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A constructive monotone iterative method for second-order BVP in the presence of lower and upper solutions. (English) Zbl 1024.65063
Summary: This paper concerns the monotone approximations of solutions of boundary value problems (BVPs) such as $$-u''+ f(t,u,u')= 0,\quad u'(0)= u'(1)= 0.$$ We consider linear iterative scheme in case $f$ is Lipschitz in $u'$ and satisfies a one-sided Lipschitz condition in $u$. The initial approximations are lower and upper solutions which can be ordered one way $(\alpha\le\beta)$ or the other $(\alpha\ge\beta)$. We also consider the periodic and the Dirichlet problems.

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B15 Nonlinear boundary value problems for ODE
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##### References:
 [1] Picard, E.: Sur l’application des méthodes d’approximations successives à l’étude de certaines équations différentielles ordinaires. J. math. 9, 217-271 (1893) · Zbl 25.0507.02 [2] C. De Coster, P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results, in: F. Zanolin (Ed.), Nonlinear Analysis and Boundary Value Problems for Ordinary Differ. Equ., C.I.S.M. Courses and Lectures, vol. 371, Springer, New York, 1996, pp. 1--79 · Zbl 0889.34018 [3] Cherpion, M.; De Coster, C.; Habets, P.: Monotone iterative methods for boundary value problems. Differ. integral equ. 12, 309-338 (1999) · Zbl 1015.34009 [4] Dragoni, G. Scorza: Il problema dei valori ai limiti studiato in grande per le equazioni differenziali del secondo ordine. Math. ann. 105, 133-143 (1931) · Zbl 57.0506.01 [5] Babkin, B. N.: Solution of a boundary value problem for an ordinary differential equation of second order by caplygin’s method. Prikl. math. Meh. akad. Nauk. SSSR 18, 239-242 (1954) [6] Gendzojan, G. V.: On two-sided Chaplygin approximations to the solution of the two point boundary value problem. Izv. SSR jiz mate nauk 17, 21-27 (1964) [7] Bernfeld, S. R.; Chandra, J.: Minimal and maximal solutions of nonlinear boundary value problems. Pacific J. Math. 71, 13-20 (1977) · Zbl 0353.34024 [8] Bellen, A.: Monotone methods for periodic solutions of second order scalar functional differential equations. Numer. math. 42, 15-30 (1983) · Zbl 0536.65065 [9] Omari, P.: A monotone method for constructing extremal solutions of second order scalar BVPs. Appl. math. Comput. 18, 257-275 (1986) · Zbl 0625.65075 [10] Amann, H.; Ambrosetti, A.; Mancini, G.: Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Math. Z. 158, 179-194 (1978) · Zbl 0368.35032 [11] De Coster, C.; Henrard, M.: Existence and localization of solutions for elliptic problem in presence of lower and upper solutions without any order. J. differ. Equations 145, 420-452 (1998) · Zbl 0908.35042 [12] Omari, P.; Trombetta, M.: Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problem. Appl. math. Comput. 50, 1-21 (1992) · Zbl 0760.65078 [13] Cabada, A.; Sanchez, L.: A positive operator approach to the Neumann problem for a second order ordinary differential equation. J. math. Anal. appl. 204, 774-785 (1996) · Zbl 0871.34014 [14] A. Cabada, P. Habets, R. Pouso, Optimal existence conditions for {$\Phi$}-Laplacian equations with upper and lower solutions in the reversed order, J. Differ. Equations 166(2) (2000) 385--401 · Zbl 0999.34011 [15] A. Cabada, P. Habets, S. Lois, Monotone method for the Neumann problem with lower and upper solutions in the reversed order, Appl. Math. Comput. 117 (2001) 1--14 · Zbl 1031.34021 [16] Kiguradze, I. T.: A priori estimates for derivatives of bounded functions satisfying second-order differential inequalities. Differentsial’nye uravnenija 3, 1043-1052 (1967) [17] Schmitt, K.: Boundary value problems for quasilinear second order elliptic equations. J. nonlinear anal. Theory meth. Appl. 2, 263-309 (1978) · Zbl 0378.35022