Kavitha, N.

Pinelas, Sandra

Thandapani, Ethiraju

Dynamic Systems and Applications

English

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Dynamic Systems and Applications 21 (2012) 83-92COMPARISON AND OSCILLATION THEOREM FORSECOND-ORDER NONLINEAR NEUTRALDIFFERENCE EQUATIONS OF MIXED TYPEE. THANDAPANI, N. KAVITHA, AND S. PINELASRamanujan Institute for Advanced Study in Mathematics,University of Madras, Chennai 600 005, IndiaDepartamento de Matema´tica, Universidade dos Ac¸ores,Ponta Delgada, PortugalABSTRACT. In this paper, we establish some comparison theorems for the oscillation of secondorder neutral difference equations of mixed type∆ (an∆(xn + bnxn−σ1 + cnxn+σ2 )α) + qnxβn−τ1+ pnxβn+τ2= 0,where α and β are ratio of odd positive integers, σ1, σ2, τ1 and τ2 are positive integers. Our resultsare new even if pn = cn = 0. Examples are provided to illustrate the results.AMS (MOS) Subject Classification. 39A10.1. INTRODUCTIONIn this paper, we shall study the oscillatory behavior of the second order nonlinearneutral difference equation of mixed type(1.1) ∆ (an∆(xn + bnxn−σ1 + cnxn+σ2)α) + qnxβn−τ1 + pnxβn+τ2 = 0,where n ≥ n0 ∈ N, subject to the following conditions:(H1) {an} is a positive sequence for all n ≥ n0 and∞∑n=n01an=∞;(H2) {bn} and {cn} are nonnegative sequences such that 0 ≤ bn ≤ b and 0 ≤ cn ≤ c,where b and c are constants;(H3) {pn} and {qn} are nonnegative real sequences and not eventually zero for manyvalues of n;(H4) σ1, σ2, τ1 and τ2 are nonnegative integers and α and β are ratio of odd positiveintegers.We put zn = (xn + bnxn−σ1 + cnxn+σ2)α. By a solution of equation (1.1), we mean areal sequence {xn} defined for all n ≥ n0−max{σ1, τ1}, and satisfies equation (1.1) forReceived January 19, 2012 1056-2176 $15.00 c©Dynamic Publishers, Inc.

Comparison and Oscillation Theorem for Second-Order Nonlinear Neutral Difference Equations of Mixed Type

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Comparison and Oscillation Theorem for Second-Order Nonlinear Neutral Difference Equations of Mixed Type

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