Positive solutions of integral equations

Builds on Integral equation Positive cone Positive operator

Definition

Let $(\Omega, \mu)$ be a measure space. Let $X$ be a space of real-valued functions on $\Omega$ , such as $C(\Omega)$ for compact $\Omega$ or $L^p(\Omega, \mu)$ . Its positive cone is $X_+ = \set{\varphi \in X : \varphi \ge 0}$ , with the inequality almost everywhere in the case of $L^p$ . Consider an integral equation $\varphi = T\varphi$ on $X$ , for example $T\varphi = f + \int_\Omega k(\cdot, y) \varphi(y) \, d\mu(y)$ , or a nonlinear one such as a Hammerstein equation $T\varphi = \int_\Omega k(\cdot, y) \, g(y, \varphi(y)) \, d\mu(y)$ .

A positive solution is a solution $\varphi \in X_+$ with $\varphi \ne 0$ .
A strictly positive solution is a solution with $\varphi(x) > 0$ for all $x \in \Omega$ (for almost all $x$ , in the case of $L^p$ ).

Remarks

Example: on $X = C[0, 1]$ take $\varphi(x) = 1 + \frac12 \int_0^1 \varphi(t) \, dt$ . The right side is constant, so $\varphi = c$ with $c = 1 + \frac{c}{2}$ , that is $c = 2$ , a strictly positive solution. More generally, if $k \ge 0$ , $f \ge 0$ , and the integral operator $K$ has spectral radius $r(K) < 1$ , the Liouville–Neumann series $\varphi = \sum_{n \ge 0} K^n f$ is a sum of nonnegative terms, so the solution is nonnegative.

Non-example: $\varphi(x) = 1 + 2 \int_0^1 \varphi(t) \, dt$ on $[0, 1]$ gives $c = 1 + 2c$ , so its only solution is $\varphi = -1$ , which is not in $X_+$ . Here $r(K) = 2 > 1$ , and the Neumann series argument fails.

For the homogeneous eigenvalue problem $\lambda\varphi = K\varphi$ , the Krein–Rutman theorem and Jentzsch's theorem give a positive eigenfunction for the eigenvalue $r(K)$ when $K$ is compact with nonnegative kernel and $r(K) > 0$ .

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Integral equation concept
Positive cone concept
Positive operator concept
Positive solutions of integral equations definition

Tag	`NC1C7EKV1V`
URI	`m2://NC1C7EKV1V`
Kind	definition
Writing	W0 stub
MSC	45M20, 47B65
Defines	Positive solutions of integral equations
Uses	Integral equation, Positive cone, Positive operator

Positive solutions of integral equations

Definition

Remarks

Reading path

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