Well-posedness of abstract integro-differential equations with state-dependent delay

Local and global existence and uniqueness of solution for abstract differential equations with state-dependent argument

“…For the case of ordinary differential equations with SDA similar to Equations (1.1)–(1.2), we cite the early papers by Cooke [5], Dunkel [9], Eder [10] and Oberg [36]. Concerning abstract problems with applications to Partial differential equations (PDEs), we cite the pioneer papers [13, 24], our recent works [21, 22, 23–26] and the interesting papers [28–30, 34, 35].…”

“…In comparison to the early works [13, 24] and the papers [1–4, 13, 16–18, 21, 22, 23–26, 31, 32], we present several novelties. To begin, we prove the existence and ‘uniqueness of a non-Lipschitz’ solution for Equations (1.1)–(1.2).…”

“…A similar regularizing property is used by Rezounenko et al in [30,38] to study the existence and 'uniqueness' of a 'Lipschitz' solution for abstract problems with state-dependent delay. Assuming, basically, that F (•) is a Lipschitz function from [0, a] × X into X, in [20][21][22][23][24][25][26], we also study the existence and uniqueness of a 'Lipschitz' solution for some different models of state-dependent delay differential equations.…”

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

“…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.…”

Existence and uniqueness of global solution for abstract second order differential equations with state‐dependent delay

“…Concerning first order ODEs on finite dimensional spaces, we mention the survey [5]. For first order ordinary abstract differential equations and first order on time partial differential equations, we cite the early paper by Hernández et al [11] and the recent interesting works [12][13][14][15][17][18][19][20].…”