Multivalued solutions of second-order differential equations

Multivalued (not set-valued as in the theory of differential inclusions!) solutions of ordinary differential equations (ODE) appear naturally in geometrical and physical problems in which the independent and dependent variablesx, y are geometric coordinates of a current point on the sought-for curve. This note contains some simple results concerning smooth multivalued solutions of real second-order ODE resolved with respect toy″; the special role of equations of the third degree with respect toy′ is underlined. The method of investigation is based on combining ODEs fory(x) andx(y). Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 871–878, December, 1999.     

Global existence and boundedness for a class of second-order nonlinear differential equations

In this paper we obtain new conditions for the global existence and boundedness of solutions for nonlinear second-order equations of the form ${\left(r\left(t\right){\left|u^{\prime }\right|}^{p?2}{u}^{\prime }\right)}^{\prime }+g(t,u,u^{\prime })u^{\prime }+a\left(t\right)f\left(u\right)=e\left(t\right)\text{,}$ where $p>1$ is a real constant. The results are applicable to well-known Emden–Fowler and Lienard type equations. An illustrative example is also provided.     

Positive periodic solutions of second-order differential equations

In this work, we have obtained existence results for single and multiple positive periodic solutions for a class of second-order differential equations. Our results are based on a fixed point theorem for cones.     

Multiple positive solutions of second-order differential equations

In this paper, by using the coincidence degree method, we obtain some sufficient conditions for the existence of multiple positive periodic solutions of a class of second-order differential equations under the existence of upper and lower solutions.     

Monotone solutions of second-order nonlinear differential equations

Necessary and sufficient conditions are obtained for existence of monotone solutions of a nonlinear differential equation. As applications, several existence criteria and comparison theorems are derived.     

Exponential boundedness of solutions for impulsive delay differential equations

For an impulsive delay differential equation

exponent estimates of solutions have been obtained.     

Positive solutions for second-order nonlinear differential equations

We prove the existence of positive solutions to the scalar equation y^″(x)+F(x,y,y^′)=0$y^{″}(x)+F(x,y,y^{\prime })=0$. Applications to semilinear elliptic equations in exterior domains are considered.     

Asymptotic representations of solutions of second-order differential equations

We establish asymptotic representations as t → ω (ω ≤ + ∞) of a class of monotone solutions of the second-order differential equation y″ = f(t, y, y′), where f:[a,ω[× Δ _Y0 × Δ _Y1 is a continuous function asymptotically close on the considered class of solutions to a function of the form ±p(t)φ ₀(y)φ ₁(y′) with functions φ ₀ and φ ₁ regularly varying as y → Y ₀ and y′ → Y ₁. Here Δ _Yi , i ∈ {0, 1}, is a one-sided neighborhood of Y _i , and Y _i is either zero or ±∞.     

On periodic solutions of second-order differential equations with attractive-repulsive singularities

Sufficient conditions for the existence of a solution to the problem

     

On the boundedness of solutions of third order differential equations

Summary This research investigates the boundedness, as t→∞, of solutions of the third order order differential equation 2.1(1). Special attention is drawn to a method in current usage which rests on the existence of appropriate Yoshizawa function with which ultimate boundedness is proved for all solutions of 2.1(1) under quite mild restraints on f, g, h and p. Entrata in Redazione il 23 maggio 1973.     

Bounded solutions of second-order systems of differential equations

Sufficient conditions for the existence, uniqueness and stability of the bounded (in ) solution of the second-order systems of nonlinear differential equations are given.     

Stability and boundedness of solutions of neutral functional differential equations with finite delay

Sufficient and necessary criteria are established for the uniform stability and uniformly asymptotic stability of solutions of neutral functional differential equations (NFDEs) with finite delay by using the Liapunov functional approach. We also prove that the uniformly asymptotic stability of solutions implies the existence of bounded solution.     

Sufficient conditions for boundedness of solutions of linear differential equations

In this paper we obtain sufficient conditions for boundedness of every solution and its derivatives of a linear differential equation. We will also show that our conditions are sharp.     

Monotone positive solutions of second-order nonlinear differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order ordinary differential equations by using the Schauder–Tikhonov fixed point theorem. The result can also be applied to prove the existence of positive solutions of certain semilinear elliptic equations in $\text{R}{}^{\mathrm{n}}\text{(n⩾3)}$.     

On the boundedness of solutions of third-order delay differential equations

We establish certain sufficient conditions guaranteeing the boundedness of all solutions of some third-order delay differential equations.