Open Access

Perturbation technique for a class of nonlinear implicit semilinear impulsive integro-differential equations of mixed type with noncompactness measure

Advances in Difference Equations20152015:11

https://doi.org/10.1186/s13662-014-0329-y

Received: 27 August 2014

Accepted: 19 December 2014

Published: 30 January 2015

Abstract

By using the Arzela-Ascoli theorem, the Bellman inequality, and a monotone perturbation iterative technique in the presence of lower and upper solutions, we discuss the existence of mild solutions for a class of nonlinear first-order implicit semilinear impulsive integro-differential equations in Banach spaces. Under wide monotone conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique mild solution between lower and upper solutions.

Keywords

nonlinear first-order implicit semilinear impulsive integro-differential equationmonotone iterative techniquemonotone condition and noncompactness measure conditionlower and upper solutionexistence and uniqueness

1 Introduction

The theory of impulsive differential equations has become an important area of investigation in recent years stimulated by their numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, etc. Various evolutionary processes undergo abrupt changes of states at certain moments of time; between intervals of continuous evolution such changes can be well approximated as being instantaneous changes at state, or in the form of impulses. These process are modeled by impulse differential equations and have been the most important research directions and connections for impulsive differential equations; see, for example, [17] and the references therein. Subsequently, many authors have investigated the existence of solutions to impulsive differential equations or (implicit) impulsive integro-differential equations with their strong applications in Banach spaces; see [127] and the references therein.

Recently, Lan and Cui [15] studied a class of initial value problems of nonlinear first-order implicit impulsive integro-differential equations in Banach space. By using the Mönch fixed point theorem, they obtained some new existence theorems of solutions for this class of nonlinear first-order implicit impulsive integro-differential equations in Banach spaces under some weaker conditions. Furthermore, some (implicit) impulsive differential equations under various initial and boundary conditions has also been studied by several authors; see, for example, [11, 13, 22, 24, 28] and the references therein. By using a monotone iterative technique in the presence of lower and upper solutions, Lan [23] discussed the existence of solutions for a new class of nonlinear first-order implicit impulsive integro-differential equations in Banach spaces. Under wide monotone conditions and the noncompactness measure conditions, he also obtained the existence of extremal solutions and a unique solution between lower and upper solutions. In [25], Chen and Li introduced and studied a class of semilinear impulsive evolution equations in Banach spaces by using a mixed monotone iterative technique. The presented results improved and extended some relevant results in ordinary differential equations and partial differential equations. For related works, see [9, 26, 27, 29, 30] and the references therein.

On the other hand, the monotone iterative technique, which is one of the approximation methods for finding solutions of a comparatively large class of impulsive differential equations, can be applied in practice easily; see, for example, [14, 16, 17, 23, 2527, 29]. Further, some nice examples of the monotone iterative technique can be found in [20, 21]. As a matter of fact, Li and Liu [16] pointed out that ‘the monotone iterative technique in the presence of lower and upper solutions is an important method for seeking solutions of differential equations in abstract spaces’. Moreover, Li and Liu [16] used a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions for the initial value problem of the impulsive integro-differential equation of Volterra type in a Banach space. Under monotone conditions and the noncompactness measure condition of the nonlinearity function f, the authors also obtained the existence of extremal solutions and a unique solution between lower and upper solutions. In [14], by using a monotone iterative technique in the presence of lower and upper solutions, we discussed the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under some monotonicity conditions and the noncompactness measure conditions, they also obtained the existence of extremal solutions and a unique solution between lower and upper solutions.

Motivated and inspired by the above works, by using the Arzela-Ascoli theorem, the Bellman inequality, and the monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of mild solutions for the following nonlinear first-order implicit semilinear impulsive differential equation problem in Banach space \({\mathbb{B}}\): Find \(u: J\to{\mathbb{B}}\) such that
$$ \begin{cases} u^{\prime}(t)=Au(t)+f(t,u(t),Tu(t), u^{\prime}(t)),\quad t\neq t_{k},\\ \triangle u|_{t=t_{k}}=I_{k}(u(t_{k})),\quad k=1, 2, \ldots, m,\\ u(t_{0})=u_{0}, \end{cases} $$
(1.1)
where \(J=[t_{0},t_{0}+a]\subset{\mathbb{R}}=(-\infty,+\infty)\) is a compact interval, the operator A is the infinitesimal generator of a positive \(C_{0}\)-semigroup \(\{G(t), t\ge t_{0}\}\) on \({\mathbb{B}}\), \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\) is a nonlinear continuous operator, \(t_{0}< t_{1}<\cdots<t_{m}<t_{0}+a<+\infty\), \(u_{0}\in{\mathbb{B}}\) is a given element, \(\hbar\in C(D,{\mathbb{R}}^{+})\), \(D=\{(t,s)\mid s,t\in J,t\ge s\}\), \(\mathbb{R}^{+}=[0,+\infty)\),
$$Tu(t)=\int_{t_{0}}^{t}\hbar(t,s)u(s)\,ds, $$
and for \(k=1, 2, \ldots, m\), \(I_{k}\in C[{\mathbb{B}},{\mathbb{B}}]\) is an impulsive function, \(\triangle u|_{t=t_{k}}\) denotes the jump of \(u(t)\) at \(t=t_{k}\), i.e., \(\triangle u|_{t=t_{k}}=u(t_{k}^{+})-u(t_{k}^{-})\), \(u(t_{k}^{-})\) and \(u(t_{k}^{+})\) represent the left and right limits of \(u(t)\) at \(t=t_{k}\), respectively. Further, under wide monotone conditions and the noncompactness measure conditions, we obtain the existence of extremal solutions and a unique mild solution between lower and upper solutions.

2 Preliminaries

Throughout this paper, let \({\mathbb{B}}\) be an ordered Banach space with the norm \(\|\cdot\|\) and partial order ≤, whose positive cone \(P=\{x\in{\mathbb{B}} \mid x\ge0\}\) is normal with normal constant N, and \(A: \operatorname{dom}(A)\subset{\mathbb{B}}\to{\mathbb{B}}\) be a closed linear operator and generate a \(C_{0}\)-semigroup \(G(t)\) (\(t\ge t_{0}\)) in \({\mathbb{B}}\). Let \(J=[t_{0},t_{0} +a]\), \(t_{0}< t_{1}<\cdots<t_{m}<t_{0}+a<+\infty\), \(J_{0}=[t_{0},t_{1}], J_{1}=(t_{1},t_{2}],\ldots, J_{k}=(t_{k},t_{k+1}], \ldots ,J_{m}=(t_{m},t_{0}+a]\), and
$$\begin{aligned} PC(J,{\mathbb{B}}) =&\bigl\{ x:J\to{\mathbb{B}} \mid x(t) \mbox{ is continuous at } t \neq t_{k}, \mbox{ and} \\ & \mbox{left continuous at } t=t_{k}, \mbox{ and } x\bigl(t_{k}^{+} \bigr) \mbox{ exists}, k = 1, 2, \ldots, m\bigr\} . \end{aligned}$$
Evidently, \(PC(J,{\mathbb{B}})\) is a Banach space with norm \(\|x\| _{PC}=\sup_{t\in J}x(t)\). Let \(J^{\prime}=J\setminus\{t_{1},t_{2}, \ldots,t_{m}\}\), and \({\mathbb{B}}_{*}\) be the Banach space generated by \(\operatorname{dom}(A)\) with norm \(\|\cdot\|_{*}=\|\cdot\|+\|A\cdot\|\). An abstract function \(x\in PC(J,{\mathbb{B}})\cap C^{1}(J^{\prime}, {\mathbb{B}})\cap C(J^{\prime}, {\mathbb{B}}_{*})\) is called a solution of problem (1.1) if \(x(t)\) satisfies all the equalities of (1.1).
Let
$$PC^{1}(J,{\mathbb{B}}) =\bigl\{ x\in PC(J,{\mathbb{B}})\cap C^{1}\bigl(J^{\prime}, { \mathbb{B}}\bigr)\cap C\bigl(J^{\prime}, {\mathbb{B}}_{*}\bigr) \mid x^{\prime}\bigl(t_{k}^{+}\bigr), x^{\prime}\bigl(t_{k}^{-}\bigr) \mbox{ exist, } k=1, 2, \ldots, m\bigr\} , $$
where \(x^{\prime}(t_{k}^{+})\) and \(x^{\prime}(t_{k}^{-})\) represent the right and left derivatives of \(x(t)\) at \(t=t_{k}\), respectively. For \(x\in PC^{1}(J,{\mathbb{B}})\), by virtue of the mean value theorem
$$x(t_{k})-x(t_{k}-\tau)\in\tau\overline{\operatorname{co}}\bigl\{ x^{\prime}(t): t_{k}-\tau< t<t_{k}\bigr\} \quad(\tau>0), $$
it is easy to see that the left derivative \(x_{-}^{\prime}(t_{k})\) exists and
$$x_{-}^{\prime}(t_{k})=\lim_{h\to0^{+}} \tau^{-1}\bigl[x(t_{k})-x(t_{k}-\tau ) \bigr]=x^{\prime}\bigl(t_{k}^{-}\bigr). $$
In the sequel, \(x^{\prime}(t_{k})\) is understood as \(x_{-}^{\prime}(t_{k})\), then \(x^{\prime}\in PC^{1}(J,{\mathbb{B}})\). If \(x\in PC(J,{\mathbb{B}})\cap C^{1}(J^{\prime}, {\mathbb{B}})\cap C(J^{\prime}, {\mathbb{B}}_{*})\) is a solution of problem (1.1), then by the continuity of f and the closed linearity of A, we know \(x\in PC^{1}(J,{\mathbb{B}})\). Evidently, \(PC^{1}(J,{\mathbb{B}})\) is a Banach space with norm \(\|x\|_{PC^{1}}=\max\{\sup_{t\in J}\|x(t)\| , \sup_{t\in J}\|x^{\prime}(t)\|\}\).
A mapping \(F: J\to{\mathbb{B}}\) is differentiable at \(t\in J\) if there exists a \(F^{\prime}(t)\in{\mathbb{B}}\) such that the limits
$$\lim_{\tau\to0^{+}}\frac{F(t + \tau)- F(t)}{\tau} \quad\mbox{and}\quad \lim _{\tau\to0^{+}}\frac{F(t)-F(t-\tau)}{\tau} $$
exist and are equal to \(F^{\prime}(t)\). Here the limits are taken in \({\mathbb{B}}\). At the endpoints of J, we consider the one-sided derivatives.
By the well-known result [31], we know that there exist \(C>0\) and \(\sigma\in\mathbb{R}\) such that \(\|G(t)\|\le Ce^{\sigma t}\). Letting
$$\delta_{0}:=\inf\bigl\{ \sigma\in\mathbb{B} \mid \exists C>0, \bigl\| G(t)\bigr\| \le Ce^{\sigma t}\bigr\} , $$
then \(\delta_{0}\) is called the increasing index of \(G(t)\). It follows from the properties of the \(C_{0}\)-semigroup that the \(C_{0}\)-semigroup \(G(t)\) (\(t\ge t_{0}\)) is exponentially stable if and only if \(\delta_{0}<0\).

Let \(C^{1}(J, {\mathbb{B}})\) denote the Banach space of all continuous differentiable \({\mathbb{B}}\)-value functions on interval J with norm \(\|x\|_{C^{1}}=\max_{t\in J}\|x^{\prime}(t)\|\). Let \(\alpha(\cdot)\) denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [32]. For any \(E\subset C^{1}(J, {\mathbb{B}})\) and \(t\in J\), set \(E(t)= \{ x(t) \mid x\in E\}\subset{\mathbb{B}}\). If E is bounded in \(C^{1}(J, {\mathbb{B}})\), then \(E(t)\) is bounded in \({\mathbb{B}}\), and \(\alpha(E(t))\le\alpha(E)\).

Lemma 2.1

Assume that the \(C_{0}\)-semigroup \(G(t)\) (\(t\ge t_{0}\)) is exponentially stable, i.e., \(\delta_{0}<0\). Then for any \(p\in PC^{1}(J, {\mathbb{B}})\) and \(v_{k},u_{0}\in\mathbb{B}\), \(k=1, 2, \ldots, m\), the initial value problem of linear impulsive evolution equation in \(\mathbb{B}\)
$$ \begin{cases} u^{\prime}(t)=Au(t)+p(t),\quad t\in J^{\prime},\\ \triangle u|_{t=t_{k}}=v_{k},\quad k=1, 2, \ldots, m,\\ u(t_{0})=u_{0}, \end{cases} $$
(2.1)
has a unique mild solution \(u\in PC^{1}(J,{\mathbb{B}})\) expressed by
$$u(t)=G(t-t_{0}) u_{0}+\int_{t_{0}}^{t}G(t-s)p(s)\,ds+ \sum_{t_{0}< t_{k}<t}G(t-t_{k})v_{k}. $$

Proof

It follows from Theorem 2.9 of [31, Chapter 4] and Lemma 2.2 in [26] that this conclusion follows directly. □

Lemma 2.2

[33]

If H is a bounded subset of \(PC^{1}(J,{\mathbb{B}})\), the element of \(H^{\prime}\) is equicontinuous at \(J_{k}\) for all \(k=0,1,2,\ldots,m\), then
$$\alpha(H)=\max\Bigl\{ \sup_{t\in J}\alpha\bigl(H(t)\bigr), \sup _{t\in J}\alpha\bigl(H^{\prime }(t)\bigr)\Bigr\} , $$
where \(H^{\prime}(t)=\{x^{\prime}(t): x\in H\}\).

Lemma 2.3

[33]

Let \(E\subset C(J, {\mathbb{B}})\) be bounded and equicontinuous. Then \(\alpha(E(t))\) is continuous on J, and
$$\alpha\biggl(\biggl\{ \int_{J}x(t)\,dt\Bigm| x\in E\biggr\} \biggr) \le\int_{J}\alpha\bigl(E(t)\bigr)\,dt. $$

Lemma 2.4

[34, Corollary 3.1(b)]

Let \(E=\{x_{n}\}\subset PC(J, {\mathbb{B}})\) be a bounded and countable set. Then \(\alpha(E(t))\) is Lebesgue integral on J, and
$$\alpha\biggl(\biggl\{ \int_{J}x_{n}(t)\,dt\biggr\} \biggr)\le2\int_{J}\alpha\bigl(E(t)\bigr)\,dt. $$

3 Existence and uniqueness theorems

In this section, we will prove our main results concerning the mild solutions of the nonlinear first-order implicit impulsive integro-differential equation (1.1) in Banach spaces.

Definition 3.1

If a function \(y\in PC^{1}(J, {\mathbb{B}})\) satisfies
$$ \begin{cases} y^{\prime}(t)\le Au(t)+f(t,u(t), Tu(t), u^{\prime}(t)), \quad t\neq t_{k},\\ \triangle y|_{t=t_{k}}\le I_{k}(u(t_{k})), \quad k=1, 2, \ldots, m,\\ y(t_{0})\le u_{0}, \end{cases} $$
(3.1)
then we call it a lower solution of problem (1.1); if all the inequalities of (3.1) are inverse, then we call it an upper solution of problem (1.1).

Definition 3.2

A \(C_{0}\)-semigroup \(G(t)\) (\(t\ge t_{0}\)) in \(\mathbb{B}\) is said to be positive, if the order inequality \(G(t)x\ge\theta\) holds for every \(x\ge\theta \), \(x\in{\mathbb{B}}\), and \(t\ge t_{0}\).

It is easy to see that for any \(M\ge0\), \(A-MI\) also generates a \(C_{0}\)-semigroup \(\Gamma(t)=e^{-M(t)}G(t)\) (\(t\ge t_{0}\)) in \(\mathbb{B}\). \(\Gamma(t)\) is a positive \(C_{0}\)-semigroup if \(G(t)\) is a positive \(C_{0}\)-semigroup for all \(t\ge t_{0}\).

Now, let us first list the following assumptions for convenience:
(H1): 
Problem (1.1) has a lower solution \(y_{0}\in PC^{1}(J, {\mathbb{B}})\) and an upper solution \(x_{0}\in PC^{1}(J, {\mathbb{B}})\) with \(y_{0}\le x_{0}\), and there exist constants \(M\in(0,1)\) such that
$$f(t, u_{2}, v_{2}, w_{2})-f(t, u_{1}, v_{1}, w_{1})\ge-M(u_{2}-u_{1}) $$
for all \(t\in J\) and \(y_{0}(t)\le u_{1}\le u_{2}\le x_{0}(t)\), \(Ty_{0}(t)\le v_{1}\le v_{2}\le Tx_{0}(t)\), and \(y_{0}^{\prime}(t)\le w_{1}\le w_{2}\le x_{0}^{\prime}(t)\).
(H2): 

\(I_{k}(x)\) is increasing on the order interval \([y_{0}(t),x_{0}(t)]\) for \(t\in J\), \(k=1, 2,\ldots, m\).

(H3): 
There exist \(0<2L<1-M\) such that
$$\alpha\bigl(\bigl\{ f\bigl(t,u_{n}(t),v_{n}(t),w_{n}(t) \bigr)\bigr\} \bigr)\le L\bigl[\alpha\bigl(\bigl\{ u_{n}(t)\bigr\} \bigr)+ \alpha\bigl(\bigl\{ v_{n}(t)\bigr\} \bigr)+\alpha\bigl(\bigl\{ w_{n}(t)\bigr\} \bigr)\bigr] $$
for all \(t\in J\), and increasing or decreasing monotonic sequences \(\{u_{n}\}\subset[y_{0}(t),x_{0}(t)]\), \(\{v_{n}\}\subset[Ty_{0}(t), Tx_{0}(t)]\) and \(\{w_{n}\}\subset[y_{0}^{\prime}(t), x_{0}^{\prime}(t)]\).

In the sequel, we prove the following main results of this paper.

Theorem 3.1

Let \({\mathbb{B}}\) be an ordered Banach space, whose positive cone P is normal, \(A: \operatorname{dom}(A)\subset{\mathbb{B}}\to{\mathbb{B}}\) be a closed linear operator, the positive \(C_{0}\)-semigroup \(G(t)\) (\(t\ge t_{0}\)) generated by A be compact in \({\mathbb{B}}\), \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\), and \(I_{k}\in C({\mathbb{B}}, {\mathbb{B}})\) for \(k=1, 2, \ldots, m\). Suppose that the conditions (H1 (H3) hold. Then problem (1.1) has minimal and maximal mild solutions between \([y_{0},x_{0}]\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) and \(x_{0}\), respectively.

Proof

Let \(M_{1}=\sup_{t\in J}\|\Gamma(t)\|\) and \(M_{2}=\sup_{t\in J}\|G^{\prime}(t)\|\). For any \(u\in PC^{1}(J, {\mathbb{B}})\), define Fu on J by the equation
$$\begin{aligned} Fu(t) =&\Gamma(t-t_{0})u_{0}+\int_{t_{0}}^{t} \Gamma(t-s)\bigl[f\bigl(s,u(s),Tu(s),u^{\prime}(s)\bigr)+Mu(s)\bigr]\,ds \\ &{} +\sum_{t_{0}< t_{k}<t}\Gamma(t-t_{k})I_{k} \bigl(u(t_{k})\bigr). \end{aligned}$$
(3.2)

It is easy to see that \(F: PC^{1}(J, {\mathbb{B}})\to PC^{1}(J, {\mathbb{B}})\) is continuous. By Lemma 2.1, we know that the mild solution of problem (1.1) is equivalent to the fixed point of F. Since \(G(t)\) (\(t\ge t_{0}\)) is a positive \(C_{0}\)-semigroup, \(G(0)=I\) ([31]) and it follows from assumptions (H1) and (H2) that F is increasing in \([y_{0},x_{0}]\) and maps any bounded set in \([y_{0},x_{0}]\) into a bounded set.

We first show that \(y_{0}\le Fy_{0}\), \(Fx_{0}\le x_{0}\). Let \(p(t)=y_{0}^{\prime}(t)-Ay_{0}(t)+My_{0}(t)\), by the definition of lower solution and (2.1), we know that \(p\in PC^{1}(J, {\mathbb{B}})\) and \(p(t)\le f(t, y_{0}(t), Ty_{0}(t), y_{0}^{\prime}(t))+My_{0}(t)\) for \(t\in J^{\prime}\). It follows from Lemma 2.1 that
$$\begin{aligned} y_{0}(t) =&\Gamma(t-t_{0})y_{0}(t_{0})+ \int_{t_{0}}^{t}\Gamma(t-s)g(s)\,ds+\sum _{t_{0}< t_{k}<t}\Gamma(t-t_{k})\triangle y_{0}|_{t=t_{k}} \\ \le&\Gamma(t-t_{0})u_{0}+\int_{t_{0}}^{t} \Gamma(t-s) \bigl[f\bigl(t, y_{0}(t), Ty_{0}(t), y_{0}^{\prime}(t)\bigr)+My_{0}(t) \bigr]\,ds \\ &{}+\sum_{t_{0}<t_{k}<t}\Gamma(t-t_{k})I_{k} \bigl(y_{0}(t_{k})\bigr) \\ =& Fy_{0}(t), \end{aligned}$$
for all \(t\in J\), i.e., \(y_{0}\le Fy_{0}\). Similarly, it can be shown that \(Fx_{0}\le x_{0}\). Combining these facts and the increasing property of F in \([y_{0},x_{0}]\), we see that F maps \([y_{0},x_{0}]\) into itself, and \(F: [y_{0},x_{0}]\to[y_{0},x_{0}]\) is a continuously increasing operator.
Secondly, we prove that \(F: [y_{0},x_{0}]\to[y_{0},x_{0}]\) is completely continuous. Let
$$ \begin{aligned} &\Phi u(t)=\int_{t_{0}}^{t}\Gamma(t-s) \bigl(f \bigl(s,u(s),Tu(s), u^{\prime}(s)\bigr)+ Mu(s)\bigr)\,ds, \\ &\Psi u(t)=\sum_{t_{0}< t_{k}<t}\Gamma(t-t_{k})I_{k} \bigl(u(t_{k})\bigr). \end{aligned} $$
(3.3)
On the one hand, for all \(t\in J\), we show that \(K(t)=\{\Phi u(t) \mid u\in[y_{0},x_{0}]\}\) is precompact in \({\mathbb{B}}\). In fact, for any \(\epsilon\in(t_{0},t)\) and \(u\in[y_{0},x_{0}]\), it follows from (3.3) that
$$\begin{aligned} \Phi_{\epsilon}u(t) =&\int_{t_{0}}^{t-\epsilon}\Gamma (t-s) \bigl(f\bigl(s,u(s),Tu(s),u^{\prime}(s)\bigr)+ Mu(s)\bigr)\,ds \\ =&\Gamma(\epsilon)\int_{t_{0}}^{t-\epsilon}\Gamma(t-\epsilon -s) \bigl(f\bigl(s,u(s),Tu(s),u^{\prime}(s)\bigr)+ Mu(s)\bigr)\,ds. \end{aligned}$$
(3.4)
It follows from the condition (H1) that
$$\begin{aligned}& f\bigl(t,y_{0}(t),Ty_{0}(t),y_{0}^{\prime}(t) \bigr)+ My_{0}(t) \\& \quad\le f\bigl(t,u(t),Tu(t),u^{\prime}(t)\bigr)+ Mu(t) \\& \quad\le f\bigl(t,x_{0}(t),Tx_{0}(t),x_{0}^{\prime}(t) \bigr)+ Mx_{0}(t). \end{aligned}$$
(3.5)
By the normality of the cone P, now we know that there exists a constant \(M_{3}>0\) such that
$$\bigl\| f\bigl(t,u(t),Tu(t),u^{\prime}(t)\bigr)+ Mu(t)\bigr\| \le M_{3},\quad \forall u\in[y_{0},x_{0}]. $$
From the compactness of \(\Gamma(\epsilon)\), we have \(K_{\epsilon}(t)=\{ \Phi_{\epsilon}u(t) \mid u\in[y_{0},x_{0}]\}\) is precompact in \(\mathbb{B}\). Since
$$\begin{aligned} \bigl\| \Phi u(t)-\Phi_{\epsilon}u(t)\bigr\| \le&\int^{t}_{t-\epsilon} \bigl\| \Gamma (t-s)\bigr\| \cdot\bigl\| f\bigl(s,u(s),Tu(s),u^{\prime}(s)\bigr)+ Mu(s)\bigr\| \,ds \\ \le& M_{1}M_{3}\epsilon, \end{aligned}$$
the set \(K(t)\) is totally bounded in \(\mathbb{B}\). Moreover, \(K(t)\) is precompact in \(\mathbb{B}\).
On the other hand, for all \(t_{1}, t_{2}\in J\), from (3.3)-(3.5), we have
$$\begin{aligned} \bigl\| \Phi u(t_{2})-\Phi u(t_{1})\bigr\| \le&\int^{t_{2}}_{t_{0}}\bigl(\Gamma(t_{2}-s)- \Gamma(t_{1}-s)\bigr) \bigl(\bigl\| f\bigl(s,u(s),Tu(s),u^{\prime}(s) \bigr)+ Mu(s)\bigr\| \bigr)\,ds \\ &{} +\int^{t_{2}}_{t_{1}}\Gamma(t_{2}-s) \bigl(\bigl\| f\bigl(s,u(s),Tu(s),u^{\prime}(s)\bigr)+ Mu(s)\bigr\| \bigr)\,ds \\ \le& M_{3}\int^{t_{1}}_{t_{0}}\bigl\| \Gamma(t_{2}-s)-\Gamma(t_{1}-s)\bigr\| \,ds+M_{1}M_{3}(t_{1}-t_{1}) \\ \le& M_{3}\int^{t_{0}+a}_{t_{0}}\bigl\| \Gamma(t_{2}-t_{1}+s)-\Gamma(s)\bigr\| \,ds+M_{1}M_{3}(t_{1}-t_{1}). \end{aligned}$$
(3.6)
The right side of (3.6) relies on \(t_{2}-t_{1}\), but it is independent of u. Since \(G(t)\) (\(t\ge t_{0}\)) is compact, \(\Gamma(t)\) is compact and continuous in the uniform operator topology for all \(t\ge t_{0}\). Thus, the right side of (3.6) tends to 0 as \(t_{2}-t_{1}\to0\). Hence, \(\Phi([y_{0},x_{0}])\) is an equicontinuous function of the cluster in \(C^{1}(J, {\mathbb{B}})\).

Similarly, we can prove the compactness of Ψ in (3.3).

For any \(t\in J\), since \(\{Fu(t) \mid u\in[y_{0},x_{0}]\}=\{\Gamma (t-t_{0})+\Phi u(t)+\Psi u(t) \mid u\in[y_{0},x_{0}]\}\), and \(Fu(t_{0})=u_{0}\) is precompact in \(\mathbb{B}\), we know that \(F([y_{0},x_{0}])\) is precompact in \(C^{1}(J_{k}, {\mathbb{B}})\) by using the Arzela-Ascoli theorem. Thus, \(F: [y_{0},x_{0}]\to[y_{0},x_{0}]\) is completely continuous.

Finally, we show that problem (1.1) has minimal and maximal mild solutions between \([y_{0},x_{0}]\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) and \(x_{0}\), respectively.

It follows from the completely continuity of F that F has minimal and maximal fixed points \(\underline{u}\) and \(\overline{u}\) in \([y_{0},x_{0}]\), and so they are the minimal and maximal mild solutions of problem (1.1) in \([y_{0},x_{0}]\), respectively.

On the other hand, from the above discussions, we know that \(F: [y_{0},x_{0}]\to[y_{0},x_{0}]\) is a continuously increasing operator. Now, we define two sequences \(\{y_{n}\}\) and \(\{x_{n}\}\) in \([y_{0},x_{0}]\) by the iterative scheme
$$ y_{n}=Fy_{n-1},\quad\quad x_{n}=Fx_{n-1},\quad n=1, 2, \ldots. $$
(3.7)
Then it follows from the monotonicity of F that
$$ y_{0}\le y_{1}\le\cdots\le y_{n}\le\cdots\le x_{n}\le\cdots\le x_{1}\le x_{0}. $$
(3.8)
We prove that \(\{y_{n}\}\) and \(\{x_{n}\}\) are uniformly convergent in J.
For convenience, let \(E=\{y_{n} \mid n\in{\mathbb{N}}\}\) and \(E_{0}=\{y_{n-1} \mid n\in{\mathbb{N}}\}\). Since \(E=F(E_{0})\), by (3.2) and the boundedness of \(E_{0}\), we easily see that E is equicontinuous in every interval \(J_{k}^{\prime}\), where \(J_{1}^{\prime}=[t_{0}, t_{1}]\) and \(J_{k}^{\prime}= (t_{k-1}, t_{k}]\), \(k=2, 3,\ldots, m\). From \(E_{0}=E\cup\{y_{0}\}\) and Lemma 2.2, it follows that \(\alpha (E_{0}(t))=\alpha(E(t))\) and
$$\alpha^{1}\bigl(E(t)\bigr)=\max\bigl\{ \alpha\bigl(E(t)\bigr), \alpha \bigl(E^{\prime}(t)\bigr)\bigr\} $$
for all \(t\in J\). Letting
$$\phi(t)= \alpha^{1}\bigl(E(t)\bigr)=\alpha^{1} \bigl(E_{0}(t)\bigr),\quad t\in J, $$
by Lemma 2.3, we know that \(\phi\in PC^{1}(J,{\mathbb{R}}^{+})\). Going from \(J_{1}^{\prime}\) to \(J_{m+1}^{\prime}\) interval by interval, we show that \(\phi(t)\equiv0\) in J.
In fact, for \(t\in J\), there exists a \(J_{k}^{\prime}\) such that \(t\in J_{k}^{\prime}\). By Lemma 2.3, we have
$$\begin{aligned} \alpha\bigl(T\bigl(E_{0}(t)\bigr)\bigr) =&\alpha \biggl( \biggl\{ \int _{t_{0}}^{t} \hbar (t,s)y_{n-1}(s)\,ds \Bigm| n\in{ \mathbb{N}} \biggr\} \biggr) \\ \le&\sum_{j=1}^{k-1}\alpha \biggl( \biggl\{ \int_{t_{j-1}}^{t_{j}} \hbar (t,s)y_{n-1}(s)\,ds \Bigm| n \in{\mathbb{N}} \biggr\} \biggr) \\ &{} +\alpha \biggl( \biggl\{ \int_{t_{k-1}}^{t} \hbar(t,s)y_{n-1}(s)\,ds \Bigm| n\in{\mathbb{N}} \biggr\} \biggr) \\ \le&\hbar_{0}\sum_{j=1}^{k-1} \int_{t_{j-1}}^{t_{j}} \alpha\bigl(E_{0}(s) \bigr)\,ds+\hbar _{0}\int_{t_{k-1}}^{t} \alpha \bigl(E_{0}(s)\bigr)\,ds \\ \le&\hbar_{0}\sum_{j=1}^{k-1} \int_{t_{j-1}}^{t_{j}} \phi(s)\,ds+\hbar_{0}\int _{t_{k-1}}^{t} \phi(s)\,ds \\ =& \hbar_{0}\int_{t_{0}}^{t} \phi(s)\,ds, \end{aligned}$$
where \(\hbar_{0}=\max\{|\hbar(t,s)|: (t,s)\in D\}\). Hence,
$$ \int_{t_{0}}^{t}\alpha\bigl(T\bigl(E_{0}(s) \bigr)\bigr)\,ds\le a\hbar_{0}\int_{t_{0}}^{t} \phi(s)\,ds. $$
(3.9)
It follows from (3.2), Lemma 2.4, assumption (H3) and (3.9) that, for \(t\in J^{\prime}_{1}\),
$$\begin{aligned}& \alpha\bigl(E(t)\bigr) = \alpha\bigl(F\bigl(E_{0}(t)\bigr)\bigr) \\& \hphantom{\alpha\bigl(E(t)\bigr)}= \alpha \biggl( \biggl\{ \int_{t_{0}}^{t}\Gamma (t-s) \bigl(f\bigl(s,y_{n-1}(s),Ty_{n-1}(s),y_{n-1}^{\prime}(s) \bigr)+ My_{n-1}(s)\bigr)\,ds \Bigm| n\in{\mathbb{N}} \biggr\} \biggr) \\& \hphantom{\alpha\bigl(E(t)\bigr)}\le 2\int_{t_{0}}^{t}\Gamma(t-s)\alpha \bigl( \bigl( \bigl(f\bigl(s,y_{n-1}(s), Ty_{n-1}(s), y_{n-1}^{\prime}(s) \bigr)+ My_{n-1}(s)\bigr) \mid n\in {\mathbb{N}} \bigr) \bigr)\,ds \\& \hphantom{\alpha\bigl(E(t)\bigr)}\le 2M_{1}\int_{t_{0}}^{t} \bigl\{ L \bigl[ \alpha\bigl(E_{0}(s)\bigr)+\alpha \bigl(T\bigl(E_{0}(s) \bigr)\bigr)+\alpha\bigl(E_{0}^{\prime}(s)\bigr) \bigr]+ M\alpha \bigl(E_{0}(s)\bigr) \bigr\} \,ds \\& \hphantom{\alpha\bigl(E(t)\bigr)}\le 2M_{1} \biggl[(L+M)\int_{t_{0}}^{t} \alpha\bigl(E_{0}(s)\bigr)\,ds+L\int_{t_{0}}^{t} \alpha \bigl(T\bigl(E_{0}(s)\bigr)\bigr)\,ds+L\int_{t_{0}}^{t} \alpha\bigl(E_{0}^{\prime}(s)\bigr)\,ds \biggr] \\& \hphantom{\alpha\bigl(E(t)\bigr)}\le 2M_{1}(M+a\hbar_{0}L +2L )\int_{t_{0}}^{t} \phi(s)\,ds, \\& \alpha\bigl(E^{\prime}(t)\bigr)=\alpha\bigl((FE_{0})^{\prime}(t) \bigr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}=\alpha \biggl(\frac{d}{dt} \biggl\{ \int_{t_{0}}^{t} \Gamma (t-s) \bigl(f\bigl(s,y_{n-1}(s),Ty_{n-1}(s),y_{n-1}^{\prime}(s) \bigr) + My_{n-1}(s)\bigr)\,ds \Bigm| n\in{\mathbb{N}} \biggr\} \biggr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}=\alpha \biggl( \biggl\{ \int_{t_{0}}^{t} \Gamma^{\prime}(t-s)f\bigl(s,y_{n-1}(s),Ty_{n-1}(s),y_{n-1}^{\prime}(s) \bigr)\,ds \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)=}{} + My_{n-1}(t)+\Gamma(0)f\bigl(t,y_{n-1}(t),Ty_{n-1}(t),y_{n-1}^{\prime}(t)\bigr) \Bigm| n\in{\mathbb{N}} \biggr\} \biggr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}=\alpha \biggl( \biggl\{ -MF\bigl(y_{n-1}(t)\bigr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)=}{} +\int_{t_{0}}^{t}G^{\prime}(t-s)e^{-M(t-s)}f\bigl(s,y_{n-1}(s),Ty_{n-1}(s),y_{n-1}^{\prime}(s) \bigr)\,ds \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)=}{} + My_{n-1}(t)+G(0)f\bigl(t,y_{n-1}(t),Ty_{n-1}(t),y_{n-1}^{\prime}(t) \bigr) \Bigm| n\in{\mathbb{N}} \biggr\} \biggr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}\le-M\alpha\bigl(F\bigl(E_{0}(t)\bigr)\bigr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)\le}{} +\alpha \biggl(\int_{t_{0}}^{t}G^{\prime}(t-s)e^{-M(t-s)}f\bigl(s,y_{n-1}(s),Ty_{n-1}(s),y_{n-1}^{\prime}(s) \bigr)\,ds \Bigm| n\in{\mathbb{N}} \biggr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)\le}{} + M\alpha\bigl(E_{0}(t)\bigr)+\alpha\bigl(\bigl\{ f \bigl(t,y_{n-1}(ts),Ty_{n-1}(t),y_{n-1}^{\prime}(t)\bigr)\bigr\} \bigr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}\le-M\alpha\bigl(F\bigl(E_{0}(t) \bigr)\bigr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)\le}{} +2\int_{t_{0}}^{t}G^{\prime}(t-s)e^{-M(t-s)} \alpha \bigl(f\bigl(s,y_{n-1}(s),Ty_{n-1}(s),y_{n-1}^{\prime}(s) \bigr) \mid n\in {\mathbb{N}} \bigr)\,ds \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)\le}{} + M\alpha\bigl(E_{0}(t)\bigr)+\alpha \bigl( \bigl\{ f \bigl(t,y_{n-1}(ts),Ty_{n-1}(t),y_{n-1}^{\prime}(t) \bigr) \bigr\} \bigr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}\le-M\alpha\bigl(F\bigl(E_{0}(t)\bigr)\bigr)+M\alpha \bigl(E_{0}(t)\bigr)+\zeta\bigl[\alpha\bigl(E_{0}(t)\bigr)+ \alpha \bigl(T(E_{0}) (t)\bigr)+\alpha\bigl(E_{0}^{\prime}(t) \bigr)\bigr] \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}\le\bigl[a\hbar_{0}\zeta-2M(M+a\hbar_{0}\zeta+2\zeta) \bigr]\int_{t_{0}}^{t} \phi (s)\,ds+(\zeta+M)\alpha \bigl(E_{0}(t)\bigr)+L \alpha\bigl(E_{0}^{\prime}(t) \bigr) \\ & \hphantom{\alpha\bigl(E^{\prime}(t)\bigr)}\le\bigl[a\hbar_{0}\zeta-2M(M+a\hbar_{0}\zeta+2\zeta) \bigr]\int_{t_{0}}^{t} \phi (s)\,ds+(M+2\zeta) \alpha^{1}\bigl(E_{0}(t)\bigr), \end{aligned}$$
and so
$$\phi(t)\le\Gamma\int_{t_{0}}^{t} \phi(s)\,ds+(M+2L ) \phi(t), $$
i.e.,
$$\phi(t)\le\Theta\int_{t_{0}}^{t} \phi(s)\,ds, $$
where \(\zeta=L(1+2aM_{2})\), \(\Gamma=\max\{2(M+a\hbar_{0}L +2L ),a\hbar_{0}\zeta-2M(M+a\hbar_{0}\zeta +2\zeta)\}\), and \(\Theta=\Gamma/(1-M-2\zeta)\). Hence, by the Bellman inequality, we know that \(\phi(t)\equiv0\) in \(J_{1}^{\prime}\). In particular, \(\alpha^{1}(E(t_{1}))=\alpha^{1}(E_{0}(t_{1}))=\phi(t_{1})=0\), and so \(\alpha (E(t_{1}))=\alpha(E_{0}(t_{1}))=0\), this means that \(E(t_{1})\) and \(E_{0}(t_{1})\) are precompact in \(\mathbb{B}\). Thus \(I_{1}(E_{0}(t_{1}))\) is precompact in \(\mathbb{B}\), and \(\alpha(I_{1}(E_{0}(t_{1})))=0\).
Now, for \(t\in J_{2}^{\prime}\), by (3.2) and the above argument for \(J_{1}^{\prime}\), we have
$$\begin{aligned} \alpha\bigl(E(t)\bigr) =&\alpha\bigl(E(t)\bigr)=\alpha\bigl(F(E_{0}) (t)\bigr) \\ =&\alpha \biggl( \biggl\{ \Gamma(t-t_{0})u_{0} +\int_{t_{0}}^{t}\Gamma(t-s) \bigl(f \bigl(s,y_{n-1}(s),Ty_{n-1}(s), y_{n-1}^{\prime}(s)\bigr)+ My_{n-1}(s)\bigr)\,ds \\ &{} +\Gamma(t-t_{1})I_{1}\bigl(y_{n-1}(t_{1}) \bigr) \Bigm| n\in{\mathbb{N}} \biggr\} \biggr) \\ \le&2(L+M+a\hbar_{0}L )\int_{t_{0}}^{t} \phi(s)\,ds+\alpha\bigl(I_{1}\bigl(E_{0}(t_{1})\bigr) \bigr) \\ \le&2(M+a\hbar_{0}L +2L )\int_{t_{0}}^{t} \phi(s)\,ds \\ =&2(M+a\hbar_{0}L +2L )\int_{t_{1}}^{t} \phi(s)\,ds \end{aligned}$$
and
$$\phi(t)\le\Theta\int_{t_{1}}^{t} \phi(s)\,ds. $$

Again by the Bellman inequality, we know that \(\phi(t)\equiv0\) in \(J_{2}^{\prime}\), from which we obtain \(\alpha(E_{0}(t_{2}))=0\) and \(\alpha(I_{2}(E_{0}(t_{2})))=0\).

Continuing such a process interval by interval up to \(J_{m+1}^{\prime}\), we can prove that \(\phi(t)\equiv0\) in every \(J_{k}^{\prime}\), \(k=1, 2, \ldots, m+1\).

For any \(J_{k}\), if we modify the value of \(y_{n}\) at \(t=t_{k-1}\) via \(y_{n}(t_{k-1})=y_{n}(t^{+}_{k-1})\), \(n\in{\mathbb{N}}\), then \(\{y_{n}\}\subset C^{1}(J_{k}, {\mathbb{B}})\) and it is equicontinuous. Since \(\alpha(\{y_{n}(t)\})\equiv0\), \(\{y_{n}(t)\}\) is precompact in \({\mathbb{B}}\) for every \(t\in J_{k}\). By the Arzela-Ascoli theorem, we know that \(\{y_{n}\}\) is precompact in \(C^{1}(J_{k}, {\mathbb{B}})\). Hence, \(\{y_{n}\}\) has a convergent subsequence in \(C^{1}(J_{k}, {\mathbb{B}})\). Combining this with the monotonicity (3.8), we easily prove that \(\{y_{n}\}\) itself is convergent in \(C^{1}(J_{k}, {\mathbb{B}})\). In particular, \(\{y_{n}(t)\}\) is uniformly convergent in \(J_{k}^{\prime}\). Consequently, \(\{y_{n}(t)\}\) is uniformly convergent over the whole of J.

Using a similar argument to that for \(\{y_{n}(t)\}\), we can prove that \(\{ x_{n}(t)\}\) is also uniformly convergent in J. Hence, \(\{y_{n}(t)\}\) and \(\{x_{n}(t)\}\) are convergent in \(PC^{1}(J, {\mathbb{B}})\). Setting
$$ \underline{u}=\lim_{n\to\infty} y_{n}, \quad\quad\overline{u}=\lim _{n\to\infty} x_{n} \quad\mbox{in } PC^{1}(J, {\mathbb{B}}) $$
(3.10)
and \(n\to\infty\) in (3.7) and (3.8), then we have \(v_{0}\le\underline {u}\le\overline{u}\le x_{0}\) and
$$ \underline{u}=F\underline{u},\quad\quad \overline{u}=F\overline{u}. $$
(3.11)

By the monotonicity of F, it is easy to see that \(\underline{u}\) and \(\overline{u}\) are the minimal and maximal fixed points of F in \([y_{0},x_{0}]\), and therefore they are the minimal and maximal mild solutions of problem (1.1) in \([y_{0},x_{0}]\), respectively. This completes the proof. □

Remark 3.1

In Theorem 3.1, if \({\mathbb{B}}\) is weakly sequentially complete, the condition (H3) holds automatically. In fact, by Theorem 2.2 of [35], any monotonic and order-bounded sequence is precompact. Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be two increasing or decreasing sequences obeying condition (H3), then, by condition (H1), \(\{f(t, x_{n}, y_{n}, z_{n})+Mx_{n}\}\) is a monotonic and order-bounded sequence. By the property of the measure of noncompactness, we have
$$\begin{aligned}& \alpha\bigl(\bigl\{ Ax_{n}(t)+f(t, x_{n}, y_{n}, z_{n})+Mx_{n}\bigr\} \bigr) \\& \quad\le\alpha\bigl(\bigl\{ Ax_{n}(t)+f(t, x_{n}, y_{n}, z_{n})+Mx_{n}\bigr\} \bigr)+M\alpha\bigl( \{x_{n}\}\bigr)=0. \end{aligned}$$
Hence, condition (H3) holds.

From Theorem 3.1, we obtain the following result.

Corollary 3.1

Let \({\mathbb{B}}\) be an ordered and weakly sequentially complete Banach space, whose positive cone P is normal, \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\), and \(I_{k}\in C({\mathbb{B}}, {\mathbb{B}})\), \(k=1, 2, \ldots, m\). If the conditions (H1) and (H2) are satisfied, then problem (1.1) has minimal and maximal mild solutions between \(y_{0}\) and \(x_{0}\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) and \(x_{0}\), respectively.

Next we discuss the uniqueness of the mild solution to problem (1.1) in \([y_{0},x_{0}]\). Assume we replace the assumption (H3) by the following assumption.

(H4) There exist positive constants \(C_{i}\) (\(i=1,2,3\)) with \(C_{3}<1\) such that
$$\begin{aligned}& f(t, u_{2}, v_{2}, w_{2})-f(t, u_{1}, v_{1}, w_{1})\le C_{1}(u_{2}-u_{1})+C_{2}(v_{2}-v_{1})+C_{3}(w_{2}-w_{1}), \\& \forall t\in J,\quad y_{0}(t)\le u_{1}\le u_{2}\le x_{0}(t),\quad\quad Ty_{0}(t)\le y_{1}\le y_{2} \le Tx_{0}(t), \end{aligned}$$
for all \(t\in J\) and \(y_{0}(t)\le u_{1}\le u_{2}\le x_{0}(t)\), \(\lambda _{1}Ty_{0}(t)\le v_{1}\le v_{2}\le\lambda_{1}Tx_{0}(t)\) and \(\lambda_{2}y_{0}^{\prime}(t)\le w_{1}\le w_{2}\le\lambda_{2}x_{0}^{\prime}(t)\). Then we have the following unique existence result.

Theorem 3.2

Let \({\mathbb{B}}\) be an ordered Banach space, whose positive cone P is normal, \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\) and \(I_{k}\in C({\mathbb{B}}, {\mathbb{B}})\), \(k=1, 2, \ldots, m\). If the conditions (H1), (H2), and (H4) hold, then problem (1.1) has a unique mild solution between \(y_{0}\) and \(x_{0}\), which can be obtained by a monotone iterative procedure starting from \(y_{0}\) or \(x_{0}\).

Proof

We first prove that (H1) and (H4) imply (H3). In fact, for \(t\in J\), let \(\{u_{n}\}\subset[y_{0}(t),x_{0}(t)]\), \(\{v_{n}\} \subset[ Ty_{0}(t), Tx_{0}(t)]\), and \(\{w_{n}\} \subset[ y_{0}^{\prime}(t), x_{0}^{\prime}(t)]\) be increasing sequences. For \(m, n\in{\mathbb{N}}\) with \(m>n\), by (H1) and (H4),
$$\begin{aligned} \theta \le&\bigl(f(t, u_{m}, v_{m}, w_{m})-f(t, u_{n}, v_{n}, w_{n})\bigr)+M(u_{m}-u_{n}) \\ \le&(C_{1}+M) (u_{m}-u_{n})+C_{2}(v_{m}-v_{n})+C_{3}(w_{m}-w_{n}). \end{aligned}$$
By this inequality and the normality of cone P, we have
$$\begin{aligned}& \bigl\| f(t, u_{m}, v_{m}, w_{m})-f(t, u_{n}, v_{n}, w_{n})\bigr\| \\& \quad\le N\bigl\| (C_{1}+M) (u_{m}-u_{n})+C_{2}(v_{m}-v_{n})+C_{3}(w_{m}-w_{n}) \bigr\| +M\|u_{m}-u_{n}\| \\& \quad\le(M+MN+NC_{1})\|u_{m}-u_{n} \|+NC_{2}\|v_{m}-v_{n}\|+NC_{3}(w_{m}-w_{n}). \end{aligned}$$
From this inequality and the definition of the measure of noncompactness, it follows that
$$\begin{aligned}& \alpha\bigl(\bigl\{ f(t, u_{n}, v_{n}, w_{n}) \bigr\} \bigr) \\& \quad\le(M+MN+NC_{1})\alpha\bigl(\{u_{n}\} \bigr)+NC_{2}\alpha\bigl(\{v_{n}\}\bigr)+NC_{3} \alpha\bigl(\{w_{n}\} \bigr) \\& \quad\le L^{\prime}\bigl[\alpha\bigl(\{u_{n}\}\bigr)+\alpha\bigl( \{v_{n}\}\bigr)+\alpha\bigl(\{w_{n}\}\bigr)\bigr], \end{aligned}$$
where \(L^{\prime}=\max\{M+NM+NC_{1}, NC_{2}, NC_{3}\}\). If \(\{u_{n}\}\), \(\{v_{n}\} \), and \(\{w_{n}\}\) are decreasing sequences, the above inequality is also valid. Hence (H3) holds.

Therefore, by Theorem 3.1, problem (1.1) has a minimal solution \(\underline{u}\) and a maximal solution \(\overline{u}\) in \([y_{0},x_{0}]\). By the proof of Theorem 3.1, (3.7), (3.8), (3.10), and (3.11) are valid. Going from \(J_{1}^{\prime}\) to \(J_{m+1}^{\prime}\) interval by interval, we show that \(\underline{u}(t)\equiv\overline {u}(t)\) in every \(J^{\prime}_{k}\), \(k=1,2,\ldots,m+1\).

Indeed, for \(t\in J_{1}^{\prime}\), by (3.11) and (3.2) and assumption (H4), we have
$$\begin{aligned} \theta \le&\overline{u}(t)-\underline{u}(t)=F\overline {u}(t)-F\underline{u}(t) \\ =&\int_{t_{0}}^{t}\Gamma(t-s) \bigl[f\bigl(s, \overline{u}(s), T\overline{u}(s), \overline{u}^{\prime}(s)\bigr) -f\bigl(s, \underline{u}(s), T\underline{u}(s), \underline{u}^{\prime}(s)\bigr) +M\bigl(\overline{u}(s)-\underline{u}(s)\bigr) \bigr]\,ds \\ \le&\int_{t_{0}}^{t}M_{1} \bigl[(M+C_{1}) \bigl(\overline{u}(s)-\underline{u}(s)\bigr)+ C_{2}\bigl(T\overline{u}(s)-T\underline{u}(s)\bigr) + C_{3}\bigl(\overline{u}^{\prime}(s)-\underline{u}^{\prime}(s) \bigr) \bigr]\,ds \\ \le& M_{1}(M+C_{1})\int_{t_{0}}^{t} \bigl(\overline{u}(s)-\underline {u}(s)\bigr)\,ds \\ &{} +M_{1}C_{2}\hbar_{0}\int_{t_{0}}^{t} \int_{t_{0}}^{s}\bigl(\overline{u}(\tau)-\underline {u}(\tau)\bigr)\,d\tau \,ds +M_{1} C_{3}\bigl(\overline{u}(t)- \underline{u}(t)\bigr) \\ \le& M_{1}(M+C_{1}+a C_{2}\hbar_{0}) \int_{t_{0}}^{t}\bigl(\overline{u}(s)-\underline {u}(s) \bigr)\,ds+M_{1} C_{3}\bigl(\overline{u}(t)-\underline{u}(t) \bigr), \end{aligned}$$
(3.12)
where \(M_{1}=\sup_{t\in J}\|\Gamma(t)\|\). It follows from (3.12) and the normality of cone P that
$$\bigl\| \overline{u}(t)-\underline{u}(t)\bigr\| \le M_{1}N(M+C_{1}+a C_{2}\hbar_{0})\int_{t_{0}}^{t} \bigl\| \overline{u}(s)-\underline{u}(s)\bigr\| \,ds+ M_{1}C_{3}\bigl\| \overline {u}(t)-\underline{u}(t)\bigr\| , $$
i.e.,
$$\bigl\| \overline{u}(t)-\underline{u}(t)\bigr\| \le\frac{M_{1}N(M+C_{1}+a C_{2}\hbar _{0})}{1-M_{1} C_{3}}\int _{t_{0}}^{t}\bigl\| \overline{u}(s)-\underline{u}(s)\bigr\| \,ds. $$
Thus, by the Bellman inequality, we obtain \(\underline{u}(t)\equiv \overline{u}(t)\) in \(J_{1}^{\prime}\).
For \(t\in J_{2}^{\prime}\), since \(I_{1}(\overline{u}(t_{1}))=I_{1}(\underline {u}(t_{1}))\), using (3.2) and by completely the same argument as above for \(t\in J_{1}^{\prime}\), we can prove that
$$\begin{aligned} \bigl\| \overline{u}(t)-\underline{u}(t)\bigr\| \le& \frac{M_{1}N(M+C_{1}+a C_{2}\hbar _{0})}{1-M_{1} C_{3}}\int _{t_{0}}^{t}\bigl\| \overline{u}(s)-\underline{u}(s)\bigr\| \,ds \\ =& \frac{M_{1}N(M+C_{1}+a C_{2}\hbar_{0})}{1-M_{1} C_{3}}\int_{t_{1}}^{t}\bigl\| \overline {u}(s)-\underline{u}(s)\bigr\| \,ds. \end{aligned}$$
Again, by the Bellman inequality, we obtain \(\underline{u}(t)\equiv \overline{u}(t)\) in \(J_{2}^{\prime}\).

Continuing such a process interval by interval up to \(J_{m+1}^{\prime}\), we see that \(\underline{u}(t)\equiv\overline{u}(t)\) over the whole of J. Hence, \(u^{*}:=\underline{u}=\overline{u}\) is the unique mild solution of problem (1.1) in \([y_{0},x_{0}]\), which can be obtained by the monotone iterative procedure (3.7) starting from \(y_{0}\) or \(x_{0}\). □

Remark 3.2

(1) Using the above argument method interval by interval from \(J_{1}^{\prime}\) to \(J_{m+1}^{\prime}\), we can also improve the main results in [19] and [21], and delete some restrictive conditions there.

(2) In this study, the equicontinuity of the semigroup \(G(t)\) (\(t\ge t_{0}\)) generated by A is not required.

4 Concluding remarks

In this paper, we introduce and study the following nonlinear first-order implicit impulsive differential equation problem in Banach space \({\mathbb{B}}\):

Find \(u: J\to{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}\) such that
$$\begin{cases} u^{\prime}(t)=Au(t)+f(t,u(t),Tu(t), u^{\prime}(t)),\quad t\neq t_{k},\\ \triangle u|_{t=t_{k}}=I_{k}(u(t_{k})),\quad k=1, 2, \ldots, m,\\ u(t_{0})=u_{0}. \end{cases} $$

By using a monotone iterative technique in the presence of lower and upper solutions, the existence of extremal solutions and a unique mild solution between the lower and upper solutions are obtained under wide monotone conditions and the noncompactness measure conditions. The results presented in this paper improved and generalized some known results concerned with the integro-differential equations and classical (abstract) differential equations.

Moreover, we remark that if the lower solution and the upper solution for problem (1.1) do not exist, then we have the following results.

Theorem 4.1

Let \({\mathbb{B}}\) be an ordered Banach space, whose positive cone P is normal, \(A: \operatorname{dom}(A)\subset{\mathbb{B}}\to{\mathbb{B}}\) be a closed linear operator and generate a positive \(C_{0}\)-semigroup \(G(t)\) (\(t\ge t_{0}\)) in \({\mathbb{B}}\), \(f\in C(J\times{\mathbb{B}}\times{\mathbb{B}}\times{\mathbb{B}}, {\mathbb{B}})\), and \(I_{k}\in C({\mathbb{B}}, {\mathbb{B}})\), \(k=1, 2, \ldots, m\). Assume that there exist \(b>0\), \(x_{0}\in \operatorname{dom}(A)\), \(x_{0}\ge\theta\), \(y_{k}\in \operatorname{dom}(A)\), \(y_{k}\ge\theta\), \(k=1, 2, \ldots, m\), \(h\in PC^{1}(J,{\mathbb{B}})\), and \(h(t)\ge\theta\) such that
$$\begin{aligned}& f\bigl(t,x,Tx,x^{\prime}\bigr)\le bx+h(t),\quad\quad I_{k}(x)\le y_{k},\quad x\ge\theta; \\& f\bigl(t,x,Tx,x^{\prime}\bigr)\ge bx-h(t),\quad\quad I_{k}(x) \ge-y_{k}, \quad x\le\theta. \end{aligned}$$
Then the following results hold:
  1. (1)

    If the \(C_{0}\)-semigroup \(G(t)\) (\(t\ge t_{0}\)) is compact in \(\mathbb{B}\), and the conditions (H1) and (H2) in Section  3 are satisfied, then problem (1.1) has minimal and maximal mild solutions.

     
  2. (2)

    Problem (1.1) has minimal and maximal mild solutions when the conditions (H1 (H3) in Section  3 are satisfied.

     
  3. (3)

    If the positive cone P is regular, and the conditions (H1) and (H2) in Section  3 are satisfied, then problem (1.1) has minimal and maximal mild solutions.

     
  4. (4)

    Problem (1.1) has a unique mild solution when the conditions (H1), (H2), and (H4) in Section  3 are satisfied.

     

Proof

Firstly, we consider the following initial value problem of the linear impulsive evolution equation in \({\mathbb{B}}\):
$$ \begin{cases} u^{\prime}(t)=Au(t)+b u(t)+h(t),\quad t\in J^{\prime},\\ \triangle u|_{t=t_{k}}=y_{k},\quad k=1, 2, \ldots, m,\\ u(t_{0})=x_{0}. \end{cases} $$
(4.1)
Since \((A+b I)\) generates a positive \(C_{0}\)-semigroup \(\Gamma (t)=e^{bt}G(t)\) (\(t\ge0\)) in \({\mathbb{B}}\), it follows from Theorem 2.9 in [31, Chapter 4] and Lemma 2.1, that problem (4.1) has a unique positive classical solution \(\hat{u}\in PC^{1}(J,E)\). Let \(y_{0}=-\hat{u}\), \(x_{0}=\hat{u}\), it is easy to see that \(y_{0}\) and \(x_{0}\) are the lower solution and the upper solution of problem (1.1), respectively. So, our conclusions (1)-(4) follow from Theorems 3.1 and 3.2. □

Declarations

Acknowledgements

We thank the referees’ valuable comments and suggestions to improve our paper.

This work was partially supported by Sichuan Province Cultivation Fund Project of Academic and Technical Leaders, and the Open Research Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2013WZJ01).

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
Institute of Nonlinear Science and Engineering Computing, Sichuan University of Science & Engineering
(2)
Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things
(3)
College of Materials and Chemical Engineering, Sichuan University of Sciences & Engineering

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