Variation formulas for solution of a nonlinear differential equation with time delay and mixed initial condition

Variation Formulas for the Solution of Delay Differential Equations with Mixed Initial Conditions and Delay Perturbations

“…If t 00 C 0 D t 10 ; then Theorem 1.1 is valid on the interval OEt 10 ; t 10 C ı 2 and Theorem 1.2 is valid on the interval OEt 10 ı 2 ; t 10 : Finally, we note that variation formulas for the solution of various classes of functional-differential equations without perturbations of delay can be found in [1][2][3][4][5][6][7][8].…”

Formulas of variation for solutions for some classes of functional differential equations and their applications

“…t 00−σ Y (ξ + σ ; t)B(ξ + σ )δg(ξ )dξ + β(t; δµ); the matrix functions Y (ξ ; t), Ψ (ξ ; t) satisfy the system Ψ ξ (ξ ; t) = −Y (ξ ; t)F x [ξ ] − (Y (ξ + τ ; t)F y 1 [ξ + τ ] Y (ξ + σ ; t)F z 1 [ξ + σ ]), Y (ξ ; t) = Ψ (ξ ; t) + (Y (ξ + τ ; t)A(ξ + τ ) Y (ξ + σ ; t)B(ξ + σ )), ξ ∈ [t 00 , t], Y (t; t) = Ψ (t; t) = I n×n ; Y (ξ ; t) = Ψ (ξ ; t) = Θ n×n , ξ > t. Theorems 1.1 and 1.2 are proved by the method described in [1,2].…”

Variation formulas of solution for a functional differential equation with delay function perturbation

“…Linear representation of the main part of the increment of a solution of an equation with respect to perturbations is called the variation formula of solution (variation formula).The variation formula allows one to construct an approximate solution of the perturbed equation in an analytical form on the one hand, and in the theory of optimal control plays the basic role in proving the necessary conditions of optimality [1][2][3]6,7,10], on the other. Variation formulas for various classes of functional-differential equations without perturbation of delay are given in [2][3][4][5]7,8,10,11]. Here we are interested in the variation formulas for the controlled delay functional-differential equatioṅ…”