On the equation Phi(x) = Integral from x to x+1 of K(xi)f[Phi(xi)]Dxi

Abstract

Let [Script K]= {K|K: R to [0,1], K measurable Lebesgue and K(x) > 0 for almost all x in R}, [Script F]= {f|f: [0,1] to [0,1], f monotone non-decreasing and continuous}, [Script C]={phi|phi: R to (0,1], and phi is continuous}, [Script K]_f = {K in [Script K]| there exists phi in [Script C] which satisfies phi(x) = integral from x to x+1 of K(xi)f[phi(xi)]dxi for all x in R}, [Script K]^0 = {K in [Script K]| there exists X in R (depending on K) such that for almost all x greater than or equal to X, K(x) =1}. The purpose of this paper is to characterize the class [Script K]_f for a given f in [Script F]. The principal results of Theorem I (in chapter IV) and Theorem II (in chapter VI). They are stated as follows, Theorem I: Suppose f in [Script F]. Then [Script K]_f = [Script K] if and only if 1/f in [Script L][0,1]; Theorem II: Suppose f in [Script F]. Then [Script K]_f = [Script K]^0 if and only if (i) 0 < f(u) less than or equal to u for all u in (0,1]; (ii) f(v) = v for some v in (0,1]; and (iii) 1/(v-f) in [Script L][0,v] for every v in (0,1]. As a complement to Theorem I, in chapter V we study [Script K]_f for a particular f in [Script F], namely f(u) = u log e/u. Since 1/f not in [Script L][0,1], it follows immediately from Theorem I that [Script K]_f subset [Script K] properly. Furthermore, we obtain the following results which partially characterize [Script K]_f; they appear in the paper as, Theorem 5: If K in [Script K] satisfies integral to infinity of K(xi)dxi < infinity, then K not in [Script K]_f; Theorem 6: Suppose K in [Script K] and K(x) = 1/x for x greater than or equal to 1. Then K is not in [Script K]_f; but Theorem 7: If K in [Script K] satisfies integral to infinity of exp(-integral from 1 to xi of K(eta)d eta)d xi < infinity and lim as x to infinity inf K(x) exp (integral from 1 to x of K(xi)d xi)>0, then K is in [Script K]_f. Finally, for this function f, f(u) = u log (e/u), f(u)>0 for u>0 and we observe that Inf{(u/f(u) : 0<u less than or equal to 1} = 0; thus by Theorem 4 in chapter III (Suppose f in [Script F] and f(u_0) greater than or equal to u_0 for some u_0 such that 0 < u_0 less than or equal to 1. If K in [Script K] satisfies K(x) greater than or equal to u_0/f(u_0) for almost all x in R, then K is in [Script K]_f.) if K in [Script K] and if K(x) greater than or equal to r >0 for all x greater than or equal to X in R, then K in [Script K]_f. An open question for this f is the following: Suppose K in [Script K] has the form ofor x large, K(x) = 1/x + 1/(x log x). Is K in [Script K]_f? Chapter VII, which is a supplementary chapter to chapter VI, is devoted to the study of a particular f in [Script F] defined by f(u) = 1 - (1-u)log(e/(1-u)). By Theorem II and lemma 11, since 1/(1-f) is not in [Script L][0,1], then [Script K]^0 is a subset of [Script K]_f properly. For this function f we derive a necessary condition for K in [Script K]_f and also we exhibit a kernel in [Script K] \ [Script K]^0. We state the results as follows: Theorem 8: If K in [Script K]_f, then integral to infinity of e^(alpha x) (1-K(x))dx < infinity for every alpha > 0; Theorem 9: If K in [Script K] satisfies K(x) = 1 - e (Summation for n in N of e(x+n))(exp(-e(x+n))) for all x greater than or equal to 0, then K is in [Script K]_f. Finally we state the following conjecture: Suppose f(u) = 1 - (1-u)log(e-(1-u)). Then K in [Script K]_f if and only if whenever F is a non-decreasing continuously differentiable solution of F'(x) = e^x (F(x) -F(x-1) ) for x greater than or equal to X in R, then ingetral from X to infinity of F'(x) e^-x (1-k(x))dx < infinity.