Existence and Uniqueness of Solutions for Abstract Neutral Differential Equations with State-Dependent Delay

Abstract neutral differential equations with state‐dependent delay and almost sectorial operators

“…Evolution equation () with linear operator $$$ A $$$, generating a strongly continuous or analytic semigroup, has received a lot of attention. We refer to the works of Hernández in [1–3] for $$$ g=0 $$$ and [4] for $$$ g\ne 0 $$$. Morales et al [5] proved the existence of mild solution for () with infinite delay and $$$ {\sigma}_1\left(t,{u}_t\right)=t $$$ with $$$ A $$$ generating a compact analytic semigroup.…”

“…For SDD differential equations in finite dimension, we refer to [9, 11–14]. For abstract evolution equations, we refer to [1, 3–5, 15, 16]. In [17–20], the authors used the theory of measure of noncompactness and fixed point theorems to discuss the existence of mild solutions.…”

“…The most crucial part of the analysis done in the recent papers [1, 3, 4, 29, 30] is that Hernández et al showed that SDD problems are not well posed in the space of continuous functions as the function $$$ u\to f\left(t,{u}_{\sigma \left(t,{u}_t\right)}\right) $$$ does not show the Lipschitz behavior. Further, they established inequality of the form $$$$ {\displaystyle \begin{array}{ll}& {\left\Vert f\left(t,{u}_{\sigma \left(t,{u}_t\right)}\right)-f\Big(t,{v}_{\sigma \left(t,{v}_t\right)}\Big)\right\Vert}_{C\left(\left[0,\tau \right],X\right)}\\ {}\le & \kern0.2em {L}_f\left(1+{\left[v\right]}_{C_{\mathrm{Lip}}\left(\left[-r,\tau \right],X\right)}{\left[\sigma \right]}_{C_{\mathrm{Lip}}\left(\left[0,\tau \right]\times {\mathfrak{B}}_X,{\mathrm{\mathbb{R}}}^{+}\right)}\right){\left\Vert u-v\right\Vert}_{C\left(\left[-r,\tau \right],X\right)},\end{array}} $$$$ when all the functions are considered to be Lipschitz.…”

“…Further, they established inequality of the form $$$$ {\displaystyle \begin{array}{ll}& {\left\Vert f\left(t,{u}_{\sigma \left(t,{u}_t\right)}\right)-f\Big(t,{v}_{\sigma \left(t,{v}_t\right)}\Big)\right\Vert}_{C\left(\left[0,\tau \right],X\right)}\\ {}\le & \kern0.2em {L}_f\left(1+{\left[v\right]}_{C_{\mathrm{Lip}}\left(\left[-r,\tau \right],X\right)}{\left[\sigma \right]}_{C_{\mathrm{Lip}}\left(\left[0,\tau \right]\times {\mathfrak{B}}_X,{\mathrm{\mathbb{R}}}^{+}\right)}\right){\left\Vert u-v\right\Vert}_{C\left(\left[-r,\tau \right],X\right)},\end{array}} $$$$ when all the functions are considered to be Lipschitz. The existence of local as well as global mild and strict solutions, their attractivity, and the well‐posedness of strict solutions in Lipschitz and Hölder spaces have been established using similar approaches as in [1, 4, 31, 32].…”

A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay

“…For this reason, research on different properties related to this class of equations has gained great interest in recent years. The literature related to this research topic is mainly dedicated to the study of functional differential systems with states in spaces of finite dimension or first-order autonomous systems (systems determined by a constant unbounded operator) with states in Banach spaces (the reader can review [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] for recent advances in this matter). Nevertheless, the study of non-autonomous abstract integro-differential systems of second-order with state-dependent delay using the properties of the resolvent operator associated with the homogeneous equation, as we will carry out in this work, does not seem to have been addressed yet.…”