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Keywords:
partially ordered space; Newton-like iteration; affine-invariant; monotone including iteration methods; systems of nonlinear ordinary differential equations
Summary:
In this paper we present a new theorem for monotone including iteration methods. The conditions for the operators considered are affine-invariant and no topological properties neither of the linear spaces nor of the operators are used. Furthermore, no inverse-isotony is demanded. As examples we treat some systems of nonlinear ordinary differential equations with two-point boundary conditions.
References:
[1] Alefeld G.: Monotone Regula-falsi-ähnliche Verfahren bei nichtkonvexen Operatorgleichungen. Beitr. Numer. Math. 8 (1980), 15-30. MR 0564583 | Zbl 0425.65034
[2] Frommer N.: Monotonicity of Brown's Method. Z. Angew. Math. Mech. 68 (1988), 101-110. DOI 10.1002/zamm.19880680211 | MR 0931771 | Zbl 0663.65047
[3] Korrnan P., Leung A. W.: A general monotone scheme for elliptic systems with applications to ecological models. Proc. Roy. Soc. Edinb. 102A (1986), 315-325.
[4] Krasnoselski M.: Positive Solutions of Operator Equations. Noordhoff, Groningen, 1964. MR 0181881
[5] McKenna P. J., Walter W.: On the Dirichlet Problem for Elliptic Systems. Appl. Anal. 21 (1986), 207-224. DOI 10.1080/00036818608839592 | MR 0840313 | Zbl 0593.35042
[6] Ortega J. M., Rheinboldt W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. Acad. Press, New York, 1970. MR 0273810
[7] Potra F. A.: Newton-like methods with monotone convergence for solving nonlinear operator equations. Nonl. Anal. Th., Meth. Appl. 11 (1987), 697-717. MR 0893775 | Zbl 0633.65050
[8] Potra F. A.: Monotone iterative methods for nonlinear operator equations. Numer. Funct. Anal. and Optimiz. 9 (1987), 809-843. MR 0910856 | Zbl 0636.65056
[9] Potra F.A., Rheinboldt W.C.: On the monotone convergence of Newton's method. Computing 36 (1986), 81-90. DOI 10.1007/BF02238194 | MR 0832932 | Zbl 0572.65034
[10] Schmidt J. W., Schneider H.: Monoton einschließende Verfahren bei additiv zerlegbaren Gleichungen. Z. Angew. Math. Mech. 63 (1983), 3-11. MR 0701830 | Zbl 0519.65036
[11] Schmidt J. W., Schneider H.: Enclosing methods in perturbated nonlinear operator equations. Comput. 32 (1984), 1-11. MR 0736257
[12] Voller R. L.: Monoton einschließende Newton-ähnliche Iterationsverfahren in halbgeordneten Räumen mit nicht notwendig regularem Kegel. Dissertation, Düsseldorf 1982.
[13] Voller R. L.: Iterative Einschließung von Lösungen nichtlinearer Differentialgleichungen durch Newton-ähnliche Iterationsverfahren. Apl. Mat. 31 (1986), 1-18. MR 0836798
[14] Voss H.: Ein neues Verfahren zur Einschließung der Lösungen von Operatorgleichungen. Z. Angew. Math. Mech. 56 (1976), 218-219. DOI 10.1002/zamm.19760560509 | MR 0408240 | Zbl 0341.65040
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