DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS
DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS
DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS
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TYPES OF SOLUTIONS AND MULTIPLICITY RESULTS<br />
FOR FOURTH-ORDER NONLINEAR BOUNDARY<br />
VALUE PROBLEMS<br />
F. SADYRBAEV AND I. YERMACHENKO<br />
The multiplicity results for the problem x (4) = f (t,x,x ′′ ), (i) x(0) = x ′ (0) = 0 = x(1) =<br />
x ′ (1), (ii) are presented, where the right side in (i) is monotone with respect to x and<br />
x ′ . Our considerations are based on the types of solutions, which are introduced using<br />
the notion of conjugate points by Leighton and Nehari. One of the main results is that<br />
the quasilinear equation x (4) − p(t)x ′′ − q(t)x = F(t,x,x ′′ ), (iii) along with the boundary<br />
conditions (ii), has a solution which possesses the oscillatory properties induced by the<br />
linear part in (iii). Results are applied to the Emden-Fowler-type equations.<br />
Copyright © 2006 F. Sadyrbaev and I. Yermachenko. This is an open access article distributed<br />
under the Creative Commons Attribution License, which permits unrestricted<br />
use, distribution, and reproduction in any medium, provided the original work is properly<br />
cited.<br />
1. Introduction<br />
In this paper we consider the fourth-order nonlinear differential equations<br />
together with the boundary conditions<br />
x (4) = f (t,x,x ′′ ), t ∈ I := [0,1], (1.1)<br />
x (4) − p(t)x ′′ − q(t)x = F(t,x,x ′′ ) (1.2)<br />
x(0) = x ′ (0) = 0 = x(1) = x ′ (1). (1.3)<br />
Function f is supposed to be continuous together with the partial derivatives f x and f x<br />
′′.<br />
Functions F, F x , F x<br />
′′ are continuous and F is bounded, that is, |F(t,x,x ′′ )| 0 are continuous functions. We prove first that quasilinear boundary value<br />
problem (1.2), (1.3) has a solution which hereditates oscillatory properties of the linear<br />
part (L 4 x)(t):= x (4) − p(t)x ′′ − q(t)x, provided that the linear part is nonresonant, that<br />
is, the homogeneous problem (L 4 x)(t) = 0, (1.3) has only the trivial solution. We consider<br />
then (1.1) (together with the boundary conditions (1.3)) under some monotonicity-type<br />
Hindawi Publishing Corporation<br />
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 989–998
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