Showing papers by "Lingju Kong published in 2009"

Existence results for nonlinear periodic boundary-value problems

Abstract: We study a class of second-order nonlinear differential equations on a finite interval with periodic boundary conditions. The nonlinearity in the equations can take negative values and may be unbounded from below. Criteria are established for the existence of non-trivial solutions, positive solutions and negative solutions of the problems under consideration. Applications of our results to related eigenvalue problems are also discussed. Examples are included to illustrate some of the results. Our analysis relies mainly on topological degree theory.

Existence of Solutions for a Higher Order Multi-Point Boundary Value Problem

Abstract: We study the n-th order multi-point boundary value problem $$ u^{(n)} + f(t, u, u', \ldots ,u^{(n - 1)} ) = \lambda p(t), \quad t \in (0,1), $$ $$\left\{ \begin{array}{ll} u^{(i)} (0) = A_{i}, \, i = 0, \ldots , n - 3,\\ u^{(n - 2)} (0) - \sum olimits_{j = 1}^m {a_j u^{(n - 2)} (t_j ) = A_{n - 2}},\\ u^{(n - 2)} (1) - \sum olimits_{j = 1}^m {b_j u^{(n - 2)} (t_{j} ) = A_{n - 1}} \end{array} \right. $$ . Sufficient conditions are obtained for the existence of one and two solutions of the problem for different values of λ. Our results extend and improve some recent work in the literature. Our analysis mainly relies on the lower and upper solution method and topological degree theory.

Nodal solutions of second order nonlinear boundary value problems

Abstract: We study the nonlinear boundary value problem (BVP) consisting of the equation −(p(t)y � ) � +q(t)y = w(t) f (y) on [a,b] and a general separated boundary condition (BC). By comparing it with a linear Sturm–Liouville problem (SLP) we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let λn,n = 0, 1, 2 ,..., be the nth eigenvalue of the corresponding linear SLP. Then the BVP has a pair of solutions with exactly n zeros in (a,b) if λn is in the interior of the range of f (y)/y; and does not have any solution with exactly n zeros in (a,b) if λn is outside this range. These conditions become necessary and sufficient when f (y)/y is monotone on (−∞, 0) and on (0, ∞). We also discuss the changes of the number of different types of nodal solutions as the equation or the BC changes. Our results are obtained without assuming the global existence and uniqueness of solutions of the corresponding initial value problems.

Positive Solutions for Third Order Boundary Value Problems with p-Laplacian

Abstract: We consider the following boundary value problem with nonhomogeneous three-point boundary condition $$ \begin{array}{l} \left( {\phi _p ( {u^\prime } )} \right)^{\prime\prime} + a( t )f( u ) = 0, \quad t \in ( {0,1} ), \\ u( 0 ) = \xi u( \eta ) + \lambda , \quad u^\prime ( 0 ) = u^\prime( 1 ) = 0 \\ \end{array}$$ We derive several existence, nonexistence, and multiplicity results for positive solutions in terms of different values of the parameter λ The uniqueness of positive solutions and the dependence of positive solutions on the parameter λ are also studied

Existence of nodal solutions of multi-point boundary value problems

Abstract: We study the nonlinear boundary value problem consisting of the equation $y^{''}+ w(t)f(y)=0$ on $[a,b]$ and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes.

Nodal solutions of nonlocal boundary value problems

Abstract: We study the nonlinear boundary value problem consisting of the second order differential equation on [a, b] and a boundary condition involving a Riemann‐Stieltjes integral. By relating it to the eigenvalues of a linear Sturm‐Liouville problem with a two‐point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes. First published online: 14 Oct 2010

Positive solutions of a nonlinear higher order boundary-value problem

Abstract: The authors study a higher order three point boundary value problem. Estimates for positive solutions are given; these estimates improve some recent results in the literature. Using these estimates, new sufficient conditions for the existence and nonexistence of positive solutions of the problem are obtained. An example illustrating the results is included.