Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations. (English) Zbl 1091.47046

The authors consider transport equations of the following type: \[ \xi \frac{\partial \psi}{\partial x}(x,\xi)+\sigma(x,\xi,\psi(x,\xi)) +\lambda \psi(x,\xi)=\int_{-1}^{1}\kappa(x,\xi,\xi') f(x,\xi',\psi(x,\xi'))\,d\xi, \] where \(x\in[-a,a]\) and \(\xi\in[-1,1]\), with boundary conditions of the form \[ \psi^-=H \psi^+, \] where \(\psi^\pm\) are the restrictions to the incoming part \(D^-\) and outgoing part \(D^+\) of the phase space boundary, respectively, and \(H\) is a suitable linear operator from \(D^+\) to \(D^-\). The proof of the existence of a solution to the above boundary value problem in an \(L^p\), \(p\in(1,\infty)\), space context was obtained by the first author in [K. Latrach, J. Math. Phys. 37, No. 3, 1336–1348 (1996; Zbl 0869.45009)].
In the paper under review, the authors consider the problem in an \(L^1\) space. To this end, they prove a new version of Krasnosel’skij’s fixed point theorem, which reads as follows.
Theorem. Suppose that \(\mathcal M\) is a nonempty closed bounded convex subset of a Banach space \(\mathcal X\). Let \(A:\mathcal M \rightarrow \mathcal X\) and \(B:\mathcal X\rightarrow \mathcal X\) obey
(i) \(A\) is continuous, \(A\mathcal M\) is relatively weakly compact and \(A\) satisfies the following condition: if \((x_n)_{n\in\mathbb N}\) is a weakly convergent sequence in \(\mathcal X\), then \((Ax_n)_{n\in\mathbb N}\) has a strongly convergent subsequence in \(\mathcal X\);
(ii) \(B\) is a contraction and satisfies the condition: let \((x_n)_{n\in\mathbb N}\) be a weakly convergent sequence in \(\mathcal X\), then \((Bx_n)_{n\in\mathbb N}\) has a weakly convergent subsequence in \(\mathcal X\);
(iii) \((x=Bx+Ay,\quad y\in\mathcal M)\Rightarrow x\in\mathcal M\).
Then there is an element \(x\in\mathcal M\) with \(Ax+Bx=x\).
This fixed point theorem is then applied to the above transport equation, showing the existence of solutions \(\psi\) in \(L^1\) under suitable assumptions on the collision frequency \(\sigma\), the scattering kernel \(\kappa\), and the function \(f\).


47H10 Fixed-point theorems
82C70 Transport processes in time-dependent statistical mechanics
47N20 Applications of operator theory to differential and integral equations
35F30 Boundary value problems for nonlinear first-order PDEs


Zbl 0869.45009
Full Text: DOI


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