On a Class of Functional Differential Equations with Symmetries

Abstract

It is shown that a class of symmetric solutions of scalar non-linear functional differential equations can be investigated by using the theory of boundary value problems. We reduce the question to a two-point boundary value problem on a bounded interval and present several conditions ensuring the existence of a unique symmetric solution.

Solvability of the symmetric nonlinear functional differential equations

“…which leads to the interval [2,4] where both ends are fixed points of 𝜓 (it is easy to verify that 2 is attractive for 𝜓 and repulsive for 𝜓 −1 , whereas 4 is attractive for 𝜓 −1 and repulsive for 𝜓). This can be justified by computing the values of 𝜓 ′ at respective points.…”

D-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations

“…Notation and definitions can be found in Sections 3 and 5, respectively. In Section 4, we summarize the conditions from [11,12] under which the symmetric functional differential Equation (1) setting on R can be studied using boundary value theory. In Section 6.1, we establish the conditions for a unique solution for the initial value linear symmetric scalar problem, where we also find conditions under which the unique solution is symmetrical (see Section 6.2).…”

“…There are many publications that study this problem (herein, we only cite some of them: see, for example, [5][6][7][8][9][10] and the references therein). Using the results from [5,6,[11][12][13][14], we investigate the D-stability of the symmetric solutions for functional differential equations. Here, the general symmetry of the solution (see (3)) is studied, which is a generalization of the periodicity of solutions.…”

“…Here, the general symmetry of the solution (see (3)) is studied, which is a generalization of the periodicity of solutions. A similar problem is studied, for example, in [11,[15][16][17].…”

“…Our investigation of functional differential equations with symmetric properties starts in [11,12]. They present the applicability of the theory of boundary value problems in the investigation of functional differential equations with symmetric properties on R. Then, the unique solvability conditions of linear functional differential equations were established in [13,14,20], which represent the unique solution in view of the functional series.…”