Triple solutions of complementary Lidstone boundary value problems via fixed point theorems
- PDF / 469,493 Bytes
- 21 Pages / 595.276 x 793.701 pts Page_size
- 4 Downloads / 158 Views
RESEARCH
Open Access
Triple solutions of complementary Lidstone boundary value problems via fixed point theorems Patricia JY Wong* * Correspondence: [email protected] School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
Abstract We consider the following complementary Lidstone boundary value problem: (–1)m y(2m+1) (t) = F (t, y(t), y (t)), y(0) = 0,
t ∈ [0, 1],
y(2k–1) (0) = y(2k–1) (1) = 0,
1 ≤ k ≤ m.
By using fixed point theorems of Leggett-Williams and Avery, we offer several criteria for the existence of three positive solutions of the boundary value problem. Examples are also included to illustrate the results obtained. We note that the nonlinear term F depends on y and this derivative dependence is seldom investigated in the literature and a new technique is required to tackle the problem. MSC: 34B15; 34B18 Keywords: positive solutions; complementary Lidstone boundary value problems; derivative-dependent nonlinearity; fixed point theorems
1 Introduction In this paper we shall consider the complementary Lidstone boundary value problem (–)m y(m+) (t) = F t, y(t), y (t) , y() = ,
t ∈ [, ],
y(k–) () = y(k–) () = ,
≤ k ≤ m,
(.)
where m ≥ and F is continuous at least in the interior of the domain of interest. It is noted that the nonlinear term F involves y , a derivative of the dependent variable. Most research papers on boundary value problems consider nonlinear terms that involve y only, and derivative-dependent nonlinearities are seldom tackled as special techniques are required. The complementary Lidstone interpolation and boundary value problems have been very recently introduced in [], and drawn on by Agarwal et al. in [, ] where they consider an (m + )th order differential equation together with boundary data at the odd order derivatives y() = a ,
y(k–) () = ak ,
y(k–) () = bk ,
≤ k ≤ m.
(.)
©2014 Wong; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wong Boundary Value Problems 2014, 2014:125 http://www.boundaryvalueproblems.com/content/2014/1/125
Page 2 of 21
The boundary conditions (.) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [–] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an mth order differential equation and the Lidstone boundary conditions y(k) () = ak ,
y(k) () = bk ,
≤ k ≤ m – .
(.)
There is a vast literature on Lidstone interpolation and boundary value problems. In fact, the Lidstone interpolation was first introduced by Lidstone [] in and further characterized in the work of [–]. More recent research on Lidstone interpolation as well as Lidstone spline can be found in
Data Loading...
