New conditions for the existence of non trivial solutions to some Volterra equations.

Okrasinski, W.. "New conditions for the existence of non trivial solutions to some Volterra equations.." Extracta Mathematicae 5.1 (1990): 7-8. <http://eudml.org/doc/39853>.

@article{Okrasinski1990,
abstract = {We consider the following Volterra equation:(1)       u(x) = ∫0x k(x-s) g(u(s)) ds,   where,k: [0, δ0] → R is an increasing absolutely continuous function such thatk(0) = 0g: [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&gt;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.},
author = {Okrasinski, W.},
journal = {Extracta Mathematicae},
keywords = {Ecuaciones de Volterra; Soluciones; Teorema de existencia},
language = {eng},
number = {1},
pages = {7-8},
title = {New conditions for the existence of non trivial solutions to some Volterra equations.},
url = {http://eudml.org/doc/39853},
volume = {5},
year = {1990},
}

TY - JOUR
AU - Okrasinski, W.
TI - New conditions for the existence of non trivial solutions to some Volterra equations.
JO - Extracta Mathematicae
PY - 1990
VL - 5
IS - 1
SP - 7
EP - 8
AB - We consider the following Volterra equation:(1)       u(x) = ∫0x k(x-s) g(u(s)) ds,   where,k: [0, δ0] → R is an increasing absolutely continuous function such thatk(0) = 0g: [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&gt;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.
LA - eng
KW - Ecuaciones de Volterra; Soluciones; Teorema de existencia
UR - http://eudml.org/doc/39853
ER -