Eigenvalues of complementary Lidstone boundary value problems

Abstract

We consider the following complementary Lidstone boundary value problemwhere l > 0. The values of l are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of l such that for any l in the interval, the existence of a positive solution of the boundary value problem is guaranteed. Some examples are also included to illustrate the results obtained. Note that the nonlinear term F depends on y' and this derivative dependence is seldom investigated in the literature.

Existence results for a fractional boundary value problem with fractional Lidstone conditions

“…It is noted that in most of the existence studies, the fractional character of the boundary value problem considered do not involve fractional derivatives of the Lidstone conditions, see for example [8,9]. To read more about Lidstone boundary value problems, we refer the reader to the interesting papers of Agarwal et al [1,2]. We can also cite the interesting papers of Webb et al [14,15], where the authors proved the existence of positive solutions by the use of fixed point theorems and in the presence of different type of boundary conditions.…”

“…Lidstone boundary value problems for ordinary differential equations have attracted considerable attention lately. Moreover, the Lidstone boundary value problems and several of its particular cases have been investigated in many papers, see [1,2,4,7,10,13,16,17]. For example the case q = 4, σ 1 = 0 and σ 2 = 2 has been studied in [4], the authors proved the existence of at least three symmetric positive solutions by using Leggett and Williams fixed point theorem, to the fourth order Lidstone boundary value problem y (4) …”

On Coupled Systems of Lidstone-Type Boundary Value Problems

“…Over the course of several years, one can find several different approaches and techniques on problems or the family of problems with boundary value problems as Lidstone. For example, in [1], Agarwal and Wong deal with the existence of a positive solution of the complementary Lidstone boundary value problem (−1) m γ (2m+1) (t) =λF (t, γ(t), γ (t)) t ∈ (0, 1),…”