Darbo-Type $\mathcal{Z}_{\rm{m}}$ and $\mathcal{L}_{\rm{m}}$ contractions and its applications to Caputo fractional integro-differential equations

1.   Introduction

Let $\mathbb{R}$ and $\mathbb{R}_{+}$ denote the sets of all real numbers and nonnegative real numbers respectively, $\mathbb{N}$ the set of all positive integers and $\overline{\mathbb{A}}$ the closure and $\overline{co}\mathbb{A}$ the convex hull closure of $\mathbb{A}$. Additionally, $\Xi$ denotes a Banach space, $\Omega = \left\{\Lambda : \Lambda\neq\emptyset, \ \text{bounded, closed and convex subset of}\ \Xi\right\}$, $\mathfrak{B}(\Xi) = \{\Lambda\neq\emptyset: \Lambda\ \text{is bounded subset of}\ \Xi\}$, $\ker {\rm{m}} = \{\Lambda\in \mathfrak{B}(\Xi) : {\rm{m}}(\Lambda) = 0\}$ be the kernel of function ${\rm{m}} : \mathfrak{B}(\Xi)\rightarrow \mathbb{R}_{+}.$

Fixed point theory has been developed in two directions. One deals with contraction mappings on metric spaces, Banach contraction principle being the first important result in this direction. In the second direction, continuous operators are dealt with convex and compact subsets of a Banach space. Brouwer's fixed point theorem and its infinite dimensional form, Schuader's fixed point theorems are the two important theorems in this second direction. In this paper, $Fix(\Upsilon)$ denotes a set of fixed points of a mapping $\Upsilon$ in $\Lambda$.

Theorem 1.1 (Brouwer's Fixed Point Theorem).^[2] Every continuous mapping from the unit ball of $\mathbb{R}^{n}$ into itself has a fixed point.

Theorem 1.2 (Schauder's Fixed Point Theorem).^[15] Let $\Upsilon:\Lambda\rightarrow \Lambda$ a compact continuous operator, where $\Lambda\in \Omega$. Then $Fix(\Upsilon)\neq\emptyset$.

In Brouwer's and Schuader's fixed point theorems, compactness of the space under consideration is required as a whole or as a part. However, later the requirement of the compactness was relaxed by making use of the notion of a measure of noncompactness (in short MNC). Using the notion of MNC, the following theorem was proved by Darbo ^[5].

Theorem 1.3. ^[5] Let $\Lambda\in \Omega$ and $\Upsilon : \Lambda\rightarrow \Lambda$ be a continuous function. If there exists $k \in [0, 1)$ such that

where $\Lambda_{0}\subset\Lambda$ and ${\rm{m}}$ is MNC defined on $\Xi$. Then $Fix(\Upsilon)\neq\emptyset$.

It generalizes the renowned Schuader fixed point result and includes the existence portion of Banach contraction principle. In the sequel many extensions and generalizations of Darbo's theorem came into existence.

The Banach principle has been improved and extended by several researchers (see ^[7,13,14,16]). Jleli and Samet ^[7] introduced the notion of $\theta$-contractions and gave a generalization of the Banach contraction principle in generalized metric spaces, where $\theta:(0, \infty)\rightarrow (1, \infty)$ is such that:

$(\theta_{1})$ $\theta$ is non-decreasing;

$(\theta_{2})$ for every sequence $\{\kappa_{j}\}\subset(0, \infty)$, we have

$(\theta_{3})$ there exists $L\in(0, \infty)$ and $\ell\in(0, 1)$ such that

Khojasteh et al. ^[9] introduced the concept of $\mathcal{Z}$-contraction using simulation functions and established fixed point results for such contractions. Isik et al. ^[6] defined almost $\mathcal{Z}$-contractions and presented fixed point theorems for such contractions. Cho ^[3] introduced the notion of $\mathcal{L}$-contractions, and proved fixed point results under such contraction in generalized metric spaces. Using specific form of $\mathscr{Z}$ and $\mathscr{L}$, we can deduce other known existing contractions. For some results concerning $\mathcal{Z}$-contractions and its generalizations we refer the reader to ^[3] and the references cited therein. In particular, Chen and Tang ^[4] generalized $\mathcal{Z}$-contraction with $\mathcal{Z}_{{\rm{m}}}$-contraction and established Darbo type fixed point results.

The aim of the present work is two fold. First we prove fixed point theorems under generalized $\mathcal{Z}_{\rm{m}}$-contraction and then we prove fixed point results under Darbo type $\mathcal{L}_{\rm{m}}$-contraction in Banach spaces. It is interesting to see that several existing results in fixed point theory can be concluded from our main results. Furthermore, as an application of our results, we have proved the existence of solution to the Caputo fractional Volterra–Fredholm integro-differential equation

under boundary conditions:

where $\wp\in(0, 1],$ $^{c}D$ is the Caputo fractional derivative, $\lambda_{1}, \lambda_{2}$ are parameters, and $a, b > 0$ are real constants, $\mu, \mathfrak{g}: \left[0, \mathfrak{T}\right]\rightarrow \mathbb{R}$, $\mathfrak{J}_{1}, \mathfrak{J}_{2}, \mathfrak{J}_{3}: \left[0, \mathfrak{T}\right]\times \left[0, \mathfrak{T}\right]\rightarrow \mathbb{R}$ and $\xi_{1}, \xi_{2}: \left[0, \mathfrak{T}\right]\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous functions. For the validity of existence result we construct an example.

2.   Preliminaries

In this section, we recall some definitions and results which are further considered in the next sections, allowing us to present the results. The concept of MNC was introduced in ^[1] as follows:

Definition 2.1. ^[1] A map ${\rm{m}} : \mathfrak{B}(\Xi)\rightarrow \mathbb{R}_{+}$ is MNC in $\Xi$ if for all $\Lambda_{1}, \Lambda_{2}\in \mathfrak{B}(\Xi)$ it satisfies the following conditions:

$\rm(i)$ $\ker{\rm{m}}\neq\emptyset$ and relatively compact in $\Xi$;

$\rm(ii)$ $\Lambda_{1}\subset \Lambda_{2}\ \Rightarrow \ {\rm{m}}(\Lambda_{1})\leq{\rm{m}}(\Lambda_{2});$

$\rm(iii)$ ${\rm{m}}(\overline{\Lambda_{1}}) = {\rm{m}}(\Lambda_{1});$

$\rm(iv)$ ${\rm{m}}(\overline{co}\Lambda_{1}) = {\rm{m}}(\Lambda_{1});$

$\rm(v)$ ${\rm{m}}(\eta \Lambda_{1} + (1-\eta)\Lambda_{2})\leq \eta{\rm{m}}(\Lambda_{1}) + (1-\eta){\rm{m}}(\Lambda_{2})\ \forall\eta\in [0, 1];$

$\rm(vi)$ if $\{{\Lambda}_{n}\}$ is a sequence of closed sets in $\mathfrak{B}(\Xi)$ with ${\Lambda}_{n+1}\subset {\Lambda_{n}}, \ \forall n\in \mathbb{N}$ and $\underset{n\rightarrow +\infty}{\lim }{\rm{m}}({\Lambda }_{n}) = 0$, then ${\Lambda}_{\infty} = \bigcap_{n = 1}^{+\infty}{\Lambda}_{n}\neq\emptyset$.

The Kuratowski MNC ^[11] is the function ${\rm{m}} : \mathfrak{B}(\Xi)\rightarrow \mathbb{R}_{+}$ defined by

where $\text{diam}(S)$ is the diameter of $S$.

Khojasteh et al. ^[9] introduced the concept of $\mathcal{Z}$-contraction using simulation functions as follows:

Definition 2.2. A function $\mathscr{Z} : \mathbb{R}_{+}\times \mathbb{R}_{+}\rightarrow \mathbb{R}$ is simulation if:

$(\mathscr{Z}_{1})$ $\mathscr{Z}(0, 0) = 0$;

$(\mathscr{Z}_{2})$ $\mathscr{Z}(\kappa_{1}, \kappa_{2}) < \kappa_{2}-\kappa_{1}$, for all $\kappa_{1}, \kappa_{2} > 0$;

$(\mathscr{Z}_{3})$ if $\{\kappa^{*}_{n}\}$ and $\{\kappa_{n}\}$ are two sequences in $(0, \infty)$ such that $\underset{n\rightarrow \infty}{\lim \kappa^{*}_{n}} = \underset{n\rightarrow \infty}{\lim \kappa_{n}} > 0$, then

Roldán-López-de-Hierro et al. ^[12] slightly modified the Definition 2.2 of ^[9] as follows.

Definition 2.3. A mapping $\mathscr{Z} : \mathbb{R}_{+}\times \mathbb{R}_{+}\rightarrow \mathbb{R}$ is simulation if:

$(\mathscr{Z}_{1})$ $\mathscr{Z}(0, 0) = 0$;

$(\mathscr{Z}_{2})$ $\mathscr{Z}(\kappa_{1}, \kappa_{2}) < \kappa_{2}-\kappa_{1}$, for all $\kappa_{1}, \kappa_{2} > 0$;

$(\mathscr{Z}_{3})$ if $\{\kappa^{*}_{n}\}$, $\{\kappa_{n}\}$ are sequences in $(0, \infty)$ such that $\underset{n\rightarrow \infty}{\lim \kappa^{*}_{n}} = \underset{n\rightarrow \infty}{\lim \kappa_{n}} > 0$ and $\kappa_{n} < \kappa^{*}_{n}$, then

Every simulation function in the original Definition 2.2 is also a simulation function in the sense of Definition 2.3, but the converse is not true, see for instance ^[12]. Note that $\mathcal{Z} = \{\mathscr{Z} :\mathscr{Z}\ \text{is a simulation function in the sense of Definition 2.3} \}$. The following are some examples of simulation functions.

Example 2.4. The mapping $\mathscr{Z}: \mathbb{R}_{+}\times\mathbb{R}_{+}\rightarrow \mathbb{R}$ defined by:

1. $\mathscr{Z}(\kappa_{1}, \kappa_{2}) = \kappa_{2}-f(\kappa_{1})-\kappa_{1}$, for all $\kappa_{1}, \kappa_{2}\in \mathbb{R}_{+}$, where $f : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is a lower semi-continuous function such that $f^{-1}(0) = \{0\},$

2. $\mathscr{Z}(\kappa_{1}, \kappa_{2}) = \kappa_{2}-\varphi(\kappa_{1})-\kappa_{1}$, for all $\kappa_{1}, \kappa_{2}\in \mathbb{R}_{+}$, where $\varphi : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is a continuous function such that $\varphi(\kappa_{1}) = \{0\}\Leftrightarrow \kappa_{1} = 0$,

3. $\mathscr{Z}(\kappa_{1}, \kappa_{2}) = \kappa_{2}\curlyvee(\kappa_{2})-\kappa_{1}$, for all $\kappa_{1}, \kappa_{2}\in \mathbb{R}_{+}$, where $\curlyvee : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is a function such that $\underset{n\rightarrow r^{+}}{\limsup}\ \curlyvee(\kappa_{1}) < 1$,

4. $\mathscr{Z}(\kappa_{1}, \kappa_{2}) = \phi(\kappa_{2})-\kappa_{1}$, for all $\kappa_{1}, \kappa_{2}\in \mathbb{R}_{+}$, where $\phi : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is an upper semi-continuous function such that $\phi(\kappa_{1}) < \kappa_{1}$, for all $\kappa_{1} > 0$ and $\phi(0) = 0$,

5. $\mathscr{Z}(\kappa_{1}, \kappa_{2}) = \kappa_{2}-\int\limits_{0}^{\kappa_{1}}\lambda(x) dx,$ for all $\kappa_{1}, \kappa_{2}\in \mathbb{R}_{+}$, where $\lambda : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is a function such that $\int\limits_{0}^{\epsilon}\lambda(x) dx$ exists and $\int\limits_{0}^{\epsilon}\lambda(x) dx > \epsilon$, for every $\epsilon > 0$,

6. $\mathscr{Z}(\kappa_{1}, \kappa_{2}) = \frac{\kappa_{2}}{\kappa_{2}+1}-\kappa_{1}$, for all $\kappa_{1}, \kappa_{2}\in \mathbb{R}_{+}$,

are simulation functions.

Cho ^[3] introduced the notion of $\mathcal{L}$-simulation function as follows:

Definition 2.5. An $\mathcal{L}$-simulation function is a function $\mathscr{L} : [1, \infty) \times [1, \infty)\rightarrow \mathbb{R}$ satisfying the following conditions:

$(\mathscr{L}_{1})$ $\mathscr{L}(1, 1) = 1$;

$(\mathscr{L}_{2})$ $\mathscr{L}(\kappa_{1}, \kappa_{2}) < \frac{\kappa_{2}}{\kappa_{1}}$, for all $\kappa_{1}, \kappa_{2} > 1$;

$(\mathscr{L}_{3})$ if $\{\kappa_{n}\}$ and $\{\kappa^{*}_{n}\}$ are two sequences in $(1, \infty)$ such that $\underset{n\rightarrow \infty}{\lim \kappa_{n}} = \underset{n\rightarrow \infty}{\lim \kappa^{*}_{n}} > 1$ and $\kappa_{n} < \kappa^{*}_{n}$, then

Note that $\mathscr{L}(1, 1) < 1$, for all $t > 1.$

Example 2.6. The functions $\mathscr{L}_{b}, \mathscr{L}_{w}:[1, \infty)\times[1, \infty)\rightarrow \mathbb{R}$ defined by

1. $\mathscr{L}_{b}(\kappa_{1}, \kappa_{2}) = \frac{\kappa_{2}^{\mathfrak{n}}}{\kappa_{1}}$, for all $\kappa_{1}, \kappa_{2}\geq1$, where $\mathfrak{n}\in(0, 1)$;

2. $\mathscr{L}_{w}(\kappa_{1}, \kappa_{2}) = \frac{\kappa_{2}}{\kappa_{1}\phi(\kappa_{2})}$ $\forall \kappa_{1}, \kappa_{2}\geq1$, where $\phi:[1, \infty)\rightarrow [1, \infty)$ is lower semicontinuous and nondecreasing with $\phi^{-1}(\{1\}) = 1$,

are $\mathcal{L}$-simulation functions.

Definition 2.7. ^[8] A continuous non-decreasing function $\phi : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ such that $\varphi(t) = 0$ if and only if $t = 0$ is called an altering distance function.

3.   Some results on $\mathcal{Z}_{\rm{m}}$-contractions

In this section, we obtain the results on generalized $\mathcal{Z}_{\rm{m}}$-contraction. First we give the following definition.

Definition 3.1. Let $\Lambda\in \Omega$. A self-mapping $\Upsilon$ on $\Lambda$ is called generalized $\mathcal{Z}_{\rm{m}}$-contraction if there exists $\mathscr{Z}\in \mathcal{Z}$ such that

where $\Lambda_{1}$ and $\Lambda_{2}$ are subsets of $\Lambda$, ${\rm{m}} (\Lambda_{1}), {\rm{m}} (\Upsilon(\Lambda_{1})), {\rm{m}} (\Upsilon(\Lambda_{2})) > 0,$ ${\rm{m}}$ is MNC defined in $\Xi$ and

Using the notion of $\mathcal{Z}_{\rm{m}}$-contraction, we establish the main result of this section.

Theorem 3.2. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function, where $\Lambda\in \Omega$. Assume that there exists $\mathscr{Z}\in \mathcal{Z}$ such that $\mathscr{Z}$ non-decreasing function and $\Upsilon$ is a generalized $\mathcal{Z}_{\rm{m}}$-contraction. Then $Fix(\Upsilon)\neq\emptyset$.

Proof. Define a sequence $\{\Lambda_{n}\}_{n = 0}^{\infty}$ such that

We need to prove that $\Lambda_{n+1}\subset \Lambda_{n}$ and $\Upsilon\Lambda_{n}\subset \Lambda_{n}$, for all $n\in \mathbb{N}$. For proof of the first inclusion, we use induction. If $n = 1$, then by $(3.2)$, we have $\Lambda_{0} = \Lambda$ and $\Lambda_{1} = \overline{co}(\Upsilon\Lambda_{0})\subset \Lambda_{0}$. Next, assume that $\Lambda_{n}\subset \Lambda_{n-1},$ then $\overline{co}(\Upsilon(\Lambda_{n}))\subset \overline{co}(\Upsilon(\Lambda_{n-1})),$ using $(3.2)$, we get the first inclusion

To obtained the second inclusion, using the inclusion $(3.3)$ we have

Thus $\Lambda_{n+1}\subset \Lambda_{n}$ and $\Upsilon\Lambda_{n}\subset \Lambda_{n}, \forall n\in \mathbb{N}$.

Now, we discuss two cases, depending on the values of ${\rm{m}}$. If we consider $m$ as a non-negative integer with ${\rm{m}} (\Lambda_{m}) = 0$, then $\Lambda_{m}$ is a compact set and hence by Theorem 1.2, $\Upsilon$ has a fixed point in $\Lambda_{m}\subset \Lambda$. Instead, assume that ${\rm{m}} (\Lambda_{n}) > 0, \ \forall n\in \mathbb{N}$. Then on setting $\Lambda_{1} = \Lambda_{n+1}$ and $\Lambda_{2} = \Lambda_{n}$ in contraction (3.1), we have

where

that is,

Using inequality (3.6) and the axiom $(\mathscr{Z}_{2})$ of $\mathscr{Z}$-simulation function, inequality $(3.5)$ becomes,

that is, ${\rm{m}} (\Lambda_{n})\geq{\rm{m}} (\Lambda_{n+1})$ and hence, $\{{\rm{m}} (\Lambda_{n})\}$ is a decreasing sequence of positive real numbers. Thus, we can find $r\geq 0$ such that $\underset{n\rightarrow \infty}{\lim}\ {\rm{m}} (\Lambda_{n}) = r$. Next, we claim that $r = 0$. To support our claim, suppose that $r\neq0$, that is, $r > 0$. Let $u_{n} = {\rm{m}} (\Lambda_{n+1})$ and $v_{n} = {\rm{m}} (\Lambda_{n})$, then since $u_{n} < v_{n}$, so by the axiom $(\mathscr{Z}_{3})$ of $\mathscr{Z}$-simulation function, we have

which is contradiction to $(3.7)$. Thus $r = 0$ and hence $\{\Lambda_{n}\}$ is a sequence of closed sets in $\mathfrak{B}(\Xi)$ with $\Lambda_{n+1}\subset \Lambda_{n}$, for all $n\in \mathbb{N}$ and $\underset{n\rightarrow \infty}{\lim}\ {\rm{m}} (\Lambda_{n}) = 0$, so the intersection set $\Lambda_{\infty} = \bigcap_{n = 1}^{+\infty}\Lambda_{n}$ is non-empty, closed and convex subset of $\Lambda$. Furthermore, since $\Lambda_{\infty}\subset \Lambda_{n}$, for all $n\in \mathbb{N}$, so by Definition 2.1(ii), ${\rm{m}} (\Lambda_{\infty})\leq{\rm{m}} (\Lambda_{n})$, for all $n\in \mathbb{N}$. Thus ${\rm{m}} (\Lambda_{\infty}) = 0$ and hence $\Lambda_{\infty}\in \ker{\rm{m}}$, that is, $\Lambda_{\infty}$ is bounded. But $\Lambda_{\infty}$ is closed so that $\Lambda_{\infty}$ is compact. Therefore by Theorem 1.2, $Fix(\Upsilon)\neq\emptyset$. From Theorem 3.2 we obtain the following corollaries. We assume that $\Lambda\in \Omega$.

Corollary 3.3. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function such that

for any non-empty subsets $\Lambda_{1}$ and $\Lambda_{2}$ of $\Lambda$, where ${\rm{m}}$ is MNC. Then $Fix(\Upsilon)\neq\emptyset$.

Corollary 3.4. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function such that

for any non-empty subsets $\Lambda_{1}$ and $\Lambda_{2}$ of $\Lambda$, where ${\rm{m}}$ is MNC and $f :\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is a lower semi-continuous function such that $f^{-1}(0) = \{0\}$. Then $Fix(\Upsilon)\neq\emptyset.$

The conclusion of Corollary 3.4 is true if $f : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is continuous function such that $f(t) = 0\ \Leftrightarrow\ \ t = 0$.

Corollary 3.5. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function such that

for any non-empty subsets $\Lambda_{1}$ and $\Lambda_{2}$ of $\Lambda$, where ${\rm{m}}$ is MNC and $\curlyvee : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is a function with $\underset{n\rightarrow r^{+}}{\limsup}\ \curlyvee(t) < 1$. Then $Fix(\Upsilon)\neq\emptyset$.

Corollary 3.6. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function such that

for any non-empty subsets $\Lambda_{1}$ and $\Lambda_{2}$ of $\Lambda$, where ${\rm{m}}$ is MNC and $\phi : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is an upper semi-continuous mapping with $\phi(t) < t, \ \forall t > 0$ and $\phi(0) = 0$. Then $Fix(\Upsilon)\neq\emptyset$.

Corollary 3.7. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function such that

for any non-empty subsets $\Lambda_{1}$ and $\Lambda_{2}$ of $\Lambda$, where ${\rm{m}}$ is MNC and $\lambda : \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is a mapping such that $\int\limits_{0}^{\epsilon}\lambda(x) dx$ exists and $\int\limits_{0}^{\epsilon}\lambda(x) dx > \epsilon$, for every $\epsilon > 0$. Then $Fix(\Upsilon)\neq\emptyset$.

Corollary 3.8. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function such that

for any non-empty subsets $\Lambda_{1}$ and $\Lambda_{2}$ of $\Lambda$, where ${\rm{m}}$ is MNC. Then $Fix(\Upsilon)\neq\emptyset$.

4.   Some results on $\mathcal{L}_{\rm{m}}$-contraction

In this section, we obtain some results on $\mathcal{L}_{\rm{m}}$-contraction. Let us denote by $\Theta$ the class of all functions $\theta:(0, \infty)\rightarrow (1, \infty)$ that satisfy conditions $(\theta_{1})$ and $(\theta_{2})$. First we introduce the notion of $\mathcal{L}_{\rm{m}}$-contraction as:

Definition 4.1. Let $\Lambda^{*}\in \Omega$. A self-mapping $\Upsilon$ on $\Lambda^{*}$ is called $\mathcal{L}_{\rm{m}}$-contraction with respect to $\mathscr{L}$ if there exist $\theta\in \Theta$ such that, for all $\Lambda\subset \Lambda^{*}$ with ${\rm{m}} (\Lambda) > 0,$

where ${\rm{m}}$ is MNC defined in $\Xi$.

Theorem 4.2. Let $\Upsilon: \Lambda\rightarrow \Lambda$ be a continuous function. Assume that there exist $\theta\in \Theta$ and $\Upsilon$ is $\mathcal{L}_{\rm{m}}$-contraction with respect to $\mathscr{L}$. Then $Fix(\Upsilon)\neq\emptyset$.

Proof. Define a sequence $\{\Lambda_{n}\}_{n = 0}^{\infty}$ such that

Then $\Lambda_{n+1}\subset \Lambda_{n}$ and $\Upsilon\Lambda_{n}\subset \Lambda_{n}, \ \forall n\in \mathbb{N}$.

Now, we discuss two cases, depending on the values of ${\rm{m}}$. If we consider $m$ as a non-negative integer with ${\rm{m}} (\Lambda_{m}) = 0$, then $\Lambda_{m}$ is a compact set and hence by Theorem 1.2, $\Upsilon$ has a fixed point in $\Lambda_{m}\subset \Lambda$. Instead, assume that ${\rm{m}} (\Lambda_{n}) > 0$, for all $n\in \mathbb{N}$. Then on setting $\Lambda = \Lambda_{n}$ in contraction (4.1), we have

that is,

Since $\theta$ is nondecreasing, so that

Now, using inequality (4.4), we have

that is, ${\rm{m}} (\Lambda_{n+1})\leq{\rm{m}} (\Lambda_{n})$ and hence, $\{{\rm{m}} (\Lambda_{n})\}$ is a decreasing sequence of positive real numbers. Thus, we can find $r\geq 0$ with $\underset{n\rightarrow \infty}{\lim}\ {\rm{m}} (\Lambda_{n}) = r$. Next, we claim that $r = 0$. To support our claim, suppose that $r\neq0$. Then in view of $(\theta_{2})$, we get

which implies that

Let $u_{n} = \theta({\rm{m}} (\Lambda_{n+1}))$ and $v_{n} = \theta({\rm{m}} (\Lambda_{n}))$, then since $u_{n}\leq v_{n}$, so by the axiom $(\mathscr{L}_{3})$ of $\mathscr{L}$-simulation function, we have

which is contradiction. Thus $r = 0$ and hence $\{\Lambda_{n}\}$ is a sequence of closed sets from $\mathfrak{B}(\Xi)$ such that $\Lambda_{n+1}\subset \Lambda_{n}$, for all $n\in \mathbb{N}$ and $\underset{n\rightarrow \infty}{\lim}\ {\rm{m}} (\Lambda_{n}) = 0$, so the intersection set $\Lambda_{\infty} = \bigcap_{n = 1}^{+\infty}\Lambda_{n}$ is non-empty, closed and convex subset of $\Lambda$. Furthermore, since $\Lambda_{\infty}\subset \Lambda_{n}$, for all $n\in \mathbb{N}$, so by Definition 2.1(ii), ${\rm{m}} (\Lambda_{\infty})\leq{\rm{m}} (\Lambda_{n})$, for all $n\in \mathbb{N}$. Thus ${\rm{m}} (\Lambda_{\infty}) = 0$ and hence $\Lambda_{\infty}\in \ker{\rm{m}}$, that is, $\Lambda_{\infty}$ is bounded. But $\Lambda_{\infty}$ is closed so that $\Lambda_{\infty}$ is compact. Therefore by Theorem 1.2, $Fix(\Upsilon)\neq\emptyset$.

By taking $\mathscr{L} = \mathscr{L}_{b}$ in Theorem 4.2, we obtain the following result.

Corollary 4.3. Let $\Upsilon: \Lambda^{*}\rightarrow \Lambda^{*}$ be a continuous functions such that, for all $\Lambda\subset \Lambda^{*}$ with ${\rm{m}} (\Lambda) > 0,$

where $\theta\in \Theta$ and $k\in (0, 1)$. Then $Fix(\Upsilon)\neq\emptyset$.

Remark 4.4. Corollary 4.3 is the Darbo type version of Theorem 2.1 in ^[7].

By taking $\mathscr{L} = \mathscr{L}_{w}$ in Theorem 4.2, we obtain the next result.

Corollary 4.5. Let $\Upsilon: \Lambda^{*}\rightarrow \Lambda^{*}$ be a continuous functions such that, for all $\Lambda\subset \Lambda^{*}$ with ${\rm{m}} (\Lambda) > 0,$

where $\theta\in \Theta$ and $\phi:[1, \infty)\rightarrow [1, \infty)$ is lower semi-continuous and nondecreasing with $\phi^{-1}(\{1\}) = 1$. Then $Fix(\Upsilon)\neq\emptyset$.

By taking $\theta(t) = e^{t}$, for all $t > 0$ in Corollary 4.5, we obtain next result.

Corollary 4.6. Let $\Upsilon: \Lambda^{*}\rightarrow \Lambda^{*}$ be a continuous functions such that, for all $\Lambda\subset \Lambda^{*}$ with ${\rm{m}} (\Lambda) > 0,$

where $\varphi:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ is lower semi-continuous and nondecreasing with $\varphi^{-1}(\{0\}) = 0$. Then $Fix(\Upsilon)\neq\emptyset$.

Remark 4.7. Corollary 4.6 is the Rhoades's Theorem of Darbo type ^[13].

5.   Some applications

Let $B(a, r)$ be the closed ball with center at $a$ and radius $r$ and $B_{r}$ be the ball $B(0, r)$. We check the existence of solution to Caputo fractional Volterra–Fredholm integro differential equation

under boundary conditions:

where $\wp\in(0, 1],$ $^{c}D$ is the Caputo fractional derivative, $\lambda_{1}, \lambda_{2}$ are parameters, and $a, b > 0$ are real constants, $\mu, \mathfrak{g}: \left[0, \mathfrak{T}\right]\rightarrow \mathbb{R}$, $\mathfrak{J}_{1}, \mathfrak{J}_{2}, \mathfrak{J}_{3}: \left[0, \mathfrak{T}\right]\times \left[0, \mathfrak{T}\right]\rightarrow \mathbb{R}$ and $\xi_{1}, \xi_{2}: \left[0, \mathfrak{T}\right]\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous functions.

Lemma 5.1. ^[10] For $p_{l}\in \mathbb{R}$, $l = 0, 1, \ldots, r-1$, we have

Using Lemma 5.1, we can easily establish the following result.

Lemma 5.2. Problem $(5.1)$ is equivalent to the integral equation

Proof. Using Lemma 5.1, we obtain

Apply boundary conditions, we deduce that

Thus by substituting the values of $c_{0}$ in $(5.4)$, we get integral equation $(5.3)$.

Notice that the solution of Eq (5.1) is equivalent to Eq (5.3). Now, we are in a position to present the existence result.

Theorem 5.3. Let $\mu, \nu\in B_{r}, \vartheta, \tau\in\left[0, \mathfrak{T}\right],$ and $a, b > 0$ be real constants. If $\mu, \mathfrak{g}: \left[0, \mathfrak{T}\right]\rightarrow \mathbb{R}$, $\mathfrak{J}_{1}, \mathfrak{J}_{2}, \mathfrak{J}_{3}: \left[0, \mathfrak{T}\right]\times \left[0, \mathfrak{T}\right]\rightarrow \mathbb{R}$ and $\xi_{1}, \xi_{2}: \left[0, \mathfrak{T}\right]\times \mathbb{R}\rightarrow \mathbb{R}$ are continuous functions satisfying the following axioms:

1. there exist $C_{i} > 0, i = 1, 2$ such that

2. there exist real numbers $\lambda_{1}$ and $\lambda_{2}$ with $\left|\lambda_{1}\right|C_{1}\mathfrak{K}_{1} +\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2} < \frac{\Gamma(\wp+1)}{2\mathfrak{T}^{p}}$ such that

where $\mathfrak{F}_{1} = {\sup} |\xi_{1}\left(t, 0\right)|, \mathfrak{F}_{2} = {\sup} |\xi_{2}\left(t, 0\right)|$ and

and

Then problem $(5.3)$ has a solution in $B_{r}$, equivalently problem $(5.1)$ has a solution in $B_{r}$.

Proof. Let $B_{r} = \left\{\mu\in \mathcal{C}\left(\left[0, \mathfrak{T}\right], \mathbb{R}\right):\left\|\mu\right\|\leq r\right\}$. Then, $B_{r}$ is a non-empty, closed, bounded, and convex subset of $\mathcal{C}\left(\left[0, \mathfrak{T}\right], \mathbb{R}\right)$. Define the operator $\Upsilon : B_{r}\rightarrow B_{r}$ by

Our first claim is $\Upsilon : B_{r}\rightarrow B_{r}$ is well-defined. Let $\mu\in B_{r}$, for some $r$. Then for all $\varkappa\in\left[0, \mathfrak{T}\right]$, we have

with the help of $(5.6)$, $(5.7)$ and $(5.8)$, we get

That is, $\left\|\Upsilon\left(\mu\right)\right\|\leq r$, for all $\mu\in B_{r}$, which implies that $\Upsilon\left(\mu\right)\in B_{r}$ and hence $\Upsilon : B_{r}\rightarrow B_{r}$ is well-defined. Now, we have to show that $\Upsilon : B_{r}\rightarrow B_{r}$ is continuous. For this, using $(5.7)$, $(5.5)$ and $(5.8)$, we have

But $\left|\lambda_{1}\right|C_{1}\mathfrak{K}_{1} +\left|\lambda_{2}\right|C_{2}\mathfrak{K}_{2} < \frac{\Gamma(\wp+1)}{2\mathfrak{T}^{p}}$, that is $\mathbb{h} = \frac{2\left(C_{1}\mathfrak{K}_{1}\left|\lambda_{1}\right| +C_{2}\mathfrak{K}_{2}\left|\lambda_{2}\right|\right)\mathfrak{T}^{\wp}}{\Gamma(\wp+1)}\in\left(0, 1\right)$. It follows from above that

That is, $\Upsilon : B_{r}\rightarrow B_{r}$ is contraction and hence continuous. Next, we have to show that $\Upsilon$ is $\mathcal{L}_{\rm{m}}$-contraction. Let $\Lambda$ be any subset of $B_{r}$ with ${\rm{m}} (\Lambda) > 0,$ and $\mu, \nu\in \Lambda$. Then from inequality $(5.9)$, we write

Now, let define $\theta:(0, \infty)\rightarrow (1, \infty)$ by $\theta(t) = e^{t}$, then clearly $\theta\in\Theta$. Using inequality $(5.10)$, we have

Consequently,

Thus for $\mathscr{L}(\kappa_{1}, \kappa_{2}) = \frac{\kappa_{2}^{\mathbb{h}}}{\kappa_{1}}$, the above inequality becomes

That is, $\Upsilon : B_{r}\rightarrow B_{r}$ is $\mathcal{L}_{\rm{m}}$-contraction and so Theorem $4.2$ ensures the existence of a fixed point of $\Upsilon$ in $B_{r}$, equivalently, the Eq $(5.3)$ has a solution in $B_{r}$.

To illustrate the Theorem $5.3$, we present an example.

Example 5.4. Consider the following Caputo fractional Volterra–Fredholm integro-differential equation

with boundary condition

Compare Eq $(5.11)$ with Eq $(5.1)$, we get

Clearly $\mathfrak{g}: [0, 2]\rightarrow \mathbb{R}$, $\mathfrak{J}_{1}, \mathfrak{J}_{2}, \mathfrak{J}_{3}: [0, 2]\times [0, 2]\rightarrow \mathbb{R}$ and $\xi_{1}, \xi_{2}: [0, 2]\times\mathbb{R}\rightarrow \mathbb{R}$ are continuous. Now, we have to verify condition $(5.5)$ of Theorem 5.3. Consider

Similarly,

Thus $\xi_{1}, \xi_{2}: \mathbb{R}\rightarrow \mathbb{R}$ are Lipschitz with $C_{1} = C_{2} = 1$.

Next, we have to verify the conditions $(5.7)$ and $(5.8)$ of Theorem 5.3. To do this, we have

and

Finally, to verify condition $(5.6)$ of Theorem $5.3$. Let $B_{2} = \left\{\mu\in \mathcal{C}\left(\left[0, 2\right], \mathbb{R}\right):\left\|\mu\right\|\leq 2\right\}$, then since $\left\|\mathfrak{g}\right\| = 0, \mathfrak{K}_{1} = 0, \mathfrak{K}_{2}\approx 2.197, \mathfrak{K}_{3} = 0.17$, $\mathfrak{F}_{1} = 2,$ and $\mathfrak{F}_{2} = \sqrt{5}$, so we have

and

Thus Theorem 5.3 ensures the existence of a solution of $(5.11)$ in $B_{2}$.

6.   Conclusions

Darbo type contractions are introduced and fixed point results are established in a Banach space using the concept of measure of non compactness. Various existing results are deduced as corollaries to our main results. Further, our results are applied to prove the existence and uniqueness of solution to the Caputo fractional Volterra–Fredholm integro-differential equation under integral type boundary conditions which is further illustrated by appropriate example. Our study paves the way for further studies on Darbo type contractions and its applications.

Acknowledgments

The authors are thankful to the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia, for supporting this research.

Conflict of interest

All authors declare no conflicts of interest in this paper.