## Abstract

An abstract unified theory of both monotone iterative and generalized quasilinearization methods is presented for operator equations of coincidence type in ordered Banach spaces. Applications are given for semilinear problems in \(C(\overline{\Omega};\mathbb{R}^k)\) and \(L^p(\Omega;\mathbb{R}^k)\).

## Authors

**Adriana Buica
**Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

**Radu Precup**

Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

## Keywords

generalized quasilinearization; monotone iterative technique; upper and lower solutions; quadratic convergence; coincidence; ordered Banach space.

## Paper coordinates

A. Buica, R. Precup, *Abstract generalized quasiliniarization method for coincidences*, Nonlinear Stud., **9** (2002), 371-387.

## About this paper

##### Journal

Nonlinear Studies

##### Publisher Name

##### DOI

##### Print ISSN

##### Online ISSN

MR 1940557, Zbl 1020.65031

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[1] B. Ahmad, J.J. Nieto and N. Shahzad, The Bellman-Kalaba-Lakshmikantham quasilinearization method for Neumann problems, J. Math. Anal. Appl. 257 (2001), 356–363.

[2] M. Balasz and I. Muntean, A unification of Newton’s methods for solving equations, Mathematica (Cluj) 44 (1979), 117–122.

[3] R. Bellman and R. Kalaba, ”Quasilinearization and Nonlinear Boundary- Value Problems”’, American Elsevier, New York, 1965.

[4] A. Buica, Some remarks on monotone iterative technique, Rev. Anal. Numer. Theor. Approx., to appear.

[5] A. Buica, Monotone iterations for the initial value problem, Seminar on Fixed Point Theory Cluj-Napoca 3 (2002), 137–148.

[6] A. Buica and R. Precup, Monotone Newton-type iterations for nonlinear equations, Proc. Tiberiu Popoviciu Itinerant Seminar on Functional Equations, Approximation and Convexity, E. Popoviciu ed., Srima, Cluj, 2002, to appear.

[7] S. Carl and S. Heikkila, Operator and differential equations in ordered spaces, J. Math. Anal. Appl. 234 (1999), 31–54.

[8] S. Carl and V. Lakshmikantham, Generalized quasilinearization for quasilinear parabolic equations with nonlinearities of DC type,J. Optim. Theory Appl. 109 (2001), 27–50.

[9] G. Goldner and R. Trımbitas, A combined method for a two-point bondary value problem, Pure Math. Appl. 11 (2000), 255–264.

[10] S. Heikkila and V. Lakshmikantham, ”Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations”, Marcel Dekker, New York, 1994.

[11] G.S. Ladde, V. Lakshmikantham and A.S. Vatsala, ”Monotone Iterative Techniques for Nonlinear Differential Equations”, Pitman, Boston, 1985.

[12] V. Lakshmikantham, Further improvement of generalized quasilinearization, Nonlinear Anal. 27(1996), 223–227.

[13] V. Lakshmikantham, S. Leela and S. Sivasundaram, Extensions of the method of quasilinearization, J. Optim. Theory Appl. 87 (1995), 379–401.

[14] V. Lakshmikantham and S. Malek, Generalized quasilinearization, Nonlinear World 1 (1994), 59–63.

[15] V. Lakshmikantham and A.S. Vatsala, ”Generalized Quasilinearization for Nonlinear Problems”, Kluwer Academic Publishers, Dordrecht, 1998.

[16] F.A. Mc Rae, Generalized quasilinearization of stochastic initial-value problems, Stochastic Anal. Appl. 13 (1995), 205–210.

[17] J.J. Nieto, Generalized quasilinearization for a second order ordinary differential equation with Dirichlet boundary conditions, Proc. Amer. Math. Soc. 125(1997), 2599–2604.

[18] F.A. Potra, W.C. Rheinboldt, On the monotone convergence of Newton’s method, Computing 36 (1986), 81–90.

[19] R. Precup, Monotone technique to the initial value problem for a delay integral equation from biomathematics, Studia Univ. Babe¸s-Bolyai Math. 40 (1995), 63–73.

[20] R. Precup, Convexity and quadratic monotone approximation in delay differential equations, Proc. Sci. Comm. Meeting of Aurel Vlaicu Univ., Gh. Halic ed., Arad, 1997, 153–158.

[21] R. Precup, ”Methods in Nonlinear Integral Equations”, Kluwer Academic Publishers, Dordrecht-Boston-London, 2002.

[22] J.W. Schmidt and H. Schneider, Monoton einschlie¯ende Verfahren bei additiv zerlegbaren Gleichungen, Z. Angew. Math. Mech. 63 (1983), 3–11.

[23] J.E. Vandergraft, Newton’s method for convex operators in partially ordered spaces, SIAM J. Numer. Anal. 4 (1967), 406–432.