NON-OSCILLATION OF SECOND ORDER LINEAR SELF-ADJOINT NONHOMOGENEOUS DIFFERENCE EQUATIONS

Abstract

In the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form ∆(pn−1∆yn−1)+qyn = 0, n ⩾ 1, where q is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type ∆(pn−1∆yn−1)+qng(yn) = fn−1, n ⩾ 1, where, unlike earlier works, fn ⩾ 0 or ⩽ 0 (but 0) for large n. Further, these results are used to obtain sufficient conditions for non-oscillation of all solutions of forced linear third order difference equations of the form yn+2 +anyn+1 +bnyn +cnyn−1 = gn−1, n ⩾ 1.