Existence of positive solutions for a neutral differential equation with positive and negative coefficients

Abstract

a b s t r a c tThis work is concerned with the existence of positive solutions for the neutral delay differential equation with positive and negative coefficients

Global existence and Mann iterative algorithms of positive solutions for first order nonlinear neutral delay differential equations

“…(1.8) with respect to all ranges of the function p, suggest some Mann iterative algorithms with errors to approximate these positive solutions and establish a few error estimates between the sequences generated by the Mann iterative algorithms with errors and these positive solutions. Our results extend, improve and unify Theorem 3.4 in [12], Theorem 1 in [14], Theorems 3.4 and 3.6 in [26] and Theorems 1-3 and A in [28]. Seven examples are given to illustrate the importance and applications of the results obtained in this paper.…”

“…Notice that in all seven examples, Theorem 3.4 in [12], Theorem 1 in [14], Theorems 3.4 and 3.6 in [26] and Theorems 1-3 and A in [28] are useless. 5 þ x 2 ðt 3 À 1Þ 1 þ t 2 À…”

“…Zhang et al [28] studied the existence of a nonoscillatory solution of the first order neutral delay differential equation with variable coefficients d dt ½xðtÞ þ pðtÞxðt À sÞ þ Q 1 ðtÞxðt À d 1 Þ À Q 2 ðtÞxðt À d 2 Þ ¼ 0; t P t 0 ; ð1:4Þ where s > 0; d i P 0; p 2 Cð½t 0 ; þ1Þ; RÞ, Q i 2 Cð½t 0 ; þ1Þ; R þ Þ and R þ1 t 0 Q i ðsÞds < þ1 for i 2 {1, 2}. Öcalan [14] discussed the existence of a positive solution for Eq. (1.4) with p 2 Cð½t 0 ; þ1Þ; R þ Þ. Shen and Debnath [19] obtained sufficient conditions for the oscillation of all solutions of Eq.…”

“…In the past twenty years there has been much research activity concerning the oscillation, nonoscillation, asymptotic behavior and existence of solutions of neutral delay differential equations, for example, see [1][2][3][4][5][6][7][8][9][10][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Liu and Huang [10] studied the existence and uniqueness of T-periodic solutions for the first order neutral functional differential equation with a deviating argument of the form d dt ½xðtÞ þ pxðt À sÞ ¼ f 1 ðt; xðtÞÞ þ f 2 ðt; xðt À dðtÞÞÞ þ qðtÞ; t P t 0 ;…”

Oscillation Criteria of First Order Neutral Delay Differential Equations with Variable Coefficients

“…Grammatikopoulos et al [6], Ladas and Schults [10], Chuanxi and Ladas [17,18], Kubiaczyk and Saker [19], and Karpuz and Ocalan [20] considered the NDDE (1) when ( ) = 1 and established some new oscillation results sorted by the value of function ( ). For further oscillation results on the oscillatory behaviour of solutions of (1), we refer the readers to the monographs by Győri and Ladas [21] and Erbe et al [22] as well as the papers of Yu et al [23], Choi and Koo [24], Ocalan [25], and Candan and Dahiya [26]. The purpose of this work is to find some sufficient conditions for the oscillation of all solutions of the first order NDDE (1).…”