We first prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in (1) satisfies the locally Lipschitz condition
where 0 [less than or equal to] x < a, 0 [less than or equal to] t < T, B(x, t) > 0 is a constant, [[psi].sub.1] (x), [[psi].sub.2](x) are smooth functions, and f(u,x,t) is a non-linear scour term that satisfies the Lipschitz condition
, that is,
for any z, w [member of] [B.sub.E], z = w, then we say that h satisfies weighted Lipschitz condition
The nonlinear function [xi](x) is continuously differentiable and satisfies Lipschitz condition
with Lipschitz constant [sigma]; that is,
Then, their first order derivative function [[partial derivative].sub.x] satisfies the Lipschitz condition
and there is a number [L.sub.1] [greater than or equalt o] 0 such that
The fractional order nonlinear system (12) is globally asymptotically stable, if it satisfies the following conditions: (1) g(x(t)) satisfies g(0) = 0 and the Lipschitz condition
with respect to x, that is, [parallel]g([x.sub.1]) - g([x.sub.2])[parallel] [less than or equal to] L[parallel][x.sub.1] - [x.sub.2][parallel]; (2) Re(eig (A)) < 0 and [omega] = -maxRe(eig (A)) > [LM.sub.3][M.sub.4][GAMMA]([alpha]), where [M.sub.3] and [M.sub.4] satisfy [parallel][e.sup.At][parallel] [less than or equal to] [M.sub.3] [e.sup.-[omega]t] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Assume that, for any r(t) = j and r([t.sup.-]) = i, i, j [member of] S, functions [I.sub.j,i] (t, x) satisfy the global Lipschitz condition
; that is, there exist constants [L.sub.j,i] i, j [member of] S, such that for arbitrary t [greater than or equal to] [t.sub.0] and arbitrary [x.sub.1], [x.sub.2] [member of] [R.sup.n],
and f: [-L, L] [right arrow] R satisfy a Lipschitz condition
with constant K > 0.
where [phi](t) is a differentiable function, [alpha](t) [member of] [C.sup.1] [0, T] is a strictly monotone increasing function and satisfies that -[tau] [less than or equal to] [alpha](t) [less than or equal to] t and [alpha](0) = -[tau], there exists [t.sub.1] [member of] [0, T] such that [alpha]([t.sub.1]) = 0, and f:D = [0,T] x R x R x R is a given continuous mapping and satisfies the Lipschitz condition
(i) g(x, t) in (1) and (2) satisfies the Lipschitz condition
about state vector x(t); namely, there exists constant [[delta].sub.1] > 0, such that
Gehring and Martio  extended HL-result to the class of uniform domains and characterized the domains D with the property that functions which satisfy a local Lipschitz condition
in D for some [alpha] always satisfy the corresponding global condition there.
(A.1) The density function of [[beta].sup.T][x.sub.ij], f(u), is bounded away from zero for u [member of] [U.sub.[omega]] and [beta] near [[beta].sub.0] and satisfies the Lipschitz condition
of order 1 on [U.sub.[omega]] where [U.sub.[omega]] is the support of [omega](u).