Principal and nonprincipal solutions of impulsive differential equations with applications

Abstract

We introduce the concept of principal and nonprincipal solutions for second order differential equations having fixed moments of impulse actions is obtained. The arguments are based on Polya and Trench factorizations as in non-impulsive differential equations, so we first establish these factorizations. Making use of the existence of nonprincipal solutions we also establish new oscillation criteria for nonhomogeneous impulsive differential equations. Examples are provided with numerical simulations to illustrate the relevance of the results.

Introduction

The concept of the principal solution was introduced in 1936 by Leighton and Morse [1] in studying positiveness of certain quadratic functionals associated with

$$(r(t)z^{\prime }{)}^{\prime }+q(t)z=0\text{,}r(t)>0\text{.}$$

Since then the principal and nonprincipal solutions have been used successfully in connection with oscillation and asymptotic theory of related equations, see for instance [1], [2], [3], [4], [5], [6], [7], [8] and the references cited therein. For some extensions to Hamiltonian systems and half-linear differential equations, we refer in particular to [4], [9].

A nontrivial solution u of (1) is said to be principal if for every solution v such that $u\ne \lambda v\text{,}\lambda \in \mathbb{R}$,

$$\underset{t\to \infty }{\lim }\frac{u(t)}{v(t)}=0\text{.}$$

It is well known that a principal solution u of (1) exists uniquely up to a multiplication by a nonzero constant if and only if (1) is nonoscillatory. A solution v linearly independent of u is called a nonprincipal solution. Roughly speaking, the terms “principal” and “nonprincipal” may be replaced by “small” and “large”. For other characterizations of principal solution and nonprincipal solutions of (1), see [5, Theorem 6.4].

Impulsive differential equations are of particular interest in many areas such as biology, physics, chemistry, control theory, medicine, etc. as they model the real processes better than differential equations. Because of the lack of smoothness property of the solutions the theory of impulsive differential equations is much richer than that of differential equations without impulse. However, due to difficulties caused by impulsive perturbations, the theory is not well-developed in comparison with that of non-impulsive differential equations. For basic theory of impulsive differential equations, we refer in particular to [10], [11].

The main purpose of this paper is to establish the concept of principal and nonprincipal solutions of second order linear impulsive differential equations of the form

$$\begin{array}{cc}\hfill & (r(t)z^{\prime }{)}^{\prime }+q(t)z=0\text{,}t\ne {\theta }_i\text{,}\hfill \\ \hfill & \mathrm{\Delta }r(t)z^{\prime }+q_{i}z=0\text{,}t={\theta }_i\text{,}\hfill \end{array}$$

where $t\in [t_0\text{,}\infty )\text{,}{t}_0\in \mathbb{R}$ is fixed; $\mathrm{\Delta }g(t)$ denotes the difference $g(t^{+})-g(t^{-})$ with $g(t^{\pm })={\lim }_{\tau \to t^{\pm }}g(\tau )$.

It turns out that one can show in a similar manner that such solutions of (2) exist if and only if (2) has an eventually positive solution. As in non-impulsive equations case these solutions may allow one to study certain related forced or perturbed impulsive equations. To illustrate the importance in this direction we derive through nonprincipal solutions of (2) new sufficient conditions for oscillation of solutions of (2) and the corresponding forced equation

$$\begin{array}{cc}\hfill & (r(t)y^{\prime }{)}^{\prime }+q(t)y=f(t)\text{,}t\ne {\theta }_i\text{,}\hfill \\ \hfill & \mathrm{\Delta }r(t)y^{\prime }+q_{i}y=f_i\text{,}t={\theta }_i\text{.}\hfill \end{array}$$

Note that in case there is no impulse Eq. (3) is reduced to

$$(r(t)z^{\prime }{)}^{\prime }+q(t)z=f(t)\text{,}$$

the qualitative theory of which has been studied by many authors, see for instance [8], [12], [13], [14], [15], [16], [17], [18]. For second order impulsive differential equations we refer in particular to [19], [20], [21], [22], [23] and the references therein.

Denote by $\mathrm{PLC}[t_0\text{,}\infty )$ the set of functions $h:[t_0\text{,}\infty )\to \mathbb{R}$ such that h is continuous on each interval $({\theta }_i\text{,}{\theta }_{i+1})\text{,}h({\theta }_i^{\pm })$ exist, and $h({\theta }_i)=h({\theta }_i^{-})$ for $i\in \mathbb{N}$.

With regard to (2), (3) we assume throughout this work that

• $r\text{,}q\text{,}f\in \mathrm{PLC}[t_0\text{,}\infty )\text{;}r(t)>0$

• $\{{\theta }_i\}$ is a strictly increasing unbounded sequence of real numbers, ${\theta }_i⩾t_0\text{;}\{q_i\}$ and $\{f_i\}$ are real sequences.

Definition 1.1

A solution to (3) on an interval $[T\text{,}\infty )\text{,}T⩾t_0$, is a function $y\in \mathrm{C}[T\text{,}\infty )$ with $y^{\prime }\text{,}({\mathit{ry}}^{\prime }{)}^{\prime }\in \mathrm{PLC}[T\text{,}\infty )$ which satisfies (3).

It is easy to see that under the conditions (i) and (ii), for any given $t_1⩾t_0$ and $y_0\text{,}{y}_1\in \mathbb{R}$ Eq. (3) has a unique solution y satisfying $y(t_1^{+})=y_0$ and $y^{\prime }(t_1^{+})=y_1$. A nontrivial solution $y(t)$ of (3) is called oscillatory if it has arbitrarily large zeros, otherwise it is called nonoscillatory. Eq. (3) is called oscillatory (nonoscillatory) if all of its solutions are oscillatory (nonoscillatory). We recall that according to the Sturm’s separation theorem [22] every solution of (2) is oscillatory (nonoscillatory) if there is one oscillatory (nonoscillatory) solution of the equation. It will be shown that principal and nonprincipal solutions of (2) exist in case the equation is nonoscillatory.

For simplicity we have assumed in this study that the solutions are continuous. It should be pointed out that the results can apply also to impulsive equations with discontinuous solutions. To see this, consider

$$\begin{array}{cc}\hfill & (r(t)x^{\prime }{)}^{\prime }+q(t)x=0\text{,}t\ne {\theta }_i\text{,}\hfill \\ \hfill & x({\theta }_i^{+})={\alpha }_{i}x({\theta }_i^{-})\text{,}({\mathit{rx}}^{\prime })({\theta }_i^{+})={\alpha }_i({\mathit{rx}}^{\prime })({\theta }_i^{-})+{\beta }_{i}x({\theta }_i^{-})\text{,}\hfill \end{array}$$

where ${\alpha }_i>0$ for all $i\in \mathbb{N}$. If we define

$$z(t)=\frac{1}{{\alpha }_1{\alpha }_2\cdots {\alpha }_i}x(t)\text{,}t\in ({\theta }_i\text{,}{\theta }_{i+1})\text{,}z({\theta }_i)=z({\theta }_i^{-})\text{,}$$

then the function z becomes continuous, and (5) is transformed into (2) with $q_i=-{\beta }_i/{\alpha }_i$. Note that x is nonoscillatory if and only if z is.

The paper is organized as follows. The main result concerning the existence of principal and nonprincipal solutions of (2) is given in the next section. The proof is based on Polya and Trench factorizations as in non-impulsive equations case, so we first establish these results in two lemmas. Section 3 contains two important applications, namely Wong and Leighton–Wintner theorems. Examples with numerical simulations are given to illustrate the relevance of the results.