Lidstone Boundary Value Problems Research Articles

Existence of nodal solutions of nonlinear Lidstone boundary value problems

Existence of nodal solutions of nonlinear Lidstone boundary value problems

Existence of positive solutions for Lidstone boundary value problems on time scales

Let mathbb{T}subseteq mathbb{R} be a time scale. The purpose of this paper is to present sufficient conditions for the existence of multiple positive solutions of the following Lidstone boundary value problem on time scales: (−1)nyΔ(2n)(t)=f(t,y(t)),t∈[a,b]T,yΔ(2i)(a)=yΔ(2i)(σ2n−2i(b))=0,i=0,1,…,n−1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &(-1)^{n} y^{\\Delta ^{(2n)}}(t) = f\\bigl(t, y(t)\\bigr), \\quad \ ext{$t\\in [a,b]_{ \\mathbb{T}}$,} \\\\ &y^{\\Delta ^{(2i)}}(a)= y^{\\Delta ^{(2i)}}\\bigl(\\sigma ^{2n-2i}(b)\\bigr)=0,\\quad i=0,1,\\ldots,n-1. \\end{aligned}$$ \\end{document} Existence of multiple positive solutions is established using fixed point methods. At the end some examples are also given to illustrate our results.

Positive solutions of the complementary Lidstone boundary value problem

The authors consider the higher order complementary Lidstone boundary value problem. Some upper and lower bounds for positive solutions of the problem are obtained. Sufficient conditions for the existence and nonexistence of positive solutions are established.

Nontrivial Solutions for the 2 n th Lidstone Boundary Value Problem

In this paper, we study the existence of nontrivial solutions for the 2 n th Lidstone boundary value problem with a sign-changing nonlinearity. Under some conditions involving the eigenvalues of a linear operator, we use the topological degree theory to obtain our main results.

Existence of Non-Spurious Solutions to Discrete Fourth-Order Lidstone Boundary Value Problems

In this paper, we study boundary value problems of difference equations which arise as discrete approximations to Lidstone boundary value problems of fourth-order ordinary differential equations. By developing the method of lower and upper solutions, we investigate the existence of solutions as well as the nonexistence of spurious solutions to the discrete problems.

Lyapunov-type inequalities for Lidstone boundary value problems on time scales

In this paper, we establish new Hartman and Lyapunov-type inequalities for even-order dynamic equations $$\begin{aligned} x^{\varDelta ^{2n}}(t)+(-1)^{n-1} q(t) x^{\sigma }(t)=0 \end{aligned}$$on time scales $${\mathbb {T}}$$ satisfying the Lidstone boundary conditions $$\begin{aligned} x^{\varDelta ^{2i}}(t_{1})=x^{\varDelta ^{2i}}(t_{2})=0;\qquad t_1,t_2\in [t_0,\infty )_{{\mathbb {T}}} \end{aligned}$$for $$i=0,1,\ldots , n-1$$. The inequalities obtained generalize and complement the existing results in the literature.

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

Our aim in this work is to study the existence of solutions for a fractional Lidstone boundary value problems. We use some fixed point theorems to show the existence and uniqueness of solution under suitable conditions. Two examples are given to ilustrate the obtained results.

Triple solutions of complementary Lidstone boundary value problems via fixed point theorems

We consider the following complementary Lidstone boundary value problem: ( − 1 ) m y ( 2 m + 1 ) ( t ) = F ( t , y ( t ) , y ′ ( t ) ) , t ∈ [ 0 , 1 ] , y ( 0 ) = 0 , y ( 2 k − 1 ) ( 0 ) = y ( 2 k − 1 ) ( 1 ) = 0 , 1 ≤ k ≤ m . By using fixed point theorems of Leggett-Williams and Avery, we offer several criteria for the existence of three positive solutions of the boundary value problem. Examples are also included to illustrate the results obtained. We note that the nonlinear term F depends on y ′ and this derivative dependence is seldom investigated in the literature and a new technique is required to tackle the problem.MSC:34B15, 34B18.

Positive Solutions for Nonlinear Lidstone Boundary Value Problems

In this paper, we consider the existence of positive solutions for nonlinear Lidstone boundary value problems. An new existence result is obtained by applying the fixed point index theorem.

Existence result for the Lidstone boundary value problem at resonance

In this paper, we consider the existence problem to some type of the Lidstone boundary value problem at resonance with discontinuity of the Carathéodory type. We prove that a solution exists if the conditions similar to the Landesman–Lazer ones are satisfied.

Eigenvalues of complementary Lidstone boundary value problems

We consider the following complementary Lidstone boundary value problem ( - 1 ) m y ( 2 m + 1 ) ( t ) = λ F ( t , y ( t ) , y ′ ( t ) ) , t ∈ ( 0 , 1 ) y ( 0 ) = 0 , y ( 2 k - 1 ) ( 0 ) = y ( 2 k - 1 ) ( 1 ) = 0 , 1 ≤ k ≤ m where λ > 0. The values of λ are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of λ such that for any λ 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.AMS Subject Classification: 34B15.

Positive solutions of complementary Lidstone boundary value problems

We consider the following complementary Lidstone boundary value problem (−1) m y (2m+1) (t) = F(t; y(t); y 0 (t)); t 2 (0;1) y(0) = 0; y (2k−1) (0) = y (2k−1) (1) = 0; 1 � km: The nonlinear term F depends on y 0 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.

Bifurcation techniques and positive solutions of discrete Lidstone boundary value problems

Abstract Let n ⩾ 1, T ⩾ n + 5 be two positive integers and T n = { n , n + 1 , … , T } . We consider the existence of solutions of the discrete Lidstone boundary value problem. ( - 1 ) n Δ 2 n u ( t - n ) = f ( t , u ( t ) , Δ 2 u ( t - 1 ) , … , Δ 2 n - 2 u ( t - n + 1 ) ) , t ∈ T n , Δ 2 i u ( n - i - 1 ) = Δ 2 i u ( T - i + 1 ) = 0 , i = 0 , 1 , … , n - 1 , where f : T n × R n → R is continuous.

Upper and lower estimates for positive solutions of the higher order Lidstone boundary value problem

We consider the higher order Lidstone boundary value problem. New upper and lower estimates for positive solutions of the problem are obtained. A discussion of the sharpness of the estimates is included.

Existence and uniqueness of positive solution for nonlinear singular 2 mth-order continuous and discrete lidstone boundary value problems

By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existence and uniqueness of positive solution to Lidstone boundary value problem for 2mth-order ordinary differential equations and 2mth-order difference equations. The nonlinear term in the differential and difference equation may be singular.

Positive Solutions for Singular Complementary Lidstone Boundary Value Problems

By using fixed‐point theorems of a cone, we investigate the existence and multiplicity of positive solutions for complementary Lidstone boundary value problems: (−1)nu(2n + 1)(t) = h(t)f(u(t)), in 0 < t < 1, u(0) = 0, u(2i + 1)(0) = u(2i + 1)(1) = 0, 0 ≤ i ≤ n − 1, where n ∈ N.

A bound for the error of the Ritz method in the case of the Lidstone problem for a singular differential equation of fourth order

For the Lidstone boundary-value problem $$ \begin{array}{*{20}{c}} {{u^{(4)}} + q(t)u = f(t),\,\,\,0 < t < 1,} \\ {u(0) = u''(0) = u(1) = u''(1) = 0} \\ \end{array} $$ with nonintegrable functions q(t) and f(t), conditions of solvability are obtained and a computable error bound for the Ritz method is established. Bibliography: 3 titles.

Positive Solutions of Singular Complementary Lidstone Boundary Value Problems

We investigate the existence of positive solutions of singular problem , , , . Here, and the Caratheodory function may be singular in all its space variables . The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used.

Global behaviour of the components of nodal solutions for Lidstone boundary value problems

By employing bifurcation techniques, we investigate the global behaviour of the components of nodal solutions for Lidstone boundary value problems depending on higher order derivatives. Our results improve on those in the literature.

A discrete fourth-order Lidstone problem with parameters

Various existence, multiplicity, and nonexistence results for nontrivial solutions to a nonlinear discrete fourth-order Lidstone boundary value problem with dependence on two parameters are given, using a symmetric Green’s function approach. An existence result is also given for a related semipositone problem, thus relaxing the usual assumption of nonnegativity on the nonlinear term.