Lipschitz stability of nonlinear systems of differential equations

Uniformly Lipschitz Stability and Asymptotic Property of Perturbed Functional Differential Systems

“…Using the nonlinear variation of constants formula of Alekseev [1], the solutions of (2.1) and (2.2) with the same initial value are related by y(t, t 0 , y 0 ) = x(t, t 0 , y 0 ) + Since x = 0 of (2.1) is ULSV, it is ULS( [8],Theorem 3.3). Using the ULSV condition of x = 0 of (2.1), together with (3.1) and (3.2), we have Thus, by (3.3), we have |y(t)| ≤ M (t 0 )|y 0 | for some M (t 0 ) > 0 whenever |y 0 | < δ.…”

“…Dannan and Elaydi introduced a new notion of uniformly Lipschitz stability (ULS) [8]. This notion of ULS lies somewhere between uniformly stability on one side and the notions of asymptotic stability in variation of Brauer [4] and uniformly stability in variation of Brauer and Strauss [3] on the other side.…”

On the ψ-exponential asymptotic stability of nonlinear systems of differential equations

“…Thus, the trivial solution of (3) is (e λt Ψ − L p e λt Ψ )-uniformly stable on R + . For every solution z(t, t 0 , z 0 ) of (3), we have (6). (5) and (7), we have…”

A CONVERSE THEOREM ON h-STABILITY VIA IMPULSIVE VARIATIONAL SYSTEMS

“…The various notions of hstability given in [17,18] include several types of known stability properties as uniform stability, exponential asymptotic stability [15] and uniform Lipschitz stability [9].…”