Asymptotic equivalence of differential equations and asymptotically almost periodic solutions

In this paper we use Rab’s lemma [M. Ráb, Über lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222–229; M. Ráb, Note sur les formules asymptotiques pour les solutions d’un systéme d’équations différentielles linéaires, Czechoslovak Math. J. 91 (1966) 127–129] to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations. Yakubovich’s result [V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1966; V.A. Yakubovich, On the asymptotic behavior of systems of differential equations, Mat. Sb. 28 (1951) 217–240] on the asymptotic equivalence of a linear and a quasilinear system is developed. On the basis of the equivalence, the existence of asymptotically almost periodic solutions of the systems is investigated. The definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained. Appropriate examples are constructed.     

Limit behavior of solutions of ordinary linear differential equations

A classification of classes of equivalent linear differential equations with respect to -limit sets of their canonical representations is introduced. Some consequences of this classification with respect to the oscillatory behavior of solution spaces are presented.     

Periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

In this paper, we study the existence of periodic solutions of the Rayleigh equations

x^″+f(x^′)+g(x)=e(t).     

Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument

The aim of this paper is to establish comparison principles on property A$A$, between a nonlinear differential equation of the third order with deviating argument (with delay, advanced or mixed argument) and the corresponding linear equation without deviating argument. On the basis of these comparison principles the sufficient conditions for delay, advanced and mixed equations to have property A$A$ are presented. The results obtained are compared with existing ones in the framework of the papers.     

Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r₂(t)((r₁(t)x^Δ(t))^Δ)^γ)^Δ+q(t)f(x(t))=0     

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.     

On monotonic solutions of systems of nonlinear second order differential equations

Bounded, monotonic, and asymptotic properties of solutions for a class of second order nonlinear differential equations are discussed. Some necessary and sufficient conditions for boundedness of all solutions are established. Moreover, all solutions are classified into four disjoint subsets which are characterized in terms of the convergence and divergence of several integrals. The obtained results extend and improve many known results of various authors.     

Lie point symmetries for systems of second order linear ordinary differential equations

Abstract

The equations for the Lie point symmetries of autonomous systems of second order linear ordinary differential equations are derived. The results for a two dimensional system are treated in detail and some consideration is extended to higher dimensional systems. The effect of the introduction of time-dependent elements into the coefficient matrix is briefly discussed.     

Qualitative behavior of global solutions to some nonlinear fourth order differential equations

We study global solutions to a fourth order semilinear ordinary differential equation. We determine sufficient conditions on the nonlinearity that ensure global continuation of the solutions. Furthermore, we discuss their qualitative behaviors such as oscillations and boundedness.     

On then-th order iterative ordinary linear differential equations

Summary It is shown that a change of the basis in the solution space of a second order linear differential equation induces a covariant change in the solution space of the corresponding iterative equation. Also studied is the problem to what extent a solution of an iterative equation determines the equation.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Fundamental systems for linear differential equations smoothly depending on a parameter

Homogeneous linear differential equations in Banach spaces are considered, which depend smoothly on a parameter. A fundamental system of solutions is generated by eigenvalues and eigenvectors of the corresponding operator pencil. The dependence on the parameter of these solutions in canonical form is, in general, not smooth because of branching eigenvalues. This means, the eigenvalues, their number and multiplicities may change in a nonsmooth way with respect to the parameter. We construct a new fundamental system depending smoothly on the parameter, wich can be represented by linear combinations of the solutions in canonical form.     

Basic theory of fractional differential equations

In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.     

Oscillation of second-order half-linear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish a few new oscillation criterions for certain second-order half-linear delay dynamic equations with damping on a time scale. Our results extend and improve some known results, but also unify the oscillation of the second-order half-linear delay differential equation with damping and the second-order half-linear delay difference equation with damping.     

The differential equations on time scales through impulsive differential equations

In this paper we investigate differential equations on certain time scales with transition conditions (DETC) on the basis of reduction to the impulsive differential equations (IDE). DETC are in some sense more general than dynamic equations on time scales [M. Bohner, A. Peterson, Dynamic equations on time scales, in: An Introduction With Applications, Birkhäuser Boston, Inc., Boston, MA, 2001, p. x+358; V. Laksmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamical Systems on Measure Chains, in: Math. and its Appl., vol. 370, Kluwer Academic, Dordrecht, 1996]. The basic properties of linear systems, the existence and stability of periodic solutions, and almost periodic solutions are considered. Appropriate examples are given to illustrate the theory.     

Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative

In this paper, we shall discuss the properties of the well-known Mittag-Leffler function, and consider the existence and uniqueness of the solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative by using the monotone iterative method.     

On the periodic boundary-value problem for systems of second-order nonlinear ordinary differential equations

The periodic boundary value problem for systems of secondorder ordinary nonlinear differential equations is considered. Sufficient conditions for the existence and uniqueness of a solution are established.     

On the equivalence of parametric contexts for linear inequality systems

In this paper we analyze the connections among different parametric settings in which the stability theory for linear inequality systems may be developed. Our discussion is focussed on the existence, or not, of an index set (possibly infinite). For some stability approaches it is not convenient to have a fixed set indexing the constraints. This is the case, for example, of discretization techniques viewed as approximation strategies (i.e., discretization regarded as data perturbation). The absence of a fixed index set is also a key point in the stability analysis of parametrized convex systems via standard linearization. In other frameworks the index set is very useful, for example if the constraints are perturbed one by one, even to measure the global perturbation size. This paper shows to what extent an index set may be introduced or removed in relation to stability.     

Analytic normalization of analytically integrable differential systems near a periodic orbit

For an analytic differential system in Rⁿ${\mathbb{R}}^n$ with a periodic orbit, we will prove that if the system is analytically integrable around the periodic orbit, i.e. it has n−1$n-1$ functionally independent analytic first integrals defined in a neighborhood of the periodic orbit, then the system is analytically equivalent to its Poincaré–Dulac type normal form. This result is an extension of analytically integrable differential systems around a singularity to the ones around a periodic orbit.