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Existence theorems for Urysohn’s integral equation

The theory of abstract Hammerstein operators is applied to obtain existence theorems for Urysohn's integral equation. Urysohn's integral equation is of the form (*) u(s) + f 1(s, t, u(t))dt = 0. Usually one assumes that Q is a subset of R', and that V(s, t, u) is a function of three variables s, t E Q, u e R, satisfying the so-called Caratheodory conditions. Urysohn's equation has been discussed by Urysohn [6], Kolomy [4], Krasnosel'skil [5] and others. Attempts have been made to apply the theory of monotone operators to get existence theorems for (*). In this paper we apply the theory of abstract Hammerstein operators to obtain existence theorems for ( *) with rather simple conditions on the function (D. We define a linear operator A: L2(Q x ) -L2(Q x Q) with range in L2(Q) and a nonlinear operator F: L2(Q) L2(Q x Q) as follows: (1) [Au](s) = u(s, t)dt, (2) [Fu](s, t) = F(s, t, u(t)). In all our considerations in this paper, Q will be a set of finite measure in Rn and (3) L 2(Q) = u: IQ u(t)| 2 dt < } (4) L2(Qx Q)={u: fflU(s t)|2dtds <4. Observe that L2(Q) is a closed subspace of L2(Q x Q). Lemma 1. A is a continuous linear map from L2(Q x Q) to L2 x Q) with range in L2((Q). One of the hypotheses of the existence theorem is the compactness of the operator AF. In the following lemmas conditions are given which assure this. Received by the editors August 30, 1973 and, in revised form, February 26, 1974. AMS (MOS) subject classifications (1970). Primary 45G99. Copyright