Abstract
The authors consider the nonlinear difference equation urn:x-wiley:01611712:media:ijmm974851:ijmm974851-math-0001 where ?yn = yn+1 − yn, {pn}, {qn}, and {rn} are real sequences, and uf(u) > 0 for u?0. Sufficient conditions for boundedness and convergence to zero of certain types of solutions axe given. Examples illustrating the results are also included.
Highlights
In this paper we obtain results on the asymptotic behavior of solutions of the forced nonlinear difference equationA[yn -6 PnYn-h] -6 qnf(Yn-k) rn (E)where Ay, y,+, y,h,k E N {0, 1, ...}, {p,}, {q,}, and {r,} are sequences of real numbers, and f" ]R ]R is continuous with uf(u) > 0 for u 0
N] > No there exists n > Na such that g,y,+ < 0; it will be called nonoscillatory if there exists a positive integer N2 such that y, has fixed sign for all n > N2; and will be called a Z-type solution if there exists a positive integer N3 such that y, does not change sign for n > N3 but y, 0 for arbitrarily large values of n
Our interest here is in obtaining results on the convergence to zero of all the nonoscillatory solutions of (E)
Summary
In this paper we obtain results on the asymptotic behavior of solutions of the forced nonlinear difference equationA[yn -6 PnYn-h] -6 qnf(Yn-k) rn (E)where Ay, y,,+,- y,,h,k E N {0, 1, ...}, {p,,}, {q,}, and {r,,} are sequences of real numbers, and f" ]R ]R is continuous with uf(u) > 0 for u 0. In this paper we obtain results on the asymptotic behavior of solutions of the forced nonlinear difference equation Notice that when rn 0, (E) may have Z-type solutions. Our interest here is in obtaining results on the convergence to zero of all the nonoscillatory solutions of (E). Our results include that type of solution as well.
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