Effect of Shoot Control on Flower Bud Differentiation,Flowering,and Fruit Setting in Zanthoxylum armatum DC.

In this study,newly sprouted shoots of Zanthoxylum armatum(Z.armatum),which were collected after the harvesting period,were used as the primary experimental specimens.A randomized block design and paraffin sectioning method were used to investigate the flower bud differentiation process and the quantity and vitality of buds.Furthermore,the study examined the response of flowering and fruiting to cultivation methods for shoot growth,including layering and plant growth regulator application.The results showed that(a)layering and application of plant growth regulators for Z.armatum accelerated the process of flower bud differentiation by approximately 20 days compared to the control group.Additionally,both shoot control methods generated more and larger bud primordia and perianth primordia during the same differentiation phase.(b)The application of plant growth regulators resulted in well-developed buds,exhibiting higher levels of flower bud differentiation than the layering method.The quality of flower bud formation for both shoot control methods was superior to that of the control group.(c)The flowering phenological period was relatively consistent between the two cultivation methods,but the fruit maturity phase for shoot-controlled trees occurred 20 days earlier than the control group.(d)Both layering and the application of plant growth regulators significantly decreased the rates of unfertilized flower shedding and fruit shedding.However,no significant difference was noted in fruit setting per inflorescence and per flower between the two methods and the control.The effect of altitude for both methods on the fruit setting was not significant.Under both shoot control methods,the Z.armatum exhibited earlier morphological differentiation of flower buds,faster differentiation process,improved flower bud quality,and significantly decreased rates of flower and fruit shedding.Thus,these cultivation methods demonstrated the potential to promote flowering,improve fruit setting,and reduce fruit shedding in Z.armatum.

On the Existence of Periodic Solutions of a Class Nonlinear Differential Systems

One method to show the existence of ω-periodic system is given. This method is based on the ultimately boundedness of the solution of the systems. By using comparing theorem and discussing some one dimensional equations the main results are obtained.

Existence of Periodic Solutions for Generalized Liénard-Typed Functional Differential Equations

The existence of periodic solutions for a kind of generalized Liénard typed functional differential equation is studied. By means of the continuation theorem of coincidence degree theory, existence criteria are established for the existence of periodic solutions and some previous results are extended.

Existence of Periodic Solutions of Nonlinear Neutral Differential Equation with Deviating Argument

Existence criteria is established for the periodic solution of the nonlinear neutral delay differential equation x′(t)=f(t,x(t),x(t-τ 1(t)),x′(t-τ 2(t)))+p(t) by means of an abstract continuous theorem of k-set contractive operator and some analysis technique.

Existence of Many and Infinitely Many Periodic Solutions for Some Types of Differential Delay Equations

J.Kaplan and J.Yorke's method is extended to establish the exis- tence of many and infinitely many periodic solutions for the DDEs (t) =±f(x(t-1))±f(x(t-2))and (t)=±f(x(t-1).

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

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))].

ON THE DEPENDENT PROPERTY OF SOLUTIONS FOR HIGHER ORDER PERIODIC DIFFERENTIAL EQUATIONS

This article investigates the property of linearly dependence of solutions f（z） and f（z ＋ 2πi） for higher order linear differential equations with entire periodic coefficients.

Periodic boundary value problem for the first order functional differential equations with impulses

This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.

THE EXISTENCE OF PERIODIC SOLUTIONS FOR A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.

Periodic Boundary Value Problem for Functional Differential Equations with Impulses

Aim To investigate the periodic boundary value problem for functional differential equations with impulses. Methods The method of upper and lower solutions and the monotone iterative technique were used to establish our results. Results and Conclusion The results of the existence of maximal and minimal solutions of the periodic boundary value problem for functional differential equations with impulses are obtained.

POSITIVE HOMOCLINIC ORBITS FOR A CLASS OF ASYMPTOTICALLY PERIODIC SECOND ORDER DIFFERENTIAL EQUATIONS

This note studies the existence of positive homoclinic orbits of the second order equation-u″+α(x)u=β(x)u q+γ(x)u p, x∈R,where 1<q<p.Assume that the coefficient functions α(x),β(x) and γ(x) are asymptotically periodic and satisfy0<a≤α(x), 0<γ(x)≤B, -M≤β(x)≤M.A positive homoclinic orbit of the equation is obtained by means of variational methods.

A PRIORI BOUNDS FOR PERIODIC SOLUTIONS OF A KIND OF THIRD-ORDER DELAY DIFFERENTIAL EQUATION

By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.

PERIODIC SOLUTIONS OF THIRD-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS

The sufficient condition for the existence of2 π- periodic solutions of the following third- order functional differential equations with variable coefficients a(t) x (t) +bx″2 k- 1(t) +cx′2 k- 1(t) + 2 k- 1 i=1 cixi(t) +g(x(t-τ) ) =p(t) =p(t+2π) is obtained.The approach is based on the abstract continuation theorem from Mawhin and the a- priori estimate of periodic solutions

Positive Periodic Solutions for Nonautonomous Differential Equations with Delay

By utilizing a fixed point theorem on cone, some new results on the existence ofpositive periodic solutions for nonautonomous differential equations with delay are derived.

Periodic Solutions for a Class of Neutral Functional Differential Equations with Distributed and Discrete Delays

Based on Krasnoselskii's fixed point theorem,matrix measure and functional analysis methods,some new sufficient conditions for the existence of periodic solutions of neutral functional differential equations with distributed and discrete delays are obtained. Moreover,we construct an example to illustrate the feasibility of our results.

Multiple Positive Periodic Solutions for Functional Differential Equations with Infinite Delay

In this paper, we investigate the existence of multiple positive periodic solutions for functional differential equations with infinite delay by applying the Krasnoselskii fixed point theorem for cone map and the Leggett-Williams fixed point theorem.

Periodic Solutions in UMD Spaces for Some Neutral Partial Functional Differential Equations

The aim of this work is to study the existence of a periodic solution for some neutral partial functional differential equations. Our approach is based on the R-boundedness of linear operators Lp-multipliers and UMD-spaces.

EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR VOLTERRA INTERGO-DIFFERENTIAL EQUATIONS

This paper deals with the existence of positive periodic solutions for a kind of nonautonomous Volterra intergo-differential equations by employing the Krasnoselskii fixed point theorem. Applying the general theorems established to several biomathematical models, the paper improves some previous results and obtains some new results.

ON A PROBLEM IN COMPLEX OSCILLATION THEORY OF PERIODIC HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

In this article, the zeros of solutions of differential equation f^（k）（z）＋A（z）f（z） = 0, are studied, where k 2, A（z） = B（e^z）, B（ζ） = g1（1/ζ） ＋ g2（ζ）, g1 and g2 being entire functions with g2 transcendental and σ（g2） not equal to a positive integer or infinity. It is shown that any linearly independent solutions f1, f2, . . . , fk of Eq.（＊） satisfy λe（f1 . . . fk） ≥ σ（g2） under the condition that fj（z） and fj（z＋ 2πi） （j = 1, . . . , k） are linearly dependent.

POSITIVE PERIODIC SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS

The nonlinear differential equationx′(t)=-δ(t)x(t)+f(t,x(t))(*)is considered,where δ(t) is a periodic function of periodic T,f(t,x) is continuous and periodic in t.It is showed that (*) has at least two positive T-periodic solutions under certain growth conditions imposed on f.Applications will be presented to illustrate the main results.