On the Existence of ¥mathrmL^¥mathrmp-Solutions of Volterra Integral Equations in Banach Spaces

This paper studies a nonlinear Volterra integral equation (the equation (1)) in a Banach space E .Most studies of (1) have been done in the space C of continuous functions over E .In contrast to this, in the paper we study (1) in the space Lᵖ over E and establish an existence theorem and a Hukuhara-type and a Kneser-type theo- rems.In our earlier paper [11], similar results were established in C .The present paper extends these results to Lᵖ under some additional restrictive conditions.To do so, we use the measure alphaₚ of noncompactness in Lᵖ , and clarify the relation between alphaₚ and the measure of noncompactness in E in Section 2. This relation plays an essential role in our paper.Using it and the assumptions 2o and (12) on g , we prove that any sequence of approximate solutions of (1) has an mathrmL^mathrmp -convergent subsequence, so that the existence theorem (Theorem 1) follows immediately.The novelty here is the use of the assumption 3 mathrmO , which is similar to that used in the