where [[absolute value of (u)].sub.p,[OMEGA]] = [([[integral.sub.[OMEGA]] [[absolute value of (u)].sup.p] dx).sup.1/p], 1 [less than or equal to] p < + [infinity], [D.sub.K]([OMEGA]) [??] [D.sub.k]([OMEGA]) x [D.sub.K]([OMEGA]), endowed with norm [mathematical expression not reproducible], and G(x, s, t) is a nonnegative Caratheodory
function from [OMEGA] x R x R to R; namely,
and we show that it is derived from the empirical version of Caratheodory
function, used in the literature on orthogonal polynomials on the unit circle.
where h [member of] [L.sup.1] (0,2[pi]) is given and g : (0,2[pi]) x R [right arrow] R is a Caratheodory
function; that is, g(x, u) is continuous in u e R, for a.e.
We say that a function f(([t.sub.1], ..., [t.sub.N]), x([t.sub.1], ..., [t.sub.N])) = f : J x S [right arrow] R satisfies Caratheodory
conditions if it is measurable in ([t.sub.1], ..., [t.sub.N]) for any x [member of] S and is continuous in x for almost all ([t.sub.1], ..., [t.sub.N]) [member of] J.
Suppose that A : [J.sub.0] x [X.sub.[mu]-1/p,p] [right arrow] B([X.sub.1], X) is a continuous map, F : [J.sub.0] x [X.sub.[mu]-1/p,p] [right arrow] X is a Caratheodory
map, and [u.sub.0] [member of] [X.sub.[mu]-1/p,p].
Later, we also consider non-Jordan domains D, where the boundary is to be understood in the sense of the Caratheodory
boundary extension theorem.
His research was elegant in the sense that it was based on a thorough understanding of classical mathematics, physics, and thermodynamics, following such great scientists as George Hadley (1685-1768), William Ferrei (1817-1891), Sir Napier Shaw (1854-1945), Constantin Caratheodory
(1873-1950), Eric Eady (1915-1966), Edward Lorenz (1917-2008), and others.
Geometrical thermodynamic method was started by Gibbs and Caratheodory
where m is a positive function  or sign-changing function  on [0, 1], g : [0, 1] x [R.sup.2] [right arrow] R satisfies the Caratheodory
condition, and g(t, 0, [mu]) [equivalent to] 0.
where E is a real Banach space, A : D(A) [subset] E [right arrow] [2.sup.E] is an m-accretive operator such that--A generates a compact semigroup, F : [0, T] X C([-r, 0]; E) [right arrow] E is a Caratheodory
function, r [greater than or equal to] 0 and C([-r, 0]; E) stands for the space of all continuous functions from [-r, 0] to E
, Theory of Functions of a Complex Variable, Vol.
The first part was proved by Caratheodory
 and the second part can be found in .