'

P(Kx +J) + t/J on [a,b] X Q-. Conversely, every solution x E L" oj (17.21) belongs to W~ and solves the boundary value problem (17.13)/(17.14). o The proof follows immediately from Lemma 17.2 with f replaced by Kx +J.• We remark that partial integral operators of Uryson type have been studied by POVOLOTSKU-KALITVIN [1985,1991], KALITvIN-GLoTov [1996], KALITVIN [1997], ApPELL-DE PASCALE-KALITVIN-ZABREJKO [1996], KALITVIN-KoRENCHUK-EvTUKHINA [1993], and CHENKALITVIN [1997, 1998J. We begin now to study the operator equation (17.21) from the viewpoint of fixed point theorems in K -normed spaces; the constructions and results will be parallel to those in Subsection 9.4. For a S !p S b, 0< t S I, and s E 8, we put

Moreover, we define four operators A,B,C, and D by

+L"ill fal i(!p, 9, t, s)n(9, t, I, T, (7, u(IJ, T, (7» dT d(7 d9,

and

+1"it!. 101 j(cp, 0, t, s)n(O, -t, s, -r, CT, v(9, r, CT» dr dCT dO. The operator equation (17.21) may then be written as a system

Suppose now that the kernels I, m, and n satisfy Lipschitz conditions

and

Moreover, assume that

(17.23)

11.!1 j(l{), 8", ')cl(8,',', T)U(T, ·)dT

+ill .!1 j(I{),8"")C2(O,.,.,T,0')U(T,0')dTdO'II ~ 'Yllull, and

1I.!1 j( I{),8"" )d1(9, ',', T)V(T,') dT

+ ill j(I{), 8, ',' )d2(O, ',',0' )v(., 0') dO' +ill .!1 j(I{),8,·,·)d3(8,·,',T,0')v(T,0')dTM ~ 611vlI,

where all norms are taken in Lp(Q+). We define a linear operator G by

(17.24)

(17.25)

satisfies the contraction condition (17.12). We take X = Lp(Q+) X Lp( Q-), equipped with the norm lIeu, 11)lIx = lIullL,. +1I11I1L,.. Moreover, let E = Lp([a,b],X) be the Bochner-Lebesgue space of all X-valued functions Ip .... z(Ip,"') = (u(Ip,·,.),l1(Ip,"'»' equipped with the norm

and the X-norm

]Izl[= (/Iu(<p,', ')111" /I11(<p,', ')111')' Thus, the X-norm takes its values in the natural cone of the Banach space Z = Lp([a,b),IR2). Our assumptions ensure that the estimate (17.12) is true for the operators (17.23) and (17.24). As in Theorem 9.7 we see that the spectral radius of G is less than 1. Consequently, Theorem 9.6 applies.•

17.3. Equations with Hammerstein operators In the preceding subsections we have proved existence and uniqueness theorems for various types of integra-differential equations of Barbashin type in Lebesgue spaces by means of topological methods. Now we are going to discuss partial integral equations of the type

in the space C(D) of continuous functions over D := T X S. Here I : D X T - IR, m : D x S - IR, n : D x D - IR, and 9 : D - IR are given measurable functions, while J : D x IR - IR is supposed to satisfy a Caratheodory condition. We restrict ourselves to the model case T = [a, b) and S = [e, d) in the sequel.