Existence of solutions of an impulsive integro-differential equation with a general boundary value condition

1.   Introduction

In recent years, impulsive integro-differential equations with boundary conditions have attracted much attentions and been studied extensively^[1,2,3]. We notice that the periodic and antiperiodic boundary value problems are very common, and they have a wide range of applications^{[4,5,6,7,8,9,10,11]}. The monotone iterative technique is a common method to prove the existence of extremal solutions for impulsive integro-differential equations ^[4]. The monotone sequences of a linear system is developed from the upper and lower solutions, and this method can prove monotone sequences converge monotonically to the extremal solutions of the original system ^[12,13]. Luo et al. ^[5] developed some new comparison principles and existence results of solutions for an impulsive integro-differential equation with the periodic boundary conditions. Recently, Kumar et al. ^[9] discussed the stability and existence of a fractional integro differential equation with the periodic boundary condition. Gou et al. ^[14] explored the existence of mild solutions for the periodic boundary conditions in a semilinear fractional evolution system. The existence result for the periodic boundary conditions in the phi-Laplacian impulsive differential equation can refer to ^[15]. Ibnelazyz et al. ^[11] studied the existence results for a fractional integro-differential equation with the antiperiodic boundary conditions. Ding et al. used the monotone iterative technique to discuss the existence of solutions for a class of impulsive functional differential equations with the anti-periodic boundary value condition^[16]. Zuo et al. ^[6] studied the existence and uniqueness of solutions of an antiperiodic boundary problem in a mixed impulsive fractional integro-differential equation. However, we note that the periodic and anti-periodic boundary values are both two special conditions. For the impulsive integro-differential equation with a more general boundary condition, such as $"w(0) = \chi w(T), \chi\in R"$, have not been involved by now.

Inspired by Luo and Nieto ^[5], in this paper, we consider the follow two-point boundary value problem (TP-BVP) for a first-order impulsive integro-differential system:

where $\xi = [0, T], f\in C(\xi\times R^3)$, $I_k\in C(R, R), 0 = \theta_0 < \theta_1 < \cdots \theta_m < \theta_{m+1} = T, \triangle w(\theta_k) = w(\theta_k^+)-w(\theta_k^-),$ $w(\theta_k^-)$ and $w(\theta_k^+)$ are the left and right limits of $w(\theta)$ at $\theta = \theta_k$,

$\Phi \in C(D, R^+), D = \{(\theta, s)\in \xi\times \xi:\theta\geq s\}, \Psi\in C(\xi\times \xi, R^+), R^+ = [0, \infty), \chi\in R.$ It should be noted that $\chi$ is an arbitrary real number and the boundary condition in Eq (1.1) is more general than the periodic or antiperiodic boundary value. Therefore, the existence result of the solution of Eq (1.1) will have a wider range of application than previous studies.

In Section 2, we establish new comparison principles. In Section 3, we discuss the existence and uniqueness of the solutions for a linear BVP. Finally, we obtain the extremal solutions for TP-BVP Eq (1.1) in Section 4.

2.   Preliminaries and some basic results

Similar to previous studies^[2,5,17,18], we give the follow spaces to define the solution of Eq (1.1): $LC(\xi) = \{w:\xi\rightarrow R:w\left|_{(\theta_k, \theta_{k+1}]}\right.\in C((\theta_k, \theta_{k+1}], R),$ $k = 0, 1, \cdots m$; $w(\theta_k^+)$ and $w(\theta_k^-)$ exist for $k = 1, 2, \cdots, m$ with $w(\theta_k^-) = w(\theta_k) \}$; $LC'(\xi) = \{w\in LC(\xi); w\left|_{(\theta_k, \theta_{k+1}]}\right. \in C'((\theta_k, \theta_{k+1}], R)$, $k = 0, 1, \cdots, m$; Limits $w'(\theta_k^-)$, $w'(\theta_k^+)$, $w'(0^+)$ and $w'(T^-)$ exist when $k = 1, 2, \cdots, m\}$. It is not difficult to verify that $LC(\xi)$ and $LC'(\xi)$ are both Bananch spaces with the following norms^[5]:

Then, a function $w\in LC'(\xi)$ is a solution of Eq (1.1) when it satisfies Eq (1.1).

Now, we prove the follow key comparison lemmas.

Lemma 2.1. (New comparison principles) Suppose that $\Lambda_k > -1 (k = 1, 2, \cdots, m)$, $\rho_1, \rho_2 \geq 0, \chi > e^{-\varepsilon T}$ and $\varepsilon > 0$, such as

or

$l_{wk} = \Lambda_kg(\theta_k)-\Delta g(\theta_k)$ and $b_w(\theta) = -g'(\theta)+\varepsilon g(\theta)+\rho_1[\Gamma g](\theta)+\rho_2[\delta g](\theta)$. Where $g\geq 0$ is a function in space $LC'(\xi)$, which satisfies $g(0)-\chi g(T)\geq \chi w(T)-w(0) > 0.$

We define $\overline {\overline {\Lambda_k } } = \min \left\{ {\Lambda_k, 0} \right\}$ for $k = 1, 2, ..., m$, and

the following inequality is assumed to be true:

Then, we can draw a conclusion that $w(\theta)\leq 0$ for $\theta\in \xi$.

Proof. For $\Lambda_k > -1$, then we define $c_k = 1+\Lambda_k > 0.$ If boundary conditions satisfy $w(0)\geq \chi w(T),$ we let $\zeta(\theta) = \left(\prod\limits_{\theta < \theta_k < T}c_k^{-1}\right)w(\theta)e^{-\varepsilon \theta},$ then the signs of w and $\zeta$ are same and it can be obtained that

Now, we complete the proof by two cases:

(i): If $\zeta\geq 0$ and $\zeta\not\equiv0$; then clearly $\zeta'(\theta)\geq 0$ for $\theta\in \xi'$ and $\zeta(T)\geq \zeta(0)\prod_{i = 1}^mc_j\geq \chi \zeta(T)e^{\varepsilon T}.$ If $\zeta(T) = 0$, it is easy to know that $\zeta(\theta)\leq 0$ by two conditions $c_k > 0$ and $\zeta'(\theta)\geq 0$, i.e., $\zeta\equiv 0$, which is inconsistent with the assumption. Moreover, if $\zeta(T) > 0,$ one can obtain that $\chi e^{\varepsilon T}\leq1$, which is a wrong conclusion.

(ii): Denote $r_1\in [0, T]$, which satisfies $\zeta(r_1) > 0$. Suppose that $\zeta(r_2) = \mbox{min}_{\theta\in [0, T]}\zeta(\theta) = n$, then clearly $n < 0$. It can be obtained that

If $r_1 > r_2$, we have

then

$\overline {\overline {c_j } } = 1+\overline{\overline {\Lambda_j}}, j = 1, 2, ...m$, it is a contradiction with the condition Eq (2.3). If $r_1 < r_2$, we have

and

For $\zeta(0)\geq \chi \zeta(T)\left(\prod\limits_{k = 1}^m c_k^{-1}\right)e^{\varepsilon T}$, it is easy to obtain that

Then, the follow inequality can be obtained with conditions $r_2 < r_1, \chi e^{\varepsilon T} > 1$ and $n < 0:$

i.e.,

It is a contradiction with Eq (2.3).

On the other hand, if boundary conditions satisfy $w(0) < \chi w(T)$, we let $b(\theta) = w(\theta)+g(\theta)$. It is easy to get that

Clearly, $b\leq 0$ from the above proof, and $w(\theta)\leq 0$.

Remark 2.1. Lemma 2.1 is a key comparison result to obtain extremal solutions of Eq (1.1). Expression and proof of Lemma 2.1 are similar to previous studies^[5]. Moreover, the boundary condition $"w(0) = \chi w(T)"$ with $\chi > e^{-\varepsilon T}$ is more general than the periodic condition $"\chi = 1"$. Our method can also generalize some known results, such as Corollary 2.1^[5,19], Corollary 2.2^[5] and Corollary 2.3^[5,20].

Corollary 2.1. Let $\varepsilon > 0, \rho_1, \rho_2\geq0, \chi > e^{-\varepsilon T}$, $\Lambda_k\geq 0, k = 1, 2, \cdots, m$, $w\in LC'(\xi)$ satisfies Eq (2.1) or (2.2), and define

if the follow inequality Eq (2.4) holds

then, we have $w(\theta)\leq0$ for $\theta\in \xi$.

Proof. We prove that Eq (2.3) holds as follows:

We know that $w(\theta)\leq0$ by Lemma 2.1.

Corollary 2.2. Suppose that $\varepsilon > 0, \rho_1, \rho_2\geq0, \chi > e^{-\varepsilon T}$, $\Lambda_k\geq 0, k = 1, 2, \cdots, m$, $w\in LC'(\xi)$ satisfies Eq (2.1) or (2.2), and let

then, we can obtain $w(\theta)\leq 0$ on $\xi$.

Proof. The condition Eq (2.3) is derived as follows:

From Lemma 2.1, we have $w(\theta)\leq 0$ on $\xi$.

Corollary 2.3. Let $\Lambda_k\geq 0, k = 1, 2, \cdots, m$, $\rho_1, \rho_2\geq0, \varepsilon > 0, \chi > e^{-\varepsilon T}$, $w\in LC'(\xi)$ satisfies Eq (2.1) or (2.2), and suppose that

where $\tau = \max \left\{ {\theta_k - \theta_{k - 1} :k = 1, 2, ...m + 1} \right\}.$ Then, we have $w(\theta)\leq 0$ on $\xi$.

Proof. We prove that the inequality Eq (2.3) holds as follows:

3.   A linear system

Now, we study the solution of a linear system (LS) with the general boundary condition:

where $I_j \in C\left({R, R} \right)$ and $\varsigma \in LC(\xi)$.

Lemma 3.1. The solution of (LS) can be described as follows:

where

Proof. Set $z(\theta) = e^{\varepsilon \theta}w(\theta), \theta\in \xi.$ Then

where $I_j^* \left(x \right) = e^{\varepsilon \theta_j } I_j^{} \left({e^{ - \varepsilon \theta_j } x} \right)$, $\varsigma ^* \left(\theta \right) = e^{\varepsilon \theta} \varsigma \left(\theta \right)$.

If $\theta \in \left({\theta_j, \theta_{j + 1} } \right], j = 1, ..., m,$ we obtain

Since

So, when $\theta\in (\theta_j, \theta_{j+1}],$ we have

Therefore,

In Eq (3.6), we let $\theta = T$, then we have

Finally, substitute z(0) into Eq (3.6), we get that

4.   Main results

Lemma 4.1 is given without proof since it is similar to Lemma 3.1.

Lemma 4.1. Let $\Lambda_k > -1, k = 1, 2, \cdots, m$, $\rho_1, \rho_2 \geq 0, \varepsilon > 0, \chi > e^{-\varepsilon T}$, $\vartheta\in LC'(\xi)$ and $\varsigma\in LC(\xi)$, then the solution ($w\in LC'(\xi)$) of the follow Eq (4.1) can be expressed as Eq (4.2):

where

Lemma 4.2. Let $\Lambda_k > -1, k = 1, 2, \cdots, m$, $\rho_1, \rho_2 \geq 0, \varepsilon > 0, \chi > e^{-\varepsilon T}$, $I_k\in C(R, R)$, $\vartheta\in LC'(\xi)$ and $\varsigma\in LC(\xi)$, if the follow inequality holds:

then the solution of Eq (4.1) is unique.

Proof. Define the operator $F:LC(\xi)\rightarrow LC(\xi),$ where $Fw$ is given by the right-hand term in Eq (4.2). Clearly, the solution of Eq (4.1) is also the fixed point of the operator equation w = Fw. Since,

Then, we know that F is a contractive mapping by condition Eq (4.3). So, according to Banach's fixed point theorem, the solution of Eq (4.1) is unique.

Finally, we can obtain the following existence theorem of extremal solutions by using Lemmas 2.1 and 4.2. The arguments of Theorem 4.1 are similar to that in ^[4] and ^[5], the proof process is omitted.

Theorem 4.1. Suppose that $\Lambda_k > -1$, $k = 1, 2, \cdots, m$, $\rho_1, \rho_2 \geq 0, \varepsilon > 0, \chi > e^{-\varepsilon T}$, and the follow four conditions satisfy:

(i) The conditions Eqs (2.3) and (4.3) hold.

(ii) There exist two functions $\nu, \mu\in LC'(\xi)$ such as $\mu(\theta) \leq \nu(\theta)$ and

or

$b_\mu(\theta) = -g_2'(\theta)+\varepsilon g_2(\theta) +\rho_1[\Gamma g_2](\theta)+\rho _2[\delta g_2](\theta)$, $l_{\mu k} = \Lambda_k g_2(\theta_k)-\Delta g_2(\theta_k)$, where $g_2\in LC'(\xi)$ with $g_2\ge 0, g_2(0)-\chi g_2(T) \geq \chi\mu(T) -\mu(0) > 0$.

and

or

$b_\nu(\theta) = -g_1'(\theta)+\varepsilon g_1(\theta) +\rho_1[\Gamma g_1](\theta)+\rho_2[\delta g_1](\theta)$, $l_{\nu k} = \Lambda_kg_1(\theta_k)-\Delta g_1(\theta_k)$, where $g_1\in LC'(\xi)$ with $g_1\geq 0, g_1(0)-\chi g_1(T)\geq\nu(0)-\chi\nu(T) > 0.$

(iii) When $\mu(\theta_k)\leq y\leq x\leq\nu(\theta_k), I_k(k = 1, 2, \cdots, m)$ meet

(iv) When $\theta \in \xi, \mu \leq \overline w\leq w\leq\nu, [\Gamma\mu](\theta)\leq \overline \zeta(\theta)\leq \zeta(\theta)\leq[\Gamma\nu](\theta), [\delta\mu](\theta) \leq \overline z(\theta)\leq z(\theta)\leq[\delta\nu](\theta),$ and $f$ meets

Then, two monotone sequences ${\nu_n}, {\mu_n}$ can be found such as $\nu = \nu _0 \geq\nu _n \geq... \geq\mu _n \geq \mu _0 = \mu$, which converge uniformly to the maximal and minimal solutions of Eq (1.1) in

Remark 4.1. If $\chi = 1$, then the Eq (1.1) is a periodic BVP. Therefore, the condition "$\chi > e^{-\varepsilon T}$" is more general than the periodic boundary condition. Existence result of solution (Theorem 4.1) obtained in this paper is more applicable than that in the periodic BVP.

5.   Conclusions

In this paper, we discuss the existence of solutions for a first-order nonlinear impulsive integro-differential equation with a general boundary value condition $"w(0) = \chi w(T)"$. Firstly, new comparison principles are developed in Section 2, which are key comparison results to obtain extremal solutions of Eq (1.1). We note that the boundary condition $"w(0) = \chi w(T)"$ with $\chi > e^{-\varepsilon T}$ is more general than the periodic condition $"\chi = 1"$. Then, the expression of solution for a linear system is given in Section 3. Finally, we obtain the existence results of extremal solutions for Eq (1.1) by using the monotone iterative technique, as shown in Theorem 4.1. Previous studies mainly focused on the periodic and antiperiodic boundary value conditions, therefore, the condition "$\chi > e^{-\varepsilon T}$" is more general. The main results in Section 4 are more general than previous studies, which may be widely applied in this field.

Acknowledgments

This research was supported by the National Science Foundation of China (No. 11602092); the China Postdoctoral Science Foundation (No. 2018M632184).

Conflict of interest

All authors declare no conflicts of interest in this paper.