Positive solutions of complementary Lidstone boundary value problems

We consider the following complementary Lidstone boundary value problem (−1)my (2m+1)(t) = F(t, y(t), y′ (t)), t ∈ [0, 1] y(0) = 0, y(2k−1)(0) = y (2k−1)(1) = 0, 1 ≤ k ≤ m. The nonlinear term F depends on y ′ and this derivative dependence is seldom investigated in the literature. Using a variety of fixed point theorems, we establish the existence of one or more positive solutions for the boundary value problem. Examples are also included to illustrate the results obtained.Published versio

Nonlinear eigenvalue problems for higher order Lidstone boundary value problems

In this paper, we consider the Lidstone boundary value problem $y^{(2m)}(t) = \lambda a(t)f(y(t), \dots, y^{(2j)}(t), \dots y^{(2(m-1))}(t), 0 0$ and $a$ is nonnegative. Growth conditions are imposed on $f$ and inequalities involving an associated Green's function are employed which enable us to apply a well-known cone theoretic fixed point theorem. This in turn yields a $\lambda$ interval on which there exists a nontrivial solution in a cone for each $\lambda$ in that interval. The methods of the paper are known. The emphasis here is that $f$ depends upon higher order derivatives. Applications are made to problems that exhibit superlinear or sublinear type growth

Positive Solutions, Existence Of Smallest Eigenvalues, And Comparison Of Smallest Eigenvalues Of A Fourth Order Three Point Boundary Value Problem

The existence of smallest positive eigenvalues is established for the linear differential equations $u^{(4)}+\lambda_{1} q(t)u=0$ and $u^{(4)}+\lambda_{2} r(t)u=0$, $0\leq t \leq 1$, with each satisfying the boundary conditions $u(0)=u\u27(p)=u\u27\u27(1)=u\u27\u27\u27(1)=0$ where $1-\frac{\sqrt{3}}{3}\le p \u3c 1$. A comparison theorem for smallest positive eigenvalues is then obtained. Using the same theorems, we will extend the problem to the fifth order via the Green\u27s Function and again via Substitution. Applying the comparison theorems and the properties of $u_0$-positive operators to determine the existence of smallest eigenvalues. The existence of these smallest eigenvalues is then applied to characterize extremal points of the differential equation $u^{(4)} + q(t)u = 0$ satisfying boundary conditions $u(0) = u\u27(p) = u\u27\u27(b) = u\u27\u27\u27(b)= 0$ where $1-\frac

Methods of extending lower order problems to higher order problems in the context of smallest eigenvalue comparisons

The theory of $u_{0}$-positive operators with respect to a cone in a Banach space is applied to the linear differential equations $u^{(4)}+\lambda_{1} p(x)u=0$ and $u^{(4)}+\lambda_{2} q(x)u=0$, $0\leq x\leq 1$, with each satisfying the boundary conditions $u(0)=u^{\prime}(r)=u^{\prime \prime}(r)=u^{\prime \prime \prime}(1)=0$, $0<r<1$. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the $n$th order problem using two different methods. One method involves finding sign conditions for the Green's function for $-u^{(n)}=0$ satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem

General Lidstone Problems: Multiplicity and Symmetry of Solutions

AbstractFor the 2mth order Lidstone boundary value problem,y(2m)t=fyt,y″t,…,y(2i)t,…,y(2(m−1))t,t∈0,1, y(2i)0=y(2i)1=0,0≤i≤m−1, where (−1)mf: Rm→[0,$thinsp;∞) is continuous, growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions. This generalizes earlier papers which have applied Avery's generalization of the Leggett–Williams theorem to Lidstone problems. We then prove the analogous result for difference equations

Eigenvalue Comparisons for Second-Order Linear Equations with Boundary Value Conditions on Time Scales

This paper studies the eigenvalue comparisons for second-order linear equations with boundary conditions on time scales. Using results from matrix algebras, the existence and comparison results concerning eigenvalues are obtained

Convolutions and Green’s Functions for Two Families of Boundary Value Problems for Fractional Differential Equations

We consider families of two-point boundary value problems for fractional differential equations where the fractional derivative is assumed to be the Riemann-Liouville fractional derivative. The problems considered are such that appropriate differential operators commute and the problems can be constructed as nested boundary value problems for lower order fractional differential equations. Green\u27s functions are then constructed as convolutions of lower order Green\u27s functions. Comparison theorems are known for the Green\u27s functions for the lower order problems and so, we obtain analogous comparison theorems for the two families of higher order equations considered here. We also pose a related open question for a family of Green\u27s functions that do not apparently have convolution representations

Odd and Even Lidstone-type polynomial sequences. Part 1: basic topics

Abstract Two new general classes of polynomial sequences called respectively odd and even Lidstone-type polynomials are considered. These classes include classic Lidstone polynomials of first and second kind. Some characterizations of the two classes are given, including matrix form, conjugate sequences, generating function, recurrence relations, and determinant forms. Some examples are presented and some applications are sketched