Abstract
This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation <svg style="vertical-align:-6.19302pt;width:554.88751px;" id="M1" height="23.549999" version="1.1" viewBox="0 0 554.88751 23.549999" width="554.88751" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x394" d="M600 0h-557v24l268 633l28 8l261 -641v-24zM497 50l-194 489l-196 -489h390z" /></g> <g transform="matrix(.012,-0,0,-.012,10.963,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> <g transform="matrix(.017,-0,0,-.017,17.287,15.775)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,23.169,15.775)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
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Highlights
Introduction and PreliminariesRecently, the oscillation, nonoscillation, asymptotic behavior, and existence of solutions of different classes of linear and nonlinear second-order difference equations have been studied by many authors; see, for example, [1–26] and the references cited therein
Using the Banach fixed point theorem and the Mann iterative schemes with errors, we discuss the existence of uncountably many unbounded positive solutions of (6), prove that the Mann iterative schemes with errors converge to these unbounded positive solutions, and compute the error estimates between the Mann iterative schemes with errors and the unbounded positive solutions
Converges to an unbounded positive solution x ∈ A(N, M) of (6) and has the error estimate (17), where {γm}m∈N0 = {γmn}(m,n)∈N0×Nβ is an arbitrary sequence in A(N, M), and {αm}m∈N0 and {βm}m∈N0 are any sequences in [0, 1] satisfying (18); (b) equation (6) possesses uncountably many unbounded positive solutions in A(N, M)
Summary
The oscillation, nonoscillation, asymptotic behavior, and existence of solutions of different classes of linear and nonlinear second-order difference equations have been studied by many authors; see, for example, [1–26] and the references cited therein. Using the Banach fixed point theorem, Jinfa [5] discussed the existence of a bounded nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients: Δ2 (xn + pxn−m) + pnxn−k − qnxn−l = 0, ∀n ≥ n0 (1). Migda [16] considered the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima: Δ2 (xn + pnxn−k) + qn max {xs : n − l ≤ s ≤ n} = 0, ∀n ≥ 1 (2). Meng and Yan [15] studied the existence of bounded nonoscillatory solutions for the second-order nonlinear nonautonomous neutral delay difference equation: m. Nothing has been done with the existence of uncountably many unbounded positive solutions for (1)∼ (5) and any other second-order neutral delay difference equations: Inspired and motivated by the results in [1–26], in this paper we introduce and study the second-order nonlinear neutral delay difference equation: Δ2 (xn + anxn−τ) + Δh
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