scholarly journals On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Yuangong Sun
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Continuous Functions ◽  
Higher Order ◽  
Oscillation Criteria ◽  
Functional Differential ◽  
Delay Dependent

We investigate the oscillation of the following higher-order functional differential equation:x(n)(t)+q(t)|x(t-τ)|λ-1x(t-τ)=e(t),whereq(t)ande(t)are continuous functions on[t0,∞),1>λ>0andτ≠0are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the caseτ>0and the caseτ<0under the assumption that the potentialsq(t)ande(t)change signs on[t0,∞).

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

Asymptotic Behavior for Nonoscillatory Solutions of Second Order Nonlinear Functional Differential Equation

2012 ◽  
Vol 616-618 ◽  
pp. 2137-2141
Author(s):  
Zhi Min Luo ◽  
Bei Fei Chen
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Functional ◽  
Nonlinear Functional Differential Equation ◽  
Functional Differential

This paper studied the asymptotic behavior of a class of nonlinear functional differential equations by using the Bellman-Bihari inequality. We obtain results which extend and complement those in references. The results indicate that all non-oscillatory continuable solutions of equation are asymptotic to at+b as under some sufficient conditions, where a,b are real constants. An example is provided to illustrate the application of the results.

On the existence of periodic solutions for autonomous retarded functional differential equations on R2

1986 ◽  
Vol 102 (3-4) ◽  
pp. 259-262 ◽  
Author(s):  
J. G. Dos Reis ◽  
R. L. S. Baroni
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Continuous Functions ◽  
Retarded Functional Differential Equations ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

SynopsisLet Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

scholarly journals Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
T. E. Govindan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Local Lipschitz Condition ◽  
Functional Differential ◽  
Special Case ◽  
Partial Functional Differential Equation

This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.

scholarly journals Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jun Zhou ◽  
Jun Shen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Positive Solutions ◽  
Functional Differential Equation ◽  
Continuous Dependence ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
State Dependence ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, this equation can be regarded as a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.</p>

scholarly journals Oscillation of a functional differential equation arising from an industrial problem

1978 ◽  
Vol 26 (3) ◽  
pp. 323-329 ◽  
Author(s):  
Hiroshi Onose
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
First Order ◽  
Functional Differential ◽  
Image Position ◽  
Industrial Problem

AbstractIn the last few years, the oscillatory behavior of functional differential equations has been investigated by many authors. But much less is known about the first-order functional differential equations. Recently, Tomaras (1975b) considered the functional differential equation and gave very interesting results on this problem, namely the sufficient conditions for its solutions to oscillate. The purpose of this paper is to extend and improve them, by examining the more general functional differential equation

Non-linear functional differential equations and abstract integral equations

1979 ◽  
Vol 84 (1-2) ◽  
pp. 71-91 ◽  
Author(s):  
F. Kappel ◽  
W. Schappacher
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Space Setting ◽  
Non Linear ◽  
Functional Differential ◽  
Order Spline ◽  
The Given

SynopsisThe equivalence between solutions of functional differential equations and an abstract integral equation is investigated. Using this result we derive a general approximation result in the state space C and consider as an example approximation by first order spline functions. During the last twenty years C1-semigroups of linear transformations have played an important role in the theory of linear autonomous functional differential equations (cf. for instance the discussion in [9, Section 7.7]). Applications of non-linear semigroup theory to functional differential equations are rather recent beginning with a paper by Webb [17]. Since then a considerable number of papers deal with problems in this direction. A common feature of the majority of these papers is that as a first step with the functional differential equation there is associated a non-linear operator A in a suitable Banach-space. Then appropriate conditions are imposed on the problem such that the conditions of the Crandall-Liggett-Theorem [5] hold for the operator A. This gives a non-linear semigroup. Finally the connection of this semigroup tothe solutions of the original differential equation has to be investigated [c.f. 8, 15, 18]. To solve thislast problem in general is the most difficult part of this approach.In the present paper we consider the given functional differential equation as a perturbation of the simple equationx = 0. The solutions of this equation generate a very simple C1-semigroup. The solutions of the original functional differential equation generate solutions of an integral equation which is the variation of constants formula for the abstract Cauchy problem associated with the equation x = 0. Under very mild conditions we can prove a one-to-one correspondence between solutions of the given functional differential equation and solutions of the integral equation in the Lp-space setting. In the C-space setting the integral equation inthe state space has to be replaced by a ‘pointwise’ integral equation. Using the pointwise integral equation together with a theorem which guarantees continuous dependence of fixed points on parameters we show under rather weak hypotheses that the original functional differential equation can be approximated by a sequence of ordinary differential equations. Using 1st order spline functions we finally get results which are very similar to those obtained in [1 and 11] in the L2-space setting.

scholarly journals Bounded and almost periodic solutions and evolution semigroups associated with nonautonomous functional differential equations

2000 ◽  
Vol 5 (4) ◽  
pp. 245-263 ◽  
Author(s):  
Bernd Aulbach ◽  
Nguyen Van Minh
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Representation Theorem ◽  
Functional Differential Equations ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
Functional Differential ◽  
Evolution Semigroups ◽  
Autonomous Evolution

We study evolution semigroups associated with nonautonomous functional differential equations. In fact, we convert a given functional differential equation to an abstract autonomous evolution equation and then derive a representation theorem for the solutions of the underlying functional differential equation. The representation theorem is then used to study the boundedness and almost periodicity of solutions of a class of nonautonomous functional differential equations.

scholarly journals One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Global Bifurcation ◽  
Periodic Functions ◽  
First Order ◽  
Functional Differential

We study one-signed periodic solutions of the first-order functional differential equationu'(t)=-a(t)u(t)+λb(t)f(u(t-τ(t))),t∈Rby using global bifurcation techniques. Wherea,b∈C(R,[0,∞))areω-periodic functions with∫0ωa(t)dt>0,∫0ωb(t)dt>0,τis a continuousω-periodic function, andλ>0is a parameter.f∈C(R,R)and there exist two constantss2<0<s1such thatf(s2)=f(0)=f(s1)=0,f(s)>0fors∈(0,s1)∪(s1,∞)andf(s)<0fors∈(-∞,s2)∪(s2,0).

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