New sufficient conditions for the oscillation of all solutions of the neutral differential equation [x(t) - p(t)x(t -τ)]' + Q(t)x(t -σ) = 0 are obtained, which further answers an open problem proposed by Gyori a...New sufficient conditions for the oscillation of all solutions of the neutral differential equation [x(t) - p(t)x(t -τ)]' + Q(t)x(t -σ) = 0 are obtained, which further answers an open problem proposed by Gyori and Ladas. An necessary and sufficient condition for the existence of positive solutions is also obtained which partially solves an open problem proposed by Chuanxi and Ladsa and which has been be proved in [9], but we give a new proof.展开更多
Consider the nonautonomous delay logistic equation △yn = pnyn(1- ynk-ln), n ≥ 0, (1)where {pn}n≥0 is a sequence of nonnegative real numbers, {ln}n≥0 is a sequence of positive integers satisfying lim n→∞(n - ln)...Consider the nonautonomous delay logistic equation △yn = pnyn(1- ynk-ln), n ≥ 0, (1)where {pn}n≥0 is a sequence of nonnegative real numbers, {ln}n≥0 is a sequence of positive integers satisfying lim n→∞(n - ln) = ∞, and k is a positive constant. Only solutions which are positive for n ≥ 0 are considered. We obtain a new sufficient for all positive solutions of (1) to oscillate about k which contains the corresponding result in [2] when i = 1.展开更多
This paper is concerned with the existence of eventually positive solutions and oscillatory solutions of first-order neutral differential equation with positive and negative coefficients. The 'sharp' condition...This paper is concerned with the existence of eventually positive solutions and oscillatory solutions of first-order neutral differential equation with positive and negative coefficients. The 'sharp' conditions of oscillation for the equation have been obtained, i.e., the conditions are sufficient and necessary when all of the coefficients are constants. These conditions are distinguished from the previous criteria in the literatures, and also weaker than those existing results.展开更多
A class of nonlinear neutral equations with positive and negative coefficients is considered. Seveal conditions for existence of eventually positive solutions are obtained,and they extend and improve some of the crite...A class of nonlinear neutral equations with positive and negative coefficients is considered. Seveal conditions for existence of eventually positive solutions are obtained,and they extend and improve some of the criteria existing in literature. Our conditions are necessary and suffcient when all the coeffcients of the equations are constants.展开更多
In this paper, a new oscillating result is established for the first order neutral delay differential equation with positive and negative coefficients, which improves and generalizes several results in the literatures.
By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(...By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].展开更多
We investigate oscillation of certain second order neutral dynamic equations of Emden-Fowler type with positive and negative coefficients. We use some different techniques and apply Riccati transformation to establish...We investigate oscillation of certain second order neutral dynamic equations of Emden-Fowler type with positive and negative coefficients. We use some different techniques and apply Riccati transformation to establish new oscillatory criteria which include two necessary and sufficient conditions. Moreover, we point out that how the power γ plays its role. Some interesting examples are given to illustrate the versatility of our results.展开更多
We establish a new oscillation condition for even order neutral type differential equation of the following formwhere f is continuous and exists 1 ≤ k ≤ m(?)(t) ≡ Tk(t), a, (?)i ∈ C([0, ∞], R), i = 1, …,m. such ...We establish a new oscillation condition for even order neutral type differential equation of the following formwhere f is continuous and exists 1 ≤ k ≤ m(?)(t) ≡ Tk(t), a, (?)i ∈ C([0, ∞], R), i = 1, …,m. such that 0 ≤ a(t) ≤ L and展开更多
In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions i...In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions in the literature.展开更多
Consider the higher order neutral differential equation[x(t)-P(t)x(t-τ)] (n) +Ni=1Q i(t)x(t-σ i(t))=0, tt 0with P(t),Q i(t),σ i(t)(i=1,2,…,N) continuous, τ >0 and n is odd. We establish se...Consider the higher order neutral differential equation[x(t)-P(t)x(t-τ)] (n) +Ni=1Q i(t)x(t-σ i(t))=0, tt 0with P(t),Q i(t),σ i(t)(i=1,2,…,N) continuous, τ >0 and n is odd. We establish several new sufficient conditions for the oscillation of all solutions. Some of these conditions including generalize and improve the known results one in . AMS Subject Classifications: 34K15, 34C10.展开更多
This paper is to obtain sufficient conditions under which the neutral functional differential equation d/dx[x(t) + integral(c)(t) x(s)d(s) mu(t, s)] + integral(c)(t) f(t, x(s))d(s) eta(t, s) = 0, t greater than or equ...This paper is to obtain sufficient conditions under which the neutral functional differential equation d/dx[x(t) + integral(c)(t) x(s)d(s) mu(t, s)] + integral(c)(t) f(t, x(s))d(s) eta(t, s) = 0, t greater than or equal to t(0) greater than or equal to c (1) has a positive solution on [c, +infinity). Some results in [1] are generalized. Then we apply our results to functional differential equations of special form and obtain sufficient conditions for those equations to have a positive solution.展开更多
In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the s...In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the solutions.展开更多
In this paper, new oscillation criteria are obtained for all solution of the firstorder neutral difference equation △(x(n) - P(n)x(n-m)) + Q(n)x(n-k) =0, n ≥n0 where m > 0, k≥ 0 are integers. p(n), Q(n)∈ C({n0...In this paper, new oscillation criteria are obtained for all solution of the firstorder neutral difference equation △(x(n) - P(n)x(n-m)) + Q(n)x(n-k) =0, n ≥n0 where m > 0, k≥ 0 are integers. p(n), Q(n)∈ C({n0, n0 + 1,...}, R+). Our results do not need the usual hypothesis ∑Q(s) = ∞. Some example are given s=N to demonstrate the advantage of our results than those in the literature.展开更多
This paper gives the rules of oscillations of two classes of neutraldifferential equations with forced terms, and some oscillation criteria undercertain conditions are presented according to the equations having diffe...This paper gives the rules of oscillations of two classes of neutraldifferential equations with forced terms, and some oscillation criteria undercertain conditions are presented according to the equations having differentcharacters.展开更多
In this paper, we further investigate a class of first order neutral Pantograph differential equations of Euler type by Guan and Shen . Some infinite-integral conditions for the oscillation of all solutions are establ...In this paper, we further investigate a class of first order neutral Pantograph differential equations of Euler type by Guan and Shen . Some infinite-integral conditions for the oscillation of all solutions are established.展开更多
The author studied the existence of positive solutions of the delay logistic difference equation Δ y n=p ny n(1-y τ(n) ), n =0,1,2,.... where { p n } is a sequence of positive real numbers, { τ(n) } is a nondecreas...The author studied the existence of positive solutions of the delay logistic difference equation Δ y n=p ny n(1-y τ(n) ), n =0,1,2,.... where { p n } is a sequence of positive real numbers, { τ(n) } is a nondecreasing sequence of integers, τ(n)<n and lim n →∞ τ(n) =∞. A sufficient condition for the existence of positive solutions of the equation was given.展开更多
A class of higher order neutral difference equations is considered and some sufficient conditions are obtained for all solutions to oscillate or tend to zero.
文摘New sufficient conditions for the oscillation of all solutions of the neutral differential equation [x(t) - p(t)x(t -τ)]' + Q(t)x(t -σ) = 0 are obtained, which further answers an open problem proposed by Gyori and Ladas. An necessary and sufficient condition for the existence of positive solutions is also obtained which partially solves an open problem proposed by Chuanxi and Ladsa and which has been be proved in [9], but we give a new proof.
文摘Consider the nonautonomous delay logistic equation △yn = pnyn(1- ynk-ln), n ≥ 0, (1)where {pn}n≥0 is a sequence of nonnegative real numbers, {ln}n≥0 is a sequence of positive integers satisfying lim n→∞(n - ln) = ∞, and k is a positive constant. Only solutions which are positive for n ≥ 0 are considered. We obtain a new sufficient for all positive solutions of (1) to oscillate about k which contains the corresponding result in [2] when i = 1.
基金Research supported by National Natural Science Foundation of P. R. China (10071016) by Foundation of the Education Department for Excellent Teacher of University.
文摘This paper is concerned with the existence of eventually positive solutions and oscillatory solutions of first-order neutral differential equation with positive and negative coefficients. The 'sharp' conditions of oscillation for the equation have been obtained, i.e., the conditions are sufficient and necessary when all of the coefficients are constants. These conditions are distinguished from the previous criteria in the literatures, and also weaker than those existing results.
文摘A class of nonlinear neutral equations with positive and negative coefficients is considered. Seveal conditions for existence of eventually positive solutions are obtained,and they extend and improve some of the criteria existing in literature. Our conditions are necessary and suffcient when all the coeffcients of the equations are constants.
基金Supported by the NSF Education Dept. (hjkj200317) and the NSF of Hainan Province (80403).
文摘In this paper, a new oscillating result is established for the first order neutral delay differential equation with positive and negative coefficients, which improves and generalizes several results in the literatures.
基金National Natural Science Foundation of China( 198710 0 5 )
文摘By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].
基金supported by National Natural Science Foundation of China (Grant No. 11271379)Guangzhou Postdoctoral Science Research Foundation Project (Grant No. gdbsh2014003)
文摘We investigate oscillation of certain second order neutral dynamic equations of Emden-Fowler type with positive and negative coefficients. We use some different techniques and apply Riccati transformation to establish new oscillatory criteria which include two necessary and sufficient conditions. Moreover, we point out that how the power γ plays its role. Some interesting examples are given to illustrate the versatility of our results.
基金This research is supported by NNSF of China(19875083,10275032).
文摘We establish a new oscillation condition for even order neutral type differential equation of the following formwhere f is continuous and exists 1 ≤ k ≤ m(?)(t) ≡ Tk(t), a, (?)i ∈ C([0, ∞], R), i = 1, …,m. such that 0 ≤ a(t) ≤ L and
基金the National Natural Science Foundation of China(1 0 0 71 0 1 8)
文摘In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions in the literature.
文摘Consider the higher order neutral differential equation[x(t)-P(t)x(t-τ)] (n) +Ni=1Q i(t)x(t-σ i(t))=0, tt 0with P(t),Q i(t),σ i(t)(i=1,2,…,N) continuous, τ >0 and n is odd. We establish several new sufficient conditions for the oscillation of all solutions. Some of these conditions including generalize and improve the known results one in . AMS Subject Classifications: 34K15, 34C10.
文摘This paper is to obtain sufficient conditions under which the neutral functional differential equation d/dx[x(t) + integral(c)(t) x(s)d(s) mu(t, s)] + integral(c)(t) f(t, x(s))d(s) eta(t, s) = 0, t greater than or equal to t(0) greater than or equal to c (1) has a positive solution on [c, +infinity). Some results in [1] are generalized. Then we apply our results to functional differential equations of special form and obtain sufficient conditions for those equations to have a positive solution.
基金This work are supported by the NNSF of China (No.10571050)the Science and Research Fund for Higher Colleges of Hunan (No.06C054).
文摘In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the solutions.
文摘In this paper, new oscillation criteria are obtained for all solution of the firstorder neutral difference equation △(x(n) - P(n)x(n-m)) + Q(n)x(n-k) =0, n ≥n0 where m > 0, k≥ 0 are integers. p(n), Q(n)∈ C({n0, n0 + 1,...}, R+). Our results do not need the usual hypothesis ∑Q(s) = ∞. Some example are given s=N to demonstrate the advantage of our results than those in the literature.
文摘This paper gives the rules of oscillations of two classes of neutraldifferential equations with forced terms, and some oscillation criteria undercertain conditions are presented according to the equations having differentcharacters.
基金supported by Scientific Research Fund of Hengyang Bureau of Science andTechnology (06KJ15)
文摘In this paper, we further investigate a class of first order neutral Pantograph differential equations of Euler type by Guan and Shen . Some infinite-integral conditions for the oscillation of all solutions are established.
文摘The author studied the existence of positive solutions of the delay logistic difference equation Δ y n=p ny n(1-y τ(n) ), n =0,1,2,.... where { p n } is a sequence of positive real numbers, { τ(n) } is a nondecreasing sequence of integers, τ(n)<n and lim n →∞ τ(n) =∞. A sufficient condition for the existence of positive solutions of the equation was given.
文摘A class of higher order neutral difference equations is considered and some sufficient conditions are obtained for all solutions to oscillate or tend to zero.
