ABSTRACT
These functions are finite for r $ R if and only if the operators (18.13) • (18.15) satisfy a local Lipschitz condition in C. Moreover, the numbers given in (18.45) - (18.47) are then the minimal Lipschitz
constants for the corresponding operators on l1r ( C). This may be stated in a more precise and convenient way:
Theorem 18.1. The operators (18.22) - (18.24) satisfy a Lipschitz condition on BR(C) if and only if the three kernels in (18.1) have second partial derivatives in the last argument
( ) 82n(t, I, 1", CT, u)n2 t,S,1",(1,U = 8u2 for all (t, s) E T x S and almost all (1", u) E T x IR., «(1, u) E S X IR., and (1", (1, u) E T X S X IR., respectively, and the three functions (18.48)
(18.49)
and
(18.50)
kt(r):= sup I sup Il~(t,,,,1",u)ld1", (t••)eTxS iT lulSr
are finite for r S R. Moreover, the numbers kt(r), km(r), and kn(r) are then the minimal Lipschitz constants for the operators (18.22), (18.23), and (18.24), respectively, on Br(C). Finally, the minimal Lipschitz constant k(r) for the operator (18.25) on l1r (C) satisfies the two-sided estimate
(18.51) max {kt(r), km(r), kn(r)} S k(r)
o The prooffor the integral operator (18.24) is contained in ApPELLDE PASCALE-ZABREJKO [1991]. Let us prove the assertion for the partial integral operator (18.22), the proof for (18.23) is similar. By what has been observed before, for this it is necessary and sufficient to show that the function (18.45) is finite for r $ R. Suppose first that (18.45) is finite for all r $ R. This means that
l Ill(t, s, T, Ul) - '1(t,.9, T, u2)1 d < k ( )sup T _ ,r < 00T IU11,Iu21Sr lUI - u21 for all (t, s) E T x S, and hence the function ,\". given by
\ ( ) 111(t,.9,T,ul)-11(t,.9,T'U2)1A". T:= sup I IIU11,Iu21Sr Ul - U2 is finite almost everywhere on T. Since 1'1(t, s, T, U1) -ll(t, s, T, u2)1 $ '\".(r)lul-u21 for lUll, IU21 $ r, the map U..... '1(t, s, T, u) is absolutely continuous. Consequently, the partial derivative 12 = Blt /8u exists for almost all u and satisfies
sup 112( t, s, r, u)\ $ '\",(r). lulSr
But this implies that
tl(r) $ sup I '\,.•(T)dr < 00. (".)ETxslT
Conversely, suppose that (18.48) is finite for r $ R. This implies that the function X". given by
is finite almost everywhere on T, for all (t, s) E T x S. Consequently, for IUtl, IU21 $ r we have
We conclude that
Of course, the proof shows also that k,(r) = k,(r) for all r S R. • Theorem 18.1 implies, in particular, that the estimate k(r) S k,(r) + km(r)+k,,(r) holds for the Lipschitz constant in (18.4) in case X = C. The problem of calculating the numbers a in (18.5) and 6 in (18.6) is quite easy. In fact, suppose that the partial integral equation (18.26) has a unique solution (18.28) in the space X = C. From the definition of the norm in the space C we obtain then the equality
(18.52)
a = sup Ig(t,s) + , r,(t,s,r)g(r,s)dr (t,.)ETxS iT
(18.53)
and from explicit formulas for the norm of a linear partial integral operator in the space C (see § 13) the equality
+is Irm(t,s,u)1 du +£is Ir,,(t,s, r,u)1 Mdr] . Here 9 is defined by (18.27), and r" rm , and r" are the resolvent kernels corresponding to It, ml, and nl, respectively. The resolvent kernels r" rm , and r" are in general difficult to compute explicitly. An exceptional case is that of degenerate kernels; we illustrate this by means of a very elementary example.
