A Study On Periodicity Of Solutions For Differential & Difference Equations With Delays

Periodicity has always been focused upon by researchers due to itsapparent practical background. By using topological degree theory andsemi-order methods, we study the periodicity of several kinds offunctional differential & difference equations. This dissertation isdeveloped in four chapters.In Chapter one, a general introduction to the dissertation has beengiven and some relevant background knowledge has been introduced.In Chapter two, the existence of periodic solutions for p-LaplacianRayleigh equation with deviating arguments (φ_p(x'(t)))'+f(t,x'(t-σ(t)))+g(t,x(t-τ(t)))=e(t)and its particular types is discussed by way of applying a theorem due toManasevinch and Mawhin. Thus a set of new sufficient conditions havebeen obtained, which generalize and improve the known results in theliterature.In Chapter three, the existence, unique and attractivity of theperiodic solutions for neutral functional differential equations (?)(t)+D(?)(t+·)=A(t,x(t+·))x(t)+f(t,x(t+·))andare studied by using the Schauder fixed point theorem. Some new results have been obtained, which extend and improve the related known worksin the literature.In Chapter four, the existence of periodic positive solutions for theperiodic difference equation with a deviating argument x(n+1)=r(n)x(n)+f(n,x(n-τ(n)))is discussed by using Krasnoselskill's fixed point theorem. Some newcriteria for the existence of periodic positive solutions of above functionaldifference equation for r(n)＞0 have been obtained, which generalizeand improve the known results in the literature.