NONLINEAR STOCHASTIC FUNCTIONAL INTEGRAL EQUATIONS IN THE PLANE

Local or global existence and uniqueness theorems for nonlinear stochastic functional integral equations are proved. The proofs are based on the successive approximations methods. The formulation includes retarded arguments and hereditary Volterra terms.

Equations of this type appear when one rewrites the partial stochastic functional differential equation Ou rz Ow (z (3) Oy' y) f(x, y u(x y)) + g(x, y, u(x Y)OxOy' y)' (x, y) +, (, ) (, ), (, ) e Zo, can be obtained from (1) by specializing given functions. Functional partial differential equations with operators of the Volterra type are also the special cases of (1) (see Section 6).

Preliminaries
Let us introduce a notation for rectangles. Suppose z-(x,y) and z'-(x',y'). We say that z<z'iffx<x'and y<y'. We also writez<z'iffx<x'andy<y'. We shall use the notation Iz z' for the closed rectangle [x,x'] x [y,y']. We denote the rectangle Io, z -[0,x] x [0, y] by I z.
Tile parameter space will be the square of the plane Ia, where a (hi, a2).
Let J-I_h,o, where h-(hi,h2) and hl,h 2 > 0. By C-C(J,R) we denote the Banach space of all continuous functions u: J--,N endowed with supremum norm Let I I_ h,a and I o I \I a. For any function u: I --R and a fixed z I a we define function Uz: J--,R by Uz() u(z + ) for e J.
Let (, , ) be a complete probability space. By w we denote a two-parameter Wiener process. We introduce a family bz, z E I, of r-algebras of subsets of f] with the following properties: (i) zC ,forz<z'<a; (ii) for every z e I a, w(z) is z-measurable; (iii) for every z Ia, the Wiener measure of the rectangle Iz, z + ) C I a, which can be intro- Let %(9) be the set of all functions v: I a x f-,R with the following conditions: (a) v(z,. ): --, is measurable for each fixed z Ia; (b) v(., w): I is continuous for a.e. fixed w ; (c) v(z, w) (z, w) for z e I 0, w e f2. It is easy to prove, similarly like in [5], by Borel-Cantelli lemma that %(9) with the norm I I (E(sup [v(z,w)[2) 1/2 z is a Banach space.
Let L2(f,C) be the set of all C(J,R)-valued, mean square integrable functions on f2 and let be the closed ball of center 7 with radius r in L2(f2, C), that is, S-S(7, r)-{v L2(, C)" v-7 I I 2 _< r}, where r > 0 and

Assumptions and Lemmas
for all z I a and all v E S; 3o 7(" ,w): I---, is continuous for a.e. w E , 7(z,. is 0-measurable, independent of the two parameter Wiener process w and 7* E(supze Now, we define the sequence of successive approximations {un(z)} as follows: ttn+l(z (Z) + f f(r, u)dv + f g(v, uT)dw (7" Proof: It is easy to show that the integrals on the right-hand side of (5) are well defined. Now, let us note, by condition 20 of H1, that the problem () 02 has a local solution with any initial functions a and /, a(0)-/(0) [3], [4], where D z -OxOu" Take c and / such that a(x)+/(y)-/(0) > 37", (x,y) I a, and let v be the local solution of (5) with the initial functions a and/. Now, we shall show that and hold for z E I, n-1, 2,... Utilizing Doob's and Schwarz's inequalities and condition (4) we obtain a 0 such that 0 < a o < a and h 1), max(O-y h2)).
Since the function H(z, v(z)) is integrable on Iao there is such that 0 < < a 0 and _< 2(4 T ala2) / H(r, v(r))dr <_ r, I Z for all z E I n Assume that (8) and (9)    On the other hand, utilizing Doob's martingale inequality and Schwartz's inequality we get E I I u I I 2 2ala2E If(r, u r )l 2dr) + 8E( g(, u )1 d) m--1 n--1 2 _< 2(4 / all2) G(r,E I I u.,. ur I I )dr Ia _ 2(4 -b all2)/-G(T, k(l)('))dv-,O as l--oc, which implies that the sequence {uS(z)} is a Cauchy sequence in Banach space %-(). Therefore, there exists a stochastic process u(z) such that .E(sup uS(z)u(z) 12)0 z I as n-.cx. As usual, we can prove that u(z)is a local solution of (1), (2). Finally, we shall show the uniqueness of the local solution. Let u(z) and v(z) be two local solutions of existing on the rectangle I with u(z) v(z) 99(z), z E I o. Then, we get E I I Uz-Vz I I 2 _< 2(4 + ala2)/G(r,E I I urvr I] 2)dr, I z for all zEIa from which using condition 20 of H2, we find Elluz-vzll2 Therefore, we must have u z-vz for all zEI, so u(z)--v(z), zI a.s. proof.
This completes the

The Global Existence of Solutions
Now, we shall present the existence and uniqueness of global solution of (1), (2). Then, the sequence {ur'(z)} defined by (5) converges uniformly on any sunrectangle I of 2+, to a unique solution of (1), (2).
Proof: Denote by I . the largest rectangle on which the sequence {un(z)} converges uni-formly. By 'Theorem 1, we have that a 1 > 0 and a 2 > 0. Now, we suppose that a 1 < cx3. Then, we can take a 1 such that a 1 > 0 and a 2 > 0. Now we suppose that a I < . Then, we can take a 1 such that a < '1 < cx3. Thus, by assumptions 1 and 2 , we have a solution v(z) of (6) with a (1, a2) which exists on I~, and estimate (8) holds on the rectangle I~. The rest of the a a proof follows as in Theorem 1, replacing a by . Similar arguments apply to the case a 2 This proves the theorem.
The following corollary gives a special case of the comparison function G. of Theorem 2 holds. Obviously, the rest of the assumptions of Theorem 2 are satisfied. Therefore, by Theorem 2, we get the desirable conclusion, which completes the proof of the corollary.