swar

The nonlinear partial integral equation (18.1) may then be rewritten as operator equation (18.2) if we put (18.12) F(x) = x - L(x) - M(x) - N(x) where X is some Banach space of real functions over T x S. Suppose now that the three kernels in (18.1) have partial derivatives in the last argument

Consider the operators L·, M* and N* defined by

(18.13)

(18.14) and

(18.15)

N*(x)(t,s,r,O'):= nl(t,s,r,O',x(r,O'». These operators are not Nemytskij operators in the usual sense, since they map functions of the two variables (t, s) into functions of three (or even four) variables; we call such operators "generalized

Nemytskij operators" in the sequel. The operators (18.13) - (18.15) may be considered between the space X and the kernel spaces Dl,(X), !Jtm(X), and 9ln(X) defined by the norms

(18.16)

(18.17)

and

(18.18) Ilrll9tn(x):= sup II' 'Ir(.,., T,u)h(T,U)1 dudT/lX,1""x~l iT is respectively. Of course, the functionals (18.16) - (18.18) are nothing else than the operator norms ofthe moduli ofthe corresponding (regular) linear partial integral operators

(18.19)

(18.20)

and

(18.21)

Rh(t, s):= his ret,s, T,u)h(T,U)dudT in the algebra .c(X) of bounded linear operators on X. We set

(18.22)

(18.23)

and

(18.25) 1"(x) =1-Ll(X) - M1(x) - Nl(x) o The assertion has been proved in ApPELL-DE PASCALE-ZABREJKO [1991] for integral operators of the form (18.11), so we have to prove it only for the partial integral operators (18.9) and (18.10). For x E BR(X) and hEX we have

= {i1[L1(X +>'h) - Ll(X)]hd>.} (t,s). Since the operator L 1 : BR(X) -+ .c(X) satisfies a Lipschitz condition, by assumption, we conclude that

which means that L'(x) = Lt(x). The equality M'(x) = Mt(x) is proved similarly. •

Applying Lemma 18.1 allows us to "find" the constants a and b for the equation (18.2), where F is given by (18.12). In fad, the function h := F'(O)-tF(O) satisfies the linear partial integral equation

h(t, s) - k It(t, s, T, O)h(T, s) dT

(18.26)

where

-Is mt(t,s,o,O)h(t,o')do - kls nl(t,s,T,O,O)h(T,O)dodT = get,s),

(18.27) g(t,s):= - kl(t,8,T,O)dT

-Is met, s, 0,0) do - k Is net, s, T, 0, 0) do dT. Suppose that the (unique!) solution of equation (18.26) may be written in the form

(18.28)

+Is rm(t, s, o)g(t, 0) do +£Is rn(t,S,T,O)g(T,0) dodT

with some resolvent kernels r" rm , and rn which are defined through the kernels It, mit and nt. Then the constant a in (18.5) is of course nothing else than the norm IIhllx of the function (18.28) in the space X. Since the explicit form of these resolvent kernels is in general hard to find, one usually looks for a representation of the form F'(O) = T - E, where T is boundedly invertible and the norm of E in .c(X)

is small. The elementary equality T - E = T(l - T-l E) implies then that, under the hypotheses of Lemma 18.1, the estimates

a = IIF'(O) F(O)II ~ 1 -I/EIIIIT-111 are true. In this way, we have proved the following

Lemma 18.2. Suppose that the hypotheses 0/ Lemma 18.1 are satisfied, and F(O) = T - E, where T is invertible in .c(X) and IIEII ~ E. Then the estimates

hold.