Positive kernels, fixed points, and integral equations
There is substantial literature going back to 1965 showing boundedness properties of solutions of the integro-differential equation \[ x'(t) = -\int^t_0 A(t-s) h(s,x(s))ds \] when $A$ is a positive kernel and $h$ is a continuous function using \[ \int^T_0 h(t,x(t))\int^t_0 A(t-s) h(s,x(s))ds dt...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2016-06-01
|
| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=6496 |
| Summary: | There is substantial literature going back to 1965 showing boundedness properties of solutions of the integro-differential equation
\[
x'(t) = -\int^t_0 A(t-s) h(s,x(s))ds
\]
when $A$ is a positive kernel and $h$ is a continuous function using
\[
\int^T_0 h(t,x(t))\int^t_0 A(t-s) h(s,x(s))ds dt \geq 0.
\]
In that study there emerges the pair:
\[\text{Integro-differential equation and Supremum norm.} \]
In this paper we study qualitative properties of solutions of integral equations using the same inequality and obtain results on $L^p$ solutions. That is, there occurs the pair:
\[ \text{Integral equations and $L^p$ norm.}\]
The paper also offers many examples showing how to use the $L^p$ idea effectively. |
|---|---|
| ISSN: | 1417-3875 1417-3875 |