Sentence examples for boundedness of a solution from inspiring English sources

In Section 5, we give a proof of Theorem 2 after we show the boundedness of a solution.

In the same way as in the homogeneous case, we show a result on the boundedness of a solution of (3).

We recall that the non-oscillation of (2.2) is equivalent to the boundedness of a solution φ of (2.21) on (mathbb{R}_{a}) (see again each one of papers [4, 15] or [19]).

Proof If boundedness of a solution of equation (1) is assumed, then by virtue of the same arguments as in Theorem 3, the thesis of the above remark is obtained.

We can also state a result on the boundedness of a solution of (13): Proposition 3.3.2 Let ρ = def 2 α ( S m W 0 sh + sup t ∈ R + ∥ I ( t ) ∥ F ), with W 0 sh = sup t ∈ J ∥ W sh ( t ) ∥ L 1.

Using nonnegative definite Lyapunov functionals, we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales.

However, the proof is incomplete without showing a boundedness of all solutions.

Here we will give some equations for which the boundedness of all solutions is clear and leave the problem of determining the boundedness of all solutions for a future study.

The question of boundedness of all solutions of x ″ + V ′ ( x, t ) = 0, (1.1).

The results on boundedness of all solutions of Eq. (1) are well known, see [2, 24].

This paper deals with the asymptotic stability and boundedness of the solution of a time-varying impulsive Volterra integro-dynamic system on time scales in which the coefficient matrix is not necessarily stable.