Periodic boundary value problems for second-order impulsive integro-differential equations with integral jump conditions

Differential equations which have impulse effects describe many evolution processes that abruptly change their state at a certain moment. In recent years, impulsive differential equations have become more important tools in some mathematical models of real processes and phenomena studied in physics, biotechnology, chemical technology, population dynamics and economics; see [1–5]. Many papers have been published about existence analysis of periodic boundary value problems of first and second order for impulsive ordinary or functional or integro-differential equations. We refer the readers to the papers [6–29]. More recent works on existence results of impulsive problems with integral boundary conditions can be found in [30–35] and the reference therein. This literature has lead to significant development of a general theory for impulsive differential equations.

The monotone iterative technique coupled with the method of upper and lower solutions has been used to study the existence of extremal solutions of periodic boundary value problems for second-order impulsive equations; see, for example, [36–41]. This method has been also used to study abstract nonlinear problems; see [42]. However, in most of these papers concerned with applications of the monotone iterative technique to second-order periodic boundary value problems with impulses, the authors assume that the jump conditions at impulse point $t_k$ of solution values and the derivative of solution values depend on the left-hand limits of solutions or the slope of solutions themselves, such as $\mathrm{\Delta }x(t_k)=I_k(x(t_k^{-}))$, $\mathrm{\Delta }x(t_k)=I_k(x^{\prime }(t_k^{-}))$, $\mathrm{\Delta }{x}^{\prime }(t_k)=I_k(x(t_k^{-}))$, $\mathrm{\Delta }{x}^{\prime }(t_k)=I_k(x^{\prime }(t_k^{-}))$.

$k(t,s)\in C(D,R^{+})$, $h(t,s)\in C(J\times J,R^{+})$, $D=\{(t,s)\in R^2,0\le s\le t\le T\}$, $R^{+}=[0,+\mathrm{\infty })$, $k_0=max\{k(t,s):(t,s)\in D\}$, $h_0=max\{h(t,s):(t,s)\in J\times J\}$.

where $0\le {\epsilon }_k\le {\delta }_k\le t_k-t_{k-1}$, $0\le {\sigma }_k\le {\tau }_k\le t_k-t_{k-1}$, $k=1,2,\dots ,m$. The integral jump condition (1.3) means that a sudden change of solution values and the derivative of solution values at impulse point $t_k$ depend on the area under the curves of $x^{\prime }(t)$ and $x(t)$ between $t=t_k-{\delta }_k$ to $t=t_k-{\epsilon }_k$ and $t=t_k-{\tau }_k$ to $t=t_k-{\sigma }_k$, respectively. It should be noticed that the impulsive effects of PBVP (1.1) have memory of the past states.

This paper is organized as follows. Firstly, we introduce a new concept of lower and upper solutions. After that, we establish some new comparison principles and discuss the existence and uniqueness of the solutions for second-order impulsive integro-differential equations with integral jump conditions. By using the method of upper and lower solutions and the monotone iterative technique, we obtain the existence of an extreme solution of PBVP (1.1). Finally, we give an example to illustrate the obtained results.

Let $J^{-}=J\setminus \{t_1,t_2,\dots ,t_m\}$, $J_0=[t_0,t_1]$, $J_k=(t_k,t_{k+1}]$ for $k=1,2,\dots ,m$. Let $PC(J,R)=\{x:J\to R;x(t)\text{ is continuous everywhere except for some }{t}_k\text{ at which }x(t_k^{+})\text{ and }x(t_k^{-})\text{ exist and}x(t_k^{-})=x(t_k),k=1,2,\dots ,m\}$, and $P{C}^1(J,R)=\{x\in PC(J,R);x^{\prime }(t)\text{ is continuous everywhere}\text{except for some }{t}_k\text{ at which }{x}^{\prime }(t_k^{+})\text{ and }{x}^{\prime }(t_k^{-})\text{ exist and }{x}^{\prime }(t_k^{-})=x^{\prime }(t_k),k=1,2,\dots ,m\}$. $PC(J,R)$ and $P{C}^1(J,R)$ are Banach spaces with the norms ${\parallel x\parallel }_{PC}=sup\{x(t):t\in J\}$ and ${\parallel x\parallel }_{P{C}^1}=max\{{\parallel x\parallel }_{PC},{\parallel x^{\prime }\parallel }_{PC}\}$. Let $E=P{C}^1(J,R)\cap C^2(J^{-},R)$. A function $x\in E$ is called a solution of PBVP (1.1) if it satisfies (1.1).

Now we are in the position to establish some new comparison principles which play an important role in the monotone iterative technique.

Then$x(t)\le 0$, $t\in J$.

Proof Suppose, to the contrary, that $x(t)>0$ for some $t\in J$. We divide the proof into two cases:

Case (i). There exists a $\tilde{t}\in J$ such that $x(\tilde{t})>0$ and $x(t)\ge 0$ for all $t\in J$.

From (2.1), we have $x^{″}(t)\ge 0$ for $t\in J^{-}$. Since $\mathrm{\Delta }{x}^{\prime }(t_k)\ge L_k^{\ast }{\int }_{t_k-{\tau }_k}^{t_k-{\sigma }_k}x(s)ds\ge 0$, then $x^{\prime }(t)$ is nondecreasing in $t\in J$ and so $x^{\prime }(0)\le x^{\prime }(T)$. However, by (2.1) $x^{\prime }(0)\ge x^{\prime }(T)$, then $x^{\prime }(0)=x^{\prime }(T)$, which implies $x^{\prime }(t)=\text{ constant}$ for all $t\in J$. Thus, $0=x^{″}(t)\ge Mx(\tilde{t})>0$, a contradiction.

Case (ii). There exists $t^{\ast },t_{\ast }\in J$ such that $x(t^{\ast })>0$, $x(t_{\ast })<0$.

Let ${inf}_{t\in J}x(t)=-\lambda <0$, then there exists $t_{\ast }\in J_i$, for some $i\in \{0,1,\dots ,m\}$, such that $x(t_{\ast })=-\lambda$ or $x(t_i^{+})=-\lambda$. Without loss of generality, we only consider $x(t_{\ast })=-\lambda$. For the case $x(t_i^{+})=-\lambda$ the proof is similar. It follows that

If $x^{\prime }(t)>0$ for all $t\in J$, then $\mathrm{\Delta }x(t_k)=L_k{\int }_{t_k-{\delta }_k}^{t_k-{\epsilon }_k}{x}^{\prime }(s)ds\ge 0$, $k=1,2,\dots ,m$. Hence, $x(t)$ is strictly increasing on J, which contradicts $x(0)=x(T)$. Then there exists a $\overline{t}\in J$ such that $x^{\prime }(\overline{t})\le 0$.

Let $\overline{t}\in J_j$, $j\in \{0,1,\dots ,m\}$. By mean value theorem, we have

Let $t^{\ast }\in J_r$ for some $r\in \{0,1,\dots ,m\}$. We first assume that $t_{\ast }<t^{\ast }$, then $i<r$. By the mean value theorem, we have

which contradicts (2.2).

For the case $t^{\ast }<t_{\ast }$, the proof is similar, and thus we omit it. This completes the proof. □

Lemma 2.2 Assume that $x\in E$ satisfies

where$M>0$, $N\ge 0$, $L\ge 0$, $L_k\ge 0$, $L_k^{\ast }\ge 0$are constants and$0\le {\epsilon }_k\le {\delta }_k\le t_k-t_{k-1}$, $0\le {\sigma }_k\le {\tau }_k\le t_k-t_{k-1}$, $k=1,2,\dots ,m$, and they satisfy (2.2). Then$x(t)\le 0$for all$t\in J$.

Then by Lemma 2.1, we get $w(t)\le 0$ for all $t\in J$, which implies that $x(t)\le 0$, $t\in J$. □

where constants $M>0$, $N\ge 0$, $L\ge 0$, $L_k\ge 0$, $L_k^{\ast }\ge 0$, ${\gamma }_k$, ${\lambda }_k$ are constants and $g\in PC(J,R)$, $0\le {\epsilon }_k\le {\delta }_k\le t_k-t_{k-1}$, $0\le {\sigma }_k\le {\tau }_k\le t_k-t_{k-1}$, $k=1,2,\dots ,m$.

This proof is similar to the proof of Lemma 2.1 in [36], and we omit it.

Lemma 2.4 Let$M>0$, $N\ge 0$, $L\ge 0$, $L_k\ge 0$, $L_k^{\ast }\ge 0$are constants and$0\le {\epsilon }_k\le {\delta }_k\le t_k-t_{k-1}$, $0\le {\sigma }_k\le {\tau }_k\le t_k-t_{k-1}$, $k=1,2,\dots ,m$. If

(2.6)

(2.7)

then (2.4) has a unique solution x in E.