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Computing positive fixed-points of decreasing Hammerstein operators by relaxed iterations. (English) Zbl 0979.65047

The authors study the iterative solution of abstract nonlinear operator equations of the type \[ u(x)= KNu(x),\quad x\in \Omega, \] where \(\Omega\) is a compact Hausdorff space, \(K\) is a linear completely continuous operator mapping the positive cone of the space of continuous real functions on \(\Omega\), into itself, and \(Nu(x):= 1/\sigma(x, u(x))\) is a Nemytskij operator with \(\sigma(\cdot,\cdot)\) being continuous and strictly positive, and – with respect to the second argument – being nondecreasing and sublinear. Examples are integral equations that arise in nuclear physics, in the theory of radiative transfer, and Chandrasekhars \(H\)-equation modelling some atmospheric heat transfer.
The equation is solved numerically by so-called stationary underrelaxation of Picard and updated Picard methods, and of Jacobi and Gauss-Seidel methods. Existence of a unique positive solution and global convergence are proved in a framework of fixed-point approximations for positive decreasing operators in ordered Banach spaces. Moreover, the methods above are compared with Newton-type methods.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65H10 Numerical computation of solutions to systems of equations
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
65R20 Numerical methods for integral equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
45G10 Other nonlinear integral equations
35K55 Nonlinear parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs

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References:

[1] E.L. Allgower, K. Böhmer, F.A. Potra and W.C. Rheinboldt, A mesh-independence principle for operator equations and their discretizations , SIAM J. Numer. Anal. 23 (1986), 160-169. JSTOR: · Zbl 0591.65043
[2] K.E. Atkinson, A survey of numerical methods for solving nonlinear integral equations , J. Integral Equations Appl. 4 (1992), 15-46. · Zbl 0760.65118
[3] C.T. Baker, The numerical treatment of integral equations , Clarendon Press, Oxford, 1977. · Zbl 0373.65060
[4] V.C. Boffi and G. Spiga, An equation of Hammerstein type arising in particle transport theory , J. Math. Phys. 24 (1983), 1625-1629. · Zbl 0526.45009
[5] V.C. Boffi, G. Spiga and J.R. Thomas, Jr., Solution of a nonlinear integral equation arising in particle transport theory , J. Comput. Phys. 59 (1985), 96-107. · Zbl 0579.65145
[6] R.K. Bose and M.C. Joshi, Some topics in nonlinear functional analysis , Wiley Eastern Limited, New Delhi, 1985. · Zbl 0596.47038
[7] P.B. Bosma and W.A. de Rooij, Efficient methods to calculate Chandrasekhar’s \(H\)-functions , Astronom. and Astrophys. 126 (1983), 283-292.
[8] S. Chandrasekhar, Radiative transfer , Dover, New York, 1960.
[9] S. Chandrasekhar and F.H. Breen, On the radiative equilibrium of a stellar atmosphere , XVI, Astrophys. J. 105 (1947), 435-440.
[10] L. Erbe, D. Guo and X. Liu, Positive solutions of a class of nonlinear integral equations and applications , J. Integral Equations Appl. 4 (1992), 179-196. · Zbl 0755.45006
[11] G.B. Folland, Real analysis , J. Wiley, New York, 1984.
[12] D. Guo, Some fixed point theorems and applications , Nonlinear Anal. 10 (1986), 77-84. · Zbl 0615.45008
[13] ——–, Existence and uniqueness of positive fixed points for mixed monotone operators and applications , Appl. Anal. 46 (1992), 91-100. · Zbl 0792.47053
[14] ——–, Positive fixed points and eigenvectors of noncompact decreasing operators with applications to nonlinear integral equations , Chinese Ann. of Math. Ser. B 4 (1993), 419-426. · Zbl 0805.47050
[15] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications , Nonlinear Anal. 11 (1987), 623-632. · Zbl 0635.47045
[16] ——–, Nonlinear problems in abstract cones , Academic Press, Inc., New York, 1988. · Zbl 0661.47045
[17] C.T. Kelley, Iterative methods for linear and nonlinear equations , Frontiers Appl. Math. 16 , SIAM, Philadelphia, 1995. · Zbl 0832.65046
[18] M.A. Krasnoselskii, Positive solutions of operator equations , P. Noordhoff, Groningen, The Netherlands, 1964.
[19] M.A. Krasnoselskii, G. Vainikko, P.P. Zabreiko, Ya.B. Rutitskii, V.Ya. Stetsenko, Approximate solution of operator equations , P. Noordhoff, Groningen, The Netherlands, 1972.
[20] R.W. Leggett, A new approach to the \(H\)-equation of Chandrasekhar , SIAM J. Math. Anal. 7 (1976), 542-550. · Zbl 0331.45012
[21] ——–, On certain nonlinear integral equations , J. Math. Anal. Appl. 60 (1977), 462-468. · Zbl 0352.45004
[22] J.M. Ortega and W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables , Academic Press, New York, 1970. · Zbl 0241.65046
[23] W.C. Rheinboldt, Methods for solving systems of nonlinear equations , SIAM, Philadelphia, 1974. · Zbl 0325.65022
[24] A. Sommariva and M. Vianello, Approximating fixed-points of decreasing operators in spaces of continuous functions , Numer. Funct. Anal. Optim. 19 (1998), 635-646. · Zbl 0907.45006
[25] ——–, Constructive analysis of purely integral Boltzmann models , J. Integral Equations Appl. 11 (1999), 393-404. · Zbl 0974.45005
[26] ——–, Constructive approximation for a class of perturbed Hammerstein integral equations , Nonlinear Anal., · Zbl 0956.45006
[27] C.A. Stuart, Positive solutions of a nonlinear integral equation , Math. Ann. 192 (1971), 119-124. · Zbl 0203.42101
[28] E. Zeidler, Nonlinear functional analysis and its applications I, Springer-Verlag, New York, 1986. · Zbl 0583.47050
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