Existence solution of a system of differential equations using generalized Darbo's fixed point theorem

1.   Introduction

The measure of noncompactness (MNC) performs an important character in real world problems. First of all, the fundamental paper of Kuratowski ^[1] in $1930$ open up a new direction of MNC to solve diffent type of Functional equations, which comes from the diffent real life problems. Using the notion of MNC, Darbo ^[2] in $1955$ ensure that the endurance of fixed points, which is obtained by the generalization of Schauder fixed point theorem (SFPT) and banach contraction principle. Many authors using the notion of MNC generalize Darbo fixed point theorem (DFPT) which ensure that the endurance of fixed point to solve various kind of integal or differentail equations. Up to now, many authors have been published several papers using the notion of generalization of DFPT and MNC ^{[3,4,5,6,7,8,9,10,11,12,13,14]}.

Our purpose of present paper is to extend the DFPT and we aaply our obtained results to find the existence of solutions of the functional differential equations.

At the beginning we provide concepts, notations, definitions and the preliminaries, which will be used all over the present paper.

2.   Definition and preliminaries

The set of real numbers is symbolize by $\mathbb{R}$, $\mathbb{R_+} = [0, \infty)$ and the set of natural numbers by $\mathbb{N}$. Let $(\Xi, \parallel.\parallel)$ be real Banach spaces. If $\Omega$ is a nonempty subset of $\Xi$ then $\bar{\Omega}$ and Conv$\Omega$, symbolize the closure and convex closure of $\Omega$ respectively. Also, let $\mathcal{M}_{\Xi}$ symbolize the set of all nonempty and bounded subsets of $\Xi$ and $\mathcal{N}_{\Xi}$ is the subset of all relatively compact sets.

Banas and Lecko ^[15] have given the definition of MNC which is given below.

Definition 2.1. A MNC is a mapping $\chi :\mathcal{M}_{\Xi} \rightarrow \mathbb{R_+}$ if it fulfills the following constraints for all $\Omega, \; \Omega_1, \; \Omega_2\in\mathcal{M}_{\Xi}.$

($M_1$) The family ker $\chi = \left\lbrace\Omega \in \mathcal{M}_{\Xi}: \chi\left(\Omega\right) = 0 \right\rbrace$ is nonempty and ker $\chi \subset \mathcal{N}_{\Xi}.$

($M_2$)$\Omega_1 \subset \Omega_2 \implies \chi\left(\Omega_1\right) \leq \chi\left(\Omega_2\right).$

($M_3$)$\chi\left(\bar{\Omega}\right) = \chi\left(\Omega\right).$

($M_4$)$\chi\left(Conv\Omega\right) = \chi\left(\Omega\right).$

($M_5$)$\chi\left(\kappa\Omega_1 +\left(1- \kappa \right)\Omega_2 \right) \leq \kappa \chi\left(\Omega_1\right)+\left(1- \kappa \right)\chi\left(\Omega_2\right)$ for $\kappa \in \left[0, 1 \right].$

($M_6$) if $\Omega_{n} \in \mathcal{M}_{\Xi}, \:\Omega_{n} = \bar{\Omega}_{n}, \: \Omega_{n+1} \subset \Omega_{n}$ for $n = 1, 2, 3, ...$ and $\lim\limits_{n \rightarrow \infty}\chi\left(\Omega_{n}\right) = 0,$ then $\bigcap\limits_{n = 1}^{\infty}\Omega_{n} \neq \emptyset.$

We are going to define the Concept of operator $S(\bullet; .)$ which was introduced by Altan and Turkoglu ^[16].

Definition 2.2. Let $A(\mathbb{R_+})$ be the set of fuctions $f:\mathbb{R_+}\rightarrow \mathbb{R_+}$ and let $Z$ be the set of functions $S(\bullet; .):A(\mathbb{R_+})\rightarrow A(\mathbb{R_+}$), which fulfills the following constraints:

($O_1$) $S(f; \sigma)\geq 0\; for\; \sigma > 0\; and \; S(h; 0) = 0$, $S(f; \sigma_1)\leq S(f; \sigma_2)\; for\; \sigma_1\leq \sigma_2$,

($O_2$) $\lim\limits_{n \rightarrow \infty}S(f; \sigma_n) = S(f; \lim\limits_{n \rightarrow \infty}\sigma_n)$,

($O_3$) $S(f; \max\lbrace \sigma_1, \sigma_2\rbrace) = \max\lbrace S(f; \sigma_1), S(f; \sigma_2)\rbrace$ for some $f\in A(\mathbb{R_+})$.

Theorem 2.1. (Schauder)^[17] A mapping $\Delta: \Omega \rightarrow \Omega$ which is compact and continuous has at least one fixed point for a nonempty, bounded, closed and convex (NBCC) subset $\Omega$ of a Banach space $\Xi$.

DFPT is generalize by resting the compactness of Schauder's mapping and theorem is known as SFPT.

Theorem 2.2. (Darbo)^[18] Let $\Delta: \Omega \rightarrow \Omega$ be a continuous mapping and $\chi$ is an MNC. Then for any nonempty subset $\wp$ of $\Omega$, there exists a $k \in \left[0, 1\right)$ having the inequality

Then the mapping $\Delta$ have a fixed point in $\Omega$.

Isik et al. ^[10] introduce a function $f$ to generalize the Banach contraction, we find various type of contraction mapping.

Theorem 2.3. Let $\Delta:\Omega\rightarrow \Omega$ be a continuous self-mapping, where $(\Omega, \rho)$ is a complete metric space. Then for all $\gamma, \delta \in\Xi$ there exists a mapping $f: \mathbb{R_+}\rightarrow \mathbb{R_+}$ such that $\lim\limits_{\tau \rightarrow 0_+}f(\tau) = 0, \; f(0) = 0,$

Then $\Delta$ contains a unique fixed point.

Parvenah et al. ^[10] generalized DFPT as follows:

Theorem 2.4. Let $\Delta: \Omega \rightarrow \Omega$ be a continuous operator defined on a NBCC subet $\Omega$ of $\Xi$ having the inequality

for all $\wp\subset\Omega$, where $f:\mathbb{R_+}\rightarrow \mathbb{R_+}$ with $\lim\limits_{\tau \rightarrow 0_+}f(\tau) = 0, \; f(0) = 0,$ and $\chi$ is an MNC. Then $\Delta$ contains a fixed point in $\Omega$.

Remark 2.1. Remember that Theorem 2.4 generalize DFPT. Since $\Delta:\wp\rightarrow \wp$ is a Darbo mapping.

Then for all $\wp\subset \Xi$ there exists $k\in[0, 1)$ having the property $\chi\left(\Delta\wp\right) < k \chi(\wp)$.

So with the help of inequality, we have

for all $\wp\subset\Xi$.

Consequently

So

Taking $f(\tau) = \frac{k}{1-\sqrt{k}}\tau$, we have $\chi(\Delta\wp)\leq f\left(\chi(\wp)\right)-f\left(\chi(\Delta \wp)\right)$ for all $\wp\subset \Xi$. Therefore the Darbo Theorem is a specific case of contraction mapping of Theorem $(2.4)$.

3.   Main results

Let us recall an important theorem in this work which extends DFPT by taking the concept of $S(h; .)$.

Theorem 3.1. Let $(\Xi, \parallel.\parallel)$ be a Banach space. Suppose $\Delta:\Xi\rightarrow \Xi$ is a continuous, nondecreasing and bounded mapping fulfills the following inequality

for all bounded $\wp$ of $\Xi$, where $\chi$ is MNC, $h\in A(\mathbb{R_+}), \; S(\bullet; .)\in Z$, $\phi, \pi:\mathbb{R_+}\rightarrow \mathbb{R_+}$ is continuous functions and $f:\mathbb{R_+}\rightarrow \mathbb{R_+}$ is a function as $\lim\limits_{\tau \rightarrow 0_+}f(\tau) = 0, \; f(0) = 0$. Then $\Delta$ contains at least one fixed point.

Proof. Assume that $\wp_n$ with $\wp_0 = \wp$ and $\wp_{n+1} = conv(\Delta\wp_n)$ for all $n\geq 0.$

Also, $\Delta\wp_0 = \Delta\wp\subseteq \wp = \wp_0, \; \; \wp_1 = conv(\Delta\wp_0)\subseteq\wp = \wp_0$. Since $\wp_n$ is a closed and bounded subset in $\Xi$ and

Following (3.1), we have

Taking the limit as $n \rightarrow \infty$ on both the sides of this inequality, we have

Therefore

By the virtue of $(iii)$ of Definition $S(h; .)$, we get

and therefore $\lim\limits_{n \rightarrow \infty}\int\limits_0^{\chi(\wp_{n})}\pi(\tau)d\tau = 0.$

But for any $\epsilon > 0, \; \; \int\limits_0^\epsilon \pi(\tau)d\tau > 0$, then $\chi(\wp_n)\rightarrow 0 \; as\; n\rightarrow \infty$.

Now since $\wp_n$ is nested sequence, by the definition of (MNC) of $(M_6)$, we conclude that $\wp_{\infty} = \cap_{n = 1}^{\infty}\wp_n$ is NBCC of $\Xi$. Also we aware that $\wp_{\infty}\in ker\chi$. Therefore $\wp_{\infty}$ is compact and invariant under the mapping $\Delta$. Therefore by the SFPT, $\Delta$ has a fixed point in $\wp_{\infty}$.

Remark 3.1. Putting $\pi(\tau) = 1$ for $\tau\in[0, \infty)$ in Theorem 3.1, we have

$S\left(h; \chi(\Delta \wp)+\phi\left(\chi(\Delta \wp)\right)\right)\leq f\left(S\left(h; \chi(\wp)+\phi\left(\chi(\wp) \right)\right) \right)-f\left(S\left(h; \chi(\Delta \wp)+\phi\left(\chi(\Delta \wp) \right)\right) \right).$

Remark 3.2. Take $\phi = 0, \; \; S(h; \tau) = \tau, \; \; h = I$ in Remark(3.1), then we have

It is a generalization of the result given by Parvenah et al.^[10].

Definition 3.1. ^[19] A mapping $\Delta:\; \Xi\times \Xi\times \Xi\rightarrow \Xi$ is said to have a TFP $(\gamma, \delta, \theta)\in \Xi^3$ if

Theorem 3.2. ^[18] Let $\chi_1, \chi_2, ..., \chi_n$ be the measure of noncompactness of $\Xi_1, \Xi_2, ..., \Xi_n$ respectively. Also assume that the function $\mathcal{B}:\mathbb{R_+}^\tau\rightarrow \mathbb{R_+}$ is convex and $\mathcal{B}(\gamma_1, \gamma_2, ..., \gamma_\tau) = 0$ if and only if $\gamma_r = 0$ for $r = 1, 2, ..., \tau$. Then

Example 3.1. ^[20] Let $\mathcal{\mathcal{B}}(\gamma, \delta, \theta) = \max\lbrace \gamma, \delta, \theta\rbrace$ for $(\gamma, \delta, \theta)\in\mathbb{R_+}^3$. Now $\mathcal{B}(\gamma, \delta, \theta) = \max\lbrace \gamma, \delta, \theta \rbrace = 0$ iff $\gamma = \delta = \theta = 0$. Then $\mathcal{\mathcal{B}}$ is convex and satisfied all conditions of Theorem 3.2. Therefore $\hat\chi(\Theta) = \mathcal{B}(\chi_1(\Theta_1), \chi_2(\Theta_2), \chi_3(\Theta_3))$ is an MNC on $\Xi_1\times \Xi_2\times \Xi_3$, where $\chi$ be an MNC in $\Xi$ and $\Theta_j$ is the natural projections of $Z$ into $\Xi_j$ for $j = 1, 2, 3$.

Example 3.2. ^[20] Let $\mathcal{B}(\gamma, \delta, \theta) = \gamma+\delta+\theta$ for $(\gamma, \delta, \theta)\in\mathbb{R_+}^3$. Now $\mathcal{B}(\gamma, \delta, \theta) = \gamma+\delta+\theta = 0$ iff $\gamma = \delta = \theta = 0$. Then $\mathcal{B}$ is convex and satisfied all conditions of Theorem 3.2. Therefore $\hat\chi(\Theta) = \mathcal{B}(\chi_1(\Theta_1), \chi_2(\Theta_2), \chi_3(\Theta_3))$ is an MNC on $\Xi_1\times \Xi_2\times \Xi_3$, where $\chi$ be an MNC in $\Xi$ and $X_j$ is the natural projections of $Z$ into $\Xi_j$ for $j = 1, 2, 3$.

Theorem 3.3. Let $C$ be a NBCC subset of a Banach space $\Xi$ and let $\Delta: C\times C\times C\rightarrow C$ be a continuous mapping such that

for all $\Theta_1, \Theta_2, \Theta_3\in C$, $\chi$ is MNC and $\omega:[0, \infty)\rightarrow [0, \infty)$ is such that $\lim\limits_{\tau \rightarrow 0_+}\omega(\tau) = 0, \; \omega(0) = 0$. Also $S(f; .)\in Z$ and $S(f; \tau_1+\tau_2+\tau_3) = S(f; \tau_1)+S(f; \tau_2)+S(f; \tau_3)$ for all $\tau_1, \tau_2, \tau_3\geq 0$. $\Delta$ has at least a triple fixed point.

Proof. We define a mapping $\hat\Delta:C_3\rightarrow C_3$ by

$\hat \Delta(\gamma, \delta, \theta) = (\Delta(\gamma, \delta, \theta), \Delta(\delta, \gamma, \theta), \Delta(\theta, \delta, \gamma))$ for all $(\gamma, \delta, \theta)\in C_3.$

$\hat \Delta$ is continuous, since $\Delta$ is continuous.

We know that $\hat\chi(\Theta) = \chi(\Theta_1)+\chi(\Theta_2)+\chi(\Theta_3)$,

where $\Theta_1, \Theta_2, \Theta_3$ denotes the natural projections of $C$. Suppose $\Theta\subset C^3$ be a nonempty subset.

Now using the Theorem 3.3, we get

Putting $\omega = \frac{1}{3}\nu$, we have

Now from the Theorem 3.1, we conclude that $\Delta$ has at least a triple fixed point.

Remark 3.3. By taking $S(f; \tau) = \tau, \; \; \; \nu(\tau) = \tau, \; \; f = I$ in Theorem 3.3, we get the corollary which is given below.

Corollary 1. Let $\Delta:C\times C\times C\rightarrow C$ be a continuous function defined on a NBCC subset $C$ of $\Xi$ in such a way that

Then $\Delta$ has a TFP.

4.   Applications

This section contains the applicability of Theorem 3.1 and Corollary 1 by using the system of equations which is defined as

where $\gamma\in[0, T]$ with the initial state $\xi(0) = \xi_0, \; \; \nu(0) = \nu_0\; \; and\; \; w(0) = w_0$.

Suppose that the space of all bounded continuous function defined on $[0, T]$ is $C[0, T]$ equipped with the standard norm

A function having Modulus of contiunity for $\gamma\in[0, T]$ is defined as

$\omega(\gamma, \epsilon)\rightarrow 0\; as\; \epsilon\rightarrow 0$, because $\gamma$ is continuously uniform on $[0, T]$. The Hausdorff MNC for every bounded subset $\wp$ of $C[0, T]$ is

Now, we construct the assumptions by which the system of integral Eq (4.1) will be studied.

(i) $\zeta, \eta: [0, T]\rightarrow [0, T]$ are the functions which are continuous.

(ii) For a continuous function $h:[0, T]\times\mathbb{R}^6\rightarrow \mathbb{R}$ there exists a continuous function $\phi:\mathbb{R_+}\rightarrow \mathbb{R_+}$ with $\phi(0) = 0\; \; and\; \; \phi(\tau) < \tau$ for all $\tau > 0$ and also satisfy

(iii) $M = \sup\lbrace S(f; |h(\tau, \xi_0, \nu_0, z_0, 0, 0)|)\rbrace < \infty$, where $\tau\in[0, T]$ and $S(f; \epsilon) < \epsilon$.

(iv) There exists $r_0$ such that

Theorem 4.1. The system (4.1) with the assumptions $(i)-(iv)$ has at least one solution which belongs to the space $\lbrace C[0, T]\rbrace^3$.

Proof. Assume that $\mathcal{U}(\tau) = \xi\prime(\tau), \; \; \mathcal{V}(\tau) = \nu\prime(\tau), \; \; \mathcal{W}(\tau) = w\prime(\tau)$. Then our system of Eq (4.1) can be written as the system of integral eqautions

where $\tau\in[0, T].$

Assume $\Delta: C[0, T]\rightarrow C[0, T]$ be a operator with

We notice for every $\tau\in C[0, T]$, the mapping $\Delta$ is continuous i.e $\Delta$ maps the space $C[0, T]$ into itself.

For fixed arbitrary $\tau\in C[0, T]$ and $f\in\mathcal{F}([0, \infty))$, we have from the assumptions $(i)-(iv)$,

Thus

and

Due to the inequality $\phi\left(S\left(f; \Delta r_0 \right)\right) +\frac{1}{2}S\left(f; 3r_0\right) +M \leq r_0$, the function $\Delta$ maps $(B_{r_0})^3$ into $(B_{r_0})$.

Now we prove that $\Delta$ is continuous on $(B_{r_0})^3$.

Let fixed arbitrary $\epsilon > 0$ and take $(\gamma, \delta, \theta), (\xi, \nu, w)\in (B_{r_0})^3$ such that

Therefore for every $t\in[0, T]$, we get

Thus, we have $\phi\left(S\left(f; \epsilon+\Delta\epsilon \right)\right) +\frac{1}{2}S\left(f; \epsilon\right)\rightarrow 0$ as $\epsilon \rightarrow 0$.

Therefore $\Delta$ is a continuous function on $(B_{r_0})^3$. Now, we shall show that $\Delta$ satisfy all the conditions of Corollary 1. To do this, let $\mathcal{U}, \; \; \mathcal{V}\; \; and\; \; W$ are nonempty and bounded subsets of $(B_{r_0})$ and $\epsilon > 0$ is constant. Moreover we take $\tau_1, \tau_2\in [0, T]$ with $|\tau_2-\tau_1|\leq\epsilon$ and $\xi\in \mathcal{U}, \; \; \nu\in \mathcal{V}\; \; and\; \; w\in W$.

Then we have

where $\omega(\eta, \epsilon) = \sup\left\lbrace|\eta(\tau_2)-\eta(\tau_1)|: \; \; \; |\tau_1-\tau_2|\leq\epsilon\; \; \;, \tau_1, \tau_2\in[0, T]\right\rbrace$,

$\omega(\zeta, \epsilon) = \sup\left\lbrace|\zeta(\tau_2)-\zeta(\tau_1)|: \; \; \; |\tau_1-\tau_2|\leq\epsilon\; \; \;, \tau_1, \tau_2\in[0, T]\right\rbrace,$

$\omega(\xi, \omega(\eta, \epsilon)) = \sup\left\lbrace|\xi(\tau_2)-\xi(\tau_1)|: \; \; \; |\tau_1-\tau_2|\leq\omega(\eta, \epsilon)\; \; \;, \tau_1, \tau_2\in[0, \eta(T)]\right\rbrace$,

$\omega(h, \epsilon) = \sup\left\lbrace|h(\tau_1, \gamma_1, ..., \gamma_6)-h(\tau_2, \gamma_1, ..., \gamma_6)|: \; \; \; |\tau_1-\tau_2|\leq\epsilon\; \; \;, \tau_1, \tau_2\in[0, T]\right\rbrace,$

and $\gamma_1, ..., \gamma_6\in[-r_0, r_0].$

We infer that

Therefore we get

$S\left(f; \omega(\Delta(\mathcal{\mathcal{U}}\times \mathcal{V}\times \mathcal{W}), \epsilon)\right)\\ \leq\frac{1}{2}S\left(f; \max\left\lbrace\omega(\mathcal{\mathcal{U}}, \omega(\eta, \epsilon)), \omega(\mathcal{V}, \omega(\eta, \epsilon)), \omega(\mathcal{W}, \omega(\eta, \epsilon))\right\rbrace\right)+S(f; \omega(h, \epsilon))+\phi(S(f; \max r_0\omega(\zeta, \epsilon))).$

Since $h, \eta, \zeta$ are uniformly continuous on $[0, T]\times [-r_0, r_0]^5, [0, T]\; \; and\; \; [0, T]$ respectively, we get $\omega(h, \epsilon)\rightarrow 0, \; \; \omega(\eta, \epsilon)\rightarrow 0\; \; and\; \; \omega(\zeta, \epsilon)\rightarrow 0\; \; as\; \; \epsilon\rightarrow 0$.

By taking $S(f; \tau) = \tau, \; \; \Theta_1 = \mathcal{\mathcal{U}}, \; \; \Theta_2 = \mathcal{V}, \; \; \Theta_3 = \mathcal{W}, \; \; f = I$ and from the MNC definition, we have $\chi(\Theta_1\times \Theta_2\times \Theta_3)\leq \frac{1}{2}\left(\max\left\lbrace\chi(\Theta_1), \chi(\Theta_2), \chi(\Theta_3)\right\rbrace\right).$

By the Corollary 1, $\Delta$ has at least a TFP.

5.   Example

Example 5.1. Let the system of differentail equations is as

with the state condition $\xi(0) = 1, \; \; \nu(0) = 3, \; \; w(0) = 2$ and $\tau\in[0, 5]$.

System of Eq (5.1) is the particular case of Eq (4.1) where $\zeta(\tau) = \tau = \eta(\tau)$,

By the definition of $\zeta$ and $\beta$ assumption $(i)$ is satisfied.

$h(\tau, \gamma_1, ..., \gamma_6) = \tau^2+\frac{\sqrt [3]{\xi{(\tau)}}+\sqrt [5]{\nu{(\tau)}}+\sqrt [7]{w{(\tau)}}}{3}+\frac{1}{6}\log(1+|\xi\prime(\tau)+\nu\prime(\tau)+w\prime(\tau)|)$.

Now assume that $\tau\in[0, T], \; \; S(f; \tau) = \tau, \; \; and\; \; \phi(\tau) = \max_{i = 3, 5, 7}\lbrace\sqrt[i]{\tau}\rbrace$,

we get

Hence assumption $(ii)$ is satisfied. Moreover

It is simple to notice every number $r\geq 75$ fulfills the inequality given in $(iii)$.

Now the inequality in assumption $(iv)$ is $\phi\left(S(f; \Delta r_0)\right)+\frac{1}{6}S\left(f; 3r_0\right)+M$ is equal to

Hence, as the number $r_0$ we can take $r_0 = 75$. Therefore, all the assumptions of Theorem 4.1 are satisfied. Hence the system of Eq (5.1) have at least one solution which belongs to $\lbrace C[0, T]\rbrace^3$ space.

6.   Conclusions

The present paper concentrated on multiple FPT which is based on the generalization of DFPT via MNC. In this work, by using the concept of operators we extend DFPT by using MNC. We demonstrate the endurance of TFP by our extended DFPT and MNC. At the last we yield an example which fulfills our findings.

Acknowledgments

The authors would like to express their deep gratitude to the journal editor and referees for their careful reviews and valuable comments which helped to improve the paper. This research is supported by Govt. of India CSIR fellowship, Program No. 09/1174(0005)/2019-EMR-I, New Delhi.

Conflict of interest

All the authors declare that there is no conflict of interest.