Further results on Ulam stability for a system of first-order nonsingular delay differential equations

Abstract This paper is concerned with a system governed by nonsingular delay differential equations. We study the β-Ulam-type stability of the mentioned system. The investigations are carried out over compact and unbounded intervals. Before proceeding to the main results, we convert the system into an equivalent integral equation and then establish an existence theorem for the addressed system. To justify the application of the reported results, an example along with graphical representation is illustrated at the end of the paper.


Introduction
Delay systems are used to characterize the evolution processes in automatic engines, physiological systems and control theory. Shuklin and Khusainov [1] introduced a notion of delayed matrix exponential and used it to derive a representation of solutions to linear delay problems under the restriction of permutable matrices. Khusainov and Diblik [2] and Wang et al. [3] used the ideas of [1] to introduce a discrete matrix delayed exponential function and to consider a representation of solutions to linear delay systems.
Among the qualitative properties of differential systems, stability is an essential property. There are different types of stabilities, but recently researchers focused on the Ulam-Hyers-type stability. The idea of the aforesaid stability was initially introduced by Ulam [4], in 1940, when he addressed a mathematical colloquium. During his talk, he raised a problem regarding the stability of group homomorphisms. In the following year, Hyers [5] responded positively to this problem under the assumption that groups are Banach spaces. Since then, this stability was named as Ulam-Hyers stability. Rassias [6], in 1978, made an extension to the result of Hyer's theorem, where the bound of norm of Cauchy difference was presented in a more general form. This stability phenomenon is termed as Hyers-Ulam-Rassias stability. For more information about the topic, we refer the reader to [7][8][9][10][11][12][13][14].
In 2019, You et al. [3] studied the exponential stability and relative controllability of nonsingular delay differential equations of the form: where A, M and N are constant permutable matrices of dimension × n n, A is a nonsingular and Motivated from [3], and using the techniques of [15], we analyze the β-Hyers-Ulam-Rassias stability [16] of solutions for the nonsingular delay differential system (1.1). We carry out our investigations in two folds: stability results over a compact interval and stability results over an unbounded interval. Before proceeding to the main results, we convert system (1.1) into an equivalent integral equation and establish an existence theorem for its solutions. To justify the application of the reported results, an example along with a graphical illustration is presented at the end of the paper.

Essential background
Here we present some basic concepts and definitions that are essential in proving the main results. Let represent the set of all real numbers, + represent the set of all nonnegative real numbers and n the space of all n-tuples of . The interval = [ ] ⊆ θ 0, and = n , ( ) , , the Banach space of all continuous functions from to with the norm , for all , . for all ∈ , 1 .
The nonsingular delay differential system , depending upon f φ β , , , such that for any > ε 0 and for any solution such that .
Let α, ϖ and U be the real valued functions defined on J. Assume that ϖ and U are continuous and that the negative part of α is integrable on every closed and bounded subinterval of J. (a) If ϖ is nonnegative and U satisfies the integral inequality

Existence result
To discuss existence result of the given system, we need some assumptions: satisfies the Caratheodory condition
( ) t is nonnegative. A 5 : Assume the negative part of ( ) η φ t ϵ φ is integrable on every closed and bounded subinterval of J and it is nondecreasing. Proof. The unique solution of the Cauchy problem   Thus, Since we know that , where , , 0 and 1.

β-Hyers-Ulam-Rassias stability on unbounded interval
Here, we study the β-Hyers-Ulam-Rassias stability on an unbounded interval. Consider some more assumptions: A 0 : The operator family { ( − ) ≥ ≥ } Z t s t s : 0 is exponentially stable, i.e., and there exists ∈ ( ) + , satisfying the Caratheodory condition for every ∈ + t and ′ ∈ ν ν , . Also, we assume that By considering inequality (2.2) and the aforementioned assumptions, we are in a position to state and prove our second result.
Proof. The unique solution of the semilinear nonautonomous differential system   Thus for each ∈ + t , we get that,

Conclusion
In the last few years and along with the explosion in studying differential equations, the notion of stability has gained extensive interest by many mathematicians. Following the trend, in this paper we discuss the β-Hyers-Ulam-Rassias stability of a nonsingular differential system over compact and unbounded intervals. Different types of conditions were established for the sake of proving the main results. An example with specific parameters and matrices and graphical representation demonstrate consistency to our theoretical findings.