Extremal Mild Solutions of Fractional Evolution Equation with Mixed Monotone Impulsive Conditions

Abstract

The well established mixed monotone iterative technique is utilized explicitly to study the existence of the extremal solutions of a class of impulsive system with Hilfer fractional order in this paper. The procedure of finding mild L-quasi solutions of such impulsive evolution equation with non-compact semigroups involves measure of non-compactness and Sadovskii’s fixed point theorem as well. An example is provided to illustrate the main results.

1 Motivation

Over the years, the basic study of differential equations mainly leads to finding its extremal solutions. Conditions under which a differential system has a unique solution is always a challenge for the researchers. Various methods are used to investigate the uniqueness of the solution for the desired differential system. Successive approximation or the iterative technique are the typical methods used to determine such a unique solution. Du and Lakshmikantham [9] constructed an iterative procedure for an initial value problem,

$$\begin{aligned} x^{'}=g(t,x); x(0)=x_{0}, \end{aligned}$$

where the unique solution was guaranteed by upper and lower solutions in a closed set. Later, the case leading to the decomposition of the function into monotonically decreasing function and monotonically increasing function guided Guo and Lakshmikantham [13] to introduce the concept of coupled fixed point. Their work focussed on the existing criteria for both continuous and discontinuous operators, defined as $A:D\times D\rightarrow E$, where D is the subset of the Banach space E, which is partially ordered by a cone N ([14] by Guo and Lakshmikantham can be referred for more details regarding cone) and the operator A(x, y) is non-decreasing in x and non-increasing in y, which was termed as mixed monotone. The mixed monotonicity property finds itself useful mainly in the convergence analysis, global stability analysis, qualitative analysis etc. In continuation, Guo [12] investigated the existence and uniqueness of a mixed monotone operator for a general case where the operator need not be continuous. Chang and Ma [3] studied the existence of coupled fixed points of set-valued operators defined as $A:D\times D\rightarrow 2^{E}$. As an application, Chang and Ma [3] considered the functional equation

$$\begin{aligned} g(x)=\sup _{y\in D}[f(x,y)+F\big (x,y,g(T(x,y))\big )],x\in S. \end{aligned}$$

(1)

Here S is the state space, D the decision space, $R=(-\infty ,\infty )$, $T:S\times D\rightarrow S$, $f:S\times D\rightarrow R$ and $F:S\times D\times R\rightarrow R$. This integral often emerges in dynamic programming. Sun and Liu [28] improved the existing results based on the conditions of the operator, where their conditions do not require the operator to be continuous or the cone N to be normal. The authors’ Sun and Liu [28] implemented their conditions to nonlinear Hammerstein integral equation given by

$$\begin{aligned} g(x)=\int _{G}k(x,y)f\big (y,g(y)\big )dy, \end{aligned}$$

with G the bounded closed subset of ${\mathbb {R}}^{n}$, $k(x,y):G\times G \rightarrow {\mathbb {R}}^{1}$ the non-negative operator and $f(y,g(y))=f_{1}(y,g(y))+f_{2}(y,g(y))$ with $f_{1}(y,g(y))$ is non-decreasing in x and $f_{2}(y,g(y))$ is non-increasing in y. Meanwhile, Chang and Guo [2] studied the existence and uniqueness of multiple mixed monotone operators. These multiple mixed monotone operators are applied, for example, in a system of a functional equation in dynamic programming of multistage decision process given as

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle f(x)=\sup _{y\in D}\big [\phi (x,y)+G\big (x,y,g(T(x,y))\big )\big ],\\ \displaystyle g(x)=\sup _{y\in D}\big [\phi (x,y)+F\big (x,y,f(T(x,y))\big )\big ], \end{array} \right. \end{aligned}$$

where the operators are defined as in (1). Zhang [34] analyzed the case when the operator is convex in nature. To be more precise, Zhang considered the operator, say A, satisfying

$$\begin{aligned} A\big (tx+(1-t)y\big )\le tAx+(1-t)Ay, \end{aligned}$$

for $x,y\in D(A)$ with $x\le y$ and $t\in [0,1]$. It is to be noted that A is concave if $-A$ is convex. Zhang [34] gave conditions for the existence and uniqueness of fixed points of such convex–concave mixed monotone operators. Application to differential equations in Banach spaces and to nonlinear integral equations in an unbounded region was also given by Zhang [34]. A general method of finding the fixed points of such $\phi$ concave-$(-\psi )$ convex operators was not developed until Xu and Jia [32] gave a typical approach for such mixed monotone operators and gave a definition for the operator A to be $\phi$ concave-($-\psi$)convex. As an application, the authors studied a nonlinear integral equation in bounded/unbounded region. Soon, the mixed monotone operators were used effectively to analyze the systems involving impulsive conditions. Chen [4] employed mixed monotone iterative technique to deal with the existence of solutions of the impulsive periodic boundary value problem in a Banach space E which is given below, with the help of coupled upper and lower L-quasi solutions.

$$\begin{aligned} \left\{ \begin{array}{ll} x^{'}(t)=g\big (t,x(t),x(t)\big ), t\in [0,\omega ], t\ne t_{k},\\ \varDelta x|_{t=t_{k}}=J_{k}\big (x(t_{k}),x(t_{k})\big ), k=1,2,\ldots ,l,\\ x(0)=x(\omega ). \end{array} \right. \end{aligned}$$

Here $g\in C([0,\omega ]\times E\times E,E)$, $0<t_{1}<t_{2}<\cdots<t_{l}<\omega$, $J_{k}\in C(E\times E,E)$ an impulsive function and $\varDelta x|_{t=t_{k}}$ is the jump at $t=t_{k}$. Chen and Li [5] studied the existence of mild L-quasi solutions for initial value problems. Their results enhanced some related theories in ordinary and partial differential equations and generalized many previous results in this direction.

Fractional order differential equation is more practical oriented, which can be realized due to their impact in problems with real-time applications. It is natural to study the existence of solutions using mixed monotone operators for fractional-order systems. Chen and Li [6] considered a fractional nonlocal evolution system of the form

$$\begin{aligned} \left\{ \begin{array}{ll} {}^{C}D^{q}_{t}x(t)+Ax(t)=g\big (t,x(t),x(t)\big ), t\in J=[0,\omega ],\\ x(0)=f(x,x), \end{array} \right. \end{aligned}$$

(2)

where ${}^{C}D^{q}_{t}$ is the Caputo derivative of order $q\in (0,1)$, $A:D(A)\subset E \rightarrow E$ a closed linear operator such that $-A$ generates Q(t) a uniformly bounded $C_{0}$-semigroup, g is a nonlinear function defined as $g\in C(J\times E\times E,E)$ and f(x, x) a nonlocal function. Using operator semigroup theory, theory of mixed monotone operator, and some perturbation methods, Chen and Li [6] obtained the coupled minimal and maximal mild L-quasi solutions of (2). It can be noted that when $L\equiv 0$, the coupled minimal and maximal mild L-quasi solutions are equivalent to minimal and maximal mild quasi solutions of (2). For semi-linear evolution equation with impulsive conditions, Li and Gou [22] studied the existence of mild L-quasi solutions of the periodic boundary value problem given by

$$\begin{aligned} \left\{ \begin{array}{ll} {}^{C}D^{q}_{t}x(t)=g\big (t,x(t),x(t)\big ), t\in [0,\omega ], t\ne t_{k},\\ \varDelta x|_{t=t_{k}}=J_{k}\big (x(t_{k}),x(t_{k})\big ), k=1,2,\ldots ,l,\\ x(0)=x(\omega ), \end{array} \right. \end{aligned}$$

where the operator, derivative and the function g are defined as in (2), $J_{k}$ an impulsive function, $0=t_{0}<t_{1}<\cdots<t_{l}<\omega$, $\varDelta x|_{t=t_{k}}= x(t_{k}^{+})-x(t_{k}^{-})$ with $x(t_{k}^{+})$ and $x(t_{k}^{-})$ correspond to right and left limit of x(t) at $t=t_{k}$. Another notable work in this direction is the work of Zhao and Wang [35] on the mixed monotone technique for the fractional impulsive system of Caputo order.

The existence of solutions of the system with Hilfer fractional derivative, which acts as an interim between the two classical fractional derivative Caputo and Riemann–Liouville derivative which was outlined by Hilfer [20] was examined by Furati et al. [10]. Consequently, Guo and Li [15] utilized the fixed point theorem and monotone iterative technique to study the existence of extremal solutions of the Hilfer fractional nonlocal evolution equation. Recently, Guo et al. [17] investigated the existence of mild L-quasi solutions using fixed point theorem along with the mixed monotone iterative technique of nonlocal Hilfer fractional evolution equation defined as

$$\begin{aligned} \left\{ \begin{array}{ll} D^{p,q}_{0+}x(t)+Ax(t)=g\big (t,x(t),G(x(t))\big ), t\in (0,\omega ],\\ I_{0+}^{(1-p)(1-q)}x(0)=x_{0}+\sum _{k=1}^{l}\lambda _{k}x(\tau _{k}), \tau _{k}\in (0,\omega ], \end{array} \right. \end{aligned}$$

where $D^{p,q}_{0+}x(t)$ denotes the Hilfer fractional derivative of order q and type p with $\frac{1}{2}<q<1$ and $0\le p\le 1$, the operator G is defined as $Gx(t)=\int _{0}^{t}K(t,s)x(s)ds$, $\tau _{k}$ are prefixed points with $k=1,2,\ldots ,l$ satisfying $0\le \tau _{1}\le \tau _{2}\le \cdots \le \tau _{l}<\omega$ and $\lambda _{k}$ are real numbers. However, as many physical problems are impulsive, it is ideal to study an impulsive system. It is challenging to work on the existence of the lower and upper solutions of impulsive fractional systems with the Hilfer fractional order. One can refer to the articles on the impulsive system with Hilfer fractional derivative by Ahmed et al. [1], Debbouche and Antonov [7] and by Du et al. [8]. Hence, it is worth studying the existence of coupled mild L-quasi solutions of the Hilfer fractional impulsive system, which is given as

$$\begin{aligned} \left\{ \begin{array}{ll} D_{0+}^{\mu ,\nu }x(t)+Ax(t)= g\big (t,x(t),x(t)\big ), t \in J=[0,T], t\ne t_{k},\\ \varDelta I_{t_{k}}^{(1-\lambda )}x(t_{k})=\phi _{k}\big (x(t_{k}),x(t_{k})\big ),k=1,2,\ldots l,\\ I_{0+}^{(1-\lambda )}[x(0)]= x_{0}, \end{array} \right. \end{aligned}$$

(3)

where $D_{0+}^{\mu ,\nu }$ denotes the Hilfer fractional derivative of order $0<\mu <1$, type $0\le \nu \le 1$ and $\lambda =\mu +\nu -\mu \nu$. $A:D(A)\subseteq E\rightarrow E$ is a closed linear operator and $-A$ generates a $C_{0}$-semigroup $Q(t)(t\ge 0)$ on the Banach space E. Let the impulse effect takes place at $t=t_{k}$, for $(k=1,2,\ldots ,l)$ and $\varDelta I_{t_{k}}^{1-\lambda }x(t_{k})$ determines the size of the jump at time $t_{k}$. In other words, the impulsive moments meet the relation $\varDelta I_{t_{k}}^{1-\lambda }x(t_{k})=I_{t_{k}^{+}}^{1-\lambda }x(t_{k}^{+})-I_{t_{k}^{-}}^{1-\lambda }x(t_{k}^{-})$, where $I_{t_{k}^{+}}^{1-\lambda }x(t_{k}^{+})$ and $I_{t_{k}^{-}}^{1-\lambda }x(t_{k}^{-})$ denotes the right and the left limit of $I_{t_{k}}^{1-\lambda }x(t)$ at $t=t_{k}$ with $0=t_{0}<t_{1}\cdots<t_{l}<t_{l+1}=T$, $g\in C(J\times E \times E,E)$, $x_{0}\in E$, $\phi _{k}\in C(E\times E,E)$.

Remark 1

The nonlinear term in the problem is of the form g(t, x, x). If, $g(t,x,x)=g_{1}(t,x)+g_{2}(t,x)$, where $g_{1}(t,x)$ is nondecreasing in x and $g_{2}(t,x)$ is non-increasing in x, the usual monotone iterative method cannot be applied. Therefore it is necessary to use a mixed monotone iterative method to study the existence of extremal solutions to such a problem. This motivated the authors to study the system using a mixed monotone iterative method, which is for a more general setup. Also, the study of extremal solutions using lower and upper solutions along with mixed monotone iterative technique has not been studied for an impulsive system with Hilfer fractional derivative. Even though, it has been studied for the impulsive system with Caputo derivative, it has not been studied for impulsive system with Riemann Liouville derivative. For values $\nu =0$, the result reduce to a system with Riemann–Liouville derivative.

The following can be given as a key point regarding this paper:

• 1. It is worth mentioning that the conditions assumed in this paper are such that the main theorem holds even if a regular positive cone replaces the normal positive cone N.

• 2. The theorem is stated for a more general case for which the operator is a positive $C_{0}$-semigroup generated by -A and need not be compact.

• 3. Our proof requires the nonlinear function to be continuous. The theory of partially ordered sets is to be used to study the convergence of iterative solutions of discontinuous nonlinear functions.

The rest of the paper is organized in the following way: Sect. 2 includes essential definitions and lemmas for the main results, while Sect. 3 contains the main result under appropriate assumptions. Section 4 possesses an example to ascertain the main results, and finally, Sect. 5 gives an outline of the case for which the system has a higher-order derivative and other open problems in this direction.

2 Essential Notions

This section includes some basic results, definitions, and lemmas relevant to this paper.

Definition 1

(see, [14]) Let E be a real Banach space. A nonempty convex closed set $N\subset E$ is called a cone if it satisfies the following two conditions:

• 1. $x\in N$, $\eta \ge \theta$ $\Rightarrow$ $\eta x\in N$.

• 2. $x\in N$, $-x \in N$ $\Rightarrow$ $x=\theta$, where $\theta$ denotes the zero element of E.

Every cone N in E defines a partial ordering in E given by $x\le y$ $\iff$ $y-x\in N$.

Definition 2

[14] A cone N is said to be normal, if there exists a positive constant ${\tilde{N}}$ such that $\forall$ $x,y\in N$,

$$\begin{aligned} \theta \le x\le y \Rightarrow \Vert x\Vert \le {\tilde{N}}\Vert y\Vert . \end{aligned}$$

It is to be noted that a normal cone is always convex, and a cone is said to be positive if for $x,y\in N$, $y-x\in N$, $\forall$ $x<y$.

Definition 3

[12] Let E be an ordered Banach space generated by the positive cone $N=\{x\in E|x(t)\ge \theta \}$. An operator $A:D\times D\rightarrow E$ with $D\subset E$ is said to be mixed monotone if A(x, y) is non-decreasing in x and non-increasing in y, that is for $x_{1},x_{2},y \in D$, if $x_{1}\le x_{2}$ then, $A(x_{1},y)\le A(x_{2},y)$. Similarly, for $y_{1},y_{2},x \in D$, if $y_{1}\le y_{2}$, then, $A(x,y_{1})\ge A(x,y_{2})$. Also, a point $({\hat{x}},{\hat{y}}) \in D\times D$ is called a coupled fixed point of A if $A({\hat{x}},{\hat{y}})={\hat{x}}$ and $A({\hat{y}},{\hat{x}})={\hat{y}}$ and ${\hat{x}}$ is the fixed point of A if $A({\hat{x}},{\hat{x}})={\hat{x}}$.

Let C(J, E) denote the space of all E-valued continuous function from J to E. Also let $C_{1-\lambda }(J,E)$ be defined as $C_{1-\lambda }(J,E)=\{x:t^{1-\lambda }x(t)\in C(J,E)\}$. Clearly $C_{1-\lambda }(J,E)$ is an ordered Banach space induced by the positive cone $N^{'}=\{x\in C_{1-\lambda }(J,E)|x(t)\ge \theta , t\in J\}$. Here N and $N^{'}$ both are normal with the same normal constant ${\tilde{N}}$. Let PC(J, E) be an ordered Banach space defined as $PC(J,E)=\{x:J\rightarrow E, x(t)\text{ is } \text{ continous } \text{ at }t\ne t_{k} \text{ and } x(t_{k}^{+}) \text{ exists },k=1,2,\ldots ,l\}$, with the norm $\Vert x\Vert _{PC}=\sup \{\Vert x(t)\Vert :t\in J\}$. Let the piecewise continuous Banach space be defined as

$$\begin{aligned} PC_{1-\lambda }(J,E)=\{x:(t-t_{k})^{1-\lambda }x(t) \in C\big ((t_{k},t_{k+1}],E\big ) \end{aligned}$$

and $\displaystyle \lim _{t\rightarrow t_{k+}}(t-t_{k})^{1-\lambda }x(t)$, $k=1,2,\ldots l$ exists} with the norm

$$\begin{aligned} \Vert x(t)\Vert _{PC_{1-\lambda }}=\max \Big \{\sup _{t\in (t_{k},t_{k+1}]}(t-t_{k})^{1-\lambda }\Vert x(t)\Vert :k=0,1,\ldots ,l\Big \}. \end{aligned}$$

The fractional integral of order $\mu$ for an integrable function g is given as [24],

$$\begin{aligned} I^{\mu }_{t}g(t):=\frac{1}{\varGamma (\mu )}\int ^{t}_{0}(t-s)^{\mu -1}g(s)ds, 0< \mu <1. \end{aligned}$$

Here $\varGamma (\cdot )$ is the well-known Gamma function. Also, the fractional derivative of Caputo and Riemann–Liouville of order $\mu$, respectively, are given by [24],

$$\begin{aligned} ^{C}D^{\mu }_{0+}g(t):=\frac{1}{\varGamma (1-\mu )}\int ^{t}_{0}\frac{g'(s)}{(t-s)^{\mu }}ds, t>0,0< \mu <1, \end{aligned}$$

and

$$\begin{aligned} ^{RL}D^{\mu }_{0+}g(t):=\frac{1}{\varGamma (1-\mu )}\left( \frac{d}{dt}\right) \int ^{t}_{0}\frac{g(s)}{(t-s)^{\mu }}ds,t>0,0< \mu <1. \end{aligned}$$

The Hilfer fractional derivative of order $0< \mu <1$ and type $0\le \nu \le 1$ of function g(t) is given as

$$\begin{aligned} \left( D^{\mu ,\nu }_{0\pm }g\right) (t)=\left( I_{0\pm }^{\nu (1-\mu )}D\big (I_{0\pm }^{(1-\nu )(1-\mu )}g\big )\right) (t) \end{aligned}$$

where $D:=\frac{d}{dt}$. Gu and Trujillo [11] can be referred for more details on Hilfer fractional derivative. Moreover, Riemann-Liouville and Caputo can be regarded as a special case of Hilfer fractional derivative, respectively, as

$$\begin{aligned} D_{0+}^{\mu ,\nu }:= \left\{ \begin{array}{ll} DI_{0+}^{1-\mu }={ }^{RL}D_{0+}^{\mu },\nu =0\\ I_{0+}^{1-\mu }D={ }^{C}D^{\mu }_{0+},\nu =1. \end{array} \right. \end{aligned}$$

The parameter $\lambda$ satisfies $\lambda =\mu +\nu -\mu \nu , 0<\lambda \le 1$.

Definition 4

[35] A $C_{0}$-semigroup $\{Q(t)\}_{t\ge 0}$ is said to be positive if the order inequality $Q(t)x\ge \theta$ holds for each $x\ge \theta$, $x\in E$, and $t\ge 0$.

It can be referred [19] that the Kuratowski measure of non-compactness measure denoted by $\alpha (\cdot )$ is defined on a bounded set. For any $t\in J$ and $B \subset C(J,E)$, define $B (t)=\{x(t):x\in B\}$. If B is bounded in C(J, E), then B is bounded in E. Also, $\alpha (B(t))\le \alpha (B)$.

In this regard, Guo et al. [18] studied the Hyers–Ulam stability of fractional differential equations using the Hausdorff measure of non-compactness. The following lemmas are necessary to prove the main theorem in the next section.

Lemma 1

[17] Let E be a Banach space and let $D\subset E$ be bounded. Then there exists a countable set $D_{0}\subset D$ such that $\alpha (D)\le 2\alpha (D_{0})$.

Lemma 2

[19] Let $B_{p}=\{x_{p}\}\subset C(J,E),(p=1,2,\ldots )$ be a bounded and countable set. Then $\alpha (B_{p}(t))$ is Lebesgue integral on J, and

$$\begin{aligned} \alpha \left( \Big \{\int _{J}x_{p}(t)dt|_{p=1,2,\ldots ,}\Big \}\right) \le 2\int _{J}\alpha (B_{p}(t))dt. \end{aligned}$$

Lemma 3

[17] Let E be a Banach space and let $D \subset C([b_{1},b_{2}],E)$ be bounded and equicontinuous. Then $\alpha (D(t))$ is continuous on $[b_{1},b_{2}]$ and

$$\begin{aligned} \displaystyle \alpha (D)=\max _{t\in [b_{1},b_{2}]}\alpha (D(t)). \end{aligned}$$

Lemma 4

[25] (Sadovskii fixed point theorem) Let E be a Banach space and $\varOmega$ be a nonempty bounded convex closed set in E. If ${\mathcal {Q}}:\varOmega \rightarrow \varOmega$ is a condensing mapping, then ${\mathcal {Q}}$ has a fixed point in $\varOmega$.

The following lemma is with reference to the generalized Gronwall inequality for fractional differential equations.

Lemma 5

[33] Suppose $b\ge 0$, $\beta > 0$ and a(t) be a nonnegative function locally integrable on $0\le t <T$ (some $T\le +\infty$), and that x(t) be nonnegative and locally integrable on $0\le t <T$ with

$$\begin{aligned} x(t)\le a(t)+b\int _{0}^{t}(t-s)^{\beta -1}x(s)ds \end{aligned}$$

on this interval. Then

$$\begin{aligned} x(t)\le a(t)+\int _{0}^{t}\Big [\sum _{n=1}^{\infty }\frac{(b\varGamma (\beta ))^{n}}{\varGamma (n\beta )}(t-s)^{n\beta -1}a(s)\Big ]ds, 0\le t <T. \end{aligned}$$

Definition 5

[7] A function $x \in PC_{1-\lambda }(J,E)$ is called the mild solution of the system (3), if for $t \in J$ it satisfies the following integral equation

$$\begin{aligned} x(t)=S_{\mu ,\nu }(t)x_{0}+&\displaystyle \sum _{i=1}^{k}S_{\mu ,\nu }(t-t_{i})\phi _{i}\big (x(t_{i}),x(t_{i})\big )\nonumber \\&+\int _{0}^{t}(t-s)^{\mu -1}P_{\mu }(t-s)g\big (s,x(s),x(s)\big )ds \end{aligned}$$

(4)

where

$$\begin{aligned}&\displaystyle S_{\mu ,\nu }(t) = I_{0+}^{\nu (1-\mu )}K_{\mu }(t),K_{\mu }(t)=t^{\mu -1}P_{\mu }(t),P_{\mu }(t)= \int _{0}^{\infty }\mu \theta \xi _{\mu }(\theta )Q(t^{\mu }\theta )d\theta , \\&\displaystyle \varpi _{\mu }(\theta )= \frac{1}{\pi } \sum _{n=1}^{\infty }(-1)^{n-1}\theta ^{-n\mu -1} \frac{\varGamma (n\mu +1)}{n!}\sin (n\pi \mu ),\theta \in (0,\infty ) \end{aligned}$$

and $\xi _{\mu }(\theta )= \frac{1}{\mu }\theta ^{-1-\frac{1}{\mu }}\varpi _{\mu }(\theta ^{-\frac{1}{\mu }})$ is a probability density function defined on $(0,\infty )$, that is

$$\begin{aligned} \xi _{\mu }(\theta )\ge 0 \text{ and } \int ^{\infty }_{0}\xi _{\mu }(\theta )d\theta =1. \end{aligned}$$

Remark 2

• 1. From [10], when $\nu =0$, the solution reduces to the solution of classical Riemann–Liouville fractional derivative, that is, $S_{\mu ,0}(t)=K_{\mu }(t)$.

• 2. Similarly when $\nu =1$, the solution reduces to the solution of classical Caputo fractional derivative, that is $S_{\mu ,1}(t)=S_{\mu }(t)$.

Lemma 6

[7] If the analytic semigroup $Q(t)(t\ge 0)$ is bounded uniformly, then the operator $P_{\mu }(t)$ and $S_{\mu ,\nu }(t)$ satisfies the following bounded and continuity conditions.

• 1. $S_{\mu ,\nu }(t)$ and $P_{\mu }(t)$ are linear bounded operators and for any $x\in E$,

  $$\begin{aligned} \Vert S_{\mu ,\nu }(t)x\Vert _{E}\le \frac{M t^{\lambda -1}}{\varGamma (\lambda )}\Vert x\Vert _{E}\text{ and }\Vert P_{\mu }(t)x\Vert _{E}\le \frac{M}{\varGamma (\mu )}\Vert x\Vert _{E}. \end{aligned}$$

• 2. $S_{\mu ,\nu }(t)$ and $P_{\mu }(t)$ are strongly continuous, which means that for any $x\in E$ and $0<t^{'}<t^{''}\le T$,

  $$\begin{aligned} \Vert P_{\mu }(t')x-P_{\mu }(t'')x\Vert _{E}\rightarrow 0 \text{ and }\Vert S_{\mu ,\nu }(t')x-S_{\mu ,\nu }(t'')x\Vert _{E}\rightarrow 0 \text{ as }t''\rightarrow t'. \end{aligned}$$

3 Main Results

To prove the main results of this paper, a perturbed equivalent system with constant $C\ge 0$ given below is taken for consideration.

$$\begin{aligned} \left\{ \begin{array}{ll} D_{0+}^{\mu ,\nu }x(t)+(A+CI)x(t)= g\big (t,x(t),x(t)\big )+Cx(t), t \in J, t\ne t_{k},\\ \varDelta I_{t_{k}}^{(1-\lambda )}x(t_{k})=\phi _{k}\big (x(t_{k}),x(t_{k})\big ),k=1,2,\ldots l,\\ I_{0+}^{(1-\lambda )}[x(0)]= x_{0}. \end{array} \right. \end{aligned}$$

(5)

With reference to [23], for any $C\ge 0$, $-(A+CI)$ generates an analytic semigroup $R(t)=e^{-Ct}Q(t)$ and for $t\ge 0$, R(t) is positive and $\displaystyle \sup _{t\in [0,\infty )}\Vert R(t)\Vert \le M^{*}$ for $M^{*}\ge 1$.

Lemma 7

With respect to the above perturbed system, the operators $S^{*}_{\mu ,\nu }(t)$ and $P^{*}_{\mu }(t)$ have the following properties.

• 1. Let $S^{*}_{\mu ,\nu }(t)$ and $P^{*}_{\mu }(t)$ for $t\ge 0$ be two families of operators defined by

  $$\begin{aligned} S^{*}_{\mu ,\nu }(t)=&I_{0+}^{\nu (1-\mu )}K^{*}_{\mu }(t),\quad K^{*}_{\mu }(t)=t^{\mu -1}P^{*}_{\mu }(t),\\&P^{*}_{\mu }(t)=\int _{0}^{\infty }\mu \theta \xi _{\mu }(\theta )R(t^{\mu }\theta )d\theta . \end{aligned}$$

• 2. The above operators are positive for $(t\ge 0)$ and for any $x\in E$,

  $$\begin{aligned} \Vert S^{*}_{\mu ,\nu }(t)\Vert \le \dfrac{M^{*} t^{\lambda -1}}{\varGamma (\lambda )},\Vert P^{*}_{\mu }(t)\Vert \le \dfrac{M^{*}}{\varGamma (\mu )}\text{ and }\Vert K^{*}_{\mu }(t)\Vert \le \dfrac{M^{*}t^{\mu -1}}{\varGamma (\mu )}. \end{aligned}$$

• 3. $S^{*}_{\mu ,\nu }(t)$ and $P^{*}_{\mu }(t)$ are strongly continuous, which means that for any $x\in E$, $0<t^{'}<t^{''}\le T$ and as $t''\rightarrow t'$,

  $$\begin{aligned} \Vert P^{*}_{\mu }(t')x-P^{*}_{\mu }(t'')x\Vert _{E}\rightarrow 0 \text{ and }\Vert S^{*}_{\mu ,\nu }(t')x-S^{*}_{\mu ,\nu }(t'')x\Vert _{E}\rightarrow 0. \end{aligned}$$

Definition 6

A function $x\in PC_{1-\lambda }(J,E)$ is said to be a mild solution of the system (5) if x satisfies the following integral equation.

$$\begin{aligned} x(t)=&S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \sum _{i=1}^{k}S^{*}_{\mu ,\nu }(t-t_{i})\phi _{i}\big (x(t_{i}),x(t_{i})\big )\\&+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,x(s),x(s)\big )+Cx(s)\Big ]ds. \end{aligned}$$

For $y,z\in PC_{1-\lambda }(J,E)$, [y, z] is used to denote the order interval $\{x\in PC_{1-\lambda }(J,E):y\le x\le z\}$ and for $t\in J$, [y(t), z(t)] denotes the order interval $\{x(t)\in PC_{1-\lambda }(J,E):y(t)\le x(t)\le z(t)\}$.

Definition 7

If $y_{0}, z_{0}\in PC_{1-\lambda }(J,E)$ satisfy all the following inequalities

$$\begin{aligned}&\left\{ \begin{array}{ll} D_{0+}^{\mu ,\nu }y_{0}(t)+Ay_{0}(t)\le g\big (t,y_{0}(t),z_{0}(t)\big )+L\big (y_{0}(t)-z_{0}(t)\big ), t \in J, t\ne t_{k}\\ \varDelta I_{t_{k}}^{(1-\lambda )}y_{0}(t_{k})\le \phi _{k}\big (y_{0}(t_{k}),z_{0}(t_{k})\big ),k=1,2,\ldots l\\ I_{0+}^{(1-\lambda )}[y_{0}]\le x_{0} \end{array} \right. \\&\left\{ \begin{array}{ll} D_{0+}^{\mu ,\nu }z_{0}(t)+Az_{0}(t)\ge g\big (t,z_{0}(t),y_{0}(t)\big )+L\big (z_{0}(t)-y_{0}(t)\big ), t \in J, t\ne t_{k}\\ \varDelta I_{t_{k}}^{(1-\lambda )}z_{0}(t_{k})\ge \phi _{k}\big (y_{0}(t_{k}),z_{0}(t_{k})\big ),k=1,2,\ldots l\\ I_{0+}^{(1-\lambda )}[z_{0}]\ge x_{0} \end{array} \right. \end{aligned}$$

for a constant $L\ge 0$, then $y_{0}$ and $z_{0}$ are called the coupled lower and upper L-quasi solutions of the problem (3). If the inequalities are replaced by equality, then $y_{0},z_{0}$ are called coupled L-quasi solutions. And when $x_{0}:=y_{0}=z_{0}$, $x_{0}$ is called the solution of the problem (3).

The following theorem guarantees the existence of the extremal mild solution of the impulsive system (3).

Theorem 1

Let E be an ordered Banach space with the positive normal cone N. Assume that $Q(t)\ge 0$ and the impulsive system (3) has both lower and upper solutions given by $y_{0}$ and $z_{0}$, respectively, where $y_{0}, z_{0}\in PC_{1-\lambda }$, and $y_{0}\le z_{0}$. By embracing the mixed monotone iterative procedure and considering the following assumptions, the impulsive system (3) has extremal solutions between $y_{0}$ and $z_{0}$.

A(1). There exist constants $C\ge 0$ and $L\le 0$ such that

$$\begin{aligned} g(t,y_{2},z_{2})-g(t,y_{1},z_{1})\ge -C(y_{2}-y_{1})-L(z_{1}-z_{2}) \end{aligned}$$

with $y_{0}(t)\le y_{1}(t)\le y_{2}(t)\le z_{0}(t)$, $y_{0}(t)\le z_{2}(t)\le z_{1}(t)\le z_{0}(t)$ for any $t\in J$.

A(2). The impulsive function for $t\in J$ satisfies

$$\begin{aligned} \phi _{k}(y_{1},z_{1})\le \phi _{k}(y_{2},z_{2}), k=1,2,\ldots ,l. \end{aligned}$$

A(3). The sequence $\{y_{p}\}\subset [y_{0}(t),z_{0}(t)]$ and $\{z_{p}\}\subset [y_{0}(t),z_{0}(t)]$ for $t\in J$ are, respectively, increasing and decreasing monotonic sequences. In particular, there exists a constant $L_{1}\ge 0$ such that

$$\begin{aligned} \alpha \Big (\{g(t,y_{p},z_{p})\}\Big )\le L_{1}\Big ( \alpha \big (\{y_{p}\}\big )+\alpha \big (\{z_{p}\}\big )\Big ),p=1,2,\ldots ,. \end{aligned}$$

A(4). Let $y_{p}={\mathcal {G}}(y_{p-1},z_{p-1})$, $z_{p}={\mathcal {G}}(z_{p-1},y_{p-1})$, $p=1,2,\ldots ,$ such that sequence $y_{p}(0)$ and $z_{p}(0)$ are convergent.

Proof

As $C>0$, the problem (3) can be presented in the form of problem (5). So the proof of the existence of an unique mild solution for the problem (5) is sufficient. Define the operator ${\mathcal {G}}:[y_{0},z_{0}]\times [y_{0},z_{0}]\rightarrow PC_{1-\lambda }(J,E)$ by

$$\begin{aligned} {\mathcal {G}}(y,z)(t)= \left\{ \begin{array}{ll} S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y(s),z(s)\big )+\\ (C+L)y(s)-Lz(s)\Big ]ds, t\in [0,t_{1}],\\ S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \sum _{i=1}^{k}S^{*}_{\mu ,\nu }(t-t_{i})\phi _{i}\big (y(t_{i}),z(t_{i})\big )\\ +\displaystyle \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y(s),z(s)\big )\\ +(C+L)y(s)-Lz(s)\Big ]ds, \quad t\in (t_{k},t_{k+1}],k=1,2,\ldots l. \end{array} \right. \end{aligned}$$

(6)

The map ${\mathcal {G}}(y,z)(t)$ is continuous since g is continuous. By Definition 5, the fixed points of the operator ${\mathcal {G}}$ are equivalent to the mild solution of the system given in (4). That is,

$$\begin{aligned} {\mathcal {G}}(x,x)=x. \end{aligned}$$

(7)

The following steps are required for the completion of the proof.

Step 1. To show ${\mathcal {G}}(y_{1},z_{1})\le {\mathcal {G}}(y_{2},z_{2})$.

The assumption A(1) is used to reduce the inequalities below, which can be applied directly in the proof of the theorem. That is, $\forall t\in J^{'}$,

$$\begin{aligned}&y_{0}(t)\le y_{1}(t)\le y_{2}(t)\le z_{0}(t), y_{0}(t)\le z_{2}(t)\le z_{1}(t)\le z_{0}(t). \nonumber \\&\quad \Longrightarrow g\big (t,y_{1}(t),z_{1}(t)\big )+C y_{1}(t)-Lz_{1}(t)\le g\big (t,y_{2}(t),z_{2}(t)\big )+Cy_{2}(t)-Lz_{2}(t). \nonumber \\&\quad \Longrightarrow g\big (t,y_{1}(t),z_{1}(t)\big )+(C+L)y_{1}(t)-Lz_{1}(t)\nonumber \\&\quad \le g\big (t,y_{2}(t),z_{2}(t)\big )+(C+L)y_{2}(t)-Lz_{2}(t). \end{aligned}$$

(8)

Considering the case for $t\in J_{0}^{'}$, for $J_{0}^{'}=[0,t_{1}]$:-

The operators $S^{*}_{\mu ,\nu }(t)$ and $P^{*}_{\mu }(t)$ are positive operators, and hence when the mild solutions are compared, using (8), the following inequality is obtained.

$$\begin{aligned}&\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{1}(s),z_{1}(s)\big )+(C+L)y_{1}(s)-Lz_{1}(s)\Big ]ds\le \\&\quad \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{2}(s),z_{2}(s)\big )+(C+L)y_{2}(s)-Lz_{2}(s)\Big ]ds. \end{aligned}$$

In the case, for $\forall t\in J_{k}^{'}=(t_{k},t_{k+1}]$, $k=1,2,\ldots l$, applying the assumption A(2) yields

$$\begin{aligned}&S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \sum _{i=1}^{k}S^{*}_{\mu ,\nu }(t-t_{i})\phi _{i}\big (y_{1}(t_{i}),z_{1}(t_{i})\big )\\&\quad +\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{1}(s),z_{1}(s)\big )+(C+L)y_{1}(s)-Lz_{1}(s)]ds\le \\&S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \sum _{i=1}^{k}S^{*}_{\mu ,\nu }(t-t_{i})\phi _{i}\big (y_{2}(t_{i}),z_{2}(t_{i})\big )\\&\quad +\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{2}(s),z_{2}(s)\big )+(C+L)y_{2}(s)-Lz_{2}(s)\Big ]ds. \end{aligned}$$

Eventually, ${\mathcal {G}}(y_{1},z_{1})(t)\le {\mathcal {G}}(y_{2},z_{2})(t)$ for $t\in J$.

Step 2. To show $y_{0}\le {\mathcal {G}}(y_{0},z_{0})$ ; ${\mathcal {G}}(z_{0},y_{0})\le z_{0}$:-

The case for which $t\in J_{0}^{'}$:-

Let $D^{\mu ,\nu }_{0+}z_{0}(t)+Az_{0}(t)+Cz_{0}(t)=\xi (t)$, $\xi (t)\in PC_{1-\lambda }(J,E)$. By Definition 7 of the coupled upper L-quasi solution, the mild solution of the system (5) can be written as

$$\begin{aligned} z_{0}(t)=&S^{*}_{\mu ,\nu }(t)z_{0}+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\xi (s)ds\\ \ge&S^{*}_{\mu ,\nu }(t)x_{0}+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,z_{0}(s),y_{0}(s)\big )\\ +&(C+L)z_{0}(s)-Ly_{0}(s)\Big ]ds. \end{aligned}$$

From (6), it can be observed that $z_{0}(t)\ge {\mathcal {G}}(z_{0},y_{0})(t).$

For the case $t\in J_{1}^{'}$:-

$$\begin{aligned} S^{*}&_{\mu ,\nu }(t)z_{0}+S^{*}_{\mu ,\nu }(t-t_{1})\phi _{1}\big (z_{0}(t_{1}),y_{0}(t_{1})\big )+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\xi (s)ds\\&\ge S^{*}_{\mu ,\nu }(t)x_{0}+S^{*}_{\mu ,\nu }(t-t_{1})\phi _{1}\big (z_{0}(t_{1}),y_{0}(t_{1})\big )+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\\&\Big [g\big (s,z_{0}(s),y_{0}(s)\big )+(C+L)z_{0}(s)-Ly_{0}(s)\Big ]ds.\\ \Longrightarrow&z_{0}(t)\ge {\mathcal {G}}(z_{0},y_{0})(t). \end{aligned}$$

Progressing in the same manner for every interval $J_{k}^{'}$ yields, $z_{0}(t)\ge {\mathcal {G}}(z_{0},y_{0})(t)$. In the same way, we can prove that $y_{0}(t)\le {\mathcal {G}}(y_{0},z_{0})(t)$. Finally, we can conclude that

$$\begin{aligned} y_{0}(t)\le {\mathcal {G}}(y_{0},z_{0})(t)\le {\mathcal {G}}(x,x)(t)\le {\mathcal {G}}(z_{0},y_{0})(t)\le z_{0}(t). \end{aligned}$$

Therefore, the conclusion may be drawn that

$$\begin{aligned} {\mathcal {G}}:[y_{0},z_{0}]\times [y_{0},z_{0}] \rightarrow PC_{1-\lambda }(J,E) \end{aligned}$$

is an increasing mixed monotonic operator. By means of the iterative pattern, two sequences $\{y_{p}\}$ and $\{z_{p}\}$ can be defined as,

$$\begin{aligned} y_{p}={\mathcal {G}} (y_{p-1},z_{p-1}); z_{p}={\mathcal {G}} (z_{p-1},y_{p-1});p=1,2,\ldots . \end{aligned}$$

(9)

Eventually, due to the monotonicity property of ${\mathcal {G}}$, an increasing sequence is derived as,

$$\begin{aligned} y_{0}\le y_{1}\le y_{2}\le \cdots \le y_{p}\le \cdots \le z_{p}\le \cdots \le z_{2}\le z_{1}\le z_{0}. \end{aligned}$$

(10)

Step 3. Convergence of the sequences $\{y_{p}\}$ and $\{z_{p}\}$ in $J^{'}$:-

Let $B_{p}=\{y_{p}|p\in {\mathbb {N}}\}+\{z_{p}|p\in {\mathbb {N}}\}$; $B_{1}=\{y_{p-1}|p\in {\mathbb {N}}\}$; $B_{2}=\{z_{p-1}|p\in {\mathbb {N}}\}$; $B_{3}=\{(y_{p-1},z_{p-1})\}|p\in {\mathbb {N}}$ and $B_{4}=\{(z_{p-1},y_{p-1})\}|p\in {\mathbb {N}}$. (9) gives the relation $B_{1}={\mathcal {G}}(B_{3}(t))$ and $B_{2}={\mathcal {G}}(B_{4}(t))$. Let $\psi (t):=\alpha (B_{p}(t))$. By proving that $\psi (t)\equiv 0$ on every interval $J^{'}_{k}$ means that $\alpha (B_{p}(t))\equiv 0$ for $k=1,2,\ldots ,l$, and hence $\{y_{p}\}+\{z_{p}\}$ is precompact in E for every $t\in J$. Ultimately, by the definition of precompact, $\{y_{p}\}$ and $\{z_{p}\}$ have converging subsequence in E. Thus, it is necessary to prove that $\psi (t)\equiv 0$.

For $t\in J_{0}^{'}=(0,t_{1}]$:-

$$\begin{aligned} \psi (t)&=\alpha (B_{p}(t))=\alpha \Big (B_{1}(t)+B_{2}(t)\Big )\\&=\alpha \Big ({\mathcal {G}}(B_{3}(t))+\mathcal {G}(B_{4}(t))\Big )=\alpha \Big ({\mathcal {G}}(y_{p-1},z_{p-1})(t)+ {\mathcal {G}}(z_{p-1},y_{p-1})(t)\Big ) \\ \psi (t)&=\alpha \Big (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{p-1},z_{p-1}(s)\big )\\&+(C+L)y_{p-1}(s)-Lz_{p-1}(s)\Big ]ds+S^{*}_{\mu ,\nu }(t)x_{0}+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\\&\Big [g\big (s,z_{p-1}(s),y_{p-1}(s)\big )+(C+L)z_{p-1}(s)-Ly_{p-1}(s)\Big ]ds\Big \}:p\in {\mathbb {N}} \Big ). \end{aligned}$$

The below inequality is a consequence of Lemma 2.

$$\begin{aligned} \psi (t)\le&2 \int _{0}^{t}\alpha \Big (\Big \{(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{p-1}(s),z_{p-1}(s)\big )\\&+g\big (s,z_{p-1}(s),y_{p-1}(s)\big )+C\big (y_{p-1}(s)+z_{p-1}(s)\big )\Big ]ds\Big \}:p=1,2,\ldots \Big ). \end{aligned}$$

Applying the assumptions along with Lemma 6 results in

$$\begin{aligned} \psi (t)&\le \dfrac{2M^{*}}{\varGamma (\mu )}\int _{0}^{t}(t-s)^{\mu -1}\Big [(2L_{1}+C)\big (\alpha B_{1}(s)+\alpha B_{2}(s)\big )\Big ]ds.\\&=\dfrac{2M^{*}}{\varGamma (\mu )}(2L_{1}+C)\int _{0}^{t}(t-s)^{\mu -1}\psi (s)ds. \end{aligned}$$

By Lemma 5, $\psi (t)\equiv 0$ on $J_{0}^{'}$. Thus $\{y_{p}(t)\}+\{z_{p}(t)\}$ is precompact and hence $\{y_{p}(t)\}$ and $\{z_{p}(t)\}$ are precompact for $t\in [0,t_{1}]$. Let $\alpha (\phi (B_{p}(t_{1})))$. Now for $t\in J_{1}^{'}=(t_{1},t_{2}]$:-

$$\begin{aligned} \psi (t)=\alpha (B_{p}(t))&=\alpha \Big ({\mathcal {G}}(B_{3}(t))+{\mathcal {G}}(B_{4}(t))\Big )\\&=\alpha \Big ({\mathcal {G}}(y_{p-1},z_{p-1})(t)+ \mathcal {G}(z_{p-1},y_{p-1})(t)\Big ) \\ \psi (t)&=\alpha \Bigg (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}+S^{*}_{\mu ,\nu }(t)\phi _{1}\Big (y_{p-1}(t_{1}),z_{p-1}(t_{1})\Big ) +\int _{0}^{t}(t-s)^{\mu -1}\\&P^{*}_{\mu }(t-s)\Big [g\big (s,y_{p-1}(s),z_{p-1}(s)\big )+(C+L)y_{p-1}(s)-Lz_{p-1}(s)\Big ]ds\\&+S^{*}_{\mu ,\nu }(t)x_{0}+S^{*}_{\mu ,\nu }(t)\phi _{1}\Big (z_{p-1}(t_{1}),y_{p-1}(t_{1})\Big )+\int _{0}^{t}(t-s)^{\mu -1}\\&P^{*}_{\mu }(t-s)\Big [g\big (s,z_{p-1}(s),y_{p-1}(s)\big )+(C+L)z_{p-1}(s)-Ly_{p-1}(s)\Big ]ds\Big \}\Bigg ). \\ \psi (t)&\le \dfrac{2M^{*}b^{1-\lambda }}{\varGamma (\lambda )}\Big [\alpha \big (\phi _{1}(B_{3}(t_{1}))+\phi _{1}(B_{4}(t_{1}))\big )\Big ]\\&+\dfrac{2M^{*}}{\varGamma (\mu )}(2L_{1}+C)\int _{0}^{t}\int _{0}^{t}(t-s)^{\mu -1}\psi (s)ds.\\&\le \dfrac{2M^{*}}{\varGamma (\mu )}(2L_{1}+C)\int _{0}^{t}(t-s)^{\mu -1}\psi (s)ds. \end{aligned}$$

By Lemma 5, $\psi (t)\equiv 0$ in $J_{1}^{'}$. By Proceeding the same way for every interval, it can be proved that $\psi (t)\equiv 0$ on every interval $J_{k}^{'}$, $k=1,2,\ldots l$. Thus $\{y_{p}\}$ and $\{z_{p}\}$ are precompact and eventually for $p=1,2,\ldots$, $\{y_{p}\}$ and $\{z_{p}\}$ have a converging subsequence. Moreover, from (10), it can be observed that $\{y_{p}\}$ and $\{z_{p}\}$ are converging sequences and hence there exists ${\underline{x}}(t)$, ${\overline{x}}(t)\in E$ such that

$$\begin{aligned} \displaystyle \lim _{p\rightarrow \infty }y_{p}(t)\rightarrow {\underline{x}}(t),\displaystyle \lim _{p\rightarrow \infty }z_{p}(t)\rightarrow {\overline{x}}(t),t\in J. \end{aligned}$$

From (9) and using the fact that $y_{p}(t)={\mathcal {G}}(y_{p-1},z_{p-1})(t)$, (6) can be represented as follows.

$$\begin{aligned} y_{p}(t)= \left\{ \begin{array}{ll} S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{p-1}(s),z_{p-1}(s)\big )\\ +(C+L)y_{p-1}(s)-Lz_{p-1}(s)\Big ]ds, t\in [0,t_{1}]\\ S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \sum _{i=1}^{k}S^{*}_{\mu ,\nu }(t-t_{i})\phi _{i}\big (y_{p-1}(t_{i}),z_{p-1}(t_{i})\big )\\ +\displaystyle \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,y_{p-1}(s),z_{p-1}(s)\big )\\ +(C+L)y_{p-1}(s)-Lz_{p-1}(s)\Big ]ds,t\in (t_{k},t_{k+1}],k=1,2,\ldots l. \end{array} \right. \end{aligned}$$

Using Lebesgue dominated convergence theorem, as $p\rightarrow \infty$, we have

$$\begin{aligned} {\underline{x}}(t)= \left\{ \begin{array}{ll} S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,{\underline{x}}(s),{\overline{x}}(s)\big )\\ +(C+L){\underline{x}}(s)-L{\overline{x}}(s)\Big ]ds, t\in [0,t_{1}],\\ S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \sum _{i=1}^{k}S^{*}_{\mu ,\nu }(t-t_{i})\phi _{i}\big ({\underline{x}}(t_{i}),{\overline{x}}(t_{i})\big )\\ +\displaystyle \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,{\underline{x}}(s),{\overline{x}}(s)\big )\\ +(C+L){\underline{x}}(s)-L{\overline{x}}(s)\Big ]ds,t\in (t_{k},t_{k+1}],k=1,2,\ldots l. \end{array} \right. \end{aligned}$$

It can be observed that ${\underline{x}}(t)\in PC_{1-\lambda }(J,E)$ and ${\underline{x}}(t)={\mathcal {G}}({\underline{x}},{\overline{x}})(t)$. In a similar manner, it can be proved that $\exists$ ${\overline{x}}(t)\in PC_{1-\lambda }(J,E)$, such that ${\overline{x}}(t)={\mathcal {G}}({\underline{x}},{\overline{x}})(t)$. With the monotonicity property of ${\mathcal {G}}$, it can be concluded that $y_{0}\le {\underline{x}}\le {\overline{x}}\le z_{0}$. This proves that there exists minimal and maximal solutions ${\underline{x}}$ and ${\overline{x}}$ respectively in $[y_{0},z_{0}]$ for the given impulsive system (3). $\square$

The existence of mild solutions of (3) can also be discussed by replacing the assumptions A(2) and A(3) by the following assumptions.

$A(2^{*}).$ The impulsive function $\phi _{k}(\cdot ,\cdot )$ satisfies

$$\begin{aligned} \phi _{k}(y_{1},z_{1})\le \phi _{k}(y_{2},z_{2}), k=1,2,\ldots ,l. \end{aligned}$$

for $t\in J$ and $y_{0}(t)\le y_{1}\le y_{2}\le z_{0}(t)$, $y_{0}(t)\le z_{2}\le z_{1}\le z_{0}(t)$. Also there exists $M_{k}>0$ satisfying the condition given by

$$\begin{aligned} \sum _{k=1}^{l}M_{k}\le \dfrac{\varGamma (\lambda )\varGamma (\mu +1)-4M^{*}(2L_{1}+C)T^{\mu }}{4M^{*}T^{\lambda -1}\varGamma (\mu -1)}, \end{aligned}$$

(11)

such that $\alpha \big (\phi _{k}(D_{1},D_{2})\big )\le M_{k}[\alpha (D_{1})+\alpha (D_{2})]$. In the condition (11), the denominator does not blow up as the term $M_{k}$ takes its value from $J_{k}^{'}$, $k>0$. That is $T\ne 0$.

$A(3^{*}).$ There exists a constant $L_{1}<0$ such that

$$\begin{aligned} \alpha (g(t,D_{1},D_{2}))\le L_{1}(\alpha (D_{1})+\alpha (D_{2})),t\in J, \end{aligned}$$

with the countable sets $D_{1}=\{y_{p}\}$ and $D_{2}=\{z_{p}\}$ in $[y_{0}(t),z_{0}(t)]$. The following theorem utilizes both the above inequalities where the existence of at least one mild solution between the coupled L-quasi upper and lower solutions is investigated. Also, the semigroup generated by the operator $-A$ is assumed to be equicontinuous.

Theorem 2

Let E be an ordered Banach space with the positive normal cone N. Assume that $Q(t)\ge 0$ is equicontinuous on E, $g\in C(J \times E \times E,E)$, $x_{0}\in E$. Let the impulsive system (3) has coupled L-quasi lower and upper solutions, given by $y_{0}$ and $z_{0}$, respectively, where $y_{0}, z_{0}\in PC_{1-\lambda }$ and $y_{0}\le z_{0}$. If the assumptions A(1), $A(2^{*})$ and $A(3^{*})$ are satisfied, then the impulsive system (3) has coupled minimal and maximal L-quasi mild solutions between $[y_{0},z_{0}]$ and at least one mild solutions in $[y_{0},z_{0}]$ between ${\underline{x}}$ and ${\overline{x}}$ such that for $p\rightarrow \infty$, $y_{p}(t)\rightarrow {\underline{x}}; z_{p}(t)\rightarrow {\overline{x}}, t\in J$. Here $y_{p}$ and $z_{p}$ are given as $y_{p}={\mathcal {G}}(y_{p-1},z_{p-1})$, $z_{p}={\mathcal {G}}(z_{p-1},y_{p-1})$, that satisfy

$$\begin{aligned} y_{0}(t)\le y_{1}(t)\le \cdots \le y_{p}(t)\le \cdots \le {\underline{x}} \le {\overline{x}} \le \cdots \le z_{p}(t)\cdots \le z_{1}(t)\le z_{0}(t). \end{aligned}$$

Proof

It can be verified that the assumption $A(3^{*})\Rightarrow A(3)$. Hence by Theorem 1, the impulsive system (3) has minimal and maximal L-quasi lower ${\underline{x}}$ and upper ${\overline{x}}$ solutions in $[y_{0},z_{0}]$. From the normality definition of the cone P, there exists ${\tilde{M}}>0$ such that

$$\begin{aligned} \Vert g(t,y(t),z(t))+(c+L)y(t)-Lz(t)\Vert \le {\tilde{M}}. \end{aligned}$$

(12)

The proof of the theorem terminates in finding at least one mild solution in $[y_{0},z_{0}]$. First, let the operator ${\mathcal {F}}$ be defined as ${\mathcal {F}}:[y_{0},z_{0}]\rightarrow [y_{0},z_{0}]$ such that ${\mathcal {F}}x={\mathcal {G}}(x,x)$. It is evident that ${\mathcal {F}}$ is continuous and the fixed point of the operator ${\mathcal {F}}$ is equivalent to the mild solution of the system (3).

The case for which $t\in J_{0}^{'}$:- Let $s_{1},s_{2}\in [0,t_{1}]$ such that $0<s_{1}<s_{2}\le t_{1}$. The following inequality determines the equicontinuous of the operator ${\mathcal {F}}$.

$$\begin{aligned} \Big \Vert s_{2}^{1-\lambda }&({\mathcal {F}}x)(s_{2})-s_{1}^{1-\lambda }({\mathcal {F}}x)(s_{2})\Big \Vert \le \Big \Vert s_{2}^{1-\lambda }{\mathcal {G}}(x,x)(s_{2})-s_{1}^{1-\lambda }{\mathcal {G}}(x,x)(s_{2})\Big \Vert \\&\le \Big \Vert s_{2}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{2})x_{0}-s_{1}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{1})x_{0}\Big \Vert +\Big \Vert \int _{0}^{s_{2}}s_{2}^{1-\lambda }(s_{2}-s)^{\mu -1}\\&P^{*}_{\mu }(s_{2}-s)\big [g\big (s,x(s),x(s)\big )+(C+L)x(s)-Lx(s)\big ]ds\Big \Vert \\&-\Big \Vert \int _{0}^{s_{1}}s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}P^{*}_{\mu }(s_{1}-s)\big [g\big (s,x(s),x(s)\big )\\&+(C+L)x(s)-Lx(s)\big ]ds\Big \Vert \end{aligned}$$

For convenience, let $g(s,x(s),x(s))+Cx(s)$ be denoted by $\zeta (s)$.

$$\begin{aligned} \Big \Vert s_{2}^{1-\lambda }&({\mathcal {F}}x)(s_{2})-s_{1}^{1-\lambda }({\mathcal {F}}x)(s_{2})\Big \Vert \le \Bigg (\Big \Vert s_{2}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{2})x_{0}-s_{2}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{1})x_{0}\Big \Vert \\&+\Big \Vert s_{2}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{1})x_{0}-s_{1}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{1})x_{0}\Big \Vert \Bigg )\\&+\Big \Vert s_{2}^{1-\lambda }\int _{s_{1}}^{s_{2}}(s_{2}-s)^{\mu -1}P_{\mu }^{*}(s_{2}-s)\big [\zeta (s)\big ]ds\Big \Vert \\&+\Big \Vert \int _{0}^{s_{1}}\big (s_{2}^{1-\lambda }(s_{2}-s)^{\mu -1}-s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}\big )P_{\mu }^{*}(s_{2}-s)\big [\zeta (s))\big ]ds\Big \Vert \\&+\Big \Vert s_{1}^{1-\lambda }\int _{0}^{s_{1}}(s_{1}-s)^{\mu -1}\big (P_{\mu }^{*}(s_{2}-s)-P_{\mu }^{*}(s_{1}-s)\big )\big [\zeta (s)\big ]ds\Big \Vert .\\&= \displaystyle \sum _{i=1}^{5}\Vert I_{i}\Vert . \end{aligned}$$

Now for $i=1,2,\ldots ,5$, $I_{i}$ can be calculated individually as follows. For $I_{1}$, using Lemma 7, it can be observed that

$$\begin{aligned} I_{1}&=\Big \Vert s_{2}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{2})x_{0}-s_{2}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{1})x_{0}\Big \Vert \\&\le \Big \Vert s_{2}^{1-\lambda }\big (S_{\mu ,\nu }^{*}(s_{2})-S_{\mu ,\nu }^{*}(s_{1})\big )\Big \Vert \Vert x_{0}\Vert \\&\longrightarrow 0, \text{ as }s_{2}\rightarrow s_{1}. \end{aligned}$$

For $I_{2}$, using Lemma 7, the following observation can be made similar to $I_{1}$.

$$\begin{aligned} I_{2}&=\Big \Vert s_{2}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{1})x_{0}-s_{1}^{1-\lambda }S_{\mu ,\nu }^{*}(s_{1})x_{0}\Big \Vert \\&\le \dfrac{M^{*}T^{\lambda -1}}{\varGamma (\lambda )}\Vert s_{2}^{1-\lambda }-s_{1}^{1-\lambda }\Vert \Vert x_{0}\Vert =\dfrac{M^{*}T^{\lambda -1}}{\varGamma (\lambda )}\Vert \big (s_{2}-s_{1}\big )^{1-\lambda }\Vert \Vert x_{0}\Vert \\&\longrightarrow 0, \text{ as }s_{2}\rightarrow s_{1}. \end{aligned}$$

$I_{3}$ can be evaluated using Lemma 7 as below, where, as $s_{2}\rightarrow s_{1}$,

$$\begin{aligned} I_{3}&=\Big \Vert s_{2}^{1-\lambda }\int _{s_{1}}^{s_{2}}(s_{2}-s)^{\mu -1}P_{\mu }^{*}(s_{2}-s)\big [\zeta (s)\big ]ds\Big \Vert \\&\le \dfrac{M^{*}{\tilde{M}}}{\varGamma (\mu )}\Big \Vert \int _{s_{1}}^{s_{2}}(s_{2}-s)^{\mu -1}ds\Big \Vert \longrightarrow 0. \end{aligned}$$

$I_{4}$ is evaluated using Lemma 7 and the bound (12), in the following manner.

$$\begin{aligned} I_{4}=&\Big \Vert \int _{0}^{s_{1}}\big (s_{2}^{1-\lambda }(s_{2}-s)^{\mu -1}-s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}\big )P_{\mu }^{*}(s_{2}-s)\big [\zeta (s)\big ]ds\Big \Vert \\&\Rightarrow I_{4}\le \dfrac{M^{*}{\tilde{M}}}{\varGamma (\mu )}\Big \Vert \int _{0}^{s_{1}}\big (s_{2}^{1-\lambda }(s_{2}-s)^{\mu -1}-s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}\big )ds\Big \Vert \\&\longrightarrow 0, \text{ as }s_{2}\rightarrow s_{1}. \end{aligned}$$

Similarly for $\epsilon \in (0,s_{1})$, $I_{5}$ can be evaluated as below.

$$\begin{aligned} I_{5}&=\Big \Vert \int _{0}^{s_{1}-\epsilon }s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}\big (P_{\mu }^{*}(s_{2}-s)-P_{\mu }^{*}(s_{1}-s)\big )\big [\zeta (s)\big ]ds\Big \Vert \\&+\Big \Vert \int _{s_{1}-\epsilon }^{s_{1}}s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}\big (P_{\mu }^{*}(s_{2}-s)-P_{\mu }^{*}(s_{1}-s)\big )\big [\zeta (s)\big ]ds\Big \Vert \\&\Longrightarrow I_{5}\le {\tilde{M}}\int _{0}^{s_{1}-\epsilon }s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}\sup _{s\in [0,s-\epsilon ]}\Big \Vert \big (P_{\mu }^{*}(s_{2}-s)-P_{\mu }^{*}(s_{1}-s)\big )\Big \Vert ds\\&+\dfrac{2{\tilde{M}} M^{*}}{\varGamma (\mu )}\int _{s_{1}-\epsilon }^{s_{1}}s_{1}^{1-\lambda }(s_{1}-s)^{\mu -1}ds.\\&\le {\tilde{M}}\int _{0}^{s_{1}-\epsilon }s_{1}^{1-\lambda }s^{\mu -1}\sup _{s\in [0,s-\epsilon ]}\Big \Vert \big (P_{\mu }^{*}(s_{2}+s-s_{1})-P_{\mu }^{*}(s)\big )\Big \Vert ds\\&+\dfrac{2{\tilde{M}} M^{*}t_{1}^{1-\lambda }\epsilon ^{\mu }}{\varGamma (\mu +1)}\\&\longrightarrow 0, \text{ as }\epsilon \rightarrow 0\text{ and } s_{2}\rightarrow s_{1}. \end{aligned}$$

Thus the following conclusion can be drawn for $J_{0}^{'}$.

$$\begin{aligned} \Big \Vert s_{2}^{1-\lambda }({\mathcal {F}}x)(s_{2})-s_{1}^{1-\lambda }({\mathcal {F}}x)(s_{2})\Big \Vert \longrightarrow 0. \end{aligned}$$

The case for which $t\in J_{k}^{'}$:-

For $J_{k}^{'}=(t_{k},t_{k+1}]$, let $s_{1},s_{2}\in (t_{k},t_{k+1}]$ such that $t_{k}<s_{1}<s_{2}\le t_{k+1}$, for which the following equality is evaluated.

$$\begin{aligned} \Big \Vert (s_{2}-t_{k})&^{1-\lambda }({\mathcal {F}}x)(s_{2})-(s_{1}-t_{k})^{1-\lambda }({\mathcal {F}}x)(s_{2})\Big \Vert \\&=\Big \Vert (s_{2}-t_{k})^{1-\lambda }{\mathcal {G}}(x,x)(s_{2})-(s_{1}-t_{k})^{1-\lambda }{\mathcal {G}}(x,x)(s_{2})\Big \Vert . \end{aligned}$$

Calculations similar to $J_{0}^{'}$ are performed to obtain the following observation.

$$\begin{aligned} \Big \Vert (s_{2}-t_{k})^{1-\lambda }{\mathcal {G}}(x,x)(s_{2})-(s_{1}-t_{k})^{1-\lambda }{\mathcal {G}}(x,x)(s_{2})\Big \Vert \rightarrow 0, \text{ as }s_{2}\rightarrow s_{1}\\ \implies \Big \Vert {\mathcal {G}}(x,x)(s_{2})-{\mathcal {G}}(x,x)(s_{2})\Big \Vert \rightarrow 0, \text{ as }s_{2}\rightarrow s_{1}. \end{aligned}$$

Consequently, $\Big \Vert ({\mathcal {F}}x)(s_{2})-({\mathcal {F}}x)(s_{2})\Big \Vert \rightarrow 0$ independently of $x\in [y_{0},z_{0}]$ as $s_{2}\rightarrow s_{1}$, which implies that $({\mathcal {F}}x):[y_{0},z_{0}]\rightarrow [y_{0},z_{0}]$ is equicontinuous. In this regard, for any $D\subset [y_{0},z_{0}]$, ${\mathcal {F}}(D)\subset [y_{0},z_{0}]$ is bounded and equicontinuous. By Lemma 1, it is evident that there exists a countable set $D_{0}=\{x_{p}\}\subset D$, such that

$$\begin{aligned} \alpha ({\mathcal {F}}(D))\le 2 \alpha ({\mathcal {F}}(D_{0})). \end{aligned}$$

From Lemma 3, it can be observed that

$$\begin{aligned} \alpha ({\mathcal {F}}(D_{0}))=\max _{t\in J}\alpha ({\mathcal {F}}(D_{0})(t)). \end{aligned}$$

For $t\in J_{0}^{'}$, by Lemma 2, equation (6) and from the assumption $A(3^{*})$, the following inequality is evaluated.

$$\begin{aligned} \alpha ({\mathcal {F}}(D_{0})(t))&=\alpha \Big (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}+\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,x_{p}(s),x_{p}(s)\big ) \\&\quad +Cx_{p}(s)\Big ]ds\Big \}\Big ).\\&\le \alpha \Big (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}\Big \}\Big )+\dfrac{2M^{*}}{\varGamma (\mu )}\int _{0}^{t}(t-s)^{\mu -1}\alpha \Big (\Big \{g\big (s,D_{0}(s),D_{0}(s)\big )\\&\quad +CD_{0}(s)\Big \}\Big )ds.\\&\le \dfrac{2M^{*}(2L_{1}+C)}{\varGamma (\mu )}\int _{0}^{t}(t-s)^{\mu -1}\alpha \Big (D_{0}(s)\Big )ds.\\&\le \dfrac{2M^{*}(2L_{1}+C)T^{\mu }}{\varGamma (\mu +1)}\alpha (D). \end{aligned}$$

For the case $J_{1}^{'}$, $t\in (t_{1},t_{2}]$, the inequality is evaluated using Lemma 2, equation (6) and the assumptions $A(2^{*})$ and $A(3^{*})$ as below.

$$\begin{aligned} \alpha ({\mathcal {F}}(D_{0})(t))&=\alpha \Big (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle S^{*}_{\mu ,\nu }(t-t_{1})\phi _{1}\big (x_{p}(t_{1}),x_{p}(t_{1})\big )\\&\quad +\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,x_{p}(s),x_{p}(s)\big )+Cx_{p}(s)\Big ]ds\Big \}\Big )\\&\le \alpha \Big (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}\Big \}\Big )+\dfrac{M^{*}T^{\lambda -1}}{\varGamma (\lambda )}\alpha \Big (\Big \{\phi _{1}(D_{0}(t_{1}),D_{0}(t_{1}))\Big \}\Big )\\&\quad +\dfrac{2M^{*}(2L_{1}+C)}{\varGamma (\mu )}\int _{0}^{t}(t-s)^{\mu -1}\alpha \Big (D_{0}(s)\Big )ds\\&\le 2M^{*}\Big (\dfrac{M_{1}T^{\lambda -1}}{\varGamma (\lambda )}+\dfrac{(2L_{1}+C)T^{\mu }}{\varGamma (\mu +1)}\Big )\alpha (D). \end{aligned}$$

For the general case, $J_{k}^{'}$, $t\in (t_{k},t_{k+1}]$, $k=1,2,\ldots ,l$ the inequality is calculated below using Lemma 2, (6) and the assumptions $A(2^{*})$ and $A(3^{*})$.

$$\begin{aligned} \alpha ({\mathcal {F}}(D_{0})(t))&=\alpha \Big (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}+\displaystyle \sum _{i=1}^{k} S^{*}_{\mu ,\nu }(t-t_{i})\phi _{i}\big (x_{p}(t_{i}),x_{p}(t_{i})\big )\\&\quad +\int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\Big [g\big (s,x_{p}(s),x_{p}(s)\big )+Cx_{p}(s)\Big ]ds\Big \}\Big ) \\ \alpha ({\mathcal {F}}(D_{0})(t))&\le \alpha \Big (\Big \{S^{*}_{\mu ,\nu }(t)x_{0}\Big \}\Big )+\dfrac{M^{*}T^{\lambda -1}}{\varGamma (\lambda )}\alpha \Big (\Big \{\sum _{i=1}^{k}\phi _{1}(D_{0}(t_{i}),D_{0}(t_{i}))\Big \}\Big )\\&\quad +\dfrac{2M^{*}(2L_{1}+C)}{\varGamma (\mu )}\int _{0}^{t}(t-s)^{\mu -1}\alpha \Big (D_{0}(s)\Big )ds\\&\le 2M^{*}\Big (\dfrac{\sum _{i=1}^{k}M_{i}T^{\lambda -1}}{\varGamma (\lambda )}+\dfrac{(2L_{1}+C)T^{\mu }}{\varGamma (\mu +1)}\Big )\alpha (D). \end{aligned}$$

By Lemma 3, since ${\mathcal {F}}(D_{0})$ is bounded and equicontinuous, the above inequality results in,

$$\begin{aligned} \alpha ({\mathcal {F}}(D_{0})(t))&\le 4M^{*}\Big (\dfrac{\sum _{i=1}^{k}M_{i}T^{\lambda -1}}{\varGamma (\lambda )}+\dfrac{(2L_{1}+C)T^{\mu }}{\varGamma (\mu +1)}\Big )\alpha (D)\le \alpha (D). \end{aligned}$$

Let $4M^{*}\Big (\dfrac{\sum _{i=1}^{k}M_{i}T^{\lambda -1}}{\varGamma (\lambda )}+\dfrac{(2L_{1}+C)T^{\mu }}{\varGamma (\mu +1)}\Big )=\eta$

• 1. If $\eta <1$, then the operator ${\mathcal {F}}:[y_{0},z_{0}]\rightarrow [y_{0},z_{0}]$ is condensing according to Lemma 4. Hence, ${\mathcal {F}}$ has a fixed point $x\in [y_{0},z_{0}]$.

• 2. If $\eta \ge 1$, that is, when $\eta$ jumps at the impulsive points, it is necessary to divide the interval [0, T] into n parts such that $\varDelta _{n}=0={\tilde{t}}_{0}<{\tilde{t}}_{1}<\cdots <{\tilde{t}}_{n}=T$. It is to be noted that the points ${\tilde{t}}_{0},{\tilde{t}}_{1},\ldots ,{\tilde{t}}_{n}$ are not the impulse points and let $\varDelta _{n}$ satisfy the below condition

  $$\begin{aligned} 4M^{*}\Big (\dfrac{\sum _{i=1}^{k}M_{i}\Vert \varDelta _{n}\Vert ^{\lambda -1}}{\varGamma (\lambda )}+\dfrac{(2L_{1}+C)\Vert \varDelta _{n}\Vert ^{\mu }}{\varGamma (\mu +1)}\Big )<1. \end{aligned}$$

In the interval $[0,{\tilde{t}}_{1}]$, according to the above two statements on $\eta$, there exists a mild solution $x_{1}(t)\in [0,{\tilde{t}}_{1}]$. Now in the interval $[{\tilde{t}}_{1},{\tilde{t}}_{2}]$ with initial condition $x({\tilde{t}}_{1})=x_{1}({\tilde{t}}_{1})$, has a mild solution $x_{2}(t)\in [{\tilde{t}}_{1},{\tilde{t}}_{2}]$. Thus, the mild solution of the equation is extended from $[0,{\tilde{t}}_{1}]$ to $[0,{\tilde{t}}_{2}]$. Subsequently, continuing this process, the mild solution of the equation is extended to [0, T]. Thus, the impulsive system (3) has a mild solution $x\in PC_{1-\lambda }(J,E)$ that satisfies $x(t)=x_{i}(t)$ such that ${\tilde{t}}_{i-1}\le t \le {\tilde{t}}_{i}$, for $i=1,2,\ldots ,n$.

Since $x={\mathcal {F}}x={\mathcal {G}}(x,x)$ for $y_{0}\le x \le z_{0}$, with respect to the mixed monotone property, the conclusion can be drawn as $y_{1}={\mathcal {G}}(y_{0},z_{0})\le {\mathcal {G}}(x,x) \le {\mathcal {G}}(z_{0},y_{0})=z_{1}$. Similarly it is true for $y_{2}\le x \le z_{2}$, and in general, $y_{p}\le x\le z_{p}$. Letting $p\rightarrow \infty$ reduces to ${\underline{x}}\le x\le {\overline{x}}$. Hence, it can be concluded that the impulsive system (3) has at least one mild solution between ${\underline{x}}$ and ${\overline{x}}$. $\square$

Corollary 1

In an ordered Banach space E, let N be the positive cone with normal constant ${\tilde{N}}$. With the assumption that the operator Q(t) is positive for $t\in J$, if the assumptions A(1) and A(2) are satisfied combined with the assumption given below, then the assumption A(3) is automatically true.

A(5). There exists a constant $C^{*}$ and $L^{*}$ such that

$$\begin{aligned} g(t,y_{2},z_{2})-g(t,y_{1},z_{1})\le C^{*}(y_{2}-y_{1})+L^{*}(z_{1}-z_{2}) \end{aligned}$$

and $y_{0}(t)\le y_{1}(t)\le y_{2}(t)\le z_{0}(t)$, $y_{0}(t)\le z_{2}(t)\le z_{1}(t)\le z_{0}(t)$ for any $t\in J$.

Proof

Let $\{y_{p}\}$,$\{y_{q}\}$ and $\{z_{p}\}$,$\{z_{q}\}$ be two set of increasing sequences such that

$$\begin{aligned} \{y_{p}\},\{y_{q}\},\{z_{p}\},\{z_{q}\} \subset [y_{0}(t),z_{0}(t)], \end{aligned}$$

for $t\in J$ and $p\le q$. By the assumptions A(1) and A(5),

$$\begin{aligned} \theta \le g(t,y_{q},z_{q})&-g(t,y_{p},z_{p})+C(y_{q}-y_{p})+L(z_{p}-z_{q})\\&\le (C^{*}+C)(y_{q}-y_{p})+(L^{*}+L)(z_{p}-z_{q}). \end{aligned}$$

From the definition of the normal cone with the normality constant ${\tilde{N}}$ of the positive cone N, the expression further reduces to,

$$\begin{aligned} \Vert g(t,y_{q},z_{q})&-g(t,y_{p},z_{p})+C(y_{q}-y_{p})+L(z_{p}-z_{q})\Vert \\&\le {\tilde{N}}\big ((C^{*}+C)(y_{q}-y_{p})+(L^{*}+L)(z_{p}-z_{q})\big ).\\ \Rightarrow \Vert g(t,y_{q},z_{q})&-g(t,y_{p},z_{p})\Vert \\&\le ({\tilde{N}}C^{*}+{\tilde{N}}C+C)\Vert y_{q}-y_{p}\Vert +({\tilde{N}}L^{*}+{\tilde{N}}L+L)\Vert z_{p}-z_{q}\Vert . \end{aligned}$$

Let $\displaystyle L_{1}={\tilde{N}}(C^{*}+C+L^{*}+L)+C+L$. By the definition of measure of non-compactness, the above expression reduces to,

$$\begin{aligned} \alpha \Big (\{g(t,y_{p},z_{p})\}\Big )\le L_{1}\Big ( \alpha \big (\{y_{p}\}\big )+\alpha \big (\{z_{p}\}\big )\Big ),p=1,2,\ldots ,. \end{aligned}$$

Thus, the assumption A(3) is reduced. $\square$

Theorem 3

An impulsive fractional system (3) is said to have a unique mild solution that lies between $[y_{0},z_{0}]$, where $y_{0}\in PC_{1-\lambda }$ and $z_{0}\in PC_{1-\lambda }$ are the coupled L-quasi lower and upper solutions with $y_{0}\le z_{0}$, if the assumptions A(1), A(2), A(4), and A(5) holds.

Proof

If ${\overline{x}}$ and ${\underline{x}}$ are the maximal and the minimal solutions of the impulsive system (3), then, to prove the uniqueness, it has to be proved that ${\overline{x}}={\underline{x}}$. Let $t\in J_{0}^{'}$. Using (7) in both the solutions results in,

$$\begin{aligned} \theta\le & {} {\overline{x}}(t)-{\underline{x}}(t)={\mathcal {G}}({\overline{x}},{\underline{x}})(t)-{\mathcal {G}}({\underline{x}},{\overline{x}})(t)\\= & {} \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\big [(g\big (s,{\overline{x}}(s),{\underline{x}}(s)\big )- g\big (s,{\underline{x}}(s),{\overline{x}}(s))\big )\\+ & {} (C+2L)({\overline{x}}(s)-{\underline{x}}(s))\big ]ds\\\le & {} \int _{0}^{t}(t-s)^{\mu -1}P^{*}_{\mu }(t-s)\big [(C^{*}+L^{*})\big ({\overline{x}}(t)-{\underline{x}}(t)\big )\\+ & {} (C+2L)\big ({\overline{x}}(t)-{\underline{x}}(t)\big )\big ]ds. \end{aligned}$$

Using the normality of the positive cone N, the above inequality reduces to,

$$\begin{aligned} \Vert {\overline{x}}(t)-{\underline{x}}(t)\Vert \le \frac{{\tilde{M}}M^{*}(C^{*}+L^{*}+C+2L)}{\varGamma (\mu )}\int _{0}^{t}(t-s)^{\mu -1}\Vert {\overline{x}}(t)-{\underline{x}}(t)\Vert ds. \end{aligned}$$

By Gronwall inequality, $\Vert {\overline{x}}(t)-{\underline{x}}(t)\Vert =0$. Which implies ${\overline{x}}(t)={\underline{x}}(t)$.

For every interval $J^{'}_{k}$, as $\phi _{k}\big ({\overline{x}}(t_{k}),{\underline{x}}(t_{k})\big )=\phi _{k}\big ({\overline{x}}(t_{k}),{\underline{x}}(t_{k})\big )$, the calculation is similar and it results in ${\overline{x}}(t)={\underline{x}}(t)$ for $t\in J_{k}^{'}$, for $k=1,2,\ldots ,l$. The uniqueness is thus proved. $\square$

Remark 3

Gou and Li [15] were the first to study the existence of extremal solutions for Hilfer fractional evolution equations with nonlocal conditions using the monotone technique. It can be observed that our results utilize the nonlinear function of the type g(t, x, x), which is a more general case. Further, Gou et al. [17] studied the existence of extremal solution for Hilfer fractional evolution equations with nonlocal conditions using mixed monotone technique. It is to be noted that when compared with the above two papers, the main theorem is proved for much weaker conditions in Corollary 1 in our paper. Also, the uniqueness of such an extremal solution is achieved in our paper. Since many real-time problems are impulsive, our paper combined with their paper [17] may lead to the study of the impulsive nonlocal system.

4 Example

An example is provided in this section which illustrates the main results.

Example 1

Let $E=L^{p}(\varLambda )$ for $1<p<\infty$ be generated by a positive cone N defined as $N=\{x\in L^{p}(\varLambda ):x(y)\ge \theta , \text{ a.e }y\in \varLambda \}$, where $\theta$ is the zero element. Here $\varLambda \subset R^{\mathbb {N}}$, ${\mathbb {N}}\ge 1$ is a bounded domain with a sufficiently smooth boundary $\partial \varLambda$. An impulsive Hilfer fractional parabolic partial differential equation with the above conditions is considered as below.

$$\begin{aligned} \left\{ \begin{array}{ll} D_{0+}^{\mu ,\nu }x(t,w)-\nabla ^{2}x(t,w)= g\big (t,w,x(t,w),x(t,w)\big ), (t,w) \in J \times \varLambda \\ \varDelta I_{t_{k}}^{(1-\lambda )}x(t_{k})=\phi _{k}\big (x(w,t_{k}),x(w,t_{k})\big ),k=1,2,\ldots l, y\in \varLambda \\ I_{0+}^{(1-\lambda )}[x(0,w)]= x_{0} \end{array} \right. \end{aligned}$$

(13)

where $D_{0+}^{\mu ,\nu }$ is the Hilfer fractional derivative with order $0<\mu <1$ and type $0\le \nu \le 1$, $t\in [0,T]$, $\nabla ^{2}$ is the Laplace operator such that $-Ax=\nabla ^{2}x$, $J=[0,T]$ with impulsive points at $t_{k}$ for $k=0,1,\ldots ,l$ such that $J^{'}=J/\{0,t_{1},t_{2},\ldots ,t_{l}\}$. Let $-A$ generates an equicontinuous analytic semigroup Q(t) for $t\ge 0$ and it is defined as $A:D(A)\subset E\rightarrow E$. Here, $D(A)=W^{2,p}\cap W^{1,p}_{0}(\varLambda )$. The continuous function g is defined as $g:J\times \varLambda \times E \times E \rightarrow E$ and the impulsive function is defined as $\phi _{k}:E \times E \rightarrow E$.

Now, Example 1 can be given as an abstract form similar to (3).

Theorem 4

Let the Hilfer fractional system given in Example 1 satisfy the following assumptions with $x_{0}\ge 0$.

E(1). Let There exists a function $z=z(t,w)\in PC_{1-\lambda }(J,\varLambda )$ such that

$$\begin{aligned} \left\{ \begin{array}{ll} D_{0+}^{\mu ,\nu }z(t,w)-\nabla ^{2}z(t,w)\ge g\big (t,w,z(t,w),z(t,w)\big ), t \in J,\\ \varDelta I_{t_{k}}^{(1-\lambda )}z(t_{k},w)\ge \phi _{k}\big (z(t_{k},w),z(t_{k},w)\big ),k=1,2,\ldots l\\ I_{0+}^{(1-\lambda )}[z(0,w)]\ge x_{0}. \end{array} \right. \end{aligned}$$

E(2). There exist constants $C\ge 0$ and $L\le 0$ such that

$$\begin{aligned} g\big (t,w,y_{2}(t,w),z_{2}(t,w)\big )&-g\big (t,w,y_{1}(t,w),z_{1}(t,w)\big )\\&\ge -C\big (y_{2}(t,w)-y_{1}(t,w)\big )-L\big (z_{1}(t,w)-z_{2}(t,w)\big ) \end{aligned}$$

and $y_{0}(t,w)\le y_{1}(t,w)\le y_{2}(t,w)\le z_{0}(t,w)$, $y_{0}(t,w)\le z_{2}(t,w)\le z_{1}(t,w)\le z_{0}(t,w)$ for any $t\in J$.

E(3). The impulsive function for $t\in J$ satisfies

$$\begin{aligned} \phi _{k}\big (y_{1}(t_{k},w),z_{1}(t_{k},w)\big )\le \phi _{k}\big (y_{2}(t_{k},w),z_{2}(t_{k},w)\big ), k=1,2,\ldots ,l. \end{aligned}$$

E(4). For $t\in J$, the sequence $\{y_{p}(t,w)\}\subset [y_{0}(t,w),z_{0}(t,w)]$ is an increasing monotonic sequence and $\{z_{p}(t,w)\}\subset [y_{0}(t,w),z_{0}(t,w)]$ is a decreasing monotonic sequences. In particular, there exists a constant $L_{1}\ge 0$ such that for $p=1,\ldots ,$

$$\begin{aligned} \alpha \Big (\{g\big (t,y_{p}(t,w),z_{p}(t,w)\big )\}\Big )\le L_{1}\Big ( \alpha \big (\{y_{p}(t,w)\}\big )+\alpha \big (\{z_{p}(t,w)\}\big )\Big ). \end{aligned}$$

Then using the monotone iterative procedure initiating from 0 to z(t, w), the system (13) has minimal and maximal solutions.

Proof

From the assumption E(1), it can be concluded that the lower and upper solutions lie between 0 and z(t, w). Also, Example 1 satisfies all the assumptions of Theorem 3. Hence it can be concluded that there exists a unique solution between 0 and z(t, w). $\square$

5 Observations

This section delivers some open problems for the readers based on this paper. Mainly an impulsive system with a higher fractional-order model is outlined along with the short comings in finding the extremal solutions along with its solutions, briefly.

5.1 Impulsive System with Hilfer Fractional-Order Derivative of Order $1<\mu <2$

It can be noted that there are very few articles available in the literature for finding the extremal solutions through the mixed monotone iterative method for the impulsive system having a higher fractional-order derivative. While studying problems with higher-order derivative, a few challenges have to be addressed.

• The properties of the solution operators $S_{\mu ,\nu }(t)$, $P_{\mu }(t)$ are not known for a higher fractional-order derivative. For example, the positivity of the operators is not known, and the probability density function may not be of use with semigroup theory.

• The procedure for the case $0<\mu <1$ may not work for a higher fractional-order in finding the lower and upper solutions.

But these difficulties can be overcome using the following theories.

• Instead of probability density functions, the properties of Mittag–Leffler function can be used to prove the positivity of the solution operators.

• The solution operators can be proved positive with the help of accretive and m-accretive operators.

• By considering operator A to be a sectorial operator and using the theory of the resolvent family.

According to the above constraints and motivated by the work of Shu and Xu [26], Gou and Li [16], and Jaiswal and Bahuguna [21], an impulsive system with Hilfer fractional derivative with order $1<\mu <2$ can be taken for consideration.

$$\begin{aligned} \left\{ \begin{array}{ll} D_{0+}^{\mu ,\nu }x(t)+Ax(t)= g\big (t,x(t),x(t)\big ), t \in J=[0,T], t\ne t_{k},\\ \varDelta I_{t_{k}}^{(1-\nu )(2-\mu )}x(t_{k})=\phi _{k}\big (x(t_{k}),x(t_{k})\big ),\\ \varDelta I_{t_{k}}^{(1-\nu )(1-\mu )}x(t_{k})=\psi _{k}\big (x(t_{k}),x(t_{k})\big ),\\ I_{0+}^{(1-\nu )(2-\mu )}[x(0)]= x_{0},I_{0+}^{(1-\nu )(1-\mu )}[x(0)]= x_{1}. \end{array} \right. \end{aligned}$$

Here, $D_{0+}^{\mu ,\nu }$ denotes the Hilfer fractional derivative of order $1<\mu <2$ of type $0\le \nu \le 1$ and $\lambda =\mu +\nu -\mu \nu$, $A:D(A)\subseteq E\rightarrow E$ is a closed linear operator and $-A$ is an infinitesimal generator of a strongly continuous resolvent family on a Banach space E. Let the impulse effect takes place at $t=t_{k}$, for $(k=1,2,\ldots ,l)$ and $\phi _{k}, \psi _{k}\in C(E\times E,E)$ determines the size of the jump at time $t_{k}$.

5.2 Other Problems

• The results can further be extended to study the case when the semigroup generated is compact and for the case when the coupled upper and lower quasi solutions does not exist.

• This article can lead to the study of impulsive systems with nonlocal conditions and the existence of extremal solution of a system with non-instantaneous impulses with Hilfer fractional derivative.

• The results can be further extended and studied for the $\psi$-Hilfer operator. For detailed work on $\psi$-Hilfer, the readers may refer to [29,30,31].

6 Conclusion

This paper is based on finding extremal solutions of impulsive system with Hilfer fractional derivative using a mixed monotone iterative technique. Theorem 1 guarantees the existence of minimal and maximal solutions for the considered system. Theorem 2 discusses the condition such that there exists at least one mild solution between the minimal and maximal solutions. Finally, Theorem 3 ensures the uniqueness of such a mild solution. The results are proved considering that the semigroup generated by the operator is non-compact and equicontinuous.