Existence of solutions for a second order abstract functional differential equation with state-dependent delay.

*(English)*Zbl 1113.47061From the Introduction: Functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received great attention in the last years. The literature devoted to this subject is concerned fundamentally with first order functional differential equations for which the state belongs to some finite-dimensional space. The problem of the existence of solutions for first order partial functional differential equations with state-dependent delay has been treated in the literature recently. To the best of our knowledge, the existence of solutions for second order abstract partial functional differential equations with state-dependent delay is an untreated topic in the literature and this fact is the main motivation of the present work.

In this note, we study the existence of mild solutions for a second order abstract Cauchy problem with state dependent delay described in the form

\[ \begin{gathered} x''(t)=Ax(t)+f(t,x_{\rho (t,x_{t})}),\quad t\in I=[0, a],\;x_0=\varphi\in\mathcal{B},\tag{1}\\ x'(0)=\zeta_{0}\in X,\tag{2}\end{gathered} \]

where \(A\) is the infinitesimal generator of a strongly continuous cosine function of the bounded linear operator \((C(t))_{t\in \mathbb{R}}\) defined on a Banach space \((X,\|\cdot\|)\); the function \(x_{s}:(-\infty,0]\to X\), \(x_{s}(\theta)=x(s+\theta)\), belongs to some abstract phase space \(\mathcal{B}\) described axiomatically and \(f:I\times\mathcal{B}\to X\), \(\rho:I\times\mathcal{B}\to (-\infty,a]\) are appropriate functions.

In this note, we study the existence of mild solutions for a second order abstract Cauchy problem with state dependent delay described in the form

\[ \begin{gathered} x''(t)=Ax(t)+f(t,x_{\rho (t,x_{t})}),\quad t\in I=[0, a],\;x_0=\varphi\in\mathcal{B},\tag{1}\\ x'(0)=\zeta_{0}\in X,\tag{2}\end{gathered} \]

where \(A\) is the infinitesimal generator of a strongly continuous cosine function of the bounded linear operator \((C(t))_{t\in \mathbb{R}}\) defined on a Banach space \((X,\|\cdot\|)\); the function \(x_{s}:(-\infty,0]\to X\), \(x_{s}(\theta)=x(s+\theta)\), belongs to some abstract phase space \(\mathcal{B}\) described axiomatically and \(f:I\times\mathcal{B}\to X\), \(\rho:I\times\mathcal{B}\to (-\infty,a]\) are appropriate functions.