Numerical verifications of solutions for nonlinear elliptic equations

H. Akima, Bivariate interpolation and smooth surface fitting based on local procedures. Comm. ACM,17 (1974), 26–31.

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary Value Problems, Theory and Computation. Academic Press, London, 1984.

P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.

J.C. Eilbeck, The pseudo-spectral method and following in reaction-diffusion bifurcation studies. SIAM J. Sci. Stat. Comput.,17 (1986), 599–610.

E.W. Kaucher and W.L. Miranker, Self-Validating Numerics for Function Space Problems. Academic Press, New York, 1984.

G. Kedem, A posteriori error bounds for two-point boundary value problems. SIAM J. Numer. Anal.,18 (1981), 431–448.

P.L. Lions, On the existence of positive solutions of semilinear elliptic equations. SIAM Review,24 (1982), 441–467.

R.J. Lohner, Enclosing the solutions of ordinary initial and boundary value problems. Computerarithmetic (eds. E. Kaucher et al.), B.G. Teubner, Stuttgart, 1987, 255–286.

M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems. Japan J. Appl. Math.,5 (1988), 313–332.

M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems II. Japan J. Appl. Math.,7 (1990), 477–488.

M.T. Nakao, A computational verification method of existence of solutions for nonlinear elliptic equations. Lecture Notes in Numer. Appl. Anal.,10 1988, 101–120. Proc. Recent Topics in Nonlinear PDE 4, Kyoto, 1988, North-Holland/Kinokuniya, Amsterdam/Tokyo, 1989.

M.T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear BVP. J. Math. Anal. Appl.,164 (1992), 489–507.

M. Plum, Numerical Existence Proofs and Explicit Bounds for Solutions of Nonlinear Elliptic Boundary Value Problems. Computing,49 (1992), 25–44.

J. Schröder, A method for producing verified results for two-point boundary value problems. Comput. Suppl.,6 (1988), 9–22.

J. Smoller, Shock Waves and Reaction-Diffusion Equations. Springer, New York, 1983.

E. Zeidler, Nonlinear Functional Analysis and its Applications 1. Springer, New York, 1986.