Sentence examples for a unique solution of the initial from inspiring English sources

Then there exists a unique solution of the initial value problem left.

It is clear that there exists a unique solution of the initial value problem a u k − 1 − c u k + b u k + 1 = φ k, 1 ≤ k ≤ N − 1, u 0 = φ, u 1 = ψ (47).

Then there exists a unique solution of the initial value problem D q, ω x i ( t ) = f i ( t, x 1 ( t ), x 2 ( t ), …, x n ( t ) ), 1 ≤ i ≤ n, t ∈ I, x i = y i ∈ X. } (4.6).

By assumption (H6), r∗<1; therefore, the operator T is a contraction in B. Hence, by Theorem 2.10 the operator T has a unique fixed point, which corresponds to the unique solution of the initial value problem (1) and (2).

Theorem 2.1 Suppose that ϕ : [ 0, 1 ] → R is a continuous function and let x ϕ be the unique solution of the initial value problem (1.1) and (1.2).

and is the unique solution of the initial value problem (24), (25) with Ξ given by (26).

If and fix, then the exponential function is for the unique solution of the initial value problem (2.2).

Based on proofs of the unique solution of the initial value problem, which is dependent continuously on all variables, we get the continuity of normalized eigenfunctions.

Let with, for all, then the delta exponential function is defined as the unique solution of the initial value problem (1.3).

For (epsilon >0) denote by (u_epsilon in C^2(Omega times (0,T cap C^1(bar{Omega }times (0,T))) the non-negative unique solution of the initial value problem (8.6).

Then the nabla exponential function e p ( ⋅, t 0 ) is the unique solution of the initial value problem y ∇ = p ( t ) y, y ( t 0 ) = 1.