Solutions for certain nonlinear Volterra integral equations

Abstract

where t belongs to a half open (possibly unbounded) interval I = (0, c), and u, + and f have values in R*, the m-dimensional Euclidean space with the norm I x I = 1 x1 / + ... + I xm I when x = (xl,..., x”). We shall give three main local existence theorems, Theorem 2.1-2.3, under very general assumptions on f. The very general assumptions are that f is BG,-, LG,or MG,-regular on 1 x I and satisfies some other proper assumptions. The definitions of BG,-, LG,and MG-regularity are given in Section 2. They are indeed somewhat similar to the definition of G-regularity given by J. Persson [4] for discussing the global existence of solutions of an ordinary differential equation. Given as a special case of Theorem 2.1, there is a local existence theorem, Theorem 3.1, with f satisfying the following assumptions (Al)-(A4). Indeed, the latter is due to R. K. Miller [2], although the assumptions on f is in the form improved recently by Z. Artstein [l]. In Theorem 2.2, the existence of Lipschitz continuous solutions will be considered. As a special case of it, the existence of Lipschitz continuous solutions will be ensured by the equi-d-continuity off, the validity of (A6) and the Lipschitz continuity of 4. The definition of equi-d-continuity off is given in Section 3. In Theorem 2.3, the existence of strictly increasing and continuous solutions are considered. As a special case of it, the existence of such solutions will be ensured by supposing that f (t, s, U) is continuous in u almost everywhere (a.e.) and measurable in s and that f and 4 satisfy other proper properties given in Theorem 3.2. In Section 4, the continuation of solutions will be discussed. And in Section 5, we shall give examples off, which will be BG,-, LG,-, MG,-regular or equi-dcontinuous and satisfies other proper conditions, so that (1.1) has solutions, but they do not satisfy the hypotheses of Theorem 3.1.