Periodic solutions for an impulsive system of integro-differential equations with maxima

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1. Problem Statement

The mathematical models of many problems of modern sciences, technology, and economics are described by differential and integro-differential equations, the solutions of which are functions with first-kind discontinuities at fixed or non-fixed times. Such differential and integro-differential equations are called equations with impulsive effects. Various publications are appearing on the study of differential and integro-differential equations with impulsive effects, describing many natural and technical processes (see for instance [1-13]).

As is known, in recent years, interest in the study of differential and integro-differential equations with periodical boundary conditions has increased. In particular, in the works [14-17], periodic solutions of differential equations with impulsive effects are studied.

In this paper, we investigate a periodical boundary value problem for a system of first-order integro-differential equations with impulsive effects, nonlinear kernel depending on construction of maxima. The questions of existence and uniqueness of the solution to the periodical boundary value problem are investigated. We note that differential and integro-differential equations with maxima have singularity in the study of the questions of solvability [18].

On the interval $\left[0,T\right]$ for $t\ne t_i$ ( $i=1,2\dots \mathrm{,}p$) we consider the questions of existence and constructive methods of calculating the periodic solutions of the system of nonlinear ordinary first-order integro-differential equations with impulsive effects and maxima

$x^{\text{'}}\left(t\right)=\left(ft\mathrm{,}x\left(t\right),{\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_{1}s{\lambda }_2\left(s\right)\right]\right\}\right)ds\right).$ (1)

We study the integro-differential equation (1) with periodic conditions

$\left\{\begin{array}{cc}x\left(t\right)=\phi \left(t\right)& t\in (-\infty ,0],\\ x\left(0\right)=x\left(T\right),& \end{array}\right.$ (2)

and nonlinear impulsive effect

$x\left(t_i^{+}\right)-x\left(t_i^{-}\right)=F_i\left(x\left(t_i\right)\right),i=1,2\dots \mathrm{,}p\mathrm{,}$ (3)

where $0<\overline{t}<T$, $\overline{t}\ne t_i$, $i=1,2\dots \mathrm{,}p$; $0=t_0<t_1<\cdots <t_p<t_{p+1}=T$; $x$, $y\in X$; $X$ is the closed bounded domain in the space ${ℝ}^n$, $\partial X$ is its border; $f\in {ℝ}^n$, $-\infty <{\lambda }_1\left(t\right)<{\lambda }_{2}t\le t$, $\phi \left(t\right)\in C\left((-\infty , 0] {\mathrm{ℝ}}^n\right)$; $\phi \left(0^{-}\right)=x\left(0^{+}\right)$; $x\left(t_i^{+}\right)=\underset{\nu \to 0^{+}}{lim}x\left(t_i+\nu \right)$, $x\left(t_i^{-}\right)=\underset{\nu \to 0^{-}}{lim}x\left(t_i-\nu \right)$ are the limits of the function on the right and left sides $x\left(t\right)$ at the point $t=t_i$, respectively. The function $f$ is $T$-periodic $F_i=F_{i+p}$, $t_{i+p}=t_i+T$,

${\int }_{-\infty }^t\left|K\left(t\mathrm{,}s\mathrm{,}x\right)\right|ds<\infty \mathrm{.}$                                         

By $C\left(\left[0,T\right],{\mathrm{ℝ}}^n\right)$ denoted the Banach space, which consists continuous vector function $x\left(t\right)$, defined on the segment $\left[0,T\right]$, with the norm

$\parallel xt\parallel =\sqrt{\sum _{j=1}^n\underset{0\le t\le T}{max}\left|x_j\left(t\right)\right|}\mathrm{.}$                                       

By $PC\left(\left[0,T\right],{\mathrm{ℝ}}^n\right)$ is denoted the following linear vector space:

$PC\left(\left[0,T\right]{\mathrm{ℝ}}^n\right)=\left\{x:\left[0,T\right]\to {\mathrm{ℝ}}^n;x\left(t\right)\in C\left(t_i\mathrm{,}{t}_{i+1}]{\mathrm{ℝ}}^n\right)i=1,2\dots \mathrm{,}p\right\},$         

where $x\left(t_i^{+}\right)$ and $x\left(t_i^{-}\right) \left(i=0,1\dots \mathrm{,}p\right)$ exist and bounded; $x\left(t_i^{-}\right)=x\left(t_i\right)$. Note, that the linear vector space $PC\left(\left[0,T\right]{\mathrm{ℝ}}^n\right)$ is Banach space with the following norm:

$\parallel xt{\parallel }_{PC}=\max \left\{\parallel x\left(t\right){\parallel }_{C\left(\left(t_i\mathrm{,}{t}_{i+1}\right]\right)}\mathrm{,}i=1,2\dots \mathrm{,}p\right\}.$                         

Formulation of problem. Find the $T$-periodic function $x\left(t\right)\in PC\left(\left[0,T\right]{\mathrm{ℝ}}^n\right)$ which for all $t\in \left[0,T\right], t\ne t_i, \left(i=1,2\dots \mathrm{,}p\right)$ satisfies the system of differential equations (1), periodic condition (2) and for $t=t_i \left(i=1,2\dots \mathrm{,}p\right)$, $0<t_1<t_2<\cdots <t_p<T$ satisfies the nonlinear limit condition (3) and goes through $x_0$  at  $t=0$.

2. Reduction to Functional-Integral Equation

Let the function $x\left(t\right)\in PC\left(\left[0,T\right],{\mathrm{ℝ}}^n\right)$ be a solution of the periodic boundary value problem (1)–(3). Then by integration of the equation (1) on the intervals $(0, t_1]$, $\left(t_1\mathrm{,}{t}_2\right]$,..., $\left(t_p\mathrm{,}{t}_{p+1}\right]$, we obtain the following:

${\int }_0^{t_1}f\left(s\mathrm{,}x\mathrm{,}y\right)ds={\int }_0^{t_1}{x}^{\text{'}}\left(s\right)ds=x\left(t_1^{-}\right)-x\left(0^{+}\right),t\in (0,t_1]$,

${\int }_{t_1}^{t_2}f\left(s\mathrm{,}x\mathrm{,}y\right)ds={\int }_{t_1}^{t_2}{x}^{\text{'}}\left(s\right)ds=x\left(t_2^{-}\right)-x\left(t_1^{+}\right),t\in \left(t_1\mathrm{,}{t}_2\right],$

${\int }_{t_p}^{t_{p+1}}f\left(s\mathrm{,}x\mathrm{,}y\right)ds={\int }_{t_p}^{t_{p+1}}{x}^{\text{'}}\left(s\right)ds=x\left(t_{p+1}^{-}\right)-x\left(t_p^{+}\right), t\in \left(t_p\mathrm{,}{t}_{p+1}\right],$

where

$f\left(s\mathrm{,}x\mathrm{,}y\right)=f\left(t\mathrm{,}x\left(t\right),{\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(s\right){\lambda }_2\left(s\right)\right]\right\}\right)ds\right).$           

Hence, taking $x\left(0^{+}\right)=x$, $x\left(t_{k+1}^{-}\right)=x\left(t\right)$ into account, on the interval $\left[0, T\right]$ we have

${\int }_0^{t}f\left(s\mathrm{,}x\mathrm{,}y\right)ds=\left[x\left(t_1\right)-x\left(0^{+}\right)\right]+\left[x\left(t_2\right)-x\left(t_1^{+}\right)\right]+\cdots +\left[x\left(t\right)-xt\left({}_p^{+}\right)\right]=$

$=-x\left(0\right)-\left[x\left(t_1^{+}\right)-x\left(t_1\right)\right]-\left[x\left(t_2^{+}\right)-x\left(t_2\right)\right]-\cdots -\left[x\left(t_p^{+}\right)-x\left(t_p\right)\right]+x\left(t\right).$

Taking into account the condition (3), we rewrite the last equality as

$x\left(t\right)=x\left(0\right)+{\int }_0^{t}f\left(s\mathrm{,}x\mathrm{,}y\right)ds+\sum _{0<t_i<t}{F}_i\left(x\left(t_i\right)\right).$ (4)

We subordinate the function $x\left(t\right)\in PC\left(\left[0,T\right]{\mathrm{ℝ}}^n\right)$ in (4) to satisfy the periodic condition (2):

$x\left(T\right)=x\left(0\right)+{\int }_0^{T}f\left(s\mathrm{,}x\mathrm{,}y\right)ds+\sum _{0<t_i<T}{F}_i\left(x\left(t_i\right)\right).$                           

Hence, taking the condition (2) into account, we obtain the following:

${\int }_0^{T}f\left(s\mathrm{,}x\mathrm{,}y\right)ds+\sum _{0<t_i<T}{F}_i\left(x\left(t_i\right)\right)=0.$                                 

Consequently, the integro-differential equation (1) can be written as

$x^{\text{'}}\left(t\right)=f\left(t\mathrm{,}x\left(t\right){\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(s\right){\lambda }_2\left(s\right)\right]\right\}\right)ds\right)-$

$-\frac{1}{T}{\int }_0^{T}f\left(t\mathrm{,}x\left(t\right){\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(s\right){\lambda }_2\left(s\right)\right]\right\}\right)ds\right)dt-$

$-\frac{1}{T}\sum _{i=1}^p{F}_i\left(x\left(t_i\right)\right).$ (5)

Then by integration of equation (5) into the intervals $(0, t_1], \left(t_1\mathrm{,}{t}_2\right], ..., \left(t_p\mathrm{,}{t}_{p+1}\right]$ instead of (4) we obtain the following system of equations:

$x\left(t\right)=x_0+{\int }_0^t[f\left(s\mathrm{,}x\left(s\right){\int }_{-\infty }^{s}K\left(s\mathrm{,}\theta \mathrm{,}\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(\theta \right){\lambda }_2\left(\theta \right)\right]\right\}\right)d\theta \right)-$

$-\frac{1}{T}{\int }_0^{T}f\left(\theta \mathrm{,}x\left(\theta \right){\int }_{-\infty }^{\theta }K\left(\theta \mathrm{,}\xi \mathrm{,}\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(\xi \right){\lambda }_2\left(\xi \right)\right]\right\}\right)d\xi \right)d\theta -$

$-\frac{1}{T}\sum _{i=1}^p{F}_i\left(x\left(t_i\right)\right)]ds+\sum _{0<t_i<t}{F}_i\left(x\left(t_i\right)\right).$ (6)

3. Preliminaries

Lemma 1. For the equation (6) the following estimate is true:

$\parallel x\left(t\right)-x_0{\parallel }_{PC}\le M_1\frac{T}{2}+2M_{2}p\mathrm{,}$ (7)

where

$M_1=\parallel f\left(t\mathrm{,}x\left(t\right)y\left(t\right)\right)\parallel \mathrm{,}{M}_2=\underset{1\le i\le p}{max}\parallel F_i\left(t\mathrm{,}x\left(t\right)\right)\parallel \mathrm{.}$                       

Proof. We rewrite the equation (6) as follows:

$x\left(t\right)-x_0={\int }_0^t[f\left(s\mathrm{,}x\left(s\right)y\left(s\right)\right)-$

$-\frac{1}{T}{\int }_0^{T}f\left(\theta \mathrm{,}x\left(\theta \right)y\left(\theta \right)\right)d\theta -\frac{1}{T}\sum _{i=1}^p{F}_i\left(x\left(t_i\right)\right)]ds+\sum _{0<t_i<t}{F}_i\left(x\left(t_i\right)\right)=$

$={\int }_0^{t}f\left(s\mathrm{,}x\left(s\right)y\left(s\right)\right)ds-\frac{t}{T}{\int }_0^{t}f\left(s\mathrm{,}x\left(s\right)y\left(s\right)\right)ds-$

$-\frac{t}{T}{\int }_t^{T}f\left(s\mathrm{,}x\left(s\right)y\left(s\right)\right)ds-\frac{t}{T}\sum _{i=1}^p{F}_i\left(x\left(t_i\right)\right)+\sum _{0<t_i<t}{F}_i\left(x\left(t_i\right)\right).$

Hence, this implies that the following estimate is true:

$\parallel x\left(t\right)-x_0{\parallel }_{PC}\le \alpha \left(t\right)\parallel f\left(t\mathrm{,}x\left(t\right)y\left(t\right)\right)\parallel +2p\underset{1\le i\le p}{max}\parallel F_i\left(t\mathrm{,}x\left(t\right)\right)\parallel \mathrm{,}$ (8)

where $\alpha \left(t\right)=2t\left(1-t/T\right)$. It is easy to check that from (8) follows (7). Lemma 1 is proved.

Remark. $T$-periodic solution $x_{\phi \left(t\right)}=\psi \left(t\right)$ of the system (1) with the initial value function $\phi \left(t\right)$ in the initial set $(-\infty ,0]$ is defined by the initial value function $\phi \left(t\right)$, which is a periodic continuation of the solution $\psi \left(t\right)$ in the initial set $(-\infty ,0]$.

Lemma 2. For the difference of two functions with maxima, we have the following estimate:

$\parallel \max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}-\max \left\{y\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}\parallel \le$

$\le \parallel x\left(t\right)-y\left(t\right)\parallel +2h\parallel \frac{\partial }{\partial t}\left[x\left(t\right)-y\left(t\right)\right]\parallel \mathrm{,}$ (9)

where

$h=\underset{-\infty t\le T}{sup}\left|{\lambda }_1\left(t\right)-{\lambda }_2\left(t\right)\right|.$                                       

Proof. We use obvious true relations:

$\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}=\max \left\{\left[\left(x\right)\tau -y\left(\tau \right)+y\left(\tau \right)\right]:\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}\le$

$\le \max \left\{\left[x\left(\tau \right)-y\left(\tau \right)\right]:\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}+\max \left\{y\left(\tau \right)\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}.$

Hence, we obtain the following:

$\max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}-\max \left\{x\left(\tau \right)\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}\le$

$\le \mathrm{ma}\left\{x\left[x\left(\tau \right)-y\left(\tau \right)\right]:\tau \in {\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right\}.$ (10)

We denote by $t_1$ and $t_2$ the points of the interval $\left[{\lambda }_1\left(t\right), {\lambda }_2\left(t\right)\right]$, on which the maximums of the functions $x\left(t\right)$ and $y\left(t\right)$ are reached:

$\max \left\{x\left(\tau \right)\left|\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right.\right\}\text{=}x\left(t_1\right), \max \left\{y\left(\tau \right)\left|\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right.\right\}\text{=}y\left(t_1\right),$

$\max \left\{\left[x\left(\tau \right)-y\left(\tau \right)\right]\left|\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right.\right\}\text{=}x\left(t_2\right)-y\left(t_2\right).$

Then, taking (10) and last equalities, we have

$\parallel \max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}-\max \left\{y\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}-x\left(t\right)+y\left(t\right)\parallel \le$

$\le \parallel \left[x\left(t\right)-y\left(t\right)\right]-\left[x\left(t_1\right)-y\left(t_1\right)\right]\parallel +\parallel \left[x\left(t_2\right)-y\left(t_2\right)\right]-\left[x\left(t_1\right)-y\left(t_1\right)\right]\parallel \mathrm{.}$ (11)

From another side, it is obvious that the estimate is valid:

$\parallel \left[x\left(\overline{t}\right)-y\left(\overline{t}\right)\right]-\left[x\overline{\left(\overline{t}\right)}-y\overline{\left(\overline{t}\right)}\right]\parallel \le h\parallel \frac{d}{dt}\left[x\left(t^{*}\right)-y\left(t^{*}\right)\right]\parallel \le h\parallel \frac{d}{dt}\left[x\left(t\right)-y\left(t\right)\right]\parallel \mathrm{,}$ (12)

where $\overline{t}$, $\overline{\overline{t}}\in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]$, $t^{*}\in \left(\overline{t}\mathrm{,}\overline{\overline{t}}\right)$. From the estimates (11) and (12) we come to the following estimate:

$\parallel \max \left\{x\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}-\max \left\{y\left(\tau \right):\tau \in \left[{\lambda }_1\left(t\right){\lambda }_2\left(t\right)\right]\right\}-x\left(t\right)+y\left(t\right)\parallel \le$

$\le 2h\parallel \frac{d}{dt}\left[x\left(t\right)-y\left(t\right)\right]\parallel \mathrm{.}$

Therefore, it is easy to check that the inequality (9) and we complete the proof of Lemma 2. 

4. Main Results

Theorem 1. Assume that for all $t\in \left[0,T\right], t\ne t_i, \left(i=1,2\dots \mathrm{,}p\right)$ the following conditions are satisfied:

• $\parallel f\left(t\mathrm{,}x\left(t\right)y\left(t\right)\right)\parallel \le M_1<\infty ,$ $\underset{1\le i\le p}{max}\parallel F\left({}_{i}t\mathrm{,}x\left(t\right)\right)\parallel \le M_2<\infty$;
• $\parallel f\left(t\mathrm{,}{x}_1\mathrm{,}{y}_1\right)-f\left(t\mathrm{,}{x}_2\mathrm{,}{y}_2\right)\parallel \le L_1\left[\parallel x_1-x_2\parallel +\parallel y_1-y_2\parallel \right]$;
• $\parallel K\left(t\mathrm{,}s\mathrm{,}{x}_1\right)-K\left(t\mathrm{,}s\mathrm{,}{x}_2\right)\parallel \le L_2\left(s\right)\parallel x_1-x_2\parallel$, $\underset{t}{0<sup}{\int }_{-\infty }^t{L}_2\left(s\right)ds<\infty$;
• $\parallel F_i\left(t\mathrm{,}{x}_1\right)-F_i\left(t\mathrm{,}{x}_2\right)\parallel \le L_3\parallel x_1-x_2\parallel$;
• the radius of the inscribed ball in $X$ is greater than $M_{1}T/2+2M_{2}p$;
• $\rho <1$, where

$\rho =\max \left\{L_1\left(1+M_3\right)\left(1+\frac{T}{2}\right)+p{L}_{3}2\left(+\frac{1}{T}\right), 2L_1{M}_3\left(T+1\right)h\right\}.$

If the system (1) has a solution for all $t\in \left[0,T\right], t\ne t_i, \left(i=1,2\dots \mathrm{,}p\right)$, then this solution can be founded by the system of nonlinear functional-integral equations

$x\left(t\mathrm{,}{x}_0\right)=x_0+$

$+{\int }_0^{t}f\left(s\mathrm{,}x\left(s\mathrm{,}{x}_0\right){\int }_{-\infty }^{s}K\left(s\mathrm{,}\theta \mathrm{,}\max \left\{x\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(\theta \right),{\lambda }_2\left(\theta \right)\right]\right\}\right)d\theta \right)-$

$-\frac{1}{T}{\int }_0^{T}f\left(\theta \mathrm{,}x\left(\theta \mathrm{,}{x}_0\right){\int }_{-\infty }^{\theta }K\left(\theta \mathrm{,}\xi \mathrm{,}\max \left\{x\left(\tau \mathrm{,}{x}_0\right):\tau \in \lambda \left[{}_1\left(\xi \right),{\lambda }_2\left(\xi \right)\right]\right\}\right)d\xi \right)d\theta -$

$-\frac{1}{T}\sum _{i=1}^p{F}_i\left(x\left(t_i\mathrm{,}{x}_0\right)\right)]ds+\sum _{0<t_i<t}{F}_i\left(x\left(t_i\mathrm{,}{x}_0\right)\right).$ (13)

Proof. The theorem we proof by the method of successive approximations, defining the iteration process as

$x_0\left(t\mathrm{,}{x}_0\right)=x_0\mathrm{,}{x}_{k+1}\left(t\mathrm{,}{x}_0\right)=x_0+$

$+{\int }_0^{t}f\left(s\mathrm{,}{x}_k\left(s\mathrm{,}{x}_0\right),{\int }_{-\infty }^{s}K\left(s\mathrm{,}\theta \mathrm{,}\max \left\{x_k\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(\theta \right),{\lambda }_2\left(\theta \right)\right]\right\}\right)d\theta \right)-$

$-\frac{1}{T}{\int }_0^{T}f\left(\theta \mathrm{,}{x}_k\left(\theta \mathrm{,}{x}_0\right),{\int }_{-\infty }^{\theta }K\left(\theta \mathrm{,}\xi \mathrm{,}\max \left\{x_k\left(\tau \mathrm{,}{x}_0\right)\tau \in \left[{\lambda }_1\left(\xi \right),{\lambda }_2\left(\xi \right)\right]\right\}\right)d\xi \right)d\theta -$

$-\frac{1}{T}\sum _{i=1}^p{F}_i\left(x_k\left(t_i\mathrm{,}{x}_0\right)\right)]ds+\sum _{0<t_i<t}{F}_i\left(x_k\left(t_i\mathrm{,}{x}_0\right)\right).$ (14)

We will show that the right-hand side of the system of equations (13) as an operator maps a ball with radius $M_{1}T/2+2M_{2}p$ into itself and is a contraction operator. So, according to Lemma 1, from (1) and (14) we have

$\parallel x_{k+1}\left(t\mathrm{,}{x}_0\right)-x_0{\parallel }_{PC}\le M_1\frac{T}{2}+2M_{2}p\mathrm{.}$ (15)

From the system of integro-differential equations

$x\left({}^{\text{'}}t\mathrm{,}{x}_0\right)=f\left(t\mathrm{,}x\left(t\mathrm{,}{x}_0\right),{\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(s\right){\lambda }_2\left(s\right)\right]\right\}\right)ds\right)-$

$-\frac{1}{T}{\int }_0^{T}f\left(t\mathrm{,}x\left(t\mathrm{,}{x}_0\right),{\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(s\right){\lambda }_2\left(s\right)\right]\right\}\right)ds\right)dt-$

$-\frac{1}{T}\sum _{i=1}^p{F}_i\left(x\left(t_i\mathrm{,}{x}_0\right)\right).$ (16)

we obtain the following:

$\parallel {x^{\text{'}}}_{k+1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}\le 2M_1+\frac{p}{T}{M}_2\mathrm{.}$ (17)

We consider a difference $x_{k+1}\left(t\mathrm{,}{x}_0\right)-x_k\left(t\mathrm{,}{x}_0\right)$ of two approximations, where the functions $x_{k+1}\left(t\mathrm{,}{x}_0\right)$ and $x_k\left(t\mathrm{,}{x}_0\right)$ are defined from the approximations of the system of equations (14). By the conditions of the theorem, from (14) we have

$\parallel x_{k+1}\left(t\mathrm{,}{x}_0\right)-x_k\left(t\mathrm{,}{x}_0\right)\parallel \le L_1{\int }_0^t\{||x_k\left(s\mathrm{,}{x}_0\right)-x_{k-1}\left(s\mathrm{,}{x}_0\right)||+$

$+{\int }_{-\infty }^s{L}_2\left(\theta \right)\parallel \max \left\{x_k\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(\theta \right),{\lambda }_2\left(\theta \right)\right]\right\}-$

$-\max \left\{x_{k-1}\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(\theta \right),{\lambda }_2\left(\theta \right)\right]\right\}\parallel d\theta +$

$+\frac{1}{T}{\int }_0^T[\parallel x_k\left(\theta \mathrm{,}{x}_0\right)-x_{k-1}\left(\theta \mathrm{,}{x}_0\right)\parallel +$

$+{\int }_{-\infty }^{\theta }{L}_2\left(\xi \right)\parallel \max \left\{x_k\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(\xi \right),{\lambda }_2\left(\xi \right)\right]\right\}-$

$-\max \left\{x_{k-1}\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(\xi \right){\lambda }_2\left(\xi \right)\right]\right\}\parallel d\xi \left]d\theta \right\}ds+$

$+2\sum _{i=1}^p{L}_3\parallel x_k\left(t_i\mathrm{,}{x}_0\right)-x_{k-1}\left(t_i\mathrm{,}{x}_0\right)\parallel \le$

$\le \alpha \left(t\right)L_1[\left(1+M_3\right)\parallel x_k\left(t\mathrm{,}{x}_0\right)-x_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}+$

$+2h{M}_3\parallel {x^{\text{'}}}_k\left(t\mathrm{,}{x}_0\right)-{x^{\text{'}}}_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}]+2p{L}_3\parallel x_k\left(t\mathrm{,}{x}_0\right)-x_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}\le$

$\le \left(L_1\left(1+M_3\right)\frac{T}{2}+2p{L}_3\right)\parallel x_k\left(t\mathrm{,}{x}_0\right)-x_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}+$

$+2L_1{M}_{3}Th\parallel {x^{\text{'}}}_k\left(t\mathrm{,}{x}_0\right)-{x^{\text{'}}}_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}\mathrm{,}$ (18)

where

$M_3\underset{t}{=sup}{\int }_{-\infty }^t{L}_2\left(s\right)ds<\infty \mathrm{.}$                                       

Similarly, by the assumptions of Theorem 1, from (16) we have the following:

$\parallel {x^{\text{'}}}_{k+1}\left(t\mathrm{,}{x}_0\right)-{x^{\text{'}}}_k\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}\le$

$\le \left(L_1\left(1+M_3\right)+L_3\frac{p}{T}\right)\parallel x_k\left(t\mathrm{,}{x}_0\right)-x_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}+$

$+2L_1{M}_{3}h\parallel {x^{\text{'}}}_k\left(t\mathrm{,}{x}_0\right)-{x^{\text{'}}}_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}\mathrm{.}$ (19)

Adding both sides of (19) to (18), we obtain

${||y_{k+1}\left(t\mathrm{,}{x}_0\right)-y_k\left(t\mathrm{,}{x}_0\right)||}_{PC}\le \rho \parallel y_k\left(t\mathrm{,}{x}_0\right)-y_{k-1}\left(t\mathrm{,}{x}_0\right){\parallel }_{PC}\mathrm{,}$ (20)

where

$\parallel y_{k+1}\left(t\mathrm{,}{x}_0\right)-y_k\left(t\mathrm{,}{x}_0\right)\parallel \parallel x_{k+1}\left(t\mathrm{,}{x}_0\right)-x_k\left(t\mathrm{,}{x}_0\right)\parallel +\parallel {x^{\text{'}}}_{k+1}\left(t\mathrm{,}{x}_0\right)-{x^{\text{'}}}_k\left(t\mathrm{,}{x}_0\right)\parallel \mathrm{,}$

$\rho =\max \left\{L_1\left(1+M_3\right)\left(1+\frac{T}{2}\right)+p{L}_3\left(2+\frac{1}{T}\right),L_1{M}_3\left(T+1\right)h\right\}.$                

According to the last condition of Theorem 1, $\rho <0$. Since

$\parallel x_{k+1}\left(t\mathrm{,}{x}_0\right)-x_k\left(t\mathrm{,}{x}_0\right)\parallel \le \parallel y_{k+1}\left(t\mathrm{,}{x}_0\right)-y_k\left(t\mathrm{,}{x}_0\right)\parallel \mathrm{,}$                        

from the estimate (20) we deduce that the operator on right-hand side of (13) is compressing. From the estimates (15), (17) and (20) implies that there exists a unique fixed point $x\left(t,x_0\right)$. Theorem 1 is proved.

From the estimate (20) it is easy to see that for $x_0, {\overline{x}}_0\in X$ holds

$\parallel x\left(t\mathrm{,}{x}_0\right)-x\left(t\mathrm{,}{\overline{x}}_0\right){\parallel }_{BD}\le \frac{\left|x_0-{\overline{x}}_0\right|}{1-\rho }\mathrm{.}$                                  

Now we will show the existence of periodic solutions of the system of impulsive integro-differential equations (1). We introduce the following designations:

$\Delta \left(x_0\right)=\frac{1}{T}\sum _{i=1}^p{F}_i\left(x_{\infty }\left(t_i\mathrm{,}{x}_0\right)\right)+$

$+\frac{1}{T}{\int }_0^{T}f\left(t\mathrm{,}{x}_{\infty }\left(t\mathrm{,}{x}_0\right){\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x_{\infty }\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(s\right),{\lambda }_2\left(s\right)\right]\right\}\right)ds\right)dt\mathrm{,}$ (21)

${\Delta }_k\left(x_0\right)=\frac{1}{T}\sum _{i=1}^p{F}_i\left(x_k\left(t_i\mathrm{,}{x}_0\right)\right)+$

$+\frac{1}{T}{\int }_0^{T}f\left(t\mathrm{,}{x}_k\left(t\mathrm{,}{x}_0\right){\int }_{-\infty }^{t}K\left(t\mathrm{,}s\mathrm{,}\max \left\{x_k\left(\tau \mathrm{,}{x}_0\right):\tau \in \left[{\lambda }_1\left(s\right){\lambda }_2\left(s\right)\right]\right\}\right)ds\right)dt\mathrm{,}$ (22)

where $x\left(t\mathrm{,}{x}_0\right)=\underset{k\to \infty }{lim}{x}_k\left(t\mathrm{,}{x}_0\right)=x_{\infty }\left(t\mathrm{,}{x}_0\right)$ is the solution of the non-linear system (13). Therefore,$x_{\infty }\left(t\mathrm{,}{x}_0\right)$ is the solution of the system of impulsive integro-differential equations (1) for $\Delta \left(x_0\right)=0$ through $x_0$ at $t=0$. Consequently, the questions of the existence of a solution of the system of impulsive integro-differential equations (1) were reduced to the questions of the existence of zeros of the function $\Delta \left(x_0\right)$ and we solve this problem by finding zeros of the function ${\Delta }_k\left(x_0\right)$.

Theorem 2. Assume that

• all the conditions of Theorem yuld:th1 1 are fulfilled;
• there is a natural number $k$ such that the function ${\Delta }_k\left(x_0\right)$ has an isolated singular point $x_0^0$ that ${\Delta }_k\left(x_0^0\right)=0$ and the index of isolated singular point $x_0^0$ is nonzero;
• there is a closed convex region $X_0\subset X\mathrm{,}$ containing a single singular point such that on its border $\partial X_0$ is an estimate fulfilled:

$\underset{x\in \partial X_0}{inf}\parallel {\Delta }_k\left(x\right)\parallel \ge \frac{M{\rho }^{k+1}}{1-\rho }\mathrm{.}$ (23)

Then the system of impulsive integro-differential equations (1) has a periodic solution for all $t\in \left[0,T\right], t\ne t_i, \left(i=1,2\dots \mathrm{,}p\right)$ that $x\left(0\right)\in X_0\mathrm{.}$

Proof. By definition, the index of an isolated singular point $x_0^0$ of continuous mapping ${\Delta }_k\left(x_0\right)$ is equal to the characteristic of the vector field, generated by mapping ${\Delta }_k\left(x_0\right)$ on a sufficiently small sphere $S^n$ with the center in $x_0^0$. Since in $X_0$ there is no other singular point, which will be different from $x_0^0$ and $X_0$ is homeomorphic to the unit ball $E^n$, then the characteristic of the vector field ${\Delta }_k\left(x_0\right)$ on the sphere $S^n$ is equal to the characteristic of this vector field on $\partial X_0$. The fields ${\Delta }_k\left(x_0\right)$ and $\Delta \left(x_0\right)$ are homotopic on $\partial X_0$. Let us consider families of everywhere continuous on $\partial X$ vector fields

$V\left(\sigma \mathrm{,}{x}_0\right)={\Delta }_k\left(x_0\right)+\sigma \Delta \left(x_0\right)-{\Delta }_k\left(x_0\right),$                              

which connect the fields

$V\left(x_0\right)={\Delta }_k\left(x_0\right),V\left(1,x_0\right)=\Delta \left(x_0\right).$                               

We note that the estimate is true:

$\parallel \Delta \left(x_0\right)-{\Delta }_k\left(x_0\right)\parallel \le \frac{M{\rho }^{k+1}}{1-\rho }\mathrm{.}$ (24)

Therefore, the vector field $V\left(\sigma \mathrm{,}{x}_0\right)$ does not vanish anywhere on $\partial X_0$. Indeed, from (23) and (24) implies that

$\parallel V\left(\sigma \mathrm{,}{x}_0\right)\parallel \ge \parallel {\Delta }_k\left(x_0\right)\parallel -\parallel \Delta \left(x_0\right)-{\Delta }_k\left(x_0\right)\parallel >0.$ (25)

The fields ${\Delta }_k\left(x_0\right)$ and $\Delta \left(x_0\right)$ are homotopic on $\partial X$ and the rotations of the homotopic fields in the compact are equal. Therefore, taking into account (25), we conclude that the rotation of the field $\Delta \left(x_0\right)$ on the $\partial X_0$ is equal to the index of the singular point $x_0$ of the field ${\Delta }_k\left(x_0\right)$ and nonzero. Consequently, the vector field $\Delta \left(x_0\right)$ on $X_0$ has a singular point $x_0$, for which $\Delta \left(x_0\right)=0$. Therefore, the system of impulsive integro-differential equations (1) has a periodic solution for all $t\in \left[0,T\right], t\ne t_i,\left( i=1,2\dots \mathrm{,}p\right)$ that $x\left(0\right)\in X_0$. In addition, we note that for $x_0, {\overline{x}}_0\in X$ from (21) and (22) we have

$\parallel \Delta \left(x_0\right){\parallel }_{PC}\le M\left(1+\frac{p}{T}\right),\parallel \Delta \left(x_0\right)-\Delta \left({\overline{x}}_0\right){\parallel }_{BD}\le \frac{\left|x_0-{\overline{x}}_0\right|}{1-\rho }\mathrm{.}$                 

Theorem 2 is proved. 

Conclusion

The theory of differential and integro-differential equations plays an important role in solving many applied problems. Especially, local and nonlocal periodical boundary value problems for differential and integro-differential equations with impulsive actions have many applications in mathematical physics, mechanics and technology, in particular in nanotechnology. In this paper, we investigate the system of first-order integro-differential equations (1) with periodical boundary value condition (2), with nonlinear kernel and with nonlinear condition (3) of impulsive effects for $t=t_i, i=1,2\dots \mathrm{,}p, 0<t_1<t_2<\dots <t_p<T$. The non-linear right-hand side of this equation consists of the construction of maxima. The questions of the existence and uniqueness of the $T$-periodic solution of the boundary value problem (1)–(3) are studied. If the system (1) has a solution for all $t\in \left[0,T\right], t\ne t_i, i=1,2\dots \mathrm{,}p,$ then this solution can be proven to be based on the system of nonlinear functional-integral equations (13). The questions of the existence of a solution of the system of impulsive differential equations (1) we reduce to the questions of the existence of zeros of function $\Delta \left(x_0\right)$ in (21) and we solve this problem by finding zeros of function ${\Delta }_k\left(x_0\right)$ in (22).

The results obtained in this work will allow us in the future to investigate other kind of periodical boundary value problem for the heat equation and the wave equation with impulsive actions.

Competing interests. I declare that I have no competing interests.

Authors’ contributions and responsibilities. I take full responsibility for submit the final manuscript to print. I approved the final version of the manuscript.

Funding. Not applicable.

About the authors

Tursun K. Yuldashev

National University of Uzbekistan named after Mirzo Ulugbek

Author for correspondence.
Email: tursun.k.yuldashev@gmail.com
ORCID iD: 0000-0002-9346-5362
SPIN-code: 1629-8554
Scopus Author ID: 24482650300
http://www.mathnet.ru/person27151

Dr. Phys. & Math. Sci., Professor, Uzbek-Israel Joint Faculty

Uzbekistan, 4, Vuzgorodok, Universitetskaya st., Tashkent, 100174

References

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