Periodic or Bounded Solutions of Carathéodory Systems of Ordinary Differential Equations

In this paper we extend the guiding function approach to show that there are periodic or bounded solutions for first order systems of ordinary differential equations of the form x′=f(t,x), a.e. t∈[a,b], where f satisfies the Carathéodory conditions. Our results generalize recent ones of Mawhin and Ward.     

Sufficient Conditions for the Existence of Bounded Solutions of Systems of Nonlinear Difference Equations

We obtain sufficient conditions for systems of nonlinear difference equations x(n + 1) = A(x(n))x(n) + f(n), n ∈ ℤ, where A(x) is a matrix function continuous on ℝ ^m , to have solutions in the space of bilateral number sequences. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 165–173, April–June, 2005.     

On a Countable-Point Nonlinear Boundary-Value Problem on a Semiaxis for Ordinary Differential Equations in the Space of Bounded Numerical Sequences

We use the idea of the Samoilenko numerical-analytic method for the investigation of a nonlinear boundary-value problem with an unbounded countable set of boundary moments on the positive semiaxis in the case where the differential equation and boundary conditions are defined in the Banach space of bounded numerical sequences.     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

Unbounded Solutions to Systems of Differential Equations at Resonance

Journal of Dynamics and Differential Equations - We deal with a weakly coupled system of ODEs of the type $$begin{aligned} x_j'' + n_j^2 ,x_j + h_j(x_1,ldots ,x_d) = p_j(t), qquad...     

C 1 Solutions for Semi-Implicit Systems of Differential Equations

The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where

${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x ₀) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function,

${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function.

The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ^?1.     

On Asymptotics of Solutions of Semiexplicit Systems of Differential Equations

We study the problem of the existence of analytic solutions of a certain semiexplicit system of differential equations and obtain sufficient conditions for the existence of analytic solutions of the Cauchy problem in the neighborhood of a singular point.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 132–144, January–March, 2005.     

Study on Exact Analytical Solutions for Two Systems of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｒｉｎｇｔｈｅｃｏｕｒｓｅｏｆｓｔｕｄｙｉｎｇｔｈｅｗａｔｅｒｗａｖｅ,ｍａｎｙｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｍｏｄｅｌｓｗｅｒｅｏｂｔａｉｎｅｄ ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ ,ｍＫｄＶｅｑｕａｔｉｏｎ ,(2 1 )＿ｄｉｍｅｎｓｉｏｎａｌＫＰｅｑｕａｔｉｏｎ ,ｃｏｕｐｌｅｄＫｄＶｅｑｕａｔｉｏｎｓ,ｖａｒｉａｎｔＢｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎｓ ,ＷＫＢｅｑｕａｔｉｏｎｓｅｔｃ .[1- 13 ].Ｉｎｏｒｄｅｒｔｏｆｉｎｄｅｘｐｌｉｔｉｃｅｘａｃｔｓｏｌｕｔｉｏ…     

Preserving One-Sided Invariance in Rn with Respect to Systems of Ordinary Differential Equations

The one-sided invariance of sets with respect to systems of ordinary differential equations in R ⁿ is investigated. We present a general class of sets that preserve such an invariance for a linear combination of differential equations. As an application of these results, we consider the case of two coupled identical systems, which is important for the synchronization problem.     

Asymptotic Solutions of Boundary Value Problems for Third-Order Ordinary Differential Equations with Turning Points

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｓｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈｎｏｔｕｒｎｉｎｇｐｏｉｎｔｈａｄｂｅｅｎｓｔｕｄｉｅｄｖｅｒｙｅａｒｌｙ ,ｂｕｔｔｈｅｓｔｕｄｙｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓｗｉｔｈｔｕｒｎｉｎｇｐｏｉｎｔｐｒｏｃｅｅｄｅｄｒａｔｈｅｒｓｌｏｗｌｙｏｗｉｎｇｔｏｔｈｅｓｉｎｇｕｌａｒｐｏｉｎｔｏｆｔｈｅｄｅｇｅｎｅｒａｔｅｄｅｑｕａｔｉｏ…     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

On Periodic Solutions of a System of Nonlinear Functional Partial Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of a system of nonlinear functional partial differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 154–158, April–June, 2005.     

A Set of Bounded Solutions of a Linear Weakly Perturbed System

For weakly perturbed systems of linear differential equations, we establish conditions for the point = 0 to bifurcate into a set of solutions bounded on the entire axis R in the case where the corresponding unperturbed homogeneous linear differential system is exponentially dichotomous on the semiaxes R ₊ and R _–. We determine the number of linearly independent solutions bounded on R and give an algorithm for finding these solutions.     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

Convergence of Global and Bounded Solutions of Some Nonautonomous Second Order Evolution Equations with Nonlinear Dissipation

In this paper we study the asymptotic behavior of solutions of the following nonautonomous wave equation with nonlinear dissipation.

$\left\{\begin{array}{ll} u_{tt}+\vert u_{t}\vert^{\alpha}u_{t}-\Delta u +f(u)=g(t,x),\quad{\rm in}\,\mathbb{R}_{+}\times\Omega,\\ \qquad\qquad u(t,x)=0,\quad\, {\rm on}\,\mathbb{R}_{+}\times\partial\Omega,\end{array}\right.$

where f is an analytic function, α is a small positive real and g(t, ·) tends to 0 sufficiently fast in L ²(Ω) as t tends to ∞.

We also obtain a general convergence result and the rate of decay of solutions for a class of second order ODE containing as a special case

$\left\{\begin{array}{ll} \ddot{U}(t)+\Vert\dot{U}(t)\Vert^{\alpha}\dot{U}(t)+\nabla F(U(t))=g(t),\quad t \in \mathbb{R}_+,\\ \qquad U(0)=U_{0}\,\in \mathbb{R}^{N},\quad\dot{U}(0)=U_{1}\in \mathbb{R}^{N}. \end{array}\right.$

     

On Bounded Solutions of a Nonlinear Volterra System

We find sufficient conditions for the boundedness of solutions of a nonlinear Volterra system in terms of the coefficients of this system.     

On the Asymptotic Behavior of Solutions of a System of Nonlinear Differential Equations

We investigate the asymptotic behavior of a system of nonlinear differential equations of a special form at infinity. We also propose a method for the reduction of more general systems of nonlinear differential equations to this form, which enables one to study their asymptotic properties.     

Asymptotic Properties of Global Solutions of Systems of Functional Differential Equations of Neutral Type with Nonlinear Deviations of an Argument

For a system of functional differential equations of the neutral type with nonlinear deviations of an argument dependent on an unknown function, we establish sufficient conditions for the existence of a solution continuously differentiable and bounded for t and study its properties.     

Iteration Method for Systems of Nonlinear Differential Equations with Delay and Restrictions

We establish a consistency condition for systems of nonlinear differential equations with delay and restrictions and justify the applicability of the iteration method to these problems.