Showing papers by "Lingju Kong published in 2008"
TL;DR: In this paper, the boundary value problem was considered in the context of boundary value maximization, where the authors considered the problem of finding a boundary value for a given set of variables.
99 citations
TL;DR: The authors establish the existence of nonnegative solutions in the case where the associated Green’s function may have zeros and illustrated with an example.
57 citations
TL;DR: In this paper, the authors studied the nonlinear boundary value problem with multi-point boundary condition ( | u | p − 1 u ] = f ( t, u, u ǫ, uǫ ) = u ( 2 i ) (1) = ∑ j = 1 m a i j u (2 i )( t j ), i = 0, 1.
Abstract: We study the nonlinear boundary value problem with multi-point boundary condition ( | u ″ | p − 1 u ″ ) ″ = f ( t , u , u ′ , u ″ ) , t ∈ ( 0 , 1 ) , u ( 2 i ) ( 0 ) = u ( 2 i ) ( 1 ) = ∑ j = 1 m a i j u ( 2 i ) ( t j ) , i = 0 , 1 . Necessary and sufficient conditions are obtained for the existence of symmetric positive solutions of this problem using fixed point theorems on cones. Applications of our results to the special case where f is a power function of u and its derivatives are also discussed. Moreover, similar conclusions for a more general higher order boundary value problem are established. Our results extend some recent work in the literature on boundary value problems for ordinary differential equations.
29 citations
01 Sep 2008
TL;DR: In this article, the authors considered classes of second order boundary value problems with a nonlinearity f (t, x) in the equations and subject to a multi-point boundary condition.
Abstract: We consider classes of second order boundary value problems with a nonlinearity f (t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship between the behavior of the quotient f (t, x)/x for x near 0 and ∞, and the largest positive eigenvalue of a related linear integral operator. Our analysis mainly relies on the topological degree and fixed point index theories.
27 citations
TL;DR: The existence and asymptotics of real eigenvalues are established for the half-linear Sturm-Liouville problem consisting of the above equation and a separated boundary condition when w changes sign.
Abstract: We study the regular half-linear Sturm-Liouville equation -(p@f"r(y^'))^'+q@f"r(y)=@lw@f"r(y)on J=(a,b), where @f"r(u)=|u|^r^-^1u, r>0, p^-^1^r,q,w@?L(a,b), and p>0 a.e. on J. Let N(@l) denote the number of zeros in J of a nontrivial solution of the equation. Asymptotic formulas are found for N(@l) when w>=0 a.e. and w changes sign, respectively. As a consequence, the existence and asymptotics of real eigenvalues are established for the half-linear Sturm-Liouville problem consisting of the above equation and a separated boundary condition when w changes sign. Our results cover the work of Atkinson and Mingarelli on second-order linear equations as a special case. The generalized Prufer transformation plays a key role in the proofs.