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Boundary value problems for ODEs

Tadeusz Jankowski (2003)

Czechoslovak Mathematical Journal

We use the method of quasilinearization to boundary value problems of ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.

Existence and non-existence of sign-changing solutions for a class of two-point boundary value problems involving one-dimensional p -Laplacian

Yūki Naito (2011)

Mathematica Bohemica

We consider the boundary value problem involving the one dimensional p -Laplacian, and establish the precise intervals of the parameter for the existence and non-existence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.

Global structure of positive solutions for superlinear 2 m th-boundary value problems

Ruyun Ma, Yulian An (2010)

Czechoslovak Mathematical Journal

We consider boundary value problems for nonlinear 2 m th-order eigenvalue problem ( - 1 ) m u ( 2 m ) ( t ) = λ a ( t ) f ( u ( t ) ) , 0 < t < 1 , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = 0 , i = 0 , 1 , 2 , , m - 1 . where a C ( [ 0 , 1 ] , [ 0 , ) ) and a ( t 0 ) > 0 for some t 0 [ 0 , 1 ] , f C ( [ 0 , ) , [ 0 , ) ) and f ( s ) > 0 for s > 0 , and f 0 = , where f 0 = lim s 0 + f ( s ) / s . We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.

Nonsymmetric solutions of a nonlinear boundary value problem

Sámuel Peres (2014)

Czechoslovak Mathematical Journal

We study the existence and multiplicity of positive nonsymmetric and sign-changing nonantisymmetric solutions of a nonlinear second order ordinary differential equation with symmetric nonlinear boundary conditions, where both of the nonlinearities are of power type. The given problem has already been studied by other authors, but the number of its positive nonsymmetric and sign-changing nonantisymmetric solutions has been determined only under some technical conditions. It was a long-standing open...

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

Qingliu Yao (2013)

Applications of Mathematics

We consider the classical nonlinear fourth-order two-point boundary value problem u ( 4 ) ( t ) = λ h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , 0 < t < 1 , u ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = u ' ' ' ( 1 ) = 0 . In this problem, the nonlinear term h ( t ) f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) contains the first and second derivatives of the unknown function, and the function h ( t ) f ( t , x , y , z ) may be singular at t = 0 , t = 1 and at x = 0 , y = 0 , z = 0 . By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

Second order boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

John R. Graef, Lingju Kong, Qingkai Kong, Bo Yang (2011)

Mathematica Bohemica

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition u ' ' + g ( t ) f ( t , u ) = 0 , t ( 0 , 1 ) , u ( 0 ) = α u ( ξ ) + λ , u ( 1 ) = β u ( η ) + μ . C r i t e r i a f o r t h e e x i s t e n c e o f n o n t r i v i a l s o l u t i o n s o f t h e p r o b l e m a r e e s t a b l i s h e d . T h e n o n l i n e a r t e r m f ( t , x ) m a y t a k e n e g a t i v e v a l u e s a n d m a y b e u n b o u n d e d f r o m b e l o w . C o n d i t i o n s a r e d e t e r m i n e d b y t h e r e l a t i o n s h i p b e t w e e n t h e b e h a v i o r o f f ( t , x ) / x f o r x n e a r 0 a n d ± , a n d t h e s m a l l e s t p o s i t i v e c h a r a c t e r i s t i c v a l u e o f a n a s s o c i a t e d l i n e a r i n t e g r a l o p e r a t o r . T h e a n a l y s i s m a i n l y r e l i e s o n t o p o l o g i c a l d e g r e e t h e o r y . T h i s w o r k c o m p l e m e n t s s o m e r e c e n t r e s u l t s i n t h e l i t e r a t u r e . T h e r e s u l t s a r e i l l u s t r a t e d w i t h e x a m p l e s .

Two-mode bifurcation in solution of a perturbed nonlinear fourth order differential equation

Ahmed Abbas Mizeal, Mudhir A. Abdul Hussain (2012)

Archivum Mathematicum

In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.

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