Integral Equations and Operator Theory Vol. 13 (1990) 0378-620X/90/050660-1151.50+0.20/0 (c) 1990 Birkh~user Verlag, Basel REGULARITY OF THE SOLUTION OF HAMMERSTEIN EQUATIONS WITH WEAKLY SINGULAR KERNEL Hideaki Kaneko, Richard Noren and Yuesheng Xu Some regularity properties of the solution to a class of weakly singular Hammer- stein equations are derived. The results obtained in this paper extend the results of C. Schneider [4], where he obtains similar properties for the solution to weakly singular Fredholm equations of the second kind. 1. INTRODUCTION In a recent paper [4], C. Schneider proved a theorem concerning the regularity properties of the solution to a class of Fredholm integral equations of the second kind. He considered the equation (1.1) where ab r - g=(t s - t f)kCs, t)r = fCs), a<s<b, s a-1 for 0<a<l g~(s) = logs for a=l, : c[a,b] and k To describe the results of Schneider and those of this paper, it is convenient to intro- duce the following definition. (1.2) DEFINITION Let 0 < a < 1 and m No = the set of nonnegative integers. Let C("'a)[a,b] be the set of all functions x Cm[a, b] such that there exists constants A > 0 and B >t a- b I with ] x(m)(s) - sCm)(t) I<_ A is t ] log if a -- 1 for all s,t [a,b]. If x C(~ the x is called a-HSlder continuous. A nonstandard way of defining HSIder continuity for a = 1 should be noted here.