Electron. J. Diff. Equ., Vol. 2012 (2012), No. 04, pp. 1-10.

Monotone iterative method and regular singular nonlinear BVP in the presence of reverse ordered upper and lower solutions

Amit K. Verma

Abstract:
Monotone iterative technique is employed for studying the existence of solutions to the second-order nonlinear singular boundary value problem
$$ -\big(p(x)y'(x)\big)'+p(x)f\big(x,y(x),p(x)y'(x)\big)=0 $$
for $0<x<1$ and $y'(0)=y'(1)=0$. Here $p(0)=0$ and $x p'(x)/p(x)$ is analytic at $x=0$. The source function $f(x,y,py')$ is Lipschitz in $py'$ and one sided Lipschitz in $y$. The initial approximations are upper solution $u_0(x)$ and lower solution $v_0(x)$ which can be ordered in one way $v_0(x)\leq u_0(x)$ or the other $u_0(x)\leq v_0(x)$.

Submitted October 19, 2011. Published January 9, 2012.
Math Subject Classifications: 34B16.
Key Words: Monotone iterative technique; lower and upper solutions; Neumann boundary conditions.

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Amit K. Verma
Department of Mathematics, BITS Pilani
Pilani - 333031, Rajasthan, India
Phone +919413789285; fax: +911596244183
email: amitkverma02@yahoo.co.in, akverma@bits-pilani.ac.in

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