## 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 $$f$$.

### MSC:

 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
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### References:

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