[For the entire collection see Zbl 0519.00009.]
This paper is a further development of the theory of interval analysis as given by the author in his monograph [Methods and applications of interval analysis (1979; Zbl 0417.65022)], and of interval integration as presented by O. Caprani, K. Madsen and L. B. Rall[SIAM J. Math. Anal. 12, 321-341 (1981; Zbl 0461.28007)] and the author [Nonlinear Anal., Theory Methods Appl. 6, 829-831 (1982; Zbl 0495.65055)]. In the 1979 and 1982 papers it was noted that if X is a space of interval valued functions, and P:\(X\to X\), then if \(B\subseteq X\) and P(B)\(\subseteq B\) it follows not only that \(P^ i(B)(=P(P^{i-1}(B)))\subseteq B\) for all i, but that \(\lim_{i\to \infty}P^ i(B)\) exists. In the present paper a similar result in the opposite direction is given: if P is a Volterra type interval integral operator, and if \(U\subseteq P(U)\) for some (bounded) interval function U on \([x_ 0,x_ 0+a]\), then \(U\subseteq P^ j(U)\) for all j and \(\lim_{j\to \infty}P^ j(U)\) exists. The technical definitions are not all given in the paper, but may be found in the above references. Here we only mention that the Volterra type operators P are of the form \[ P[Y](x):=H(x)+\int^{x}_{x_ 0}F(x,t,Y(t),\Lambda)dt,\quad x_ 0\leq x\leq x_ 0+a, \] and the notation \(U\subseteq P(U)\) means U(x)\(\subseteq P[U](x)\) for all \(x\in [x_ 0,x_ 0+\alpha]\) for some \(\alpha\in(0,a]\). The author claims (without giving any details) that the (complete) result includes a number of Gronwall type integral inequalities as special cases.