We consider the following Volterra equation:
(1) u(x) = ∫0 x k(x-s) g(u(s)) ds, where,
k: [0, δ0] → R is an increasing absolutely continuous function such that
k(0) = 0
g: [0,+ ∞) → [0,+ ∞) is an increasing absolutely continuous function such that g(0) = 0 and g(u)/u → ∞ as u → 0+ (see [3]).
Let us note that (1) has always the trivial solution u = 0.
Some necessary and sufficient conditions for the existence of nontrivial solutions to (1) with k(x) = xα - 1 (α>0) are given in [1], [2] and [4]. In [3] and [5] conditions for more general kernels are presented. But those results do not give answers about nontrivial solutions in the case of kernels which are very smooth near the origin. Now we are able to show a necessary condition.
@article{urn:eudml:doc:39853,
title = {New conditions for the existence of non trivial solutions to some Volterra equations.},
journal = {Extracta Mathematicae},
volume = {5},
year = {1990},
pages = {7-8},
zbl = {1231.45005},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:39853}
}
Okrasinski, W. New conditions for the existence of non trivial solutions to some Volterra equations.. Extracta Mathematicae, Tome 5 (1990) pp. 7-8. http://gdmltest.u-ga.fr/item/urn:eudml:doc:39853/