C1+-strict solutions and wellposedness of abstract differential equations with state dependent delay

-Asymptotically Periodic Solutions for Abstract Equations With State-Dependent Delay

“…This paper continues our study [5] of abstract differential equations with statedependent delay. Specifically, we study the existence and uniqueness of Sasymptotically ω-periodic solutions for abstract problems of the form u (t) = Au(t) + F(t, u σ(t,u t ) ), t ≥ 0, (1.1) u 0 = ϕ with ϕ ∈ B X = C([−p, 0]; X)), (1.2) where A : D(A) ⊂ X → X is the generator of an analytic semigroup of bounded linear operators (T (t)) t≥0 defined on a Banach space (X, • ) and F, σ are functions to be specified later.…”

“…The theory of differential equations with state-dependent delay is a field of intense research because of its many applications and the fact that the qualitative theory is different from those for equations with discrete and time-dependent delays. For differential equations on finite-dimensional spaces, we cite the survey by Hartung et al [3], and for related differential equations on abstract Banach spaces, the recent papers [4,5,[7][8][9]12]. For global solutions of problems on unbounded intervals, see [6,11,13] for equations on finite-dimensional spaces, [1,10,20] for abstract and partial differential equations and [1,10] for problems with almost periodic solutions.…”

Wellposedness of abstract differential equations with state‐dependent delay

“…, Walther and the references in these works. Concerning first order ordinary abstract differential equations and partial differential equations with state‐dependent delay, we refer the reader to the early paper by Hernandez, Prokopczyk & Ladeira , the works by Rezounenko , Rezounenko & Wu , Kosovalic, Magpantay, Chen & Wu and some recent interesting papers by Krisztin & Rezounenko , Yunfei, Yuan & Pei , Kosovalic, Chen & Wu and Hernandez, Pierri & Wu .…”

“…We note that in most of the works involving PDE (with exception of ), the uniqueness of solution (which is a relevant topic with important implications) is not discussed. This fact is not surprising in the field of state‐dependent equations, since functions of the form are not (in general) Lipschitz on spaces of continuous functions, see for additional comments.…”

“…By using some ideas in our recent work , we study the existence and uniqueness of solutions, and the “wellposedness” of the problem –. In Theorem we prove the existence and uniqueness of a mild solution in .…”