Three-point Boundary Value Problem Research Articles

Existence of multiple unbounded solutions for a three-point boundary value problems on an infinite time scales

Abstract In this paper, we consider the second-order three point boundary value problem on time scales with integral boundary conditions on a half-line. We will use the upper and lower solution method along with the Schauder’s fixed point theorem to establish the existence of at least one solution which lies between pairs of unbounded upper and lower solutions. Further, by assuming two pairs of unbounded upper and lower solutions, the Nagumo condition on the nonlinear term involved in the first-order derivative, we will establish the existence of multiple unbounded solutions on an infinite interval by using the topological degree theory. The results of this paper extend the results of Akcan and Çetin (2018), Akcan and Hamal (2014), Eloe, Kaufmann and Tisdell (2006), and generalize the results of Lian and Geng (2011). Examples are included to illustrate the validation of the results.

On the Solution of Caputo Fractional High-Order Three-Point Boundary Value Problem with Applications to Optimal Control

This research paper establishes the existence and uniqueness of solutions for a non-integer high-order boundary value problem, incorporating the Caputo fractional derivative with a non-local type boundary condition. The analytical approach involves the introduction of the fractional Green’s function. To analyze our findings effectively, we apply the Banach contraction fixed point theorem as the primary principle. Furthermore, we illustrate our results through the presentation of various examples.

Existence theory for nabla fractional three-point boundary value problems via continuation methods for contractive maps

In this article, we analyse an $\alpha$-th order, $1 < \alpha \leq 2$, nabla fractional three-point boundary value problem (BVP). We construct the Green's function associated to this problem and derive a few of its important properties. We then establish sufficient conditions on existence and uniqueness of solutions for the corresponding nonlinear BVP using the modern ideas of continuation methods for contractive maps. Our results extend recent results on nabla fractional BVPs. Finally, we provide an example to illustrate the applicability of main results.

Existence and multiplicity of solutions for three-point boundary value problems with instantaneous and noninstantaneous impulses

In this paper, three-point boundary value problems for second-order p-Laplacian differential equations with instantaneous and noninstantaneous impulses are studied. The existence of at least one classical solution and infinitely many classical solutions is obtained by using variational methods and critical point theory. In addition, some examples are given to illustrate our main results.

Existence and Uniqueness of Solutions to Three-point Boundary Value Problems associated with Third Order Non-Linear Fuzzy differential Equations

Existence and Uniqueness of Solutions to Three-point Boundary Value Problems associated with Third Order Non-Linear Fuzzy differential Equations

Green functions for three-point boundary value problems governed by differential equation systems with applications to Timoshenko beams

The present paper is devoted to the issue of the Green function matrices that belongs to some three-point boundary- and eigenvalue problems. A detailed definition is given for the Green function matrices provided that the considered boundary value problems are governed by a class of ordinary differential equation systems associated with homogeneous boundary and continuity conditions. The definition is a constructive one, i.e., it provides the means needed for calculating the Green function matrices. The fundamental properties of the Green function matrices—existence, symmetry properties, etc.—are also clarified. Making use of these Green functions, a class of three-point eigenvalue problems can be reduced to eigenvalue problems governed by homogeneous Fredholm integral equation systems. The applicability of the novel findings is demonstrated through a Timoshenko beam with three supports.

NEW APPROACH BASED ON COLLOCATION AND SHIFTED CHEBYSHEV POLYNOMIALS FOR A CLASS OF THREE-POINT SINGULAR BVPS

In the recent decades, variety of real-life problems arises in astrophysics have been mimic using the class of three-point singular boundary value problems (BVPs). Finding an effective and accurate approach for a class of three-point BVPs is still a difficult problem, though. The goal of this paper is to design a numerical strategy for approximating a class of three-point singular boundary value problems using the collocation technique and shifted Chebyshev polynomials. Utilizing shifted Chebyshev polynomials, the problem is reduced to a matrix form, which is then converted into a system of nonlinear algebraic equations by employing the collocation points. The key advantages of the new approach are (a) it is a straightforward mathematical formulation, which makes it effortless to code, and (b) it is easily adaptable to solve various classes of three-point singular boundary value problems. The convergence analysis is carried out to ensure the viability of the proposed scheme. Various examples are considered and tested in order to illustrate its applicability and efficiency. The results show excellent accuracy and efficiency compared to the other existing methods.

Three solutions for a three-point boundary value problem with instantaneous and non-instantaneous impulses

<abstract><p>In this paper, we consider the multiplicity of solutions for the following three-point boundary value problem of second-order $ p $-Laplacian differential equations with instantaneous and non-instantaneous impulses:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ {\begin{array}{l} -(\rho(t)\Phi_{p} (u'(t)))'+g(t)\Phi_{p}(u(t))=\lambda f_{j}(t,u(t)),\quad t\in(s_{j},t_{j+1}],\; j=0,1,...,m,\\ \Delta (\rho (t_{j})\Phi_{p}(u'(t_{j})))=\mu I_{j}(u(t_{j})), \quad j=1,2,...,m,\\ \rho (t)\Phi_{p} (u'(t))=\rho(t_{j}^{+}) \Phi_{p} (u'(t_{j}^{+})),\quad t\in(t_{j},s_{j}],\; j=1,2,...,m,\\ \rho(s_{j}^{+})\Phi_{p} (u'(s_{j}^{+}))=\rho(s_{j}^{-})\Phi_{p} (u'(s_{j}^{-})),\quad j=1,2,...,m,\\ u(0)=0, \quad u(1)=\zeta u(\eta), \end{array}} \right. \end{equation*} $\end{document} </tex-math> </disp-formula></p> <p>where $ \Phi_{p}(u): = |u|^{p-2}u, \; p > 1, \; 0 = s_{0} < t_{1} < s_{1} < t_{2} < ... < s_{m_{1}} < t_{m_{1}+1} = \eta < ... < s_{m} < t_{m+1} = 1, \; \zeta > 0, \; 0 < \eta < 1 $, $ \Delta (\rho (t_{j})\Phi_{p}(u'(t_{j}))) = \rho (t_{j}^{+})\Phi_{p}(u'(t_{j}^{+}))-\rho (t_{j}^{-})\Phi_{p}(u'(t_{j}^{-})) $ for $ u'(t_{j}^{\pm}) = \lim\limits_{t\to t_{j}^{\pm}}u'(t) $, $ j = 1, 2, ..., m $, and $ f_{j}\in C((s_{j}, t_{j+1}]\times\mathbb{R}, \mathbb{R}) $, $ I_{j}\in C(\mathbb{R}, \mathbb{R}) $. $ \lambda\in (0, +\infty) $, $ \mu\in\mathbb{R} $ are two parameters. $ \rho(t)\geq 1 $, $ 1\leq g(t)\leq c $ for $ t\in (s_{j}, t_{j+1}] $, $ \rho(t), \; g(t)\in L^{p}[0, 1] $, and $ c $ is a positive constant. By using variational methods and the critical points theorems of Bonanno-Marano and Ricceri, the existence of at least three classical solutions is obtained. In addition, several examples are presented to illustrate our main results.</p></abstract>

Three-point boundary value problems for implicit Caputo tempered fractional differential equations in b-metric spaces

Three-point boundary value problems for implicit Caputo tempered fractional differential equations in b-metric spaces

Existence of Positivity of the Solutions for Higher Order Three-Point Boundary Value Problems involving p-Laplacian

The present study focusses on the existence of positivity of the solutions to the higher order three-point boundary value problems involving $p$-Laplacian
 $$[\phi_{p}(x^{(m)}(t))]^{(n)}=g(t,x(t)),~~t \in [0, 1],$$
 $$
 \begin{aligned}
 x^{(i)}(0)=0, &\text{~for~} 0\leq i\leq m-2,\\
 x^{(m-2)}(1)&-\alpha x^{(m-2)}(\xi)=0,\\
 [\phi_{p}(x^{(m)}(t))]^{(j)}_{\text {at} ~ t=0}&=0, \text{~for~} 0\leq j\leq n-2,\\
 [\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=1}&-\alpha[\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=\xi}=0,
 \end{aligned}
 $$
 where $m,n\geq 3$, $\xi\in(0,1)$, $\alpha\in (0,\frac{1}{\xi})$ is a parameter.
 The approach used by the application of Guo--Krasnosel'skii fixed point theorem to determine the existence of positivity of the solutions to the problem.

Consistency of an Infinite system of third order three-point boundary value problem in the $bv_0$ space by the theory of measure of noncompactness

Several authors have examined the solvability conditions for an infinite system of differential equations in different Banach spaces using the concept of measure of noncompactness. In all these studies, they have considered differential equations where the boundary conditions are defined on two points. In this paper, we have studied the solvability conditions for an infinite system of third-order three point boundary value problem in the sequence space of bounded variation $bv_0$ with the help of the theory of measure of noncompactness and have given a suitable example to illustrate the result.

A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem

In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz’s formula, Rolle and Cauchy’s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

Solvability of infinite systems of fractional differential equations in the double sequence space $$ 2^c(\triangle )$$

In this work, we first introduce the concept of double sequence space $$2^c(\triangle )$$ . Then, we construct a Hausdorff measure of noncompactness on this space. Furthermore, by employing this measure of noncompactness we discuss the existence of solutions for infinite systems of third-order three-point boundary value problem in the double sequence space $$2^c(\triangle )$$ . Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.

Vibration of an axially loaded heterogeneous fixed–fixed beam with an intermediate roller support

The present paper investigates how an axial load can change the natural frequencies of heterogeneous fixed–fixed beams with an intermediate roller support. The problem is treated as a three-point boundary value problem (eigenvalue problem) that is paired with homogeneous boundary conditions. The Green functions are determined for the unloaded and pre-loaded beams as well—in the later case, both for compression and tension. With these, the eigenvalue problems can be transformed into eigenvalue problems governed by a homogeneous Fredholm integral equations. It is then replaced by an algebraic eigenvalue problem, that is solved numerically with an effective solution algorithm which is based on the boundary element method.

BUCKLING OF BEAMS WITH A BOUNDARY ELEMENT TECHNIQUE

The present work is devoted to the buckling study of non-homogeneous fixed- fixed beams with intermediate spring support. The stability issue of these beams leads to three-point boundary value problems. If the Green functions of these boundary value problems are known, the differential equations of the stability problems that contain the critical load sought can be turned into eigenvalue problems given by homogeneous Fredholm integral equations. The kernel function of these equations can be calculated from the associated Green functions. The eigenvalue issues can be reduced to algebraic eigenvalue problems, which are subsequently solvable numerically with the use of an efficient algorithm from the boundary element method. Within this article, the critical load findings of these beams are compared to those obtained using commercial finite element software, and the results are in excellent correlation.

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

This paper discusses a kind of coupled nonlocal Laplacian evolution equation with Caputo time-fractional derivatives and proportional delays. Green function and mild solution are firstly established by employing the approach of eigenvalues’ expansions and Fourier analysis technique. By the properties of eigenvalues and Mittag–Leffler functions, several vital estimations of Green functions are presented. In view of these estimations and some appropriate assumptions, the existence and uniqueness of the mild solution are studied by utilizing the Leray–Schauder fixed-point theorem and the Banach fixed-point theorem. Finally, an example is provided to illustrate the effectiveness of our main results.

Bounded solutions to a three-point fourth-order iterative boundary value problem

The aim of this paper is to deal with a fourth-order three-point boundary value problem with an iterative source term. By using a technique based on Green’s function method, the contraction mapping principle and Schauder’s fixed point theorem, we establish some sufficient conditions for proving the existence, uniqueness and continuous dependence of a bounded solution on parameters. Furthermore, two examples are provided to illustrate the main findings, which are completely new, and extend some relevant works to some degree.

Stability of heterogeneous beams with three supports through Green functions

The present paper is devoted to the issue how the critical load of some heterogeneous beams with three supports can be determined by using Green functions. The stability problems of these beams are equivalent to three-point boundary value problems, paired with homogeneous boundary conditions. If the Green functions of these boundary value problems are known, the eigenvalue problems that provide the critical load can be transformed into eigenvalue problems governed by homogeneous Fredholm integral equations. The later eigenvalue problems can be reduced to algebraic eigenvalue problems which then can be solved numerically with effective algorithms.

Dynamics of three-point boundary value problems with Gudermannian neural networks

Dynamics of three-point boundary value problems with Gudermannian neural networks

Applications of variational methods to some three-point boundary value problems with instantaneous and noninstantaneous impulses

In this paper, we study the multiple solutions for some second-order p-Laplace differential equations with three-point boundary conditions and instantaneous and noninstantaneous impulses. By applying the variational method and critical point theory the multiple solutions are obtained in a Sobolev space. Compared with other local boundary value problems, the three-point boundary value problem is less studied by variational method due to its variational structure. Finally, two examples are given to illustrate the results of multiplicity.