Periodic And Subharmonic Solutions Of A Class Of Difference Equations

This paper mainly concern with the existence and multiplicity of periodic andsubharmonic solutions for a class of non-linear second-order difference equations. Inchapter 2, some sufficient conditions are obtained for the existence of periodic so-lutions of asymptotically linear second order equation. Secondly we improve someknown results under the assumptions of superlinear and get one solution. Following,some sufficient conditions, which are weaken than sublinear in the sense of Rabi-nowitz, are obtained for the existence of one periodic solution. One main result isthat some sufficient conditions which are asymptotically linear, are obtained for theexistence of multiplicity of periodic solutions. Also we associate the eigenvalues ofthe matrix of functional with the number of periodic solutions. In chapter 3, somesufficient conditions are obtained for the existence of subharmonic sulotions of non-linear difference equation. By employing an effective technique to estimate the energyof the subharmonic solutions, some sufficient conditions for the existence of subhar-monic solution are obtained. Applying to the pendulum equation, some sufficientconditions for the existence of subharmonic solutions of pendulum difference systemare obtained which improve some known results in the literature. Also, the conditionsin our paper are easy to check.