Study of multi term delay fractional order impulsive differential equation using fixed point approach

1.   Introduction

Fractional calculus has gotten considerable attention recently. Derivatives and integrals of non-integer order are increasingly using for different analysis of various problems. Fractional differential operators are global in nature and preserve greater degree of freedom. Therefore, researchers now give preference to use fractional order differential equations (FODEs) in mathematical modelings of various real world process and phenomenons over classical order differential equation. FODEs have multi-dimensional applications in the variety of fields of modern sciences, such as to control the phase difference in oscillators, to accomplish the high frequency oscillation and in electric engineering DC converter models are used to obtain good assessment of the power conversion efficiency. Recently various biological models have been investigated by using fractional order differential equations. In the mentioned study, researchers have established more good results than those already derived for ordinary differential equations. For some more applications of fractions derivatives, for theory and applications of FODEs see ^[1,2], for integro-FODEs, see ^[3]. For basic theory and applications we refer ^[4]. Some generalized type FODEs have been analyzed in ^[5,6]. Some engineering applications have been investigated in ^[7]. For real world applications by using FODEs, see ^[8]. Those differential equations which observe impulsive conditions at points of discontinuity of solution are known as impulsive differential equations. Impulsive differential equations are the tools used for modeling of those evolutionary and physical phenomena that containing sudden changes and discontinuous jumps. Therefore, the proposed types of impulsive differential equations play a significant role to models such phenomena, in this regards (see ^[9]). Some stability results about the said area has been studied in ^[10]. Also some conformable impulsive FODEs have been studied in ^[11]). In some circumstance physical problems depends on preceding states of problem and cannot be describes by current time. In order to avoid such circumstance, researches introduced an important class of differential equations (DEs) known as Delay Differential Equations (DDEs). There are verities of DDEs including proportional (pantograph), continuous and discrete type DDEs. The concern types of DEs are widely using to formulate various real world phenomena in different fields, such as dynamics, quantum mechanics ^[12], biology ^[13], and electrodynamics ^[14].

Important aspects of mathematical analysis are existence theory and stability analysis. Researchers have used various tools of nonlinear analysis for investigating the existence and uniqueness of solution to various problems of FODEs. Various fixed point results and degrees theories have been developed to investigate the said area for existence of solutions. In same line stability analysis of FODEs has also been given proper attention recently. The mentioned analysis is important for developing various numerical methods. Stability results have been investigated by using various methods including exponential method, Mittag-Leffler method and Hyers-Ulam method (see some detail in ^[15]). Here, we remark that the Ulam's type stability analysis has given more attention recently. The aforementioned stability results have been derived for various problems of FODEs in last few years (for instance see ^[16]). Hyers and Ulam had been introduced the mentioned stability for the first time for functional equations in 1940 (see some detail ^[17]). Rassias ^[18] extended the mentioned stability analysis for linear equations. Also Jung ^[19] extended the Rassias stability results for functional equations in nonlinear analysis.

Motivated from the mentioned work, researchers have been given much attention to investigate the aforesaid stability analysis for various dynamical problems (we refer few as ^[20,21]). For boundary value problems of FODEs, the mentioned stability has been studied very well (we refer few results as ^[22,23]). Furthermore, results related to existence theory of solutions to various problems of fractional order mathematical models of epidemiology have been investigated very well (for instance see ^[24]). The said results have been investigated for TB models in ^[27]. The Green functions theory using FODEs has been established in ^[25,26]. Also the mentioned analysis has been studied for those FODEs involving non-singular derivatives (for instance see ^[28]).

The existence theory has been developed very well for FODEs in last few years. As an example the reader can look at the second order FODE with non local boundary condition on the independent variable ^[29]. Researchers have been used fixed point theory together with topological degree theory to develop necessary condition for existence of solution for various problems of fractional order differential equations. Furthermore, they have also derived various results related to Ulam type stability for said problems. Here we recall a suitable example which has been studied in ^[33] as

where $\mathfrak{Z}_1, \mathfrak{Z}_2, \mathfrak{Z}_3$ and $\mathfrak{Z}_4$ are members of the set of real numbers, which satisfy the condition given as: $\mathfrak{Z}_1+\mathfrak{Z}_2\neq0$ and $\mathfrak{Z}_3+\mathfrak{Z}_4\neq0$. The function $\mathfrak{F}, \mathfrak{P}_1$ and $\mathfrak{P}_1$ are continuous. The authors initially utilized the tools of fractional calculus as well as nonlinear analysis to transform the aforementioned FDE to corresponding integral equation and then used fixed theory to achieve their aims.

Similarly, in ^[30] uniqueness and existence of solution have been studied by utilizing the tools of fixed point theory. Authors have investigated the following system of FODEs with anti periodic coupled with non local subsidiary conditions as

The author have used standard Caputo derivative in the consider problem, where parameters $\alpha_1, \alpha_2, \alpha_3, \alpha_4, \kappa_j, \kappa_j^*, \lambda_j$ and $\lambda_j^*$ are real numbers for $j = 1, 2, \dots, m$ and $0 < \beta_j < 1.$

Inspired from by the above discussion, in this research work, we take the following system by extending the problem (1.1) utilizing the concept of ^[30,31] as

In the consider problem (1.2), $\sigma_i\in R$ for $i = 1, 2, \dots, p$ with $\sigma_1\neq0$, and functions $g_1, g_2: PC([0, \tau]), R)\mapsto R$ and non linear function $\mathscr{H}:[0, \tau]\times R\times R\mapsto R$ are continuous and $\tau > 0$ is real constant. Furthermore, impulsive operators $\mathcal{I}_k$ and $\mathcal{I^*}_k$ are also continuous. In this article, we use tools of fixed point theory and functional analysis to obtain the desired results. Results devoted to the existence and uniqueness of solution are derived by using Banach and Krasnoselskii's fixed point theorems. Also, the results devoted to stability analysis of Ulam type are established by using tools of nonlinear functional analysis. For verification of the obtain results, we give appropriate example.

The rest of the paper is organized as follows: In Section 2, we recall some basic concepts of fractional calculus, while the main results, relying on Krasnoselski's fixed point theorem and Banach contraction principal are presented in Section 3. Section 4 is devoted to stability analysis of the proposed problem (1.2). Section 5 contains illustrative examples for the obtained results. In Section 6, we present conclusion of our findings.

2.   Auxiliary definitions and results

This section of research, is devoted to basic results, theorems and lemmas of FPT and non-linear analysis, which we need for investigation of the main work.

In the present work, we use the following space and norm

with norm define as

where

Definition 2.1. The integral of fractional order $\alpha$ of a function $y(t)\in L[0, d]$ is denoted by $I^{\alpha}y(t),$ and defined as

Definition 2.2. ^[32] Fractional order Caputo derivative for a function $y(t)\in L^1([0, d], R_+)$ on the interval $[0, d]$ is defined as

where $n = \lceil{\alpha}\rceil$ and $\lceil{\alpha}\rceil$ is defined to be the smallest integer equal or greater than $\alpha$.

Lemma 2.1. ^[34] The relation between fractional order integral and derivative is given as

where $A_i\in R$ for $i = 1, 2, \dots, n.$

Definition 2.3. The mapping $T:X\longrightarrow Y$ on norm linear spaces is continuous and complete, if for each bounded $M \in X$, $\overline{T(M)} \in Y$ is compact.

Definition 2.4. ^[35] Let $F(X)$ be the collection of function (real valued) on $(X, d)$ metric space, be equi-continuous $x\in X$, if for each $\epsilon > 0,$ we can find $\delta > 0,$ such that for every function $f\in F(X)$ and $x_0\in X$, we have $|f(x_0)-f(x)| < \epsilon$, whenever $d(x_0-x) < \delta$.

Definition 2.5. ^[35] An operator $\mathcal{T}$ on $(X, d)$ metric space into itself is Lipschitz, if $\exists \; c\geq 0,$ and $d(\mathcal{T}(x_1), \mathcal{T}(x_2))\leq cd(x_2, x_1),$ for each $x_2, x_1\in X,$ where $c$ is called Lipschitz constant and contraction, if $0 < c < 1.$

Definition 2.6. ^[35] An operator $\mathcal{T}$ from a metric space $(X, d)$ into itself is contraction, if $\exists\; \; 0 < c < 1,$ such that $d(\mathcal{T}(x_1), \mathcal{T}(x_2))\leq cd(x_1, x_2), \forall x_1, x_2 \in X.$

Theorem 2.7. ^[35]every mapping(self contraction) $\mathcal{T}$ in complete $(X, d)$ metric space has unique fixed point.

Theorem 2.8. ^[36] Assume that $\mathcal{H}$ is a non empty, convex, bounded and closed convex bounded subset of a Banach space $\mathcal{X}$. let $\mathscr{J}_1$ and $\mathscr{J}_1$ be two operator provide that $\mathscr{J}_1\mathscr{U}_1+\mathscr{J}_2\mathscr{U}_2\in \mathcal{H}$ whenever $\mathscr{U}, \mathscr{U}_2\in\mathcal{H}$, $\mathscr{J}_1$ is continuous and compact and $\mathscr{J}_2$ is contraction map. Then there is $\mathscr{U}\in \mathcal{H}$ provide that, $\mathscr{U} = \mathscr{J}_1\mathscr{U}+\mathscr{J}_2\mathscr{U}$.

3.   Qualitative analysis

This section of research work is committed, to integral representation and existence results for the consider class of multi-term fractional delay differential equations. The authors established the expression for integral representation of proposed problem. In order to obtain results for existence and stability analysis the authors used the tools of analysis and fixed point theory.

3.1.   Integral representations for the proposed problem

This subsection of the research work, is devoted the integral representation of the consider model (1.2).

Theorem 3.1. Assumed that $\mathscr{Y}(t)\in C(J, R)$, then the solution of multi-term impulsive fractional delay differential equation,

is equivalent to the integral equation

Where $g_i(i = 1, 2):C(J, R)\mapsto R$ are continuous function, $a_l+b_l\neq 0$ for $l = 1, 2$ and $\alpha_1-\alpha_i-1 > 0$ for $i = 2, 3, \dots, p$ and

Proof. Applying fractional order integral $I^{\alpha_{1}}$ on (3.1) and in-view of Lemma 2.1 for $t\in[0, t_1],$ we get

By differentiating (3.3), we get

Similarly, for $t\in (t_1, t_2]$, the system (3.1) become

By differentiating (3.5), we get

Now to compute $\sigma_1\mathscr{U}(t_1^-)$, $\sigma_1\mathscr{U}'(t_1^-)$, $\sigma_1\mathscr{U}(t_1^+)$ and $\sigma_1\mathscr{U}'(t_1^+)$ using (3.3)–(3.6), we obtain

Using the impulsive condition

and

we get

Using (3.8) in (3.5), we obtain

On the same fashion, for $t\in (t_k, t_{k+1}]$, (3.1) becomes

By using the boundary conditions involve in (3.1), we obtain

and

One can obtain the desired integral form of solution (3.2), by using (3.10) and (3.11) in (3.9) and (3.3).

Corollary 3.1. In view of Theorem 3.1 the solution of the given multi-terms fractional delay differential equation $(1.2)$ is given by

Where $(g_i(i = 1, 2) : C(J, R)\mapsto R)$ are continuous functions, $a_l+b_l\neq 0$ for $l = 1, 2$ and $\alpha_1-\alpha_i-1 > 0$ for $i = 2, 3, \dots, p$ and

3.2.   Data dependence results for proposed problem

In this subsection of the research work, we represent the desired solution for MIFDDE (1.2), in the form of operator equation and provides some assumptions for investigation of existence results for proposed problem.

Lets define $\mathcal{T}:X\mapsto X$, such that

We consider the following assumptions, which we needs for further correspondence in this work. $(H_1)$ For $\mathscr{U}_1, \mathscr{U}_2\in X$, there exist $\mathscr{L}_{\mathfrak{F_1}}, \mathscr{L}_{\mathfrak{F_2}}\geq 0$, i.e

$(H_2)$ For $\mathscr{U}_1, \mathscr{U}_2\in X$, there exist $\mathscr{L}_{g_i} > 0$, for i = 1, 2, such that

$(H_3)$ For $\mathscr{U}_1, \mathscr{U}_2\in X$, there exist $\mathscr{L}_{I} > 0$, such that

$(H_4)$ For $\mathscr{U}_1, \mathscr{U}_2\in X$, there exist $\mathscr{L}_{I^*} > 0$, such that

$(H_5)$ For any $\mathscr{U}\in X$, there exist $\mathscr{B}_{\mathscr{F}}:C(J, R^+)$ such that

$(H_6)$ For any $\mathscr{U}\in X$, there exist $\mathscr{B}_{g_i}:C(J, R^+)$ for $i = 1, 2,$ such that

$(H_7)$ For any $\mathscr{U}\in X$, there exist $\mathscr{B}_{I}:C(J, R^+)$, such that

$(H_8)$ For any $\mathscr{U}\in X$, there exist $\mathscr{B}_{I^*}:C(J, R^+)$, such that

For computational convenience, we introduce the following notation:

Theorem 3.2. Consider $(H_1)$–$(H_4)$ holds and $\mathscr{L} < 1$, then problem $(1.2)$ has at most one fixed point, where $\mathscr{L}$ is defined by (3.27).

Proof. Consider that $\mathscr{U}, \mathscr{U}^*\in X$ and $t\in J_k$ where k = 1, 2…, m.

Let us assume $\mathscr{L}_{\mathfrak{F}} = \mathscr{L}_{\mathfrak{F}_1}+\mathscr{L}_{\mathfrak{F}_2}$ and evaluate the integral involve in (3.30), we have

It is quite clearly that $t_{j-1}-t_j\leq \tau$, hence by using the aforementioned inequality equation $(3.31)$ can be express as

By rearranging term containing $\mathscr{L}_{I}$, $\mathscr{L}_{I^*}$, $\mathscr{L}_{g_1}$, $\mathscr{L}_{g_2}$ and $\mathscr{L}_{\mathfrak{F}}$ in Eq (3.32), we obtain

Now using (3.22)–(3.27) in (3.33), we have

Therefore, by Banach contraction principal the mapping $\mathcal{T}$ has fixed point. Thus, the consider problem (1.2) has solution, which is unique.

Theorem 3.3. The consider problem (1.2) has at least one solution, if $(H_1)$ and $(H_5)$–$(H_8)$ holds and $\mathscr{L}_d < 1$, where $\mathscr{L}_d$ is defined in $(3.28)$.

Proof. In order to prove existence of at least one solution, we define operator $\mathfrak{T}_1, \mathfrak{T}_2:PC(J, R)\mapsto PC(J, R)$ given by

and

Moreover, we construct a ball $\mathcal{H} = \{\mathscr{U}(t)\in PC(J, R) : \|\mathscr{U}\|\leq \mathcal{R}\},$ with positive radius $\mathcal{R}$ chosen as

Step 1: We claim that $\mathfrak{T}_1\mathscr{U}_1(t)+\mathfrak{T}_2\mathscr{U}_2(t)\in \mathcal{H}\subseteq PC(J, R)$ for every $\mathscr{U}_1(t), \mathscr{U}_2(t)\in \mathcal{H}\subseteq PC(J, R).$

Now for the proof of desired results, consider

By using $(H_5)$–$(H_8)$, and evaluating the integral in (3.34), we obtain

By rearranging term containing $\mathscr{L}_{I}$, $\mathscr{L}_{I^*}$, $\mathscr{L}_{g_1}$, $\mathscr{L}_{g_2}$, and $\mathscr{L}_{\mathfrak{F}}$ in (3.34), we obtain

By using (3.22)–(3.26) in (3.35), we obtain

Hence it show that $\mathfrak{T}_1\mathscr{U}_1(t)+\mathfrak{T}_2\mathscr{U}_2(t)\in \mathcal{H}\subseteq PC(J, R).$

Step 2: Here we claim that $\mathfrak{F_1}$ is uniformly bounded for confirmation, we proceed as

As we clearly see that $t-t_m\leq \tau$. So substitute $\tau$ instead of $t-t_m$ in (3.36), then we obtain

Using (3.22)–(3.24) in (3.37), we obtain

Hence $\mathfrak{T}_1$ is uniformly bounded.

Step 3: Suppose $\mathscr{U}_n(t)$ is a sequence in $\mathcal{H}$ which converge to $\mathscr{U}\in\mathcal{H}$ for the continuity of $\mathfrak{T_1}$ we have to prove $\mathfrak{F_1}(\mathscr{U}_n(t))\mapsto \mathfrak{T_1}(\mathscr{U}(t))$. For the proof we precede as

Hence clearly from the continuity of $g_1, \; g_2, \; \mathcal{I}_j$ and $\mathcal{I}_j^*,$ we get that $\mathfrak{T}_1$ is continuous.

Step 4: To prove $\mathfrak{T}_1$ is equi-continuous. Consider

clearly as $t_1 \mapsto t_2$ we have $||\mathfrak{T}_1\mathscr{U}(t_2)-\mathfrak{T}_1\mathscr{U}(t_1)|| = 0$. Hence $\mathfrak{T}_1$ is equi-continuous.

Step 5: To prove $\mathfrak{T}_2$ is contraction, one can get help from Theorem 3.2 and obtain the following expression,

By using (3.25), (3.26) and (3.28) in (3.38), we obtain

Thus all assumption of Krasnoselskii's fixed point theorem are satisfied. So the problem (1.2) has at least one solution.

4.   Stability analysis

The authors motivated by the literature ^[10,29] and present some specific findings for the stability analysis of the proposed problem (1.2).

Definition 4.1. The solution $\mathscr{U}(t)$ of proposed problem will be Ulam-Hyres (UH) stable, if for unique solution $\mathscr{U}^*(t)$ one can find $\mathcal{B}_1 > 0$, such that for each solution $\mathscr{U}\in PC(J, R)$ of the following differential inequality and $\epsilon > 0$

and a unique solution $\mathscr{U}^* \in PC(J, R)$ of the given problem (1.2), such that $|\mathscr{U}-\mathscr{U}^*|\leq \mathcal{B}_1 \epsilon$ and solution will be generalized Ulam-Hyers (GUH) stable, if there exist a positive function $\mathcal{K}:(0, \infty)\mapsto (0, \infty)$ with $\mathcal{K}(0) = 0,$ such that $|\mathscr{U}-\mathscr{U}^*|\leq \mathcal{B}_1 \mathcal{K}(t).$

Definition 4.2. The solution of consider problem is UH Rassias stable, with respect to continuous function $\chi\in X$ and a positive constant $\psi > 0$ if we have $\mathcal{B}_2$ (positive constant) $> 0$, and $\epsilon > 0$, for each solution $\mathscr{U}\in PC(J, R)$ of the following differential inequality

and a unique solution $\mathscr{U}^* \in PC(J, R)$ of the given problem (1.2), such that $|\mathscr{U}-\mathscr{U}^*|\leq \mathcal{B}_2(\chi(t)+\psi)\epsilon.$

Definition 4.3. The solution of consider problem is GUH Rassias stable, with respect to continuous function $\chi\in X$ and a positive constant $\psi > 0$ if we have $\mathcal{B}_2$(positive constant) $> 0$, for each solution $\mathscr{U}\in PC(J, R)$ of the following differential inequality

and a unique solution $\mathscr{U}^* \in PC(J, R)$ of the given problem (1.2), such that $|\mathscr{U}-\mathscr{U}^*|\leq \mathcal{B}_2\chi(t)\epsilon.$

Remark 1. The solution of the inequality (4.1) is $\mathscr{U}^* \in PC(J, R)$, iff one can find a function $\zeta \in PC(J, R)$, and a sequence $\zeta_k, \; k = 1, 2, \dots, m.$ Depend on $\mathscr{U}$, such that

Remark 2. Let $\mathscr{U}^* \in PC(J, R)$ be the solution of (4.2), iff one can find a function $\zeta \in PC(J, R)$, and a sequence $\zeta_k, \; k = 1, 2, \dots, m.$ Depend on $\mathscr{U}$, such that

Remark 3. Let $\mathscr{U}^* \in PC(J, R)$ be the solution of (4.3), iff one can find a function $\xi \in PC(J, R)$, and a sequence $\zeta_k, \; k = 1, 2, \dots, m.$ Depend on $\mathscr{U}$, such that

Lemma 4.1. Consider $\mathscr{U}\in PC(J, R)$ is solution of FDDE,

satisfy the following relation,

Proof. In light of Theorem 3.1, the solution of (4.7), is given as

By taking absolute on (4.8) and using Remark 1, we get

By using (3.22), (3.23) and (3.25), we obtain

Which prove the required result.

Theorem 4.4. Under the assumptions ($H_1$–$H_4$), the problem (1.2) is UH as well as GUH stable, if $\mathscr{L} < 1,$ where $\mathscr{L}$ is defined in (3.27).

Proof. For any solution $\mathscr{U}\in PC(J, R)$, and unique solution $\mathscr{U}^*$ of the the given problem (1.2), then

Using Theorem 3.2 and Lemma 4.1 we have

Let $\mathcal{B}_1 = \frac{\mathscr{L}_U}{1-\mathscr{L}}$, then the solution of the consider problem (1.2) is UH stable further set $\mathcal{K}(\epsilon) = \epsilon$ then the consider problem (1.2) is GUH stable.

To prove the next stability result we need the following assumption given as $(H_9)$. For any non decreasing function $\chi\in PC(J, R^+)$ there exist a positive constant $\mathscr{G}$, such that

Lemma 4.2. If assumption ($H_9$) holds then for any solution $x\in PC(J, R)$ the muti-term fractional delay differential equation

satisfy the following relation,

Proof. We omit the proof as it is straightforward and may be derived like Lemma 4.1.

Theorem 4.5. Under the assumption ($H_1$–$H_4$) and $H_9$, the problem (1.2) is UHR stable and GUHR stable, if $\mathscr{L} < 1$.

Proof. For any solution $\mathscr{U}\in PC(J, R)$, and unique solution $\mathscr{U}^*$ of the the given problem (1.2), then

Using Theorem 3.2 and Lemma 4.2, we have

Let $\mathcal{B}_1 = \frac{max\Big(\mathscr{B}_I +\mathscr{B}_{I^*} +\mathscr{B}_{\mathscr{L}_\mathscr{R}}, \frac{\mathscr{G}}{|\sigma_1|}\Big)}{1-\mathscr{L}}$, then the solution of the consider problem (1.2) is UHR stable further set $\epsilon = 1$ then the consider problem (1.2) is GUHR stable.

5.   Example

Here we present an example to demonstrate our results.

Example 1. Consider the following multi-term impulsive fractional delay differential equation

here

Hence, we obtain $\mathscr{L}_{\mathscr{F}} = \frac{L}{44^2}\Big(\frac{2}{cos(1)}+1\Big),$ and similarly by simple computation, one can calculate,

Hence, we can calculate that $\mathfrak{L} < 1$ if $G < 24.608599$. Now as a consequence of Theorem 3.2, we conclude that example 1, has unique solution.

Set $\chi(t) = (4t-1)^2$, and $\psi = 3 \; \forall t\in J$. since

condition $H_9$ is satisfied with $\mathscr{G} = \frac{2}{6.3103}$. Hence all assumption of Theorem 4.5 are satisfied so as a conclusion the solution of problem 1.2 is UHR and GUHR stable. Moreover problem $(1.2)$ is UH and GUH stable due to Theorem 4.4.

6.   Conclusions

In this research work we have established a detailed analysis about existence theory of at least one solution to a class of multi term impulsive FODEs. The required theory has been established by using Krasnoselskii's fixed-point theorem and Banach contraction principle. Also keeping in mind the importance of stability, we have developed some results regarding different kinds of Ulam stability including HU, GHU, HUR and GHUR. The obtained analysis has been demonstrated by using pertinent examples.

Acknowledgments

Authors Kamal Shah, Nabil Mlaiki and Thabet Abdeljawad would like to thank Prince Sultan University for paying the APC and support through research Lab TAS.

Conflict of interest

No conflict of interest exists.