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


[1] Appell, J., The superposition operator in function spaces—a survey, Expo. Math., 6, 209-270 (1988) · Zbl 0648.47041
[2] Appell, J.; De Pascale, E., Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Un. Mat. Ital. B(6), 3, 497-515 (1984) · Zbl 0507.46025
[3] Barroso, C. R., Krasnoslskii’s fixed point theorem for weakly continuous maps, Nonlinear Anal., 55, 25-31 (2003) · Zbl 1042.47035
[4] Brezis, H., Analyse Fonctionnelle. Théorie et Applications (1983), Masson: Masson Paris · Zbl 0511.46001
[5] Burton, T. A., A fixed point theorem of Krasnosel’skii, Appl. Math. Lett., 11, 85-88 (1998) · Zbl 1127.47318
[6] Cercignani, C., Mathematical Methods in Kinetic Theory (1990), Plenum Press: Plenum Press New York · Zbl 0726.76083
[7] Chow, S. N.; Hale, J. K., Methods of Bifurcation Theory (1982), Springer: Springer Berlin · Zbl 0487.47039
[8] Dautray, R.; Lions, J. L., (Analyse Mathématique et Calcul Numérique, vol. 9 (1988), Masson: Masson Paris)
[9] De Blasi, F. S., On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. Roumante, 21, 259-262 (1977) · Zbl 0365.46015
[10] Dunford, N.; Schwartz, J. T., Linear Operators, Part I: General Theory (1958), Interscience: Interscience New York · Zbl 0084.10402
[11] Greenberg, W.; Van der Mee, C.; Protopopescu, V., Boundary Value Problems in Abstract Kinetic Theory (1987), Birkhäuser: Birkhäuser Basel · Zbl 0624.35003
[12] Krasnoselskii, M. A., Integral Operators in Space of Summable Functions (1976), Noordhoff: Noordhoff Leyden · Zbl 0312.47041
[13] Latrach, K., Compactness properties for linear transport operator with abstract boundary conditions in slab geometry, Transport Theory Statist. Phys., 22, 39-64 (1993) · Zbl 0774.45006
[14] Latrach, K., Time asymptotic behaviour for linear mono-energetic transport equations with abstract boundary conditions in slab geometry, Transport Theory Statist. Phys., 23, 633-670 (1994) · Zbl 0816.45008
[15] Latrach, K., Quelques remarques sur les équations de transport avec des opérateurs de collision du type Hammerstein, C. R. Acad. Sci. Paris, Série I, 321, 1431-1436 (1995) · Zbl 0845.35019
[16] Latrach, K., On a nonlinear stationary problem arising in transport theory, J. Math. Phys., 37, 1336-1348 (1996) · Zbl 0869.45009
[17] Latrach, K.; Dehici, A., Relatively strictly singular perturbations, essential spectra and applications, J. Math. Anal. Appl., 252, 767-789 (2000) · Zbl 0976.47008
[18] O’Regan, D., Fixed-point theory for weakly sequentially continuous mappings, Math. Comput. Model., 27, 1-14 (1998) · Zbl 1185.34026
[19] O’Regan, D., Weak solutions of ordinary differential equations in Banach spaces, Appl. Math. Lett., 12, 101-105 (1999) · Zbl 0933.34068
[20] Smart, D. R., Fixed Point Theorems (1980), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0427.47036
[21] Zeidler, E., Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems (1993), Springer: Springer New York · Zbl 0794.47033
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