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Improved convergence theorems of Newton’s method designed for the numerical verification for solutions of differential equations. (English) Zbl 1112.65101
Improved theorems on existence and uniqueness of solution of nonlinear elliptic equations are proved. A verification procedure for solutions is presented.

65N12Stability and convergence of numerical methods (BVP of PDE)
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65G20Algorithms with automatic result verification
35J65Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI
[1] Kawanago, T.: A symmetry-breaking bifurcation theorem and some related theorems applicable to maps having unbounded derivatives. Japan J. Indust. appl. Math. 21, 57-74 (2004) · Zbl 1054.37030
[2] Kawanago, T.: Computer assisted proof to symmetry-breaking bifurcation phenomena in nonlinear vibration. Japan J. Indust. appl. Math. 21, 75-108 (2004) · Zbl 1129.37338
[3] T. Kawanago, Some bifurcation theorems applicable to existence proofs of bifurcation points by computer, in: Lecture Notes for the 43rd Joint Symposium of Real and Functional Analysis at Iwate University, 2004, pp. 42 -- 56 (in Japanese).
[4] Nakao, M. T.: Numerical verification methods for the existence of solutions for functional equations. Sugaku expositions 5, 71-91 (1992)
[5] M.T. Nakao, K. Hashimoto, Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, preprint. · Zbl 1151.35337
[6] Nakao, M. T.; Watanabe, Y.: An efficient approach to the numerical verification for solutions of the elliptic differential equations. Numer. algorithms 37, 311-323 (2004) · Zbl 1114.65152
[7] M.T. Nakao, N. Yamamoto, The numerical methods with guaranteed accuracy (in Japanese), Nihon-Hyōron-sha, 1998.
[8] S. Oishi, The numerical methods with guaranteed accuracy (in Japanese), Corona-sha, 2000.
[9] Watanabe, Y.; Nakao, M. T.: Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. appl. Math. 10, 165-178 (1993) · Zbl 0784.65082