| Recent years have witnessed a rapid progress for the theory of impulsive differentialequations which provide a natural description of the motion of several real world processessubject to short time perturbations. Due to many applications in physics, chemistry,population dynamics, ecology, biological systems, control theory, etc, an increasing numberscholars begin to study the theory oscillation of impulsive differential equations.In this paper, both El-Sayed type and Kamenev type oscillation criteria for impulsivedifferential equations with Riemann-Stieltjes integral are established. This thesis arranges forthe content as follows:In chapter one, we briefly describe the development of the theory of impulisivedifferential equations, and introduce the main work of this paper.In chapter two, both El-Sayed type and Kamenev type oscillation criteria for a forcedimpulsive differential equation with Riemann-Stieltjes integral are established. By using ageneralized El-Sayed type function and Kong’s technique in terms of the number of impulsemoments on a series of intervals, we not only drop the restriction on the impulse constantsckandd kthatd k≥ckin the literature, but also extend some existing results to the case ofRiemann-Stieltjes integral. The main results are published in Computers and Mathematicswith Applications which is reviewed by Science Citation Index.In chapter three, we study the oscillation problem of certain impulsive differentialequation with delay and Riemann-Stieltjes integral which contains many equations studied inthe literature. First, in the case of oscillatory potentials, we present an estimation onx (τ (t,s)/x(t)in a bounded interval, which plays a key role in the proof of the main results.Second, the redundant restriction on impulse constants c_k and d_k that d_k≥c_k is removedby introducing particular El-Sayed type functions and using Kong’s technique many timesbased on the number of impulse moments in a bounded interval. Finally, both impulse, delayand Riemann-Stieltjes integral are taken into consideration in this paper. Therefore, most of mixed type Emden-Fowler equations considered in the literature are included as special cases.In chapter four, we investigate the oscillatory behavior of certain impulsive dynamicequation with delay and Riemann-Stieltjes integral on time scales. Some sufficient oscillationconditions are established by overcoming the difficulty caused by impulses and time scales inthe estimation of the delayed argument. In this part, we use the geometric mean inequalitytechnique to solve the interval oscillation of impulsive differential equations. Both delay andRiemann-Stieltjes integral are taken into consideration, meanwhile we limited the functionx (t)is continuous, but its derivative has pulse disturbance. The main results include somerecent oscillation results for dynamic equations on time scales as special cases. Even for thespecial case when Τ=R, our results also generalize some oscillation results for differentialequations.In chapter five, the main results are summarized and the future research work isprospected. |
