Hyers-Ulam stability of integral equations with infinite delay (Q6614065)

A differential equation or a difference equation is said to be Hyers-Ulam stable, if in a neighborhood of an approximate solution, a true solution is found. In this paper, the authors study this problem for a general class of integral equations with infinite delay of the form\N\[\Nx(t)=\int_{-\infty}^{t}K(t-s)x(s)ds,\qquad t\geq 0,\tag{1}\N\]\Nwhere \(K:\mathbb R^{+}\rightarrow \mathbb C^{m\times m}\) is a measurable matrix-valued function satisfying the following assumptions:\N\N(i) \(\| K\|_{1,\rho}:= \int_{0}^{\infty}| K(t)| e^{\rho t}dt <\infty\);\N\N(ii) \(\| K\|_{\infty,\rho}:= \mathrm{esssup} \{| K(t)| e^{\rho t}: t\geq 0\} <\infty\), where \(\rho > 0\) is fixed. \N\NThe phase space for (1) is \(X:= L_{1,\rho} (\mathbb R^{-},\mathbb C^{m})\), the space of equivalent classes of measurable functions \(\phi:\mathbb R^{-} \rightarrow \mathbb C^{m}\) such that \(\|\phi\| _{X}:=\int_{-\infty}^{0}| \phi(\theta)| e^{\rho\theta}d\theta<\infty\). Clearly, \((X,\| \cdot \|_{X} )\) is a Banach space. The authors also consider nonlinear perturbations of (1) of the form\N\[\Nx(t)=\int_{-\infty}^{t} K(t-s)x(s)ds + f(t,x_{t}),\qquad t\geq 0,\tag{2}\N\]\Nwhere \(f:\mathbb R^{+} \times X \rightarrow \mathbb C^{m}\) is a continuous function satisfying the global Lipschitz condition \(| f(t,\phi)-f(t,\psi)| \leq \gamma \| \phi -\psi \|_{X}\) for all \(t\geq 0\) and \(\phi, \psi \in X\) where \(\gamma\geq 0\). Here, for any function \(x:\mathbb R\rightarrow \mathbb C^{m}\) and \(t\in \mathbb R \), the \(t\)-segment \(x_{t}:\mathbb R^{-}\rightarrow \mathbb C^{m}\) is defined by \(x_{t}(\theta)=x(t+\theta)\), for \(\theta\in \mathbb R^{-}\).\N\NIn this paper, the authors establish two types of stability criteria:\N\N(i) The Hyers-Ulam stability of the linear homogeneous equation (1) can be completely characterized in terms of its characteristic values;\N\N(ii) The Hyers-Ulam stability of the linear equation (1) is preserved for for the nonlinear equation(2) whenever \(\gamma\) is sufficiently small.\N\NNote that the characteristic values of (1) are the complex roots of the characteristic equation \[\det\Delta (z)=0 ,\qquad \Delta (z):= I_{m}-\int_{0}^{\infty}K(s)e^{-zs}ds, \qquad \mathrm{Re} z >-\rho,\] \Nwhere \(I_{m}\) is the \(m\times m\) identity matrix.