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

Abstract

We study the global existence and uniqueness of solutions and wellposedness of a general class of abstract second order differential equations with state dependent delay. Some examples are presented.

Abstract random differential equations with state-dependent delay using measures of noncompactness

“…In [20], Hernandez studied the global existence and uniqueness of solutions and well posedness of the following general class of abstract second-order differential equations with state dependent delay:…”

“…Specifically, our investigation focuses on second-order differential equations involving a random variable, as opposed to considering first-order differential equations in [13]. Furthermore, our research builds upon the work done in [20] by exploring differential equations on an unbounded domain in conjunction with random variables.…”

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

“…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].…”

“…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].…”