The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｅｘｉｓｔｅｎｃｅｏｆ2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｔｏｔｈｅｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｏｆｆｉｒｓｔ＿ｏｒｄｅｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈａｄｅｖｉａｔｉｎｇａｒｇｕｍｅｎｔ ｘ(ｔ) =Ｂｘ(ｔ) Ｆ(ｘ(ｔ-τ) ) ｐ(ｔ) ,( 1 )ｗｈｅｒｅｘ(ｔ)∈Ｒ2 , ｘ(ｔ) =ｄｄｔｘ(ｔ) ,τ∈Ｒ ,Ｂ∈Ｒ2×2 ,Ｆ :Ｒ2 →Ｒ2 ｉｓｂｏｕｎｄｅｄａｎｄｐ∈Ｃ(…     

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

ＴＨＥＥＸＩＳＴＥＮＣＥＯＦＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＦＯＲＡＣＬＡＳＳＯＦＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＡＮＤＴＨＥＩＲＡＰＰＬＩＣＡＴＩＯＮＺｈａｏＪｉｅ－ｍｉｎ（赵杰民）;ＨｕａｎｇＫｅ－ｌｅｉ（黄克累）...     

The existence of periodic solutions of impulsive differential equations of mixed type

1　ＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＩｍｐｕｌｓｉｖｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｉｓａｎｅｗｉｍｐｏｒｔａｎｔｂｒａｎｃｈｏｆｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ.Ｉｎ1989,[1],[2]ｓｙｓｔｅｍａｔｉｃａｌｌｙｓｕｍｍａｒｉｚｅｄｒｅｓｅａｒｃｈｗｏｒｋａｂｏｕｔｉｍｐｕｌｓｉｖｅｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ｉｎｒｅｃｅｎｔｙｅａｒｓ,ｔｈｅｒｅａｒｅｍａｎｙｌｉｔｅｒａｔｕｒｅｓｄｅａｌｉｎｇｗｉｔｈｔｈｅｏｓｃｉｌｌａｔｉｏ…     

Criteria for the existence of solutions to random integral and differential equations

In [1], we proved a general rondom fixed point theorem and gave some applications. In this paper, we shall give further applications of the theorem. We first obtain a rondom Darbo’s fixed point theorem, using the theorem, we give the criteria for the existence of solutions under compactness hypotheses to nonlinear rondom volterra integral equations and the Cauchy problem of nonlinear rondom differential equations. Our theorems improve andgeneralize some main results of Lakshmikanthem^[2], Vaughn^[3,4] as well as De Blasi and Myjak^[5].     

Global existence of solutions to the periodic initial value problems for two-dimensional Newton-Boussinesq equations

A class of periodic initial value problems for two-dimensional Newton- Boussinesq equations are investigated in this paper. The Newton-Boussinesq equations are turned into the equivalent integral equations. With iteration methods, the local existence of the solutions is obtained. Using the method of a priori estimates, the global existence of the solution is proved.     

On positive periodic solutions of nonlinear impulsive functional differential equations

Using the Krasnosel’skii theorem on a fixed point of a mapping in a cone, we obtain conditions for the existence of positive, piecewise-smooth, periodic solutions of impulsive functional differential equations. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 501–511, October–December, 2008.     

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R ² → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008.     

On the existence and stability of periodic solutions for hopfield neural network equations with delay

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｒｅｃｅｎｔｙｅａｒｓ,ｗｉｔｈａｌｏｔｏｆａｐｐｌｉｃａｔｉｏｎｓｏｆｎｅｕｒａｌｎｅｔｗｏｒｋｍｏｄｅｌｓ,ｍａｎｙａｕｔｈｏｒｓ［１～３］ａｒｅｉｎｔｅｒｅｓｔｅｄｉｎｔｈｅｒｅｓｅａｒｃｈｏｆｔｈｅｓｔｒｕｃｔｕｒｅａｎｄｐｅｒｆｏｒｍａｎｃｅｆｏｒｔｈｅｓｅｎｅｔｗｏｒｋｓ．ＢｅｃａｕｓｅＨｏｐｆｉｅｌｄｎｅｕｒａｌｎｅｔｗｏｒｋｗｅｌｌｓｉｍｕｌａｔｅｔｈｅｅｃｏｌｏｇｉｃａｌｓｙｓｔｅｍ,ｍａｎｙｓｔｕｄｉｅｓａｒｅｃｏｎｃｅｎｔｒａｔｅｄｏｎｔｈ…     

On the existence of periodic solutions to higher dimensional periodic system with delay

Ｃｏｎｓｉｄｅｒｉｎｇｔｈｅｈｉｇｈｅｒｄｉｍｅｎｓｉｏｎａｌｐｅｒｉｏｄｉｃｓｙｓｔｅｍｓｗｉｔｈｄｅｌａｙｏｆｔｈｅｆｏｒｍｘ′（ｔ）＝Ａ（ｔ,ｘ（ｔ））ｘ（ｔ）＋ｆ（ｔ,ｘ（ｔ－τ））,（１）ｘ′（ｔ）＝ｇｒａｄＧ（ｘ（ｔ））＋ｆ（ｔ,ｘ（ｔ－...     

ON PERIODIC SOLUTIONS OF SEVERAL CLASSES 0F RICCATI'S EQAUTION AND SECOND一ORDER DIFFERENTIAL EQUATIONS

In this paper, the problem on periodic solutions of several classes of Riccati's equation with periodic coefficients is discussed, and the conditions, under which several classes of secondorder equations with periodic coefficients have periodic solutions, are given.     

Boundedness of solutions of linear differential systems with periodic coefficients

Archive for Rational Mechanics and Analysis -     

Averaging method for the solution of non-linear differential equations with periodic non-harmonic solutions

While Krylov and Bogolyubov used harmonic functions in their averaging method for the approximate solution of weakly non-linear differential equations with oscillatory solution, we apply a similar averaging technique using Jacobi elliptic functions. These functions are also periodic and are exact solutions of strongly non-linear differential equations. The method is used to solve non-linear differential equations with linear and non-linear small dissipative terms and/or with time dependent parameters. It is also shown that quite general dissipative terms can be transformed into time-dependent parameters. As a special example, the Langevin (collisional) equation of motion of electrons in a neutralizing ion background under the influence of a time and space-dependent electric field is presented. The method may also be used for non-linear control theory, dynamic and parametric stabilization of non-linear oscillations in plasma physics, etc.     

Periodic solutions of systems of functional differential equations with linear deviations of argument and their properties

We establish sufficient conditions for the existence of periodic solutions of systems of linear and nonlinear functional differential equations with linear deviations of the argument and investigate their properties.     

Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp-function method

In this paper, the Exp-function method with the aid of the symbolic computational system Maple is used to obtain the generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics, namely, (2+1)-dimensional Konopelchenko–Dubrovsky equations, the (3+1)-dimensional Jimbo–Miwa equation, the Kadomtsev–Petviashvili (KP) equation, and the (2+1)-dimensional sine-Gordon equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.