The Interplay Between Linear Functional Equations and Inequalities

In this paper, we give a characterization of spectral abscissa of positive linear functional differential equations. Then the obtained result is applied to derive necessary and sufficient conditions for the exponential stability of positive linear functional differential equations. Finally, we give an extension of the classical Perron–Frobenius theorem to positive linear functional differential equations.

We are dealing with those continuous solutions $ \varphi $ of the functional equation¶¶ $ \varphi\circ f=g\cdot \varphi+h $ ¶that are asymptotically comparable at the origin (the fixed point of f) with the function h. Connections with a linear iterative functional inequality of second order are also mentioned.

Linear functional equations on an arbitrary piecewise smooth curve are considered. Such equations are studied in connection with the theory of singular integral equations, which are a mathematical tool in the study of mathematical models of elasticity theory in which the conjugation conditions contain a boundary shift. Such equations also arise in the mathematical modeling of the transfer of charged particles and ionized radiation. The shift function is assumed to act cyclically on a set of simple curves forming a given curve, with only the ends of simple curves being periodic points. The aim of the work is to find the conditions for the existence and cardinality of a set of continuous solutions to such equations in the classes of Helder functions and primitive Lebesgue integrable functions with coefficients and the right side of the equation from the same classes. The solutions obtained have the form of convergent series and can be calculated with any degree of accuracy. The method of operation consists in reducing this equation to an equation of a special type in which all periodic points are fixed, which allows you to use the results for the case of a simple smooth curve.

The present paper deals with special solutions for linear functional differential equations with small delays. The importance of special solutions relies on the fact that, under some conditions, they can be used to describe the asymptotic behaviour of all solutions. Sufficient conditions for the existence of special solutions for nonautonomous equations in $\mathbb R^n$ are given. For autonomous equations in $\mathbb R^n$, results relating special solutions and characteristic values are presented. We also consider linear autonomous functional differential equations in Banach spaces, and use special solutions to study the asymptotic behaviour of their solutions. Several applications are given.

A Stability Criterion for Linear Autonomous Functional Differential Equations

Some sufficient conditions for exponential stability of linear neutral functional differential equations

In this paper we consider periodic functional differential and integral equations. Massera proved that the existence of a bounded solution implies the existence of a periodic solution for linear (inhomogeneous) ordinary differential equations. Chow proved a similar result for linear functional differential equations with finite delay. We give a new proof for Chow's result that works for linear functional differential equations with infinite delay and also for integral equations. We also give examples that do not satisfy uniform boundedness or uniform ultimate boundedness, the conditions frequently used to prove that there is a periodic solution.

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.

In this paper we consider the well-posedness of linear functional differential equations on product spaces. Let L and D be linear $\mathbb{R}^n $-valued functions with domains $\mathfrak{D}(L)$ and $\mathfrak{D}(D)$ subspaces of the Lebesgue measurable $\mathbb{R}^n $-valued functions on $[ - r,0]$ and such that $W^{1,p} ([ - r,0];\mathbb{R}^n ) \subseteq \mathfrak{D}(L) \cap \mathfrak{D}(D)$. Under weak conditions on D and L we establish the equivalence between generalized solutions to the functional differential equation \[ \frac{d}{{dt}}Dx_t = Lx_t + f(t)\] and mild solutions to the Cauchy problem in $\mathbb{R}^n \times L_p ([ - r,0];\mathbb{R}^n )$\[ \dot z(t) = \mathfrak{a} z(t) + (f(t),0), \] where $\mathfrak{a}$ is the operator defined on \[ \mathfrak{D}(\mathfrak{a}) = \left\{ {(\eta ,\varphi ) \in \mathbb{R}^n \times L_p {{\left( {[ - r,0];\mathbb{R}^n } \right)} / \varphi } \in W^{1,p} \left( {[ - r,0];\mathbb{R}^n } \right),D\varphi = \eta } \right\},\] by \[ \mathfrak{a}(\eta ,\varphi ) = (L\varphi ,\dot \varphi ). \] The results are applicable to neutral functional differential equations and certain singular integral equations.

General linear non-autonomous functional differential equations are considered. Explicit criteria for exponential stability are given. Furthermore, the authors present an explicit robust stability bound for systems subject to time-varying perturbations. Two examples are given to illustrate the obtained results.

Stability and asymptotic behavior of solutions for some linear partial functional differential equations in critical cases

An identification problem is studied for linear retarded functional differential equations (FDEs) in Banach spaces. The unique identification of solution states and scalar forcing functions from the given data is proved.

Nonoscillations for odd-dimensional systems of linear retarded functional differential equations

The problem on solutions with specified growth for linear functional differential equations with negative coefficients is treated by using two-sided monotone iterations. New theorems on the existence and localisation of such solutions are established.

The sufficient conditions of well-posedness of the weighted Cauchy problem for higher order linear functional differential equations with deviating arguments, whose coefficients have nonintegrable singularities at the initial point, are found.