×

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

Citations:

Zbl 0869.45009
PDF BibTeX XML Cite
Full Text: DOI

References:

[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 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 New York · Zbl 0726.76083
[7] Chow, S.N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer Berlin
[8] Dautray, R.; Lions, J.L., ()
[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 New York
[11] Greenberg, W.; Van der Mee, C.; Protopopescu, V., Boundary value problems in abstract kinetic theory, (1987), Birkhäuser Basel · Zbl 0624.35003
[12] Krasnoselskii, M.A., Integral operators in space of summable functions, (1976), Noordhoff Leyden
[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 · Zbl 0427.47036
[21] Zeidler, E., Nonlinear functional analysis and its applications I: fixed-point theorems, (1993), Springer New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.