| We consider the mth order nonlinear difference equations (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where {dollar}m ge 1, n,k,l in {lcub}rm I!N{rcub}sb0 = {lcub}0,1,2,...{rcub}, {lcub}asbsp{lcub}n{rcub}{lcub}1{rcub}{rcub},..., {lcub}asbsp{lcub}n{rcub}{lcub}m{rcub}{rcub}, {lcub}bsb{lcub}n{rcub}{rcub}, {lcub}rsb{lcub}n{rcub}{rcub},{dollar} and {dollar}{lcub}psb{lcub}n{rcub}{rcub}{dollar} are sequences of real numbers, {dollar}asbsp{lcub}n{rcub}{lcub}i{rcub} > 0,{dollar} for i = 1,2, ..., m and {dollar}n in {lcub}rm I!N{rcub}sb0, Sigmasbsp{lcub}s=N{rcub}{lcub}n{rcub} {lcub}1over asbsp{lcub}s{rcub}{lcub}i{rcub}{rcub} to infty{dollar} as {dollar}n to infty{dollar} for {dollar}N in {lcub}rm I!N{rcub}sb{lcub}0{rcub}, i = 1,2,..., m, Delta{dollar} is the forward difference operator {dollar}Delta ysb{lcub}n{rcub} = ysb{lcub}n+1{rcub} - ysb{lcub}n{rcub}, Dsb0ysb{lcub}n{rcub} = ysb{lcub}n{rcub}, Dsb{lcub}i{rcub}ysb{lcub}n{rcub} = asbsp{lcub}n{rcub}{lcub}i{rcub}Delta(Dsb{lcub}i-1{rcub}ysb{lcub}n{rcub}), i = 1,2,..., m, asbsp{lcub}n{rcub}{lcub}m-1{rcub} equiv 1{dollar} for equation (A), {dollar}asbsp{lcub}n{rcub}{lcub}m{rcub} equiv 1{dollar} for equations (B) and (C) and {dollar}F : {lcub}rm I!N{rcub}sb0 times {lcub}rm IR{rcub}sp2 to {lcub}rm IR{rcub}{dollar} is continuous. Notice that if {dollar}asbsp{lcub}n{rcub}{lcub}i{rcub} equiv 1{dollar}, for i = 1,2, ..., m - 1, then {dollar}Dsb{lcub}i{rcub} = Deltasp{lcub}i{rcub} = Delta(Deltasp{lcub}i-1{rcub}), i = 1,2,..., m{dollar}.; In this thesis, we obtain a nonoscillation result for all solutions of (A). We also find sufficient conditions for the boundedness of solutions as well as sufficient conditions for every oscillatory solution to converge to zero. Eventually, these results are used to prove the nonoscillation result. For equation (B), we discuss the problem of obtaining sufficient conditions for the oscillation and nonoscillation of all its solutions.; For the neutral equation, (C), we derive results on the classification of the set of nonoscillatory solutions of a special case, that is, when {dollar}Dsb{lcub}m{rcub} = Deltasp{lcub}m{rcub}{dollar} and F is linear. Here, we use fixed point techniques to show the existence of solutions having certain types of asymptotic behavior. We give sufficient conditions and some necessary conditions for the nonoscillatory solutions to posses certain types of asymptotic behavior. Also, we consider the asymptotic behavior of solutions of (C) when {dollar}F : {lcub}rm I!N{rcub}sb0 times {lcub}rm IR{rcub} to {lcub}rm IR{rcub}{dollar} with emphasis on the sequence {dollar}{lcub}psb{lcub}n{rcub}{rcub}{dollar} in the neutral term of (C). We also discuss how {dollar}psb{lcub}n{rcub} equiv{dollar} -1 behaves as a bifurcation point of the nonoscillatory solutions of the equation. These results extend previously known results to the case where {dollar}psb{lcub}n{rcub}{dollar} is allowed to oscillate about -1 in some regular fashion.; We also establish comparison results for the special case of equation (B) when {dollar}Dsb{lcub}m{rcub} = Deltasp{lcub}m{rcub}{dollar} and the right hand side of (B) is zero. In particular, we compare the equation {dollar}{dollar}Deltasp{lcub}m{rcub}xsb{lcub}n{rcub} + delta h(n,xsb{lcub}n-w{rcub}) = 0{dollar}{dollar}with{dollar}{dollar}Deltasp{lcub}m{rcub}ysb{lcub}n{rcub} + delta H(n,ysb{lcub}n-W{rcub}) = 0,{dollar}{dollar}where {dollar}delta = pm1, w, W in {lcub}rm I!N{rcub}sb0,{dollar} and {dollar}h, H : {lcub}rm N{rcub}sb0 times {lcub}rm IR{rcub} to {lcub}rm IR{rcub}{dollar} are continuous.; Examples are also included to illustrate all the main results in this thesis. In equations (A) - (C), the function F is generally nonlinear. It is interesting to note that some of the results we obtain here are new even in the case when F is linear and there is no delay term in F, that is, {d... |
