Multiple periodic solutions for system of first-order differential equation

Sufficient conditions have been obtained for the existence of at least two non-negative periodic solutions to a system of first-order nonlinear functional differential equations. Applications to some ecological models are given.     

Exact multiplicity for periodic solutions of a first-order differential equation

Consider the differential equation

(1)     

Stability of solutions of a system of linear differential equations with periodic stochastic coefficients

Systems of equations for first and second moments are investigated and transformed. Stability of solutions of a first-order linear differential equations is analyzed. Stability of solutions of the stochastic Mathieu equation is investigated and the boundaries of the instability region are determined.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 119–126, 1986.     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green’s function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.     

Stability properties of solutions of linear second order differential equations with random coefficients

The equation

     

Impulsive periodic solutions of first-order singular differential equations

In this work, we study the impulsive periodic solutions of first-ordersingular ordinary differential equations. The proof is basedon a nonlinear alternative principle of Leray–Schauder,together with a truncation technique. Some recent results inthe literature are generalized and improved.     

Positive periodic solutions of first-order functional differential equations with parameter

We prove the existence and multiplicity of positive T$T$-periodic solution(s) for T$T$-periodic equation x^′(t)=h(t,x)−λb(t)f(x(t−τ(t)))$x^{\prime }\left(t\right)=h(t,x)-\lambda b\left(t\right)f\left(x(t-\tau \left(t\right))\right)$ by Krasnoselskii fixed point theorem, where f(x)$f\left(x\right)$ may be singular at x=0$x=0$. Our results improve some recent results in previous literature.     

On the existence and multiplicity of positive periodic solutions for first-order vector differential equation

In this paper, we consider the existence and multiplicity of positive periodic solutions for first-order vector differential equation x^′(t)+f(t,x(t))=0, a.e. t∈[0,ω] under the periodic boundary value condition x(0)=x(ω). Here ω is a positive constant, and is a Carathéodory function. Some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones.     

Investigation of a system of linear differential equations with random coefficients

We investigate a system of linear differential equations with random coefficients that depend on a periodic Markov process. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1137–1143, August, 1998.     

Optimization of a system of linear differential equations with random coefficients

We consider a system of differential equations with controls that are linearly contained in the right-hand sides. We establish a necessary condition for the optimal control that minimizes a quadratic functional. Kiev National Economic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 556–561, April, 1999.