Sturm-Liouville Type Research Articles

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

On the solutions of a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions

We consider a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions and we obtain an existence result for this problem in the case when the set-valued maps have nonconvex values.

Countably Many Positive Symmetric Solutions For Sturm Liouville Type Boundary Conditions Of Second Order Iterative System

In this paper we consider second order iterative boundary value problem with Sturm Liouville type boundary conditions and establish the existence of countably many positive symmetric solutions by using Krasnoselskii’s fixed point theorem for operator on a cone.

Second regularized trace of a differential operator of Sturm–Liouville type

We consider a Sturm–Liouville‐type differential operator defined on a separable Hilbert space with bounded operator coefficient and periodic boundary conditions given on a finite interval. We obtain an explicit formula which we call the second regularized trace formula of the mentioned operator.

Wintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations

In this study, we addressed the nonoscillation of th Sturm-Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm-Liouville type.

On a complete parametric Sturm-Liouville problem with sign changing coefficients

<abstract><p>In this paper we study a complete second order differential equation of Sturm-Liouville type under Dirichlet boundary condition and where the variable coefficients are allowed to be sign changing. Through critical point theory, we obtain the existence of two nontrivial generalized solutions by requiring a specific growth on the nonlinearity. Moreover, the solutions turn out to be nonnegative and with opposite energy sign.</p></abstract>

Critical Point Approaches for a Class of Differential Equations with Sturm-Liouville Type Nonhomogeneous Boundary Conditions

A class of p-Laplacian equations with Sturm-Liouville type nonhomogeneous boundary value problem with nonlinear derivative depending on two control parameters is investigated. Existence and multiplicity of solutions are discussed by means of variational methods and critical point theory. Two examples supporting our theoretical results are also presented.

Boundary value problems of quaternion-valued differential equations: solvability and Green’s function

This paper is associated with Sturm–Liouville type boundary value problems and periodic boundary value problems for quaternion-valued differential equations (QDEs). Employing the theory of quaternionic matrices, we prove the conditions for the solvability of the linear boundary valued problem and find Green’s function.

Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary–Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti–Prodi alternative has been obtained for such systems with different parameters.

On the existence and uniqueness of a positive solution to a boundary-value problem of the Sturm-Liouville type for a nonlinear ordinary differential equation

Using the fixed point theorem in partially ordered sets, we obtain sufficient conditions for the existence of a unique positive solution to a boundary-value problem of the Sturm-Liouville type for a nonlinear ordinary differential equation, and give an example illustrating the results obtained.

The Dirac equation with /non--symmetric potentials in curved spacetime backgrounds

In this paper, we have presented the Dirac equation in the frame of position-dependent mass on two dimensional gravitational static background in the presence of /non--symmetric potential interactions. The exponential metric component have formed the reduced Dirac operator into the general supersymmetric model within mass changing with coordinates. We have obtained the eigenvalues of the Dirac operator for the complex Morse and trigonometric complex Scarf-II potentials SL(2, C) using Lie algebras and the supersymmetric quantum mechanical approaches. Moreover, after obtaining a general Sturm-Liouville type equation using a convenient mapping, the system have become available to be investigated within η- pseudo-Hermiticity. With this context, η operator is found for the examples of complex trigonometric Rosen-Morse potential and complex Morse potentials with real and complex parameters of the initial system and finally the solutions are obtained for each model with the graphics of energy values and probability densities.

On Recovering Sturm–Liouville-Type Operators with Global Delay on Graphs from Two Spectra

We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, which is measured in the direction of a specific boundary vertex, called the root, propagates through the internal vertex to other edges. Now, we intend to recover the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except the root. For simplicity, we focus on star graphs with equal edges when the delay parameter is not less than their length. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained.

Asymptotic Properties of Solutions to Discrete Sturm-Liouville Monotone Type Equations

Abstract We investigate the discrete equations of the form Δ ( r n Δ x n ) = a n f ( x σ ( n ) ) + b n . \Delta \left( {{r_n}\Delta {x_n}} \right) = {a_n}f\left( {{x_{\sigma \left( n \right)}}} \right) + {b_n}. Using the Knaster-Tarski fixed point theorem, we study solutions with prescribed asymptotic behaviour. Our technique allows us to control the degree of approximation. In particular, we present the results concerning harmonic and geometric approximations of solutions.

Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point

Abstract In this article, we focus on the scattering analysis of Sturm-Liouville type singular operator including an impulsive condition and turning point. In the classical literature, there are plenty of papers considering the positive values of the weight function in both continuous and discontinuous cases. However, this article differs from the others in terms of the impulsive condition appearing at the turning point. We generate the scattering function, resolvent operator, and discrete spectrum of the operator using the hyperbolic representations of the fundamental solutions. Finally, we create an example to show the article’s primary conclusions.

Sturm–Liouville Differential Inclusions with Set-Valued Reaction Term Depending on a Parameter

In this paper we study the controllability for a Cauchy problem governed by a nonlinear differential inclusion driven by a Sturm–Liouville type operator. In particular, the considered second order differential inclusion involves a set-valued reaction term depending on a parameter. The key tool in the proof of the controllability result we provide is a multivalued version of the theorem recently proved by Haddad–Yarou, here established for an initial conditions problem monitored by a nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side. We thereby deduce the existence of a local admissible pair for the considered control problem, that is the existence of a couple of functions consisting of a control, which is a measurable function, and the correspondent trajectory, which is an absolutely continuous function with absolutely continuous derivative. Secondly, under appropriate assumptions on the involved multimaps, we obtain an increased regularity for the solutions produced by our existence result. This regularity is the same of that recently tested by Bonanno, Iannizzotto and Marras for a different type of problem, which however involves the Sturm–Liouville operator.

Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

<p style='text-indent:20px;'>In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.</p>

FRACTIONAL DISSIPATIVE STURM-LIOUVILLE PROBLEMS WITH DISCONTINUITY AND EIGEN-DEPENDENT BOUNDARY CONDITIONS

This paper studies two main fractional discontinuous dissipative Sturm-Liouville type boundary-value problems with boundary conditions and transmission conditions. In both types of research, with the aid of the operator theory, we define different classes of new inner products by combining the parameters in the boundary and transmission conditions, then the boundary value problems are transferred to operators in the Hilbert spaces such that the eigenvalues and eigenfunctions of the main problem coincide with those of operators. And we prove those of operators are dissipative. Moreover, the fundamental solutions are constructed and the uniqueness of the solutions of the problem is also given.

Positive solutions of multipoint ϕ-Laplacian BVPs with first-order derivative dependence

This paper concerns existence of positive solutions for a second-order boundary value problem of Sturm–Liouville type associated with a [Formula: see text]-Laplacian operator and posed on a bounded interval. Existence results are obtained by an adapted version of the Krasnosel’skii’s fixed point theorem of cone expansion and compression. Some examples illustrate our results.

Multiple solutions for nonlinear Sturm–Liouville differential equations with possibly negative variable coefficients

The aim of this paper is to study a parameter-dependent nonlinear differential problem with Sturm–Liouville type equation and Dirichlet boundary conditions. In particular, by requiring a suitable behavior on the nonlinearity, we determine an interval of parameter λ for which the problem admits three or infinitely many solutions.

Operator-valued Camassa–Holm systems and their integrability

We study a Sturm–Liouville-type operator with operator-valued coefficients and its iso-spectral deformations, related to a new two-component Camassa–Holm-type completely integrable dynamical systems. Based on a specially devised gradient-holonomic scheme, generalizing the one before developed for studying a Sturm–Liouville-type spectral problem on a spatially multidimensional Hilbert–Schmidt operator-valued Hilbert space, we constructed the related two compatible Poisson structures and an infinite hierarchy of commuting to each other conservation laws of the derived two-component Camassa–Holm-type Hamiltonian system. The latter makes it possible to state under some additional constraints its complete integrability, and in particular, to develop the corresponding inverse spectral-type-based method for constructing its exact solutions.