Nonlinear boundary value problems with multiple solutions

The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

“…(2) With the exception of a few, for example, [1,3], most of the results on the existence of solutions to multipoint boundary value problems are based upon fixed-point theorems of Krasnoselskii's type; see Krasnoselskii [2], Guo and Lakshmikantham [19], and recent articles of Ma [12,14,20], Liu [18], and Sun et al [13]. Other methods in nonlinear functional analysis such as Leray-Schauder continuation theorem, nonlinear alternative of Leray-Schauder fixed point theorem, and coincidence degree theory have also been used, see Mawhin [21,22].…”

“…This approach has an advantage over the traditional method of using fixed point theorems on cones by Krasnosel'skiȋ [2]. It has come to our attention after the publication of [1] that Baxley and Haywood [3] had also used similar ideas to study Dirichlet boundary value problems.…”

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

“…In a recent paper, Baxley and Haywood [5] consider the second order problem (3)-(4) and extend Henderson and Thompson's result to any odd number of symmetric nonnegative solutions. As they point out, "positive" in the conclusions of Theorems A and B above really means nonnegative.…”

“…They note that if the nonlinear function f is identically zero for 0 ≤ x ≤ a (see Theorem A above), then the smaller solution must in fact be the trivial one. While the analysis used by Baxley and Haywood [5] to obtain their results is quite nice, it does not extend to higher order equations.…”

“…They are interesting as well from a theoretical perspective. There has been a great deal of research work on the existence of positive solutions for BVPs, and we cite as recent examples the papers of Agarwal and Wong [3], Avery, Davis, and Henderson [4], Baxley and Haywood [5,6], Eloe et al [7,8], Erbe, Hu, and Wang [9], Henderson and Thompson [10], Ma, Zhang, and Fu [12], and Wong and Agarwal [13]. Additional results and extensive bibliographies can be found in the recent monographs by Agarwal [1] and Agarwal, O'Regan, and Wong [2].…”