Oscillatory and periodic solutions for systems of two first order linear differential equations with piecewise constant argument

Oscillatory, nonoscillatory, and periodic solutions of re-tarded functional differential equations are investigated. The study concerns a system of two first order linear equations with piecewise constant argument.     

Monotone method for first order functional differential equations with retardation and anticipation

This paper discusses first order functional differential equations with retardation and anticipation. By using the method of upper and lower solutions coupled with monotone iterative technique, new existence results are obtained which improve known results in the literature.     

Boundary value problems for first order differential equations of mixed type

We apply the monotone iterative method to formulate existence results for integro-differential equations with nonlinear boundary conditions. Some results for integro-differential inequalities are also given. Examples illustrate obtained results.     

On the distribution of zeros of solutions of first order delay differential equations

This paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation

x^′(t)+p(t)x(t−τ)=0     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

On the solvability of perturbations of linear boundary value problems at resonance for functional differential equations

Necessary and sufficient conditions of the unique solvability of perturbations of linear boundary value problems at resonance for functional differential equations are obtained. These conditions are effective and expressed in terms of norms of the positive and negative parts of regular operators.     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.     

Analytical and numerical investigation of mixed-type functional differential equations

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x(t), defined for t∈[−1,k], (k∈N), that satisfies this equation almost everywhere on [0,k−1] and assumes specified values on the intervals [−1,0] and (k−1,k]. We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

Oscillatory criteria for nonlinearnth-order differential equations with quasiderivatives

Sufficient conditions are given for the existence of oscillatory proper solutions of a differential equation with quasiderivativesL _n y=f(t,L ₀ y, ..., L _n–1 y) under the validity of the sign conditionf(t,x ₁ ,...,x _n )x ₁0,f(t,0,x ₂ ,...,x _n )=0 on ₊ x ⁿ .     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments

The state-dependent delay differential equation

     

On stability of linear periodic differential difference equations

The paper considers the equation

where the operator-valued bounded functions a_j and b_j are 2π-periodic, and the operator-valued kernels m and n are 2π-periodic with respect to the first argument. The connection between the input-output stability of the equation and the invertibility of a family of operators acting on the space of periodic functions is investigated.     

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.     

The existence of multiple positive solutions for singular functional differential equations with sign-changing nonlinearity

In this paper, we study the existence of multiple positive solutions for boundary value problems based on second-order functional differential equations with the form

     

First- and second-order discontinuous functional differential equations with impulses at fixed moments

We give sufficient conditions for the existence of extremal solutions to discontinuous and functional differential equations with impulses. Our main results are new even for ordinary differential equations without impulses.