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.     

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(…     

Existence and uniqueness of periodic solutions for fourth-order nonlinear periodic systems

In this paper, we study the existence, uniqueness and stability of the periodic solutions for fourth-order nonlinear nonhomogeneous periodic systems with slowly changing coefficients by using the method of Liapunor Function. We obtain some sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions of these systems and estimate the extent to which the coefficients are allowed to change.     

Computable error bounds for approximate periodic solutions of autonomous delay differential equations

In this paper, we prove a result that says: Given an approximate solution and frequency to a periodic solution of an autonomous delay differential equation that satisfies a certain noncriticality condition, there is an exact periodic solution and frequency in a neighborhood of the approximate solution and frequency and, furthermore, numerical estimates of the size of the neighborhood are computed. Methods are outlined for estimating the parameters required to compute the errors. An application to a Van der Pol oscillator with delay in the nonlinear terms is given.     

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 periodic solutions of differential inclusions and applications

The periodic problem of evolution inclusion is studied and its results are used toestablish existence theorems of periodic solutions of a class of semi-linear differential inclusion.Also existence theorem of the extreme solutions and the strong relaxation theorem are given forthis class of semi-linear differential inclusion. An application to some feedback control systems isdiscussed.     

Boundedness of solutions of linear differential systems with periodic coefficients

Archive for Rational Mechanics and Analysis -     

Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators,in which the perturbation procedure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived.The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method.Several strongly nonlinear oscillators with quadratic,cubic,and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.     

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.     

On the existence of periodic solutions for nonlinear system with multiple delays

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…