Quasilinearization method for an impulsive integro-differential system with delay

1.   Introduction

In this paper, we employ the monotone iterative ^[1,2] and quasilinearization method ^[3,4] to discuss the existence, uniform and quadratic convergence of solution sequences for an antiperiodic boundary value problem (BVP) of impulsive integro-differential system with delay ^[5]:

where $f\in C(I\times R^4, R),$ $I = [0, T], I^+ = [-r, T], r > 0, h-r\leq \tau(h)\leq h,$ $h_0 = 0 < h_1 < h_2 < \cdots < h_m < T = h_{m+1}, I_k\in C(R, R),$ $\triangle y(h_k) = y(h_k^+)-y(h_k^-),$

$K\in C(L, R^+), L = \{(h, s)\in I\times I:h\geq s\},$ $H\in C(I\times I, R^+), R^+ = [0, \infty).$ We denote $k_0 = \max \left\{ {K\left({h, s} \right):\left({h, s} \right) \in L} \right\},$ $h_0 = \max \left\{ {H\left({h, s} \right):\left({h, s} \right) \in I \times I} \right\}, \rho = \max\{h_{u+1}-h_u\}, u = 0, 1, ..., m.$

Impulsive differential equation is a basic mathematical model to describe real world phenomena which suddenly alter states at some moments ^[6,7]. It is widely used in physics, population dynamics, ecology, industrial robotic, etc ^[8,9,10]. Note that, in recent years, there are many authors interest in impulsive integro-differential equations ^[11,12,13]. And, the existence and approximate controllability for neutral differential equations with delay have been widely concerned ^{[14,15,16,17]}. Nisar and Vijayakumar discussed approximate controllability for a class of Sobolev-type Hilfer fractional neutral delay differential equations ^[18]. The controllability result for a fuzzy delay differential system can refer to ^[19]. The existence and controllability for fractional integro-differential delay equations of order $1 < r < 2$ have been considered in ^[20] and ^[21].

The monotone iterative method is effective to get solution sequences, which uniformly converge to extreme solutions of equations ^[1]. Moreover, the quasilinearization(QSL) method is often used to get solution sequences, which are square convergent ^[3,22,23,24]. The QSL method, whose iterations are constructed to yield rapid convergence, has been used for solving a series of problems and obtained many excellent results ^[25,26,27]. The application of the QSL method in functional differential equations, can see ^[3,28,29]. However, the application of the QSL method in impulsive integro-differential systems with delay has been little discussed.

Similar to previous studies ^[1,3,5], we introduce some spaces for the following use:

Letting $I^- = I^+\backslash \{h_1, h_2, \cdots, h_m\}$, $PC(I^+, R) = \left\{y: I^+\rightarrow R; y(h)\right.$ is a continuous function in $I^-$, $h_k (k = 1, \cdots, m)$ are some jumping points, $y(h_k^+)$ and $y(h_k^-)$ exist at $h_k$, and $\left.y(h_k^-) = y(h_k) \right\}$;

$PC'(I^+, R)$ = $\{ y\in PC(I^+, R); y'$ is continuous in $I^-$, $y'(h_k^+)$, $y'(h_k^-)$, $y'(0^+)$ and $y'(T^-)$ exist$\}$;

$E_0 = \{y \in PC(I^+, R):y(h) = y(0), h\in [-r, 0]\}$, then the norm of $E_0$ is defined as $\|y\|_{E_0} = \mathop {\sup }\limits_{h\in I^+}|y(h)|$;

$E = {PC(I^+, R)\bigcap PC'(I^+, R)}.$ Then $y\in E$ is a solution of system (1.1) if and only if y satisfies system (1.1).

2.   Some basic concepts and conclusions

Firstly, we give the definition of upper and lower solutions.

Concept 2.1. ^1001[5] A function $\phi_0\in E\bigcap E_0$ is a lower solution of system (1.1) if and only if

Concept 2.2. ^1001[5] A function $\varphi_0\in E\bigcap E_0$ is an upper solution of system (1.1) if and only if

Then, we give some lemmas ^[5] to the following problem:

where $A > 0, B_1, B_2, B\geq 0, 0\leq L_k < 1$ and $\sigma (h)\in PC(I, R)$, $\eta (h)\in PC'(I^+, R).$

Lemma 2.1. ^1001[5] Let $A > 0, B_1, B_2, B\geq 0$ and $0\leq L_k < 1$ satisfy the inequality:

Then $y\in E$ is the unique solution of system (2.1) if $y\in E_0$ satisfies:

where

Lemma 2.2. ^1001[5] Suppose that $y\in E$ satisfies

where constants $A > 0, B_1, B_2, B\geq 0, 0\leq L_k < 1,$ with

Then $y(h)\leq 0$ on $I^+$.

Lemma 2.3. ^1001[5] If the functions $\phi_0, \varphi_0$ are lower and upper solutions of BVP system (1.1), which satisfy $\phi_0(h)\leq \varphi_0(h)$ in $I^+$, and both the $f$ and $I_k$ satisfy one-sided Lipschitz condition, then we can find sequences ${\phi_n}, {\varphi_n}\subset [\phi_0, \varphi_0]$ that uniformly converge to minimal and maximal solutions of the BVP system (1.1).

3.   A main theorem

Theorem 3.1. Suppose that the following assumptions are true:

$(S_1)$ Functions $\phi_0(h), \varphi_0(h)$ are lower and upper solutions of the BVP system (1.1), which satisfy $\phi_0(h)\leq \varphi_0(h)$ in $I^+$;

$(S_2)$ $f$ satisfies $f_y(h, y(h), [\Gamma y](h), [\delta y](h), y(\tau(h))) < 0, f_{\Gamma y}(h, y(h),$ $[\Gamma y](h), [\delta y](h), y(\tau(h)))\leq0,$ $f_{\delta y}(h, y(h), [\Gamma y](h), [\delta y](h), y(\tau(h)))\leq0,$ and $f_{y\tau}(h, y(h), [\Gamma y](h), [\delta y](h), y(\tau(h)))\leq0.$ And, the quadratic form $K(f(h, y, k, z, p))$ is

and $K(f)\leq0$ on $I\times R^4$, where $\phi_0\leq v\leq y_1\leq y\leq \varphi_0,$ $\phi_0\leq u\leq y_2\leq k\leq \varphi_0,$ $\phi_0\leq w\leq y_3\leq z\leq \varphi_0,$ $\phi_0\leq q\leq y_4\leq p\leq \varphi_0, \quad h\neq h_k, h\in I$;

$(S_3)$ The functions $I_k$ satisfy $-1\leq I_k^{'}(.)\leq 0$ and $I_k^{''}(.)\geq 0$, $k = 1, 2, \cdots, m$.

Then we can find monotone solution sequences $\{\phi_n(h)\}$ and $\{\varphi_n(h)\}$, which quadratically and uniformly converge to extremal solutions of system (1.1) on $[\phi_0, \varphi_0]$.

Proof: By using $(S_2)$ and the Taylor's formula, we can obtain

where

and employing the Taylor's formula together with $(S_3)$, we obtain

where $\phi_0(h_k)\leq b(h_k)\leq a(h_k)\leq \varphi_0(h_k)$.

Now, solution sequences ${\phi_i(h)}$ and ${\varphi_i(h)}$ are constructed to satisfy:

Obviously, by Lemma 2.1, we know that system (3.1) or (3.2) has an unique solution, respectively. We will finish our proof in four steps:

1. We prove that $\phi_0\leq \phi_{1}$ and $\varphi_1\leq \varphi_0$.

Let $i = 1$, then by system (3.1), we have

Setting $\varpi(h) = \phi_0(h)-\phi_1(h),$ we get

We can easily prove that

So it is clear that $\varpi(h)\leq0$ (from Lemma 2.2), i.e., $\phi_0\leq\phi_1$. Similarly, we can get that $\varphi_1\leq\varphi_0$ for all $h\in I^+$.

2. We show that $\phi_1\leq\varphi_1$ on $I^+$.

Letting $\varpi(h) = \phi_1-\varphi_1$ and by $(S_1)-(S_ 3)$, we have

From Lemma 2.2, we get $\varpi(h)\leq 0$, i.e., $\phi_1\leq\varphi_1$ for all $h\in I^+$. So we have $\phi_0(h)\leq\phi_1(h)\leq\varphi_1(h)\leq\varphi_0(h)$ in $I^+$. Then, by mathematical induction, we obtain ${\phi_n(h)}$ and ${\varphi_n(h)}$ satisfying

and each $\phi_i(h), \varphi_i(h)\in E\bigcap E_0 (i = 1, 2, \cdots)$ satisfy system (3.1) or (3.2), respectively. We can easily prove that the sequences ${\phi_n(h)}$ and ${\varphi_n(h)}$ are uniformly bounded and equi-continuous, then by Ascoli-Arzela criterion^[6], they uniformly converge to two solutions of system (1.1):

3. We verify that $\varsigma(h)$ and q(h) are minimum and maximum solutions of system (1.1) in $[\phi_0, \varphi_0]$, respectively.

Suppose y(h) is an arbitrary solution of system (1.1), which satisfies $\phi_0(h)\leq y(h)\leq \varphi_0(h)$ in $I^+$. Now, we assume that $\phi_n(h)\leq y(h)\leq \varphi_n(h)$ is hold for a positive integer n, in what follows we prove that $\phi_{n+1}(h)\leq y(h)\leq \varphi_{n+1}(h)$.

Letting $\varpi(h) = \phi_{n+1}(h)-y(h)$, we obtain

Clearly, by Lemma 2.2, we have $\varpi(h)\leq 0$, i.e., $\phi_{n+1}(h)\leq y(h)$ on $I^+$. Similarly, it can be proved that $y(h)\leq \varphi_{n+1}(h)$ on $I^+$. So $\phi_{n+1}(h)\leq y(h)\leq \varphi_{n+1}(h)$. Then, by taking $n\rightarrow \infty$, it is clear that $\varsigma(h)\leq y(h)\leq q(h)$.

4. Finally, we prove that the quadratic convergence of ${\phi_n}$ and ${\varphi_n}$.

First of all, letting $\varpi_n(h) = \varsigma(h)-\phi_n(h)\geq 0$, then

We write the above expression as follows:

where

and

where $\phi_{n-1}(h_k)\leq \xi\leq \varsigma(h_k), \eta = \varphi_{n-1}(T)-\phi_n(T).$ By Lemma 2.1, the solution of the above system is

where $M(h) = -\int_0^h f_y(v, \phi_{n-1}(v), [\Gamma \phi_{n-1}](v),$ $[\delta \phi_{n-1}](v), \phi_{n-1}(\tau(v)))dv.$ Letting $|f_{yy}|\leq \delta_1, |f_{\Gamma y \Gamma y}|\leq \delta_2, |f_{\delta y \delta y}|\leq \delta_3,$ $|f_{ y\tau y\tau}|\leq \delta_4, |f_{y \Gamma y}|\leq \delta_5, |f_{y\delta y}|\leq \delta_6, |f_{ y y\tau}|\leq \delta_7,$ $|f_{\Gamma y \delta y}|\leq \delta_8, |f_{\Gamma yy\tau}|\leq \delta_9, |f_{\delta y y\tau}|\leq \delta_{10},$ we have

Taking the norm of $\varpi_{n-1}$ on $I^+$ is $\|\varpi_{n-1}\|_{E_0} = \mbox{max}_{I^+ }\{\varpi_{n-1}(h), [\Gamma \varpi_{n-1}](h), [\delta \varpi_{n-1}](h), \varpi_{n-1}(\tau(h))\}$. Obviously, from the expression of $\varpi_{n}(h)$ we know that the follow formula is established for a constant $\zeta$:

Therefore, the ${\varpi_n}$ is quadratic convergent.

4.   Conclusions

In this paper, we mainly use the monotone iterative and quasilinearization method to study the quadratic convergence of the extremal solution for a class of integro-differential equations with delay. The results obtained are new and more general than previous studies, which can be applied to several special cases: 1) If $\tau(h) = h$, then the Bvp(1.1) is an impulsive integro-differential system with anti-periodic boundary value conditions; 2) If $\tau(h) = h+\theta$, $\theta\in[-r, 0],$ then the Bvp(1.1) becomes a delay impulsive integro-differential equation; 3) If $I_k(y(h_k)) = 0,$ then the Bvp(1.1) is a non-impulsive integro-differential equation; 4) If $K(h, s) = 0, H(h, s) = 0$, then the Bvp(1.1) is reduced to an impulsive functional differential equation with delay; 5) If $K(h, s) = 0, H(h, s) = 0, \tau(h) = h$, then the Bvp(1.1) becomes an impulsive ordinary differential equation.

Acknowledgments

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

Conflict of interest

The authors declare there is no conflict of interest.