Hypertranscendency of meromorphic solutions of a linear functional equations

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.

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.

In this paper, we obtain exact non-exponential rates of growth and decay of solutions of linear functional differential equations with unbounded delay. As a by-product, exponential asymptotic stability is ruled out for asymptotically stable solutions of these equations. Equations with maximum functionals on the right hand side, as well as perturbed equations, are considered. We also give examples showing how the rate of growth or decay of solutions depends on the rate of growth of the unbounded delay. The results established here are obtained by constructing carefully functions which satisfy upper and lower differential inequalities and one aim of the paper is to elucidate this constructive comparison technique.

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.

In this note we establish a criterion to obtain classical solutions for a quasi-linear abstract retarded functional differential equation with infinite delay and we apply this result to characterize the infinitesimal generators of several strongly continuous semigroup of linear operators that arise in the theory of linear abstract retarded functional differential equations with infinite delay on a phase space defined axiomatically.

This paper presents a method of establishing the D-stability terms of the symmetric solution of scalar symmetric linear and nonlinear functional differential equations. We determine the general conditions of the unique solvability of the initial value problem for symmetric functional differential equations. Here, we show the conditions of the symmetric property of the unique solution of symmetric functional differential equations. Furthermore, in this paper, an illustration of a particular symmetric equation is presented. In this example, all theoretical investigations referred to earlier are demonstrated. In addition, we graphically demonstrate two possible linear functions with the required symmetry properties.

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.

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Periodic solutions of first-order nonlinear functional differential equations

A Stability Criterion for Linear Autonomous Functional Differential Equations

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

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

In this paper, a collocation method based on the Dejdumrong polynomial matrix approach was used to estimate the solution of higher-order pantograph-type linear functional differential equations. The equations are considered with hybrid proportional and variable delays. The proposed method transforms the functionaltype differential equations into matrix form. The matrices were converted into a system of algebraic equations containing the Dejdumrong polynomial. The coefficients of the Dejdumrong polynomial were obtained by solving the system of algebraic equations. Moreover, the error analysis is performed, and the residual improvement technique is presented. The presented methods are applied to three examples. Finally, the obtained results are compared with the results of other methods in the literature and were found to be better compared. All results in this study have been calculated using Matlab R2021a.

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.