Abstract:

It is well known that all the results in integral inequalities of BellmanGronwallReid type demand an assumption of monotony on the functions involved. Since the corresponding theory of differential inequalities does not require the monotonic assumption, it is believed that this extra condition is due to the technics employed rather than the necessity.
It is also well known that in proving the convergence of successive approximations, it becomes crucial to suppose an additional restriction of monotony on the functions satisfying uniqueness criteria. The question whether this additional assumption is really needed has been open for many years. This problem was discussed by the author
in [5] where a partial answer was given, namely, it is sufficient if the function g(t,u) satisfying, for example, Kamke's uniqueness condition (without the monotonic assumption) dominates the function
[see pdf for notation]
Later Olech and Pliss [3] also gave a partial answer showing that the monotony can be dropped if g(t,u) is of a special type. Very recently Deimling [1] has given the complete answer to the problem by using [see pdf for notation] type function judiciously and working around the critical point. Since
he considers an abstract Cauchy problem and consequently [see pdf for notation] may not be continuous in r, his proof needs an equivalent uniqueness criteria.
The known results [4,6,7] that give sufficient conditions for the existence of a limit as [see pdf for notation] of solutions also assume monotonic condition on the comparison function and it is felt desirable to dispense with this requirement.
In this paper, we develop results that can be applied to show the redundancy of the assumption of monotony in several situations. Employing our auxiliary results, we prove that the monotonicity assumption is superfluous in
(i) the theory of integral inequalities of BellmanGronwall
Reid type;
(ii) the convergence of successive approximations including the infinite dimensional situation;
and
(iii) the sufficient conditions for the existence of a limit as [see pdf for notation] of solutions.
We believe that our method can profitably be used in other situations and in vectorial integral inequalities. 