Related papers: Parabolic partial differential equations with disc…

Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of…

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} $X\subset C^1([-h,0],\mathbb{R}^n)$. For systems with discrete…

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on the associated {\it solution manifold} in the Banach space $C^1_n=C^1([-h,0],\mathbb{R}^n)$. For a…

Differential equations with state-dependent delays define a semiflow of continuously differentiable solution operators in general only on an associated submanifold of the Banach space $C^1([-h,0],\mathbb{R}^n)$. We extend a recent result on…

Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional…

For a differential equation with a state-dependent delay we show that the associated solution manifold $X_f$ of codimnsion 1 in the space $C^1([-r,0],\mathbb {R})$ is an almost graph over a hyperplane, which implies that $X_f$ is…

A wide class of non-autonomous nonlinear parabolic partial differential equations with delay is studied. We allow in our investigations different types of delays such as constant, time-dependent, state-dependent (both discrete and…

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent…

We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…

We show that for a system $$ x'(t)=g(x(t-d_1(Lx_t)),\dots,x(t-d_k(Lx_t))) $$ of $n$ differential equations with $k$ discrete state-dependent delays the solution manifold, on which solution operators are differentiable, is nearly as simple…

We analyze a differential equation with a state-dependent delay that is implicitly defined via the solution of an ODE. The equation describes an established though little analyzed cell population model. Based on theoretical results of…

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one…

This work is the first attempt to treat partial differential equations with discrete (concentrated) state-dependent delay. The main idea is to approximate the discrete delay term by a sequence of distributed delay terms (all with…

We discuss the non-uniqueness of continuous solutions to differential equations with a {\it discrete } state-dependent delay and continuous initial functions. We are interested not only in the fact (conditions) of non-uniqueness, but in…

The paper concerns classical solution of path-dependent partial differential equations (PPDEs) with coefficients depending on both variables of path and path-valued measure, which are crucial to understanding large-scale mean-field…

We deal with a class of second order in time nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in the theory ofnonlinear plates. Our first result states well-posedness in a…

We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that,…

We establish variants of existing results on existence, uniqueness and continuous dependence for a class of delay differential equations (DDE). We apply these to continue the analysis of a differential equation from cell biology with…

A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss…

Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant…