Oscillation Criteria of Second Order Forced Delay Dynamic Equations

Abstract

This thesis in overall consists of five chapters, distributed as follows: Chapter 1 contains the basic concepts of the theory of functional differentialequations and some preliminary results of the theory of second order forced delaydifferential equations. Chapter 2 we give an introduction to the theory of dynamic equations on time scales, differentiation, integration, and various properties of the exponentialfunction on arbitrary time scale. Also, we present the most important studiesfor the oscillation theory of second order forced delay dynamic equations on timescales. Chapter 3 we establish some new oscillation criteria for the second-order nonlinear functional dynamic equation with forced term 〖(r(t) x^(∆ ) (t))〗^∆+p(t)f(x(τ(t) ) )=e(t), 〖(r(t) x^(∆ ) (t))〗^∆-p(t)f(x(τ(t) ) )=e(t) on a time scale 𝕋 by using a class ofPhilos type functions. The following cases are taken into consideration: (i) p(t)>0,τ(t)≤t(≥t) and τ(t)≤σ(t)(≥σ(t) ). (ii) p(t) change its sign,τ (t)≤t(≥t),τ: T→Tis a strictly increasing differentiable function and lim┬(t→∞)⁡〖τ(t)=∞〗. The obtained results are given in [4].

Chapter 4 presents some new oscillation criteria for the second-order forcednonlinear functional dynamic equations with damping term 〖(r(t) x^(∆ ) (t))〗^∆+q(t) x^(∆ ) (t)+p(t)f(x(σ(t) ) )=e(t), 〖(r(t) x^(∆ ) (t))〗^∆+q(σ(t) ) x^(∆ ) (t)+p(t)f(x(τ(t) ) )=e(t), on a time scale 𝕋, where r(t), p(t), q(t) and e(t) are real-valued right-dense continuousfunctions with p(t) < 0 and no restrictions imposed on the forcing terme(t) to satisfy Kartsatos condition. Our results improve and extend some resultsestablished by Sun andWong [44] and also answer their question for the oscillationwhen 0 <ν< 1. These results are given in [6]. Chapter 5 we use Riccati transformation technique to establish some newoscillation criteria for the second-order nonlinear forced dynamic equation withdamping 〖(r(t)g(x^(∆ ) (t)))〗^∆+p(t)〖g(x〗^(∆ ) (t))+q(t)f(x(σ(t) ) )=G(t,(x(σ(t) ) ), on time scale 𝕋, where r(t), p(t) and q(t) are real-valued right-dense continuousfunctions and no sign conditions imposed on these functions. Our results extendand improve some previous results established by Sun et al. [4]. These results are given in [5]. This thesis in overall consists of five chapters, distributed as follows: Chapter 1 contains the basic concepts of the theory of functional differentialequations and some preliminary results of the theory of second order forced delaydifferential equations. Chapter 2 we give an introduction to the theory of dynamic equations on time scales, differentiation, integration, and various properties of the exponentialfunction on arbitrary time scale. Also, we present the most important studiesfor the oscillation theory of second order forced delay dynamic equations on timescales. Chapter 3 we establish some new oscillation criteria for the second-order nonlinear functional dynamic equation with forced term 〖(r(t) x^(∆ ) (t))〗^∆+p(t)f(x(τ(t) ) )=e(t), 〖(r(t) x^(∆ ) (t))〗^∆-p(t)f(x(τ(t) ) )=e(t) on a time scale 𝕋 by using a class ofPhilos type functions. The following cases are taken into consideration: (i) p(t)>0,τ(t)≤t(≥t) and τ(t)≤σ(t)(≥σ(t) ). (ii) p(t) change its sign,τ (t)≤t(≥t),τ: T→Tis a strictly increasing differentiable function and lim┬(t→∞)⁡〖τ(t)=∞〗. The obtained results are given in [4].

Chapter 4 presents some new oscillation criteria for the second-order forcednonlinear functional dynamic equations with damping term 〖(r(t) x^(∆ ) (t))〗^∆+q(t) x^(∆ ) (t)+p(t)f(x(σ(t) ) )=e(t), 〖(r(t) x^(∆ ) (t))〗^∆+q(σ(t) ) x^(∆ ) (t)+p(t)f(x(τ(t) ) )=e(t), on a time scale 𝕋, where r(t), p(t), q(t) and e(t) are real-valued right-dense continuousfunctions with p(t) < 0 and no restrictions imposed on the forcing terme(t) to satisfy Kartsatos condition. Our results improve and extend some resultsestablished by Sun andWong [44] and also answer their question for the oscillationwhen 0 <ν< 1. These results are given in [6]. Chapter 5 we use Riccati transformation technique to establish some newoscillation criteria for the second-order nonlinear forced dynamic equation withdamping 〖(r(t)g(x^(∆ ) (t)))〗^∆+p(t)〖g(x〗^(∆ ) (t))+q(t)f(x(σ(t) ) )=G(t,(x(σ(t) ) ), on time scale 𝕋, where r(t), p(t) and q(t) are real-valued right-dense continuousfunctions and no sign conditions imposed on these functions. Our results extendand improve some previous results established by Sun et al. [4]. These results are given in [5].