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Steepest descent and differential equations
J. W. NEUBERGER
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J. W. NEUBERGER
Department of Mathematics North Texas State University
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1985 Volume 37 Issue 2 Pages 187-195
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- Published: 1985 Received: January 31, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Correction information
Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : In memory of my teacher, H.S. Wall
Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. A. Adams, Sobolev Spaces, Academic Press, 1975.
2) S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.
3) J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math., 4(1978), 1-9.
4) C. Beasley, Finite element solution to nonlinear partial differential equations, Dissertation, North Texas State Univ., 1981.
5) M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.
6) H. Brezis, Opéxateurs maximaux monotones et semi-groupes nonlinéares, Math. Studies 5, North Holland, 1973.
7) F. E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis, Proc. Sympos. Pure Math., 16 (1970), Amer. Math. Soc., 1-60.
8) P. L. Cauchy, Méthode générale pour la résolution des systemes d'équations simultanées, C. R. Acad. Sci. Paris, 25 (1847), 536-538.
9) H. B. Curry, The method of steepest descent for nonlinear minimization problems, Quart. Appl. Math., 2(1944), 258-261.
10) R. Fletcher, Unconstrained Optimization, John Wiley, 1980.
11) M. Hestenes, Conjugate Direction Methods in Optimization, Springer-Verlag, 1980.
12) L. Hörmander, Linear Partial Differential Operators, Academic Press, 1963.
13) M. Liaw, The steepest descent method using finite elements for systems of nonlinear partial differential equations, Dissertation, North Texas State University, 1981.
14) J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.
15) M. Z. Nashed, Aspects of Generalized Inverses in Analysis and Regularization, Generalized Inverses and Applications, Academic Press, 1976.
16) J. W. Neuberger, A type-independent numerical method for solving systems of nonlinear partial differential equations: Applications to the problem of transonic flow, Comput. Math. Appl., 6(1980), 67-78.
17) J. W. Neuberger, Finite dimensional potential theory applied to numerical analysis, Linear Algebra Appl., 35 (1981), 193-202.
18) J. W. Neuberger, Steepest descent for systems of nonlinear partial differential equations, Oak Ridge National Laboratory CSD/TM-161 (1981).
19) J. W. Neuberger, Steepest descent for general systems of linear differential equations, Lecture Notes in Math., 1032 (1983), Springer-Verlag, 390-406.
20) J. W. Neuberger, Some global steepest descent results for nonlinear systems, Trends Th. Appl. Duff. Eq., Marcel Dekker, 1984.
21) J. W. Neuberger and R. J. Renka, A portable software package for nonlinear partial differential equations, Oak Ridge National Laboratory, CSD-102 (1982).
22) L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 4(1981), 267-302.
23) J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, 1970.
24) R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc., 70 (1964), 165-172.
25) A. Pazy, Strong convergence of semigroups on nonlinear contractions in Hilbert space, J. Analyse Math., 34 (1978), 1-35.
26) P. C. Rosenblum, The method of steepest descent, Proc. Sympos. Appl. Math. VI, Amer. Math. Soc.
27) P. H. Rabinowitz, A minimax principle and applications to elliptic partial differential equations, Lecture Notes in Math., 648 (1978), Springer-Verlag, 97-115.
28) F. Treves, Locally Convex Spaces and Linear Partial Differential Equations, Springer-Verlag, 1967.
29) K. Yosida, Functional Analysis, Springer-Verlag, 1965.
Right : [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
[2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math., 12 (1959), 623-727.
[3] J. B. Baillon, R. E. Bruck and S. Reich, On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math., 4 (1978), 1-9.
[4] C. Beasley, Finite element solution to nonlinear partial differential equations, Dissertation, North Texas State Univ., 1981.
[5] M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.
[6] H. Brezis, Opérateurs maximaux monotones et semi-groupes nonlinéares, Math. Studies 5, North Holland, 1973.
[7] F. E. Browder, Existence theorems for nonlinear partial differential equations, Global Analysis, Proc. Sympos. Pure Math., 16 (1970), Amer. Math. Soc., 1-60.
[8] P. L. Cauchy, Méthode générale pour la résolution des systemes d'équations simultanées, C. R. Acad. Sci. Paris, 25 (1847), 536-538.
[9] H. B. Curry, The method of steepest descent for nonlinear minimization problems, Quart. Appl. Math., 2 (1944), 258-261.
[10] R. Fletcher, Unconstrained Optimization, John Wiley, 1980.
[11] M. Hestenes, Conjugate Direction Methods in Optimization, Springer-Verlag, 1980.
[12] L. Hörmander, Linear Partial Differential Operators, Academic Press, 1963.
[13] M. Liaw, The steepest descent method using finite elements for systems of nonlinear partial differential equations, Dissertation, North Texas State University, 1981.
[14] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.
[15] M. Z. Nashed, Aspects of Generalized Inverses in Analysis and Regularization, Generalized Inverses and Applications, Academic Press, 1976.
[16] J. W. Neuberger, A type-independent numerical method for solving systems of nonlinear partial differential equations: Applications to the problem of transonic flow, Comput. Math. Appl., 6 (1980), 67-78.
[17] J. W. Neuberger, Finite dimensional potential theory applied to numerical analysis, Linear Algebra Appl., 35 (1981), 193-202.
[18] J. W. Neuberger, Steepest descent for systems of nonlinear partial differential equations, Oak Ridge National Laboratory CSD/TM-161 (1981).
[19] J. W. Neuberger, Steepest descent for general systems of linear differential equations, Lecture Notes in Math., 1032 (1983), Springer-Verlag, 390-406.
[20] J. W. Neuberger, Some global steepest descent results for nonlinear systems, Trends Th. Appl. Diff. Eq., Marcel Dekker, 1984.
[21] J. W. Neuberger and R. J. Renka, A portable software package for nonlinear partial differential equations, Oak Ridge National Laboratory, CSD-102 (1982).
[22] L. Nirenberg, Variational and topological methods in nonlinear problems, Bull. Amer. Math. Soc. (N. S.), 4(1981), 267-302.
[23] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, 1970.
[24] R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc., 70 (1964), 165-172.
[25] A. Pazy, Strong convergence of semigroups on nonlinear contractions in Hilbert space, J. Analyse Math., 34 (1978), 1-35.
[26] P. C. Rosenblum, The method of steepest descent, Proc. Sympos. Appl. Math. VI, Amer. Math. Soc.
[27] P. H. Rabinowitz, A minimax principle and applications to elliptic partial differential equations, Lecture Notes in Math., 648 (1978), Springer-Verlag, 97-115.
[28] F. Treves, Locally Convex Spaces and Linear Partial Differential Equations, Springer-Verlag, 1967.
[29] K. Yosida, Functional Analysis, Springer-Verlag, 1965.
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