Formulas of Variation of Solution for Quasi-linear Controlled Neutral Differential Equations

Let J = [a, b] be a finite interval, O ⊂ Rn, G ⊂ Rr be open sets. Let the function f : J ×Os ×G → Rn satisfy the following conditions: for almost all t ∈ J the function f(t, ·) : Os ×G → Rn is continuously differentiable; for any (x1, . . . , xs, u) ∈ Os ×G the functions f(t, x1, . . . , xs, u), fxi(·), i = 1, . . . , s, fu(·) are measurable on J ; for arbitrary compacts K ⊂ O, N ⊂ G there exists a function mK,N (·) ∈ L(J,R+), R+ = [0,∞), such that for any (x1, . . . , xs, u) ∈ Ks ×N and for almost all t ∈ J, the following inequality is fulfilled