The existence and uniqueness of solutions to the equation u(t) + f'0a(t -s)Au(s) ds Bf(t), where A is a maximal monotone operator, is proved under various restrictions on A and /. I. Introduction. This paper discusses the existence and uniqueness of solutions to the abstract Volterra integral equation where A is a possibly multiple valued maximal monotone operator from the real Hilbert space 77 into 77, / maps the interval [0, T] into 77, and a(t) is a real valued differentiable function

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... on [0, T], such that a(0) > 0. Equations similar to (1.1) arise in the study of heat transfer subject to nonlinear boundary conditions, and in other areas. We refer the reader to [5] and [7]. There has been some recent work on this problem [1], [3], [4], [6] and the results are basically of two types. The first insists that f(t) is smooth, i.e., / G H71'2^, T; 77], while the second allows / to be an arbitrary element of L2[0, T; 77], but requires that the nonlinear operator A have at most linear growth. We will show that if A takes bounded subsets of 77 into bounded sets and/ is in L°°[0, T; 77], then (1.1) has a unique local solution. By further restricting the kernel function a(t) we are able to show the existence of global solutions. We are also able to show that by further restricting the growth of A, i.e., \Ax\ grows no faster than some polynomial in \x\, then, even for unbounded /(?), (1.1) still has a unique solution. This result (Proposition 2.6) extends the results of [1], [4] . A result concerning the asymptotic behavior of the solution u(t) is also included and an example shows this is best possible. In § §II and III we state our results and give their proofs, respectively. In the last section we give several examples. We will use the following notation throughout: