ASYMPTOTIC-BEHAVIOR OF SOLUTIONS OF HIGHER-ORDER DIFFERENCE AND PARTIAL DIFFERENCE-EQUATIONS WITH DISTRIBUTED DEVIATING ARGUMENTS

We shall study the asymptotic behaviour of solutions of the difference equation \Delta(m)[y(j) + g(j)y(j - sigma)]\(alpha-1)Delta(m)[y(j) g(j)y(j - sigma)] + Sigma(xi=a)(b)T(j, xi)f(y(h(j, xi))) = 0, j greate r than or equal to j(0) where alpha > 0. As an application of the resu lts obtained, oscillation theorems are established for the partial dif ference equation Delta(j)(m)[y(i,j) + g(j)y(i,j - sigma)] + Sigma(xi=a )(b)Lambda(i, j, xi) f(y(i, h(j, xi))) = d(j)Ly(i,j) + Sigma(iota=1)(m u)d(iota)(j)Ly(i, tau(iota)(j)), i is an element of Omega, j greater t han or equal to j(0) subject to two different types of boundary condit ions. We have also included examples to dwell upon the importance of t he results obtained. (C) 1998 Elsevier Science Inc. All rights reserve d.