \int _{0}^{1} \varsigma ^{\rho -1}(1-\varsigma ) \phi ( \varsigma ) \,d\varsigma, \qquad\sigma _{4}:= \int _{0}^{1} \varsigma ^{\varrho -1}(1-\varsigma ) \varphi (\varsigma ) \,d \varsigma >0. \end{aligned}$$

Lemma 2.4

([41])

If $\int _{0}^{1}\phi (\varsigma )\varsigma ^{\rho -1}\,d\varsigma \neq 1$, then, for any $\sigma \in CI$, the unique solution of the following boundary value problem:

$$ \textstyle\begin{cases} D_{0^{+}}^{\rho } u(\varsigma ) +\sigma (\varsigma )=0, \quad 0< \varsigma < 1, \\ u(0)=0, \qquad u(1)=\int _{0}^{1}\phi (\varsigma )u(\varsigma )\,d \varsigma,\end{cases} $$

is given by

$$ u(\varsigma ) = \int _{0}^{1}G_{1\rho }(\varsigma,\eta ) \sigma (\eta )\,d \eta, $$

where

$$ G_{1\rho }(\varsigma,\eta )=G_{2\rho }(\varsigma,\eta )+G_{3\rho }( \varsigma,\eta ), $$

$$\begin{aligned} \begin{aligned}& G_{2\rho } (\varsigma,\eta )=\frac{1}{\varGamma (\rho )} {\textstyle\begin{cases} \varsigma ^{\rho -1}(1-\eta )^{\rho -1}-(\eta -\varsigma )^{\rho -1}, & 0 \leq \varsigma \leq \eta \leq 1, \\ \varsigma ^{\rho -1}(1-\eta )^{\rho -1}, & 0\leq \eta \leq \varsigma \leq 1, \end{cases}\displaystyle } \\ &G_{3\rho } (\varsigma,\eta )= \frac{\varsigma ^{\rho -1}}{1-\int _{0}^{1}\phi (\varsigma )\varsigma ^{\rho -1}\,d\varsigma } \int _{0}^{1}\phi (\varsigma )G_{2\rho }( \varsigma,\eta )\,d\varsigma, \end{aligned} \end{aligned}$$

(2)

Then $G(\varsigma,\eta )=(G_{1\rho }(\varsigma,\eta ),G_{1\varrho }( \varsigma,\eta ))$ is a Green’s function of the system (1).

Lemma 2.5

([41])

Let $\rho,\varrho \in (1,2]$. Assume that $(Q)$holds. Then the functions $G_{1\rho }(\varsigma,\eta ), G_{1\varrho }(\varsigma,\eta )$have the following properties:

$$\begin{aligned} &\frac{(\rho -1)\sigma _{3}\eta (1-\eta )^{\rho -1}\varsigma ^{\rho -1}}{(1-\sigma _{1})\varGamma (\rho )} \leq G_{1\rho }\leq \frac{(1-\eta )^{\rho -1}\varsigma ^{\rho -1}}{\varGamma (\rho )(1-\sigma _{1})}, \\ &\frac{(\varrho -1)\sigma _{4}\eta (1-\eta )^{\varrho -1}\varsigma ^{\varrho -1}}{(1-\sigma _{2})\varGamma (\varrho )} \leq G_{1\varrho }\leq \frac{(1-\eta )^{\varrho -1} \varsigma ^{\varrho -1}}{\varGamma (\varrho )(1-\sigma _{2})}, \quad\eta,\varsigma \in I. \end{aligned}$$

Lemma 2.6

([41])

Assume that $(Q)$holds and $f(\kappa,x), g(\kappa,x)$are continuous, then $(u,v)\in X\times X$is a solution of the system (1) if and only if it is a solution of the integral equations

$$ \textstyle\begin{cases} u(\kappa )=\int _{0}^{1}G_{1\varrho }(\kappa,\varkappa )f(\varkappa,v( \varkappa ))\,d\varkappa, \\ v(\kappa )=\int _{0}^{1}G_{1\rho }(\kappa,\varkappa )g(\varkappa,u( \varkappa ))\,d\varkappa. \end{cases} $$

3 Main results

Now, we are ready to state and prove our main results.

The following key lemma is essential to proceed in proving the main results. It states that the admissibility of a pair of multi-valued functions will guarantee the existence of a sequence of points with diameter greater than 1.

Lemma 3.1

Let $T_{1},T_{2}:X\rightarrow CB (X)$be two multi-valued mappings such that the pair $(T_{1},T_{2})$is triangular $\rho _{*}$-admissible. Assume that there exists $\varrho _{0}\in X$with $\rho _{*}(\varrho _{0},T_{1}\varrho _{0})\geq 1$. Define a sequence $\{\varrho _{n}\}$in X by $\varrho _{2i+1}\in T_{1}\varrho _{2i}$and $\varrho _{2i+2}\in T_{2}\varrho _{2i+1}$, where $i=0, 1, 2,\ldots$ . Then, for $m,n\in \mathbb{N}\cup \{0\}$with $m > n$, we have $\rho (\varrho _{n},\varrho _{m})\geq 1$.

Proof

From $\rho _{*}(\varrho _{0},T_{1}\varrho _{0})\geq 1$ we get $\rho (\varrho _{0},\varrho _{1})\geq 1$. Since $(T_{1},T_{2})$ is $\rho _{*}$-admissible, we obtain $\rho _{*}(T_{1}\varrho _{0},T_{2}\varrho _{1})\geq 1$, hence $\rho (\varrho _{1},\varrho _{2})\geq 1$ and so $\rho _{*}(T_{2}\varrho _{1},T_{1}\varrho _{2})\geq 1$, then $\rho (\varrho _{2},\varrho _{3})\geq 1$, with continuing this process we obtain, $\rho (\varrho _{m},\varrho _{m+1})\geq 1$.

By (ii) from definition of triangular $\rho _{*}$-admissible and regarding as; $\rho (\varrho _{n},\varrho _{n+1})\geq 1$ and $\rho (\varrho _{n+1},\varrho _{n+2})\geq 1$, deduce $\rho (\varrho _{n},\varrho _{n+2})\geq 1$. Again with continuing this process and from $m>n$, we find $\rho (\varrho _{n},\varrho _{m})\geq 1$. □

The following theorem gives the existence of a common fixed point for two mappings $T_{1}$ and $T_{2}$ under less hypotheses than the results existing in the literature.

Theorem 3.2

Let $(X,d)$be an ρ-complete b-metric space (with $s\geq 1$), and $\rho:X\times X\rightarrow [0,+\infty )$be a function. Suppose that $T_{1},T_{2}:X\rightarrow CB(X)$are mappings such that

$$\begin{aligned} \rho (\varsigma,\eta )\varphi \bigl(s^{3} H(T_{1}\varsigma,T_{2}\eta )\bigr) \leq \vartheta \bigl( \varphi \bigl(M(\varsigma,\eta )\bigr)\bigr)\varphi \bigl(M(\varsigma, \eta ) \bigr)+L\phi \bigl(N(\varsigma,\eta )\bigr), \end{aligned}$$

(3)

where

$$\begin{aligned} \begin{aligned} &M(\varsigma,\eta )=\max \biggl\{ d(\varsigma,\eta ),D(\varsigma,T_{1} \varsigma ),D(\eta,T_{2}\eta ), \frac{D(\eta,T_{1}\varsigma ) +D(\varsigma,T_{2}\eta )}{2s}\biggr\} \quad\textit{and} \\ & N(\varsigma,\eta )=\min \bigl\{ D( \eta,T_{1}\varsigma ),D( \eta,T_{2}\eta )\bigr\} , \end{aligned} \end{aligned}$$

(4)

for $\vartheta \in \varOmega$and $\varphi,\phi \in \chi$. Moreover, suppose

• (i) $(T_{1},T_{2})$is triangular $\rho _{*}$-admissible;

• (ii) there exists $\varrho _{0}\in X$with $\rho _{*}(\varrho _{0},T_{1}\varrho _{0})\geq 1$;

• (iii) if for every sequence $\{\varrho _{n}\}$in X with $\rho (\varrho _{n},\varrho _{n+1})\geq 1$for all $n\in \mathbb{N}\cup \{0\}$and $\varrho _{n}\rightarrow \varrho \in X$, then there exists a subsequence $\{\varrho _{n(k)}\}$of $\{\varrho _{n}\}$with $\rho (\varrho _{n(k)},\varrho )\geq 1$.

Then $T_{1}$and $T_{2}$have a common fixed point $\varrho \in X$.

Proof

Let $\varrho _{0}\in X$ with $\rho _{*}(\varrho _{0},T_{1}\varrho _{0})\geq 1$. Choose $\varrho _{1}\in T_{1}\varrho _{0}$ such that $\rho (\varrho _{0},\varrho _{1})\geq 1$ and $\varrho _{1}\neq \varrho _{0}$. Let $q= \frac{1}{\sqrt{\vartheta (\varphi (d(\varrho _{0},\varrho _{1})))}}$. By (3) and considering that $q>1$, we have

$$ 0< \varphi \bigl(D(\varrho _{1},T_{2}\varrho _{1})\bigr)\leq \rho (\varrho _{0}, \varrho _{1})\varphi \bigl(H(T_{1}\varrho _{0},T_{2} \varrho _{1})\bigr)< q\rho ( \varrho _{0},\varrho _{1})\varphi \bigl(s^{3}H(T_{1}\varrho _{0},T_{2} \varrho _{1})\bigr). $$

Hence, there exists $\varrho _{2}\in T_{2}\varrho _{1}$ such that

$$\begin{aligned} \varphi \bigl(d(\varrho _{1},\varrho _{2}) \bigr)&< q\rho (\varrho _{0},\varrho _{1}) \varphi \bigl(s^{3}H(T_{1}\varrho _{0},T_{2} \varrho _{1})\bigr) \\ & \leq q\vartheta \bigl(\varphi \bigl(M(\varrho _{0},\varrho _{1})\bigr)\bigr) \varphi \bigl(M(\varrho _{0},\varrho _{1})\bigr)+qL\phi \bigl(N(\varrho _{0}, \varrho _{1})\bigr) \\ & =\sqrt{\vartheta \bigl(\varphi \bigl(d(\varrho _{0}, \varrho _{1})\bigr)\bigr)} \varphi \bigl(M(\varrho _{0}, \varrho _{1})\bigr)+qL\phi \bigl(N(\varrho _{0}, \varrho _{1})\bigr), \end{aligned}$$

(5)

where

$$\begin{aligned} M(\varrho _{0},\varrho _{1})&=\max \biggl\{ d(\varrho _{0},\varrho _{1}),D( \varrho _{0},T_{1} \varrho _{0}),D(\varrho _{1},T_{2}\varrho _{1}),\frac{D( \varrho _{1},T_{1}\varrho _{0})+D(\varrho _{0},T_{2}\varrho _{1}) }{2s}\biggr\} \\ &\leq \max \biggl\{ d(\varrho _{0},\varrho _{1}),D(\varrho _{1},T_{2} \varrho _{1}),\frac{D(\varrho _{0},T_{2}\varrho _{1})}{2s}\biggr\} \\ &\leq \max \biggl\{ d(\varrho _{0},\varrho _{1}),D(\varrho _{1},T_{2} \varrho _{1}),\frac{D(\varrho _{0},T_{2}\varrho _{1})}{2s}\biggr\} \end{aligned}$$

and

$$\begin{aligned} N(\varrho _{0},\varrho _{1})&=\min \bigl\{ D(\varrho _{0},T_{2}\varrho _{0}),D( \varrho _{1},T_{1}\varrho _{0})\bigr\} \\ &\leq \min \bigl\{ d(\varrho _{0},\varrho _{1}),d(\varrho _{1},\varrho _{1}) \bigr\} =0. \end{aligned}$$

Since

$$ \frac{D(\varrho _{0},T_{2}\varrho _{1})}{2}\leq \frac{[d(\varrho _{0},\varrho _{1})+D(\varrho _{1},T_{2}\varrho _{1})]}{2} \leq \max \bigl\{ d(\varrho _{0},\varrho _{1}),D(\varrho _{1},T_{2} \varrho _{1}) \bigr\} , $$

we get

$$\begin{aligned} M(\varrho _{0},\varrho _{1})\leq \max \bigl\{ d(\varrho _{0},\varrho _{1}),D( \varrho _{1},T_{2} \varrho _{1})\bigr\} . \end{aligned}$$

If $\max \{d(\varrho _{0},\varrho _{1}),D(\varrho _{1},T_{2}\varrho _{1}) \}=D(\varrho _{1},T_{2}\varrho _{1})$, then by (5), we have

$$ \varphi \bigl(D(\varrho _{1},T_{2}\varrho _{1}) \bigr)\leq \varphi \bigl(d(\varrho _{1}, \varrho _{2})\bigr)< \sqrt{\vartheta \bigl(\varphi \bigl(d(\varrho _{0},\varrho _{1})\bigr)\bigr)} \varphi \bigl(D(\varrho _{1},T_{2} \varrho _{1})\bigr)< \varphi \bigl(D(\varrho _{1},T_{2} \varrho _{1})\bigr), $$

which is a contradiction. Hence, we obtain $\max \{d(\varrho _{0},\varrho _{1}),D(\varrho _{1},T_{2}\varrho _{1}) \}=d(\varrho _{0},\varrho _{1})$ and so by (5),

$$ \varphi \bigl(d(\varrho _{1},\varrho _{2})\bigr)\leq \sqrt{ \vartheta \bigl(\varphi \bigl(d( \varrho _{0},\varrho _{1}) \bigr)\bigr)}\varphi \bigl(d(\varrho _{0},\varrho _{1}) \bigr). $$

Knowing that $\varphi \in \chi$ and regarding the fact that $\sqrt{\vartheta (\varphi (d(\varrho _{0},\varrho _{1})))}<1$, we have

$$\begin{aligned} \varphi \biggl( \frac{1}{\sqrt{\vartheta (\varphi (d(\varrho _{0},\varrho _{1})))}}\,d( \varrho _{1}, \varrho _{2})\biggr)\leq \frac{1}{\sqrt{\vartheta (\varphi (d(\varrho _{0},\varrho _{1})))}} \varphi \bigl(d(\varrho _{1},\varrho _{2})\bigr)< \varphi \bigl(d(\varrho _{0}, \varrho _{1})\bigr). \end{aligned}$$

(6)

Since φ is increasing,

$$\begin{aligned} d(\varrho _{1},\varrho _{2})\leq \sqrt{\vartheta \bigl( \varphi \bigl(d(\varrho _{0}, \varrho _{1})\bigr)\bigr)}\,d( \varrho _{0},\varrho _{1}). \end{aligned}$$

Recall that $\varrho _{2}\in T_{2}\varrho _{1}$ and $\varrho _{1}\notin T_{2}\varrho _{1}$, so it is clear that $\varrho _{2}\neq \varrho _{1}$. Put

$$ q_{1}=\frac{\sqrt{\vartheta (\varphi (d(\varrho _{0},\varrho _{1})))} \varphi (d(\varrho _{0},\varrho _{1}))}{\varphi (d(\varrho _{1}, \varrho _{2}))}. $$

By (6), we have $q_{1}>1$. Then

$$ 0< \varphi \bigl(d(\varrho _{2},T_{1}\varrho _{2}) \bigr)\leq \rho (\varrho _{1}, \varrho _{2})\varphi \bigl(H(T_{2}\varrho _{1},T_{1}\varrho _{2})\bigr)< q_{1} \rho (\varrho _{1},\varrho _{2})\varphi \bigl(H(T_{2}\varrho _{1},T_{1} \varrho _{2})\bigr). $$

Hence, there exists $\varrho _{3}\in T_{1}\varrho _{2}$ such that

$$\begin{aligned} \varphi \bigl(d(\varrho _{2},\varrho _{3}) \bigr)&< q_{1}\rho (\varrho _{1}, \varrho _{2}) \varphi \bigl(H(T_{2}\varrho _{1},T_{1}\varrho _{2})\bigr) \\ &\leq q_{1}\vartheta \bigl(\varphi \bigl(M(\varrho _{1}, \varrho _{2})\bigr)\bigr)\varphi \bigl(M( \varrho _{1}, \varrho _{2})\bigr)+q_{1}L\phi \bigl(N(\varrho _{1},\varrho _{2})\bigr) \end{aligned}$$

Similarly, $M(\varrho _{1},\varrho _{2})\leq d(\varrho _{1},\varrho _{2})$ and $N(\varrho _{1},\varrho _{2})=0$. So by (5) we have

$$\begin{aligned} \varphi \bigl(d(\varrho _{2},\varrho _{3})\bigr)&\leq \sqrt{\vartheta \bigl(\varphi \bigl(d( \varrho _{1},\varrho _{2})\bigr)\bigr)}\varphi \bigl(d(\varrho _{1},\varrho _{2})\bigr) \\ &\leq \sqrt{\vartheta \bigl(\varphi \bigl(d(\varrho _{1},\varrho _{2})\bigr)\bigr)}\sqrt{ \vartheta \bigl(\varphi \bigl(d(\varrho _{1},\varrho _{2})\bigr)\bigr)}\varphi \bigl(d( \varrho _{0},\varrho _{1})\bigr) \\ &=\bigl(\sqrt{\vartheta \bigl(\varphi \bigl(d(\varrho _{1},\varrho _{2})\bigr)\bigr)}\bigr)^{2} \varphi \bigl(d(\varrho _{0},\varrho _{1})\bigr). \end{aligned}$$

Again by (6), we obtain

$$\begin{aligned} d(\varrho _{2},\varrho _{3})\leq \bigl(\sqrt{\vartheta \bigl(\varphi \bigl(d( \varrho _{1},\varrho _{2})\bigr) \bigr)} \bigr)^{2}\,d(\varrho _{0},\varrho _{1}). \end{aligned}$$

It is clear that $\varrho _{2}\neq \varrho _{1}$. Put

$$ q_{2}= \frac{ (\sqrt{\vartheta (\varphi (d(\varrho _{1},\varrho _{2})))} )^{2}\varphi (d(\varrho _{0},\varrho _{1}))}{\varphi (d(\varrho _{2},\varrho _{3}))}. $$

Then $q_{2}>1$ and we have

$$ 0< \varphi \bigl(d(\varrho _{3},T_{2}\varrho _{3})\bigr)\leq \rho (\varrho _{2}, \varrho _{3}) \varphi \bigl(H(T_{1}\varrho _{2},T_{2}\varrho _{3})\bigr)< q_{2} \rho (\varrho _{2},\varrho _{3})\varphi \bigl(H(T_{1}\varrho _{2},T_{2} \varrho _{3})\bigr). $$

Thus, there exists $\varrho _{4}\in T_{2}\varrho _{3}$ such that

$$\begin{aligned} \varphi \bigl(d(\varrho _{3},\varrho _{4}) \bigr)&< q_{2}\rho (\varrho _{2}, \varrho _{3}) \varphi \bigl(H(T_{1}\varrho _{2},T_{2}\varrho _{3})\bigr) \\ & \leq q_{2}\vartheta \bigl(\varphi \bigl(M(\varrho _{2},\varrho _{3})\bigr)\bigr) \varphi \bigl(M(\varrho _{2},\varrho _{3})\bigr)+q_{2}L\phi \bigl(N( \varrho _{2}, \varrho _{3})\bigr). \end{aligned}$$

(7)

Similarly, $M(\varrho _{2},\varrho _{3})\leq d(\varrho _{2},\varrho _{3})$ and $N(\varrho _{2},\varrho _{3})=0$. So by (7),

$$ \varphi \bigl(d(\varrho _{3},\varrho _{4})\bigr)\leq \sqrt{\vartheta \bigl(\varphi \bigl(d( \varrho _{2},\varrho _{3})\bigr)\bigr)}\varphi \bigl(d(\varrho _{2},\varrho _{3})\bigr) \leq \bigl(\sqrt{\vartheta \bigl(\varphi \bigl(d(\varrho _{0},\varrho _{1})\bigr)\bigr)} \bigr)^{3} \varphi \bigl(d(\varrho _{0},\varrho _{1})\bigr). $$

Similarly, from (6), we obtain

$$\begin{aligned} d(\varrho _{3},\varrho _{4})\leq \bigl(\sqrt{\vartheta \bigl(\varphi \bigl(d( \varrho _{0},\varrho _{1})\bigr) \bigr)} \bigr)^{3}\,d(\varrho _{0},\varrho _{1}). \end{aligned}$$

It is clear that $\varrho _{3}\neq \varrho _{2}$. Put

$$ q_{3}= \frac{ (\sqrt{\vartheta (\varphi (d(\varrho _{0},\varrho _{1})))} )^{3}\varphi (d(\varrho _{0},\varrho _{1}))}{\varphi (d(\varrho _{2},\varrho _{3}))}. $$

Then $q_{3}>1$. By continuing this process and by Lemma 3.1 we obtain a sequence $\{\varrho _{n}\}$ in X such that

$$\begin{aligned} \varrho _{2i+1}\in T_{1}\varrho _{2i} \quad\text{and}\quad \varrho _{2i+2} \in T_{2}\varrho _{2i+1}, \quad\text{where } i=0, 1, 2,\ldots. \end{aligned}$$

Also, $d(\varrho _{n},\varrho _{n+1})< (\sqrt{\vartheta (\varphi (d( \varrho _{0},\varrho _{1})))} )^{n}\,d(\varrho _{0},\varrho _{1})$ for all n. Let $t=\sqrt{\vartheta (\varphi (d(\varrho _{0},\varrho _{1})))}$, then $0< t<1$ for $n< m$, by the triangle inequality we have

$$\begin{aligned} d(\varrho _{n},\varrho _{m})\leq{}& d(\varrho _{n}, \varrho _{n+1})+d( \varrho _{n+1},\varrho _{n+2})+ \cdots \\ &{}+d(\varrho _{m-2},\varrho _{m-1})+d(\varrho _{m-1}, \varrho _{m}) \\ \leq{}& t^{n}\bigl(1+t+t^{2}+\cdots \bigr)\,d(\varrho _{0},\varrho _{1}) \\ ={}&\biggl(\frac{t^{n}}{1-\varsigma }\biggr)\,d(\varrho _{0},\varrho _{1})\rightarrow 0\quad \text{as }n \to +\infty. \end{aligned}$$

Therefore, for $n< m$, we obtain

$$\begin{aligned} d(\varrho _{n},\varrho _{m})\rightarrow 0\quad \text{as }n\to +\infty. \end{aligned}$$

Therefore

$$\begin{aligned} \lim_{m,n\rightarrow \infty }\,d(\varrho _{n},\varrho _{m})=0. \end{aligned}$$

We deduce that $\{\varrho _{n}\}$ is a Cauchy sequence in $(X,d)$. Since $(X,d)$ is a complete b-metric space, so there exists $\varrho ^{*}\in X$ such that $\lim_{n\rightarrow \infty }\varrho _{n}=\varrho ^{*}$. Since $\rho (\varrho _{n},\varrho _{n+1})\geq 1$, so there exists a subsequence $\{\varrho _{2n_{k}}\}$ of $\{\varrho _{n}\}$ such that

$$\begin{aligned} \rho \bigl(\varrho _{2n_{k}},\varrho ^{*}\bigr) \geq 1, \end{aligned}$$

(8)

for all k. By the triangular inequality

$$\begin{aligned} D\bigl(\varrho ^{*},T_{2}\varrho ^{*}\bigr)&\leq sd\bigl(\varrho ^{*},\varrho _{2n_{k}+1}\bigr)+sD\bigl( \varrho _{2n_{k}+1},T_{2}\varrho ^{*}\bigr) \\ &\leq sd\bigl(\varrho ^{*},\varrho _{2n_{k}+1}\bigr)+sH \bigl(T_{1}\varrho _{2n_{k}},T_{2} \varrho ^{*}\bigr). \end{aligned}$$

Letting k tend to infinity

$$\begin{aligned} D\bigl(\varrho ^{*},T_{2}\varrho ^{*}\bigr)\leq \lim_{k\rightarrow \infty } sH\bigl(T_{1} \varrho _{2n_{k}},T_{2}\varrho ^{*}\bigr). \end{aligned}$$

(9)

Having $\varphi \in \chi$, (8) and (9),

$$\begin{aligned} &\varphi \bigl(s^{2}D\bigl(\varrho ^{*},T_{2} \varrho ^{*}\bigr)\bigr) \\ &\quad\leq \lim_{k \rightarrow \infty }\varphi \bigl(s^{3}H\bigl(T_{1}\varrho _{2n_{k}},T_{2} \varrho ^{*}\bigr)\bigr) \\ &\quad\leq \lim_{k\rightarrow \infty } \rho \bigl(\varrho _{2n_{k}},\varrho ^{*}\bigr) \varphi \bigl(s^{3}H\bigl(T_{1}\varrho _{2n_{k}},T_{2} \varrho ^{*}\bigr)\bigr) \\ &\quad\leq \lim_{k\rightarrow \infty }\bigl[\vartheta \bigl(\varphi \bigl(M\bigl( \varrho _{2n_{k}}, \varrho ^{*}\bigr)\bigr)\bigr)\varphi \bigl(M \bigl(\varrho _{2n_{k}},\varrho ^{*}\bigr)\bigr)+L\phi \bigl(N \bigl( \varrho _{2n_{k}},\varrho ^{*}\bigr)\bigr)\bigr]. \end{aligned}$$

(10)

We have

$$\begin{aligned} &M\bigl(\varrho _{2n_{k}},\varrho ^{*}\bigr)\\ &\quad=\max \biggl\{ d \bigl(\varrho _{2n_{k}}, \varrho ^{*}\bigr),D(\varrho _{2n_{k}},T_{1}\varrho _{2n_{k}}),D\bigl(\varrho ^{*},T_{2} \varrho ^{*}\bigr), \frac{D(\varrho _{n_{k}},T_{2}\varrho ^{*})+D(\varrho ^{*},T_{1}\varrho _{2n_{k}})}{2s} \biggr\} \\ &\quad\leq \max \biggl\{ d\bigl(\varrho _{2n_{k}},\varrho ^{*} \bigr),d(\varrho _{2n_{k}}, \varrho _{2n_{k+1}}),D\bigl(\varrho ^{*},T_{2}\varrho ^{*}\bigr), \frac{D(\varrho _{2n_{k}},T_{2}\varrho ^{*})+d(\varrho ^{*},\varrho _{2n_{k}+1})}{2s} \biggr\} \end{aligned}$$

and

$$\begin{aligned} N\bigl(\varrho _{2n_{k}},\varrho ^{*}\bigr)&=\min \bigl\{ D \bigl(\varrho ^{*},T_{1} \varrho _{2n_{k}}\bigr),D \bigl(\varrho ^{*},T_{2}\varrho ^{*}\bigr)\bigr\} \\ &\leq \min \bigl\{ d\bigl(\varrho ^{*},\varrho _{2n_{k}+1}\bigr),d \bigl(\varrho ^{*},T_{2} \varrho ^{*}\bigr)\bigr\} . \end{aligned}$$

Recall that

$$ \frac{D(\varrho _{2n_{k}},T_{2}\varrho ^{*})+d(\varrho ^{*},\varrho _{2n_{k}+1})}{2s} \leq \frac{sd(\varrho _{2n_{k}},\varrho ^{*})+sD(\varrho ^{*},T_{2}\varrho ^{*})+d(\varrho ^{*},\varrho _{2n_{k}+1})}{2s}. $$

Then

$$ \limsup_{k\to \infty } \frac{D(\varrho _{2n_{k}},T_{2}\varrho ^{*})+d(\varrho ^{*},\varrho _{2n_{k}+1})}{2s} \leq \frac{D(\varrho ^{*},T_{2}\varrho ^{*})}{2}. $$

When k tends to infinity, we deduce

$$\begin{aligned} \lim_{k\rightarrow \infty }M\bigl(\varrho _{n_{k}},\varrho ^{*}\bigr)=D\bigl( \varrho ^{*},T_{2}\varrho ^{*}\bigr) \end{aligned}$$

and

$$\begin{aligned} \lim_{k\rightarrow \infty }N\bigl(\varrho _{n_{k}},\varrho ^{*}\bigr)=0. \end{aligned}$$

Since $\lim_{k\rightarrow \infty }\varrho (\varphi (M( \varrho _{n_{k}},\varrho ^{*})))\leq \frac{1}{s^{2}}$, by (10)

$$ \varphi \bigl(s^{2}D\bigl(\varrho ^{*},T_{2} \varrho ^{*}\bigr)\bigr)\leq \frac{1}{s^{2}} \varphi \bigl(D\bigl( \varrho ^{*},T_{2}\varrho ^{*}\bigr)\bigr). $$

Since $\varphi \in \chi$, the above holds unless $D(\varrho ^{*},T_{2}\varrho ^{*})=0$, that is, $\varrho ^{*}\in T_{2}\varrho ^{*}$. Similarly, we can prove that $\varrho ^{*}\in T_{1}\varrho ^{*}$, so $\varrho ^{*}$ is a common fixed point of $T_{2}$ and $T_{1}$. □

Corollary 3.3

Let $(X,d)$be an ρ-complete b-metric space, and $\rho:X\times X\rightarrow [0,+\infty )$be a function. Suppose that $T_{1},T_{2}:X\rightarrow X$are mappings such that

$$\begin{aligned} \rho (\varsigma,\eta )\varphi \bigl(s^{3} H(T_{1}\varsigma,T_{2}\eta )\bigr) \leq \vartheta \bigl( \varphi \bigl(d(\varsigma,\eta )\bigr)\bigr)\varphi \bigl(d(\varsigma, \eta ) \bigr), \end{aligned}$$

(11)

for $\varrho \in \varOmega$and $\varphi \in \chi$. Moreover, suppose

• (i) $(T_{1},T_{2})$is triangular $\rho _{*}$-admissible;

• (ii) there exists $\varrho _{0}\in X$with $\rho (\varrho _{0},T_{1}\varrho _{0})\geq 1$;

• (iii) if for every sequence $\{\varrho _{n}\}$in X with $\rho (\varrho _{n},\varrho _{n+1})\geq 1$for all $n\in \mathbb{N}\cup \{0\}$and $\varrho _{n}\rightarrow \varrho \in X$, then there exists a subsequence $\{\varrho _{n(k)}\}$of $\{\varrho _{n}\}$with $\rho (\varrho _{n(k)},\varrho )\geq 1$for all k.

Then $T_{1}$and $T_{2}$have a common fixed point $\varrho \in X$.

Moreover, if the following condition holds:

$H_{1}$: Either $\rho (u,v)\geq 1$or $\rho (v,u)\geq 1$whenever $T_{1}u =T_{2}u=u$and $T_{1}v =T_{2}v=v$, then $T_{1}$and $T_{2}$have a unique common fixed point.

Proof

The proof of the existence of a common fixed point of $T_{1}$ and $T_{2}$ was shown in Theorem 3.2.

We claim that, if $T_{1}u =T_{2}u=u$ and $T_{1}v =T_{2}v=v$, then $u =v$. By hypothesis, if $u\neq v$, then either $\rho (u,v)\geq 1$ or $\rho (v,u)\geq 1$. Suppose that $\rho (u,v)\geq 1$, then

$$\begin{aligned} \psi \bigl(d(u,v)\bigr)&=\psi \bigl(d(Su,Tv)\bigr)\leq \psi \bigl(s^{3}\,d(Su,Tv)\bigr)\leq \rho (u,v) \psi \bigl(s^{3} \,d(Su,Tv)\bigr) \\ &\leq \vartheta \bigl(\psi \bigl(d(u,v)\bigr)\bigr)\psi \bigl(d(u,v)\bigr)< \psi \bigl(d(u,v)\bigr), \end{aligned}$$

which is contradiction. So $u =v$. Similarly, if $\rho (v,u)\geq 1$, we can prove $u =v$. □

Theorem 3.4

Suppose for $\kappa \in I$ and $\eta,z\in C(I)$ there exists $\xi:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $\varphi \in \chi$ such that

• (i)
  $$ \bigl\vert g\bigl(\kappa,\eta (\kappa )\bigr)-f\bigl(\kappa,z(\kappa ) \bigr) \bigr\vert \leq \frac{1}{2\sqrt{2}} \frac{\varGamma (\max \{\rho,\varrho \})}{(1-\varkappa )^{\min \{\rho,\varrho \}-1}\kappa ^{\min \{\rho,\varrho \}-1}} \frac{\varphi ( \vert (\kappa )-z(\kappa ) \vert ^{2})}{\sqrt{4 \Vert (\eta -z)^{2} \Vert _{\infty }+1}}, $$

• (ii) ∃ $\eta _{0}\in C(I)$with $\xi (\eta _{0}(\kappa ),\int _{0}^{1}G(\kappa,\varkappa )f( \varkappa,\eta _{0}(\varkappa ))\,d\varkappa \geq 0$.

  If we set

  $$\begin{aligned} &\eta _{1}=T_{1}\eta _{0}= \int _{0}^{1}G_{1\rho }(\kappa,\varkappa )g \bigl( \varkappa,\eta _{0}(\varkappa )\bigr)\,d\varkappa; \\ &\eta _{2}=T_{2}\eta _{1}= \int _{0}^{1}G_{1\varrho }(\kappa, \varkappa )f \bigl(\varkappa,\eta _{1}(\varkappa )\bigr)\,d\varkappa ; \\ &\eta _{3}=T_{1}\eta _{2}= \int _{0}^{1}G_{1\rho }(\kappa,\varkappa )g \bigl( \varkappa,\eta _{2}(\varkappa )\bigr)\,d\varkappa ; \\ &\vdots \\ &\eta _{2n}=T_{2}\eta _{2n-1}= \int _{0}^{1}G_{1\varrho }(\kappa, \varkappa )f \bigl(\varkappa,\eta _{2n-1}(\varkappa )\bigr)\,d\varkappa ; \\ &\eta _{2n+1}=T_{1}\eta _{2n}= \int _{0}^{1}G_{1\rho }(\kappa, \varkappa )g \bigl(\varkappa,\eta _{2n}(\varkappa )\bigr)\,d\varkappa ; \\ &\vdots \end{aligned}$$

  we may assume the following conditions are met:

• (iii) $\xi (\eta _{2n-1},\eta _{2n})\geq 0$and $\xi (\eta _{2n},\eta _{2n+1})\geq 0$, respectively, imply $\xi (\eta _{2n},\eta _{2n+1})\geq 0$and $\xi (\eta _{2n+1},\eta _{2n+2})\geq 0$, respectively;

• (iv) $\xi (\eta _{2n-1},\eta _{2n})\geq 0$and $\xi (\eta _{2n},\eta _{2n+1})\geq 0$implies $\xi (\eta _{2n-1},\eta _{2n+1})\geq 0$;

• (v) if for every sequence $\{\eta _{n}\}$in X with $\xi (\eta _{n},\eta _{n+1})\geq 0$for all $n\in \mathbb{N}\cup \{0\}$and $\eta _{n}\rightarrow \eta \in X$, then there exists a subsequence $\{\eta _{n(k)}\}$of $\{\eta _{n}\}$with $\xi (\eta _{n(k)},\eta )\geq 1$for all k.

Then the system (1) has a solution in $C(I)$.

Moreover, if the following conditions hold:

$H_{2}$: Either $\xi (\eta ^{*},\zeta ^{*})\geq 0$or $\xi (\zeta ^{*},\eta ^{*})\geq 0$whenever

$$ \int _{0}^{1}G_{1\rho }(\kappa,\varkappa )g \bigl(\varkappa,\eta ^{*}( \varkappa )\bigr)\,d\varkappa = \int _{0}^{1}G_{1\varrho }(\kappa,\varkappa )f \bigl( \varkappa,\eta ^{*}(\varkappa )\bigr)\,d\varkappa =\eta ^{*} $$

and

$$ \int _{0}^{1}G_{1\rho }(\kappa,\varkappa )g \bigl(\varkappa,\zeta ^{*}( \varkappa )\bigr)\,d\varkappa = \int _{0}^{1}G_{1\varrho }(\kappa,\varkappa )f \bigl( \varkappa,\zeta ^{*}(\varkappa )\bigr)\,d\varkappa \zeta ^{*}=\zeta ^{*}, $$

then the system (1) has a unique solution in $C(I)$.

Proof

By Lemma 2.6$\eta \in C(I)$ is a solution of (1) if and only if it is a solution of

$$ \textstyle\begin{cases} z=\int _{0}^{1}G_{1\varrho }(\kappa,\varkappa )f(\varkappa,\eta ( \varkappa ))\,d\varkappa, \\ \eta =\int _{0}^{1}G_{1\rho }(\kappa,\varkappa )g(\varkappa,z( \varkappa ))\,d\varkappa. \end{cases} $$

We define $T_{1},T_{2}: C(I)\rightarrow C(I)$ by

$$\begin{aligned} &T_{1}\eta _{2n}= \int _{0}^{1}G_{1\rho }(\kappa,\varkappa )g \bigl( \varkappa,\eta _{2n}(\varkappa )\bigr)\,d\varkappa,\\ &T_{2}\eta _{2n-1}= \int _{0}^{1}G_{1\varrho }(\kappa,\varkappa )f \bigl(\varkappa,\eta _{2n-1}( \varkappa )\bigr)\,d\varkappa, \end{aligned}$$

for all $\kappa \in I$. For this purpose, we find a common fixed point of $T_{1}$ and $T_{2}$. Let $\eta,z\in C(I)$ with $\xi (\eta (\kappa ),z(\kappa ))\geq 0$ for all $\kappa \in I$. By using (i), we get

$$\begin{aligned} \bigl\vert T_{1}\eta (\kappa )-T_{2}z(\kappa ) \bigr\vert ^{2}&= \biggl\vert \int _{0}^{1} G_{1 \rho }(\kappa,\varkappa ) \biggl(g\bigl(\varkappa,\eta (\varkappa )\bigr)- \int _{0}^{1} G_{1\varrho }(\kappa,\varkappa )f \bigl(\varkappa,z(\varkappa )\bigr)\biggr)\,d \varkappa \biggr\vert ^{2} \\ &\leq \biggl[ \int _{0}^{1} \frac{(1-\varkappa )^{\min \{\rho,\varrho \}-1}\kappa ^{\min \{\rho,\varrho \}-1}}{\varGamma (\max \{\rho,\varrho \})} \bigl\vert h \bigl(\varkappa,\eta (\varkappa )\bigr)-h\bigl(\varkappa,z(\varkappa )\bigr) \bigr\vert \,d \varkappa \biggr]^{2} \\ & \leq \biggl[ \int _{0}^{1} \frac{1}{2\sqrt{2}} \frac{\varphi ( \vert \eta (\varkappa )-z(\varkappa ) \vert ^{2})}{\sqrt{4 \Vert (\eta -z)^{2} \Vert _{\infty }+1}} \,d \varkappa \biggr]^{2} \\ &\leq \frac{1}{8} \frac{(\varphi ( \Vert (\eta -z)^{2} \Vert _{\infty }))^{2}}{4 \Vert (\eta -z)^{2} \Vert _{\infty }+1}. \end{aligned}$$

Hence, for $\eta,z\in C(I)$, $\kappa \in I$ with $\xi (\eta (\kappa ),z(\kappa ))\geq 0$, we have

$$\begin{aligned} \bigl\Vert (T_{1}\eta -T_{2}z)^{2} \bigr\Vert _{\infty }\leq \frac{1}{8} \frac{(\varphi ( \Vert (\eta -z)^{2} \Vert _{\infty }))^{2}}{4 \Vert (\eta -z)^{2} \Vert _{\infty }+1}. \end{aligned}$$

Put $\rho: C(I)\times C(I)\rightarrow [0,+\infty )$ by

$$\begin{aligned} \rho (\eta,z)=\textstyle\begin{cases} 1&\xi (\eta (\kappa ),z(\kappa ))\geq 0, \text{ for all }\kappa \in I, \\ 0&\text{else}. \end{cases}\displaystyle \end{aligned}$$

Setting $\vartheta: [0,+\infty )\rightarrow [0,\frac{1}{4})$ with $\vartheta (q)=\frac{q}{4q+1}$ and $s=2$ we can obtain

$$\begin{aligned} \rho (\eta,z)\varphi \bigl(8\,d(T_{1}\eta,T_{2}z)\bigr)& \leq 8\rho (\eta,z) \varphi \bigl(d(T_{1}\eta,T_{2}z)\bigr) \\ &\leq \frac{(\varphi (d(\eta,z)))^{2}}{4d(\eta,z)+1} \\ &\leq \frac{(\varphi (d(\eta,z)))^{2}}{4\varphi (d(\eta,z))+1} \\ &= \frac{1}{\vartheta (\varphi (d(\eta,z)))}\vartheta \bigl(\varphi \bigl(d( \eta,z)\bigr)\bigr) \frac{(\varphi (d(\eta,z)))^{2}}{4\varphi (d(\eta,z))+1} \\ & \leq \vartheta \bigl(\varphi \bigl(d(\eta,z)\bigr)\bigr)\varphi \bigl(d( \eta,z)\bigr), \quad\vartheta \in \varOmega. \end{aligned}$$

So $T_{1}$ and $T_{2}$ obey all the conditions of Theorem 3.2. We find $\eta ^{*}\in C(I)$ with $\eta ^{*}=T_{1}\eta ^{*}=T_{2}\eta ^{*}$.

Now we claim that the solution of coupled system (1) is unique. By the condition $H_{2}$ we have the following.

Either $\rho (\eta ^{*},\zeta ^{*})\geq 0$ or $\rho (\zeta ^{*},\eta ^{*})\geq 0$ whenever

$$ T_{1} \eta ^{*} =T_{2} \eta ^{*}=\eta ^{*} $$

and

$$ T_{1} \zeta ^{*} =T_{2} \zeta ^{*}= \zeta ^{*}. $$

Now, using the condition of $H_{1}$ in Corollary 3.3, we obtain $\eta ^{*}=\zeta ^{*}$ □

4 Conclusion

Fixed point theory is one of the main tools of pure mathematics that are used to serve in the development in the qualitative theory of differential and integral equations. For the last few years, researchers have not only used the traditional fixed point theorems in proving the existence and uniqueness of solutions to various types of fractional differential equation, but have developed new fixed point theorems and applied them as well. In this article, we have developed a new fixed point theorem and utilized it to prove the local existence of positive solution to a coupled system of differential equations where the Riemann–Liouville fractional derivative is used. We believe that the new fixed point theorem considered here can be used to handle fractional differential equations in the setting of other derivatives, not only containing singular kernels, but nonsingular kernels as well.