Three-point Boundary Value Problem Research Articles

Existence of fixed points in conical shells of a Banach space for sum of two operators and application in ODEs

In this work a new functional expansion-compression fixed point theorem of Leggett--Williams type is developed for a class of mappings of the form $T+F,$ where $(I-T)$ is Lipschitz invertible map and $F$ is a $k$-set contraction. The arguments are based upon recent fixed point index theory in cones of Banach spaces for this class of mappings. As application, our approach is applied to prove the existence of nontrivial nonnegative solutions for three-point boundary value problem.

On the Ulam-Hyers-Rassias stability of two structures of discrete fractional three-point boundary value problems: Existence theory

<abstract><p>We prove existence and uniqueness of solutions to discrete fractional equations that involve Riemann-Liouville and Caputo fractional derivatives with three-point boundary conditions. The results are obtained by conducting an analysis via the Banach principle and the Brouwer fixed point criterion. Moreover, we prove stability, including Hyers-Ulam and Hyers-Ulam-Rassias type results. Finally, some numerical models are provided to illustrate and validate the theoretical results.</p></abstract>

MODELLING OF VARIABLE BAND-GAP SUPERLATTICES WITH USING A GENETIC ALGORITHM

The numerical calculation of the coordinate distributions of the electrostatic potential in an asymmetric variable-band superlattice with piecewise linear coordinate profiles of the gap width and electron affinity was performed using a genetic algorithm. A mathematical model is formulated, represented by two nonlinear Poisson equations and four boundary conditions, which reflect the continuity of the electrostatic potential and electric field strength at the boundaries of the variable band-gap superlattice, as well as the spatial periodicity of its physical quantities. The solution of the three-point boundary value problem for the electrostatic potential is reduced to the solution in an iterative cycle of two Cauchy problems with initial conditions determined using a genetic algorithm. The search space was chosen in the form of a rectangular region centered at a point that corresponded to the values of the electrostatic potential and electric field strength on one of the boundaries of the superlattice. These values were obtained by analytically solving the corresponding Poisson equations. An initial population of 50-100 trial solutions (chromosomes) was created by selection with equal probability in the search area. The coding of chromosomes and their modification under the action of crossover and mutation operations were used. The genetic algorithm was implemented in the Matlab environment using the Genetic Algorithm package. Numerical experiments showed that the initial population of trial solutions no later than after 20 -30 generations approaches to the desired initial values for the Cauchy problems. The latter were solved by the Runge-Kutta method of the 4th order. The obtained dependences of the electrostatic potential show that its maximum value, which corresponds to the zero intensity of the internal electric field, in contrast to symmetric variable band-gap superlattices, is observed not at the interface of layers, but in a layer of greater thickness. The results of the calculations indicate a substantial quantitative difference in the values of the electrostatic potential for the nonlinear and linearized models. Key words: variable band-gap semiconductors, superlattices, modelling, genetic algorithms.

Existence and Multiplicity of Positive Solutions for a Singular Third-Order Three-Point Boundary Value Problem with a Parameter

In this paper, we investigate the existence of positive solutions for a singular third-order three-point boundary value problem with a parameter. By using fixed point index theory, some existence, multiplicity and nonexistence results for positive solutions are derived in terms of different values of λ.

Existence and uniqueness of solutions to discrete,third-order three-point boundary value problems

The purpose of this article is to move towards a more complete understanding of the qualitative properties of solutions to discrete boundary value problems. In particular, we introduce and develop sufficient conditions under which the existence of a unique solution for a third-order difference equation subject to three-point boundary conditions is guaranteed. Our contributions are realized in the following ways. First, we construct the corresponding Green’s function for the problem and formulate some new bounds on its summation. Second, we apply these properties to the boundary value problem by drawing on Banach’s fixed point theorem in conjunction with interesting metrics and appropriate inequalities. We discuss several examples to illustrate the nature of our advancements.

DISCRETE METHOD FOR SOLVING A THREE-POINT BOUNDARY-VALUE PROBLEM FOR A THIRD-ORDER EQUATION

DISCRETE METHOD FOR SOLVING A THREE-POINT BOUNDARY-VALUE PROBLEM FOR A THIRD-ORDER EQUATION

Application of Fixed-Point Index Theory for a Nonlinear Fractional Boundary Value Problem with an Advanced Argument

In this paper, we investigate the existence of positive solutions of a class of fractional three-point boundary value problem with an advanced argument by using fixed-point index theory. Our results improve and extend some known results in the literature. Two examples are given to demonstrate the effectiveness of our results.

Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function

Abstract In this paper, we discuss the existence of positive solutions to a discrete third-order three-point boundary value problem. Here, the weight function a ( t ) a\left(t) and the Green function G ( t , s ) G\left(t,s) both change their sign. Despite this, we also obtain several existence results of positive solutions by using the Guo-Krasnoselskii’s fixed-point theorem in a cone.

Research of the three-point boundary value problems for nonlinear second-order difference equation

Research of the three-point boundary value problems for nonlinear second-order difference equation

A New Approach of Morgan-Voyce Polynomial to Solve Three Point Boundary Value Problems

In this paper, a new procedure is introduced to estimate the solution for the three-point boundary value problem which is instituted on the use of Morgan-Voyce polynomial. In the beginning, Morgan-Voyce polynomial along with their important properties is introduced. Next, this polynomial with aid of the collocation method utilized to modify the differential equation with boundary conditions to the algebraic system. Finally, the examples approve the validity and accuracy of the proposed method.

Existence Results for Fourth Order Non-Homogeneous Three-Point Boundary Value Problems

The present paper focuses on establishing the existence and uniqueness of solutions to the nonlinear differential equations of order four y(4)(t) + g(t, y(t)) = 0, t ∈ [a, b], together with the non-homogeneous three-point boundary conditions y(a) = 0, y′(a) = 0, y′′(a) = 0, y(b) − αy(ξ ) = λ, where 0 ≤ a < b, ξ ∈ (a, b), α, λ are real numbers and the function g: [a, b] × R→R is a continuous with g(t, 0) ≠ 0. With the aid of an estimate on the integral of kernel, the existence results to the problem are proved by employing fixed point theorem due to Banach.

The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation

An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The findings are obtained by employing properties of discrete fractional equations, Banach contraction, and Brouwer fixed-point theorems. Further, we discuss our problem’s results concerning Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam–Rassias (HUR), and generalized Hyers–Ulam–Rassias (GHUR) stability. Specific examples with graphs and numerical experiment are presented to demonstrate the effectiveness of our results.

Existence of three solutions for Kirchhoff-type three-point boundary value problems

The present paper is an attempt to investigate the multiplicity results of solutions for a three-point boundary value problem of Kirchhoff-type. Indeed, we will use variational methods for smooth functionals, defined on the reflexive Banach spaces in order to achieve the existence of at least three solutions for the equation. Finally, by presenting one example, we will ensure the applicability of our results.

Solving nonlinear third-order three-point boundary value problems by boundary shape functions methods

For nonlinear third-order three-point boundary value problems (BVPs), we develop two algorithms to find solutions, which automatically satisfy the specified three-point boundary conditions. We construct a boundary shape function (BSF), which is designed to automatically satisfy the boundary conditions and can be employed to develop new algorithms by assigning two different roles of free function in the BSF. In the first algorithm, we let the free functions be complete functions and the BSFs be the new bases of the solution, which not only satisfy the boundary conditions automatically, but also can be used to find solution by a collocation technique. In the second algorithm, we let the BSF be the solution of the BVP and the free function be another new variable, such that we can transform the BVP to a corresponding initial value problem for the new variable, whose initial conditions are given arbitrarily and terminal values are determined by iterations; hence, we can quickly find very accurate solution of nonlinear third-order three-point BVP through a few iterations. Numerical examples confirm the performance of the new algorithms.

Discrete Method for Solving a Three-Point Boundary-Value Problem for a Third-Order Equation

Discrete Method for Solving a Three-Point Boundary-Value Problem for a Third-Order Equation

A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to α and β are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.

Positive Solutions of a Three-Point Boundary-Value Problem for the p-Laplacian Dynamic Equation on Time Scales

Positive Solutions of a Three-Point Boundary-Value Problem for the p-Laplacian Dynamic Equation on Time Scales

Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations via Common Fixed-Point Techniques in Complex Valued Fuzzy Metric Spaces

The main purpose of this paper is to study the existence theorem for a common solution to a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations. In this respect, we study the fuzzy version of some essential common fixed-point results from metric spaces in the newly introduced notion of complex valued fuzzy metric spaces. Also, we provide an illustrative example to demonstrate the validity of our derived results.

A unique solution to a fourth-order three-point boundary value problem

In this study, it is aimed to examine the solutions of the following nonlocal boundary value problem \begin{equation*} y^{(4)}+g(x,y)=0,x\in [{c,d}], y(c)=y'(c)=y''(c)=0,y(d)=\lambda y(\xi). \end{equation*} Here, $\xi\in ({c,d}),\lambda \in \mathbb{R},g\in C([{c,d}]\times \mathbb{R},\mathbb{R})$ and $g(x,0)\neq 0.$ It is concentrated on applications of Green's function that corresponds to the above problem to derive existence and uniqueness results for the solutions. One example is also given to demonstrate the results.

A Peek into Theory of Automation with Applications to Various Branches of Science

In this paper, we shall be concerned with the existence and uniqueness of solution to three- point boundary value problems associated with a system of first order matrix difference system. Shortest and Closest Lattice vector methods are used as a tool to obtain the best least square solution of the three-point boundary value problems when the characteristic matrix D is rectangular. In this paper, we shall be concerned with the existence and uniqueness of solution to three- point boundary value problems associated with a system of first order matrix difference system. Shortest and Closest Lattice vector methods are used as a tool to obtain the best least square solution of the three-point boundary value problems when the characteristic matrix D is rectangular.