On Three-Point Boundary Value Problem

Three-point boundary value problems for the second order nonlinear ordinary differential equations are considered. Existence of solutions are established by using the quasilinearization approach. As an application, the Emden-Fowler type problems with nonresonant and resonant linear parts are considered to demonstrate our results

Fuzzy Solutions to Second Order Three Point Boundary Value Problem

In this manuscript, the proposed work is to study the existence of second-order differential equations with three point boundary conditions. Existence is proved using fuzzy set valued mappings of a real variable whose values are normal, convex, upper semi continuous and compactly supported fuzzy sets. The sufficient conditions are also provided to establish the existence results of fuzzy solutions of second order differential equations for three point boundary value problem. By using Banach fixed point principle, a new existence theorem of solutions for these equations in the metric space of normal fuzzy convex sets with distance given by the maximum of the Hausdorff distance between level sets is obtained. Then to further establish the existence, fixed point theorem for absolute retracts is used by taking consideration that space of fuzzy sets can be embedded isometrically as a cone in Banach space. Finally, an example is provided to illustrate the result

The method of quasilinearization and a three-point boundary value problem

The method of quasilinearization generates a monotone iteration scheme whose iterates converge quadratically to a unique solution of the problem at hand. In this paper, we apply the method to two families of three-point boundary value problems for second order ordinary differential equations: Linear boundary conditions and nonlinear boundary conditions are addressed independently. For linear boundary conditions, an appropriate Green\u27s function is constructed. For nonlinear boundary conditions, we show that these nonlinearities can be addressed similarly to the nonlinearities in the differential equation

Discrete first-order three-point boundary value problem

We study difference equations which arise as discrete approximations to three-point boundary value problems for systems of first-order ordinary differential equations. We obtain new results of the existence of solutions to the discrete problem by employing Euler’s method. The existence of solutions are proven by the contraction mapping theorem and the Brouwer fixed point theorem in Euclidean space. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. We also give some examples to illustrate the existence of a unique solution of the contraction mapping theorem

Positive solutions to a nonlinear three-point boundary value problem with singularity

In this paper, we discuss the existence and uniqueness of positive solutions to a singular boundary value problem of fractional differential equations with three-point integral boundary conditions. The nonlinear term f possesses singularity and also depends on the first-order derivative u′. Our approach is based on Leray-Schauder fixed point theorem and Banach contraction principle. Examples are presented to confirm the application of the main results

Eigenvalue problems for a three-point boundary-value problem on a time scale

Let $\mathbb{T}$ be a time scale such that $0, T \in \mathbb{T}$. We us a cone theoretic fixed point theorem to obtain intervals for $\lambda$ for which the second order dynamic equation on a time scale, \begin{gather*} u^{\Delta\nabla}(t) + \lambda a(t)f(u(t)) = 0, \quad t \in (0,T) \cap \mathbb{T},\\ u(0) = 0, \quad \alpha u(\eta) = u(T), \end{gather*} where $\eta \in (0, \rho(T)) \cap \mathbb{T}$, and $0 < \alpha <T/\eta$, has a positive solution

Positive solutions of nonlinear fractional three-point boundary-value problem

In this paper, we study the existence of positive solutions to the boundary-value problem with fractional order\begin{eqnarray*} \begin{split}(^{C}_{a}D^{\alpha}y)(t)+q(t)f(y)&amp;=0, \hskip 0.5cm 0\leq a&lt;t&lt;b, \hskip 0.5cm 1&lt;\alpha &lt;2,\\\\&amp;y(a)=0, \hskip 0.5 cm y(b)=\beta y(\eta),\end{split}\end{eqnarray*}where a<\eta<b and $\beta (\eta-a)-b+a\neq 0$. We prove the existence of at least one positivesolution when $f$ is either superlinear or sublinear using the well-known Guo's fixed point theorem incones. Moreover, the convexity and concavity of the solutions are investigated with respect to the behavior of the function $q$