Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays

In this paper, we study oscillatory properties of solutions for the nonlinear impulsive hyperbolic equations with several delays. We establish sufficient conditions for oscillation of all solutions.


Introduction
The theory of partial functional differential equations can be applied to many fields, such as biology, population growth, engineering, control theory, physics and chemistry, see the monograph [1] for basic theory and applications. The oscillation of partial functional differential equations has been studied by many authors see, for example [2]- [7], and the references cited therein.
The theory of impulsive partial differential systems makes its beginning with the paper [8] in 1991. In recent years, the investigation of oscillations of impulsive partial differential systems has attracted more and more attention in the literature see, for example [9]- [13]. Recently, the investigation on the oscillations of impulsive partial differential systems with delays can be found in [14]- [19].
To the best of our knowledge, there is little work reported on the oscillation of second order impulsive partial functional differential equation with delays. Motivated by this observation, in this paper we study the oscillation of nonlinear forced impulsive hyperbolic partial differential equation with several delays of the form ,0 , .
In the sequal, we assume that the following conditions are fulfilled: , u x t and their derivatives ( ) , t u x t are piecewise continuous in t with discontinuities of first kind only at , 1, 2, , k t t k = =  and left continuous at , , ,  and there exist positive constants * * , , , By a solution of problem (1), (2) ((1),(3)) with initial condition (4), we mean that any function ( ) , u x t for which the following conditions are valid: , u x t coincides with the solution of the problem (1) and (2) ((3)) with initial condition.
Here the number i k is determined by the equality .  (3)) is called nonoscillatory in the domain G if it is either eventually positive or eventually negative. Otherwise, it is called oscillatory. This paper is organized as follows: Section 2, deals with the oscillatory properties of solutions for the problem (1) and (2). In Section 3, we discuss the oscillatory properties of solutions for the problem (1) and (3). Section 4 presents some examples to illustrate the main results.

Oscillation Properties of the Problem (1) and (2)
To prove the main result, we need the following lemmas. , be a positive solution of the problem (1), (2) in G. Then the functions x g x t S η ∂Ω = ∫ has an eventually positive solution.

Proof. Let ( )
, u x t be a positive solution of the problem (1), (2) in G. Without loss of generality, we may assume that there exists a 0 0, , which is the same as that in Lemma 2.1 and then integrating (1) with respect to x over Ω yields By Green's formula, and the boundary condition we have where dS is the surface element on ∂Ω . Also from condition (H2), and Jenson's inequality we can easily obtain Hence we obtain the following differential inequality (1) and condition (H4), we obtain Hence, we obtain that ( ) 0 v t > is a positive solution of impulsive differential inequalities (5)- (7). This completes the proof.  be a positive solution of the problem (1), (2) in G. If we further 0, u ∈ +∞ and the impulsive differential inequality (5), and have no eventually positive solution, then each nonzero solution of the problem (1)-(2) is oscillatory in the domain G.

Proof. Let ( )
, u x t be a positive solution of the problem (1), (2) in G. Without loss of generality, we may assume that there exists a 0 0, is a positive solution of the following impulsive hyperbolic equation (1) and condition (H4), we obtain . , is a positive solution of the inequality (8)-(10) for 0 t T > which is also a contradiction. This completes the proof.  Now, if we set 0 g ≡ in the proof of Lemma 2.3, then we can obtain the following lemma.
be a positive solution of the problem (1), (2) in G. If we further 0, u ∈ +∞ and the impulsive differential inequality (5), and has no eventually positive solution, then each nonzero solution of the problem (1), satisfying the boundary condition Proof. Let ( )  (1) and condition (H4), we obtain

x t a t u x t p x t f u x t t t q x t f u x t f x t t t x t G
is a positive solution of the inequality (11)-(13) for 0 t T > which is also a contradiction. This completes the proof.  Lemma 2.5. Assume that The proof of the lemma can be found in [21].  Lemma 2.6. Let ( ) v t be an eventually positive (negative) solution of the differential inequality (11)- (13). Assume that there exists Proof. The proof of the lemma can be found in [22].  We begin with the following theorem. Theorem 2.1. If condition (14), and the following condition then every solution of the problem (1), (2) oscillates in G.

Proof. Let ( )
, u x t be a nonoscillatory solution of (1), (2). Without loss of generality, we can assume that , . x t t ∈Ω × ∞ From Lemma 2.4, we know that ( ) v t is a positive solution of (11)- (13). Thus from Lemma 2.6, we can find the last inequality contradicts condition (15). This completes the proof. 

Oscillation Properties of the Problem (1) and (3)
Next we consider the problem (1) and (3). To prove our main result we need the following lemmas. 1500 , which is the same as that in Lemma 3.1 and then integrating (1) with respect to x over Ω yields The proof is similar to that of Lemma 2.1 and therefore the details are omitted.   0, u ∈ +∞ and the impulsive differential inequality (19), and have no eventually positive solution, then each nonzero solution of the problem (1) ,