Sturm-Liouville Boundary Conditions Research Articles

Precise eigenvalues in the solutions of generalized Sturm–Liouville problems

In the paper, we determine the eigenvalues and eigenfunctions of the generalized Sturm–Liouville problems, whose potentials may be nonlinear functions of eigen-parameter, by developing new iterative algorithms based on the fictitious time integration method (FTIM) and half-interval method (HIM). We derive two eigen-parameter dependent linear shape functions, from which we can transform the generalized Sturm–Liouville problem to an initial value problem for a new variable, and automatically preserve the prescribed eigen-parameter dependent Sturm–Liouville boundary conditions. A nonlinear equation in terms of the relative norm of two consecutive right-end values of the new variable is derived for iteratively determining the eigenvalue by using the FTIM. The resultant sequence of the iterated eigenvalues are monotonically convergent to the desired eigenvalue, and meanwhile the unknown initial values of the eigenfunction can be determined for computing the eigenfunction by integrating the Sturm–Liouville equation. We propose a high precision point target method by using the HIM to determine the eigenvalue and eigenfunction of the generalized Sturm–Liouville problem, which is transformed to a definite initial value problem with a simpler Dirichlet or Neumann boundary condition on the right-end as a point target equation. Several new theoretical results are proved such that more simple generalized Sturm–Liouville problem can be derived with a single point target on the right-end. Depending on different transformation techniques and the derived target equations, seven types FTIM and four types HIM are developed. Numerical examples confirm that the present FTIM and HIM are effective and accurate.

Study on a new [formula omitted]-Laplacian fractional differential model generated by instantaneous and non-instantaneous impulsive effects

In this paper, we discuss a new p-Laplacian fractional differential equation involving instantaneous and non-instantaneous impulses, supplemented with Sturm–Liouville boundary conditions. To study the stated problem, we introduce a new norm for fractional space, denoted by Eα,p. Under this framework, using the critical point theorem proposed by Bonanno and Marano, we prove the existence result of multiple solutions. Finally, an illustrative example is presented.

On a class of one-dimensional superlinear semipositone $ (p, q) $ -Laplacian problem

<abstract><p>We study the existence of positive solutions for a class of one-dimensional superlinear $ (p, q) $ -Laplacian with Sturm-Liouville boundary conditions. We allow the reaction term to be singular at 0 with infinite semipositone behavior. Our approach depends on Amann's fixed point theorem.</p></abstract>

Periodic solutions of two-dimensional wave equations with x-dependent coefficients and Sturm–Liouville boundary conditions

This paper considers the periodic solutions of a two-dimensional nonlinear wave equation with x-dependent coefficients subject to the Sturm–Liouville boundary conditions where x = (x 1, x 2) ∈ (0, π) × (0, π), and for i, j = 1, 2. Such a model arises from the forced vibrations of a nonhomogeneous membrane and the propagation of waves in nonhomogeneous media. By using the duality principles of Brézis (1983 Bull. Am. Math. Soc. 8 409–26), we first set up the theoretical framework and give an abstract theorem for the existence of periodic solution. Then, by analysing the spectral asymptotic behaviours of the weighted Sturm–Liouville problem and classifying the boundary conditions, we obtain the fundamental properties of the variable coefficients wave operator under appropriate assumptions for different class of boundary conditions. Finally, based on these properties and with the help of the abstract theorem, we establish some results on the existence and regularity of periodic solutions. To the best of our knowledge, the results are entirely new, and this is the first time to obtain such results for the two-dimensional nonlinear wave equation with x-dependent coefficients.

The Green's function for Caputo fractional boundary value problem with a convection term

<abstract><p>By using the operator theory, we establish the Green's function for Caputo fractional differential equation under Sturm-Liouville boundary conditions. The results are new, the method used in this paper will provide some new ideas for the study of this kind of problems and easy to be generalized to solving other problems.</p></abstract>

Two-point boundary value problems for planar systems: A lower and upper solutions approach

We extend the theory of lower and upper solutions to planar systems of ordinary differential equations with separated boundary conditions, both in the well-ordered and in the non-well-ordered cases. We are able to deal with general Sturm–Liouville boundary conditions in the well-ordered case, and we analyze the Dirichlet problem in the non-well-ordered case. Our results apply in particular to scalar second order differential equations, including those driven by the mean curvature operator. Higher dimensional systems are also treated, with the same approach.

Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

A note on Lyapunov-type inequalities for fractional boundary value problems with Sturm-Liouville boundary conditions

In this note, we consider fractional Strum-Liouville boundary value problem containing Caputo derivative of order $\alpha$, $ 1<\alpha\leq 2$ with mixed boundary conditions. We establish Cauchy-Schwarz-type inequality to determine a lower bound for the smallest eigenvalues. We give comparison between the smallest eigenvalues and its lower bounds obtained from the Lyapunov-type and Cauchy-Schwarz-type inequalities. Result shows that the Lyapunov-type inequality gives the worse and Cauchy-Schwarz-type inequality gives better lower bound estimates for the smallest eigenvalues. We then use these inequalities to obtain an interval where a linear combination of certain Mittag-Leffler functions have no real zeros.

Lyapunov-type inequalities for Hadamard fractional differential equation under Sturm-Liouville boundary conditions

&lt;abstract&gt;&lt;p&gt;In this paper, we establish new Lyapunov-type inequalities for a Hadamard fractional differential equation under Sturm-Liouville boundary conditions. Our conclusions cover many results in the literature.&lt;/p&gt;&lt;/abstract&gt;

Existence and uniqueness of mild solutions for a fractional differential equation under Sturm-Liouville boundary conditions when the data function is of Lipschitzian type

AbstractIn this article, we present a sufficient condition about the length of the interval for the existence and uniqueness of mild solutions to a fractional boundary value problem with Sturm-Liouville boundary conditions when the data function is of Lipschitzian type. Moreover, we present an application of our result to the eigenvalues problem and its connection with a Lyapunov-type inequality.

Variational approaches to systems of Sturm–Liouville boundary value problems

In this paper, we consider the existence of one solution and three solutions for the boundary value system with Sturm–Liouville boundary conditions [Formula: see text] for [Formula: see text]. Our technical approach is based on variational methods. In addition, examples are provided to illustrate our results.

Time Periodic Solutions of One-Dimensional Forced Kirchhoff Equation with Sturm–Liouville Boundary Conditions

This paper is concerned with the time periodic solutions for the Sturm–Liouville boundary value problem of a one-dimensional Kirchhoff equation in presence of a time periodic external forcing with period $$2\pi /\omega $$ and amplitude $$\epsilon $$. Such a model arises from the forced vibrations of a bounded string in which the dependence of the tension on the deformation cannot be neglected. By using the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period $$2\pi /\omega $$ and order $$\epsilon $$. Such results hold for parameters $$(\omega ,\epsilon )$$ in a positive measure Cantor set that has asymptotically full measure as $$\epsilon $$ goes to zero.

A Lyapunov-type inequality for a fractional differential equation under Sturm-Liouville boundary conditions

A Lyapunov-type inequality for a fractional differential equation under Sturm-Liouville boundary conditions

Countably Infinitely Many Positive Solutions for Even Order Boundary Value Problems with Sturm-Liouville Type Integral Boundary Conditions on Time Scales

In this paper, we establish the existence of countably infinitely many positive solutions for a certain even order two-point boundary value problem with integral boundary conditions on time scales by using Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.

Existence of positive solutions for second-order undamped Sturm–Liouville boundary value problems

In this paper, we consider the second-order differential equations of the form [Formula: see text] satisfying the Sturm–Liouville boundary conditions [Formula: see text] where [Formula: see text]. By an application of Avery–Henderson fixed point theorem, we establish conditions for the existence of multiple positive solutions to the boundary value problem.

Three solutions for second-order impulsive differential inclusions with Sturm-Liouville boundary conditions via nonsmooth critical point theory

A second-order impulsive differential inclusion with Sturm-Liouville boundary conditions is studied. By using a nonsmooth version of a three critical point theorem of Ricceri, the existence of three solutions is obtained.

Existence of positive solutions for a generalized and fractional ordered Thomas-Fermi theory of neutral atoms

The singular boundary value problem we discuss is as follows: $$\begin{aligned}& {}^{\mathrm{C}}D_{0^{+}}^{\alpha}u(t)=\lambda q(t)f \bigl(t,u(t)\bigr),\quad 0< t< 1, \\& \alpha_{1}u(0)+\alpha_{2}u'(0)=a,\qquad \beta_{1}u(1)+\beta_{2}u'(1)=b, \end{aligned}$$ where $1<\alpha\leq2$ , $\lambda>0$ is a parameter, ${}^{\mathrm{C}}D_{0^{+}}^{\alpha}$ is the Caputo fractional derivative. We present the existence of positive solutions for a fractional boundary value problem modeled from the Thomas-Fermi equation subjected to Sturm-Liouville boundary conditions.

EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SYSTEM OF EVEN ORDER DYNAMIC EQUATION ON TIME SCALES

We determine interval of two eigenvalues for which there existence and nonexistence of positive solution for a system of even-order dynamic equation on time scales subject to Sturm-Liouville boundary conditions.

TYPES AND MULTIPLICITY OF SOLUTIONS TO STURM–LIOUVILLE BOUNDARY VALUE PROBLEM

We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.

Upper and lower solution method for nth-order BVPs on an infinite interval

This work is devoted to the study of n th-order ordinary differential equations on a half-line with Sturm-Liouville boundary conditions. The existence results of a solution, and triple solutions, are established by employing a generalized version of the upper and lower solution method, the Schäuder fixed point theorem, and topological degree theory. In our problem the nonlinearity depends on derivatives, and we allow solutions to be unbounded, which is an extra interesting feature.MSC:34B15, 34B40.