The matrix functional-differential equation y′(x)=Ay(λx)+By(x)

The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps

“…, taking limits on both sides of (4) and letting n → ∞, we can show that x(t) is a solution of equation (3). On the other hand, similar to the proof of lemma 3.1, we can easily obtain that…”

“…Now define x(t) by x(t) = x n 0 (t) for t ∈ [t 0 , T ]. Next to verify that x(t) is a solution of equation (3). By (34), x(t ∧ τ n ) = x n (t ∧ τ n ), and by (27), it follows that…”

“…Equation (1) arises in different fields of pure and applied mathematics such as dynamical systems, probability, quantum mechanics and electrodynamics (see [1,2]) and its generalisations possess a wide range of applications. In [3,4], the authors studied the existence, uniqueness and asymptotic properties of solution to equation (1). Numerical analysis for solution to equation (1) are studied by [5,6].…”

Eigenfunctions Arising From a First-Order Functional Differential Equation in a Cell Growth Model

“…https://doi.org/10.1017/S1446181111000575 [8] Eigenfunctions arising from a first-order functional differential equation 53…”

“…Heard [21] studied the equation with the functional argument αx replaced by the argument x α , and the complex version of this modified equation was studied in [28]. Second-order versions of the pantograph equation were studied in [24,32,33], and a matrix version was considered by Carr and Dyson [8]. More recently, attention has turned to pantograph equations with several functional arguments (see [7,9]).…”