In this paper, the forced odd order neutral differential equations of the form are considered d n d t n[x(t)-R(t)x(t-τ)]+P(t)x(t-σ)=f(t),t≥t 0.A sufficient condition for the oscillation of all solutions is...In this paper, the forced odd order neutral differential equations of the form are considered d n d t n[x(t)-R(t)x(t-τ)]+P(t)x(t-σ)=f(t),t≥t 0.A sufficient condition for the oscillation of all solutions is obtained.展开更多
A kind of extended first order impulses nonlinear FDE with forcing term is studied in this paper. Several criteria on the oscillations of solutions are given. We find some suitable impulse functions such that all the ...A kind of extended first order impulses nonlinear FDE with forcing term is studied in this paper. Several criteria on the oscillations of solutions are given. We find some suitable impulse functions such that all the solutions of the equation are oscillatory under the impulse control.展开更多
A smooth, compact and strictly convex hypersurface evolving in R^n＋1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is...A smooth, compact and strictly convex hypersurface evolving in R^n＋1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.展开更多
The authors consider the forced nonlinear neutral delay equation [y(t) + P(t)y(g(t)]' - Q(t)f(y(h(t)) = R(t) (E)where P,Q,R,g,h: [t0,∞)→R are continuous, g(t)≤t,h(t)≤t,g(t)→∞ and h(t)→ ∞ as t →∞, Q(t)≥0...The authors consider the forced nonlinear neutral delay equation [y(t) + P(t)y(g(t)]' - Q(t)f(y(h(t)) = R(t) (E)where P,Q,R,g,h: [t0,∞)→R are continuous, g(t)≤t,h(t)≤t,g(t)→∞ and h(t)→ ∞ as t →∞, Q(t)≥0, f:R →R is continuous, and uf(u) > 0 for u ≠0. Sufficient conditions are placed on P,Q,f, and R so that certain classes of solutions of (E) tend to zero as t →∞. Some suggestions for further research are also indicated.1991 Mathematics Subject Classification. Primary 34K40, 34K15; Secondary 34C11,34C15.展开更多
In this paper we analyze the qualitative behaviour of the equation ε+q(X) +εX=bp(t), where e is a small parameter.We divide the interval of parameter b into four sets of subintervals,A, B,C and D.For bA,B or D,we di...In this paper we analyze the qualitative behaviour of the equation ε+q(X) +εX=bp(t), where e is a small parameter.We divide the interval of parameter b into four sets of subintervals,A, B,C and D.For bA,B or D,we discuss the different structures of the attractors of the equation and their stabilities.When chaotic phenomena appear,we also estimate the entropy.For bC,the set of bifurcation intervals,we analyze the bifurcating type and get a series of consequences from the results of Newhouse and Palis.展开更多
At first,by means of Kartsatos technique,we reduce the impulsive differential equation to a second order nonlinear impulsive homogeneous equation.We find some suitable impulse functions such that all the solutions to ...At first,by means of Kartsatos technique,we reduce the impulsive differential equation to a second order nonlinear impulsive homogeneous equation.We find some suitable impulse functions such that all the solutions to the equation are oscillatory.Several criteria on the oscillations of solutions are given.At last,we give an example to demonstrate our results.展开更多
文摘In this paper, the forced odd order neutral differential equations of the form are considered d n d t n[x(t)-R(t)x(t-τ)]+P(t)x(t-σ)=f(t),t≥t 0.A sufficient condition for the oscillation of all solutions is obtained.
基金This work is supported by the Science Foundation of Educational Department of Guangdong Province(Z03052).
文摘A kind of extended first order impulses nonlinear FDE with forcing term is studied in this paper. Several criteria on the oscillations of solutions are given. We find some suitable impulse functions such that all the solutions of the equation are oscillatory under the impulse control.
基金supported by National Natural Science Foundation of China(Grant No.10971055)Funds for Disciplines Leaders of Wuhan(Grant No.Z201051730002)+1 种基金Project of Hubei Provincial Department of Education(Grant No.T200901)supported by Fundao Ciênciae Tecnologia(FCT)through a doctoral fellowship SFRH/BD/60313/2009
文摘A smooth, compact and strictly convex hypersurface evolving in R^n＋1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.
文摘The authors consider the forced nonlinear neutral delay equation [y(t) + P(t)y(g(t)]' - Q(t)f(y(h(t)) = R(t) (E)where P,Q,R,g,h: [t0,∞)→R are continuous, g(t)≤t,h(t)≤t,g(t)→∞ and h(t)→ ∞ as t →∞, Q(t)≥0, f:R →R is continuous, and uf(u) > 0 for u ≠0. Sufficient conditions are placed on P,Q,f, and R so that certain classes of solutions of (E) tend to zero as t →∞. Some suggestions for further research are also indicated.1991 Mathematics Subject Classification. Primary 34K40, 34K15; Secondary 34C11,34C15.
文摘In this paper we analyze the qualitative behaviour of the equation ε+q(X) +εX=bp(t), where e is a small parameter.We divide the interval of parameter b into four sets of subintervals,A, B,C and D.For bA,B or D,we discuss the different structures of the attractors of the equation and their stabilities.When chaotic phenomena appear,we also estimate the entropy.For bC,the set of bifurcation intervals,we analyze the bifurcating type and get a series of consequences from the results of Newhouse and Palis.
基金supported by the National Natural Science Foundation of China (No.1097123131071560)+1 种基金the NSF of Guangdong Province (No.101510225010000048151027501000053)
文摘At first,by means of Kartsatos technique,we reduce the impulsive differential equation to a second order nonlinear impulsive homogeneous equation.We find some suitable impulse functions such that all the solutions to the equation are oscillatory.Several criteria on the oscillations of solutions are given.At last,we give an example to demonstrate our results.
