Nonlinear Three-point Boundary Value Problem Research Articles

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”.

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.

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.

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.

Existence of solution for a nonlinear fifth-order three-point boundary value problem

Existence of solution for a nonlinear fifth-order three-point boundary value problem

Solutions to nonlinear second-order three-point boundary value problems of dynamic equations on time scales

In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; 24: 295-304].

New unique existence criteria for higher-order nonlinear singular fractional differential equations

In this paper, a nonlinear three-point boundary value problem of higher-order singular fractional differential equations is discussed. By applying the properties of Green function and some fixed point theorems for sum-type operator on cone, some new criteria on the existence and uniqueness of solutions are obtained. Moreover, two iterative sequences are given for uniformly approximating the positive solution, which are important for practical application. At last, we give two examples to illustrate the main results.

The Existence and Uniqueness of Positive Solutions for a Singular Nonlinear Three-Point Boundary Value Problems

Using the method of lower and upper solutions, we study the following singular nonlinear three-point boundary value problems: , where K ∈ C[0,1] ,0 α η λ is a positive parameter and present the existence, uniqueness, and the dependency on parameters of the positive solutions under various assumptions. Our result improves those in the previous literatures.

Existence of solution for nonlinear fourth-order three-point boundary value problem

In this paper, we study the existence of nontrivial solution for the fourth-order three- point boundary value problem having the following formu(4) (t) + f (t, u(t)) = 0, 0 < t < 1,u(0) = α(η), u'(0) = u''(0) = 0, u(1) = βu(η),where η ∈ (0, 1), α, β ∈ R, f ∈ C ([0, 1] × R, R). We give sufficient conditions that allow us to obtain the existence of a nontrivial solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.

On the existence of positive solutions for nonlinear three-point boundary value problems

In this paper,considering the nonlinear three-point boundary value problem new method for some properties of Green’s function was studied.By using Guo-Krasnoselskii fixed point theorem,The existence of solutions was established which improved and generalize some existing results in corresponding literatures.

Existence and approximation of solution for a nonlinear second-order three-point boundary value problem

A nonlinear second-order ordinary differential equation with four cases of three-point boundary value conditions is studied by investigating the existence and approximation of solutions. First, the integration method is proposed to transform the considered boundary value problems into Hammerstein integral equations. Second, the existence of solutions for the obtained Hammerstein integral equations is analyzed by using the Schauder fixed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of Hammerstein integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.

Existence of positive pseudo-symmetric solution for second-order three-point boundary value problems

The existence results of at least one positive pseudo-symmetric solution for a nonlinear second-order three-point boundary value problem are established by means of the fixed point theorem due to Avery, Anderson and Krueger.

A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems

In this work an iterative method is proposed to predict and demonstrate the existence and multiplicity of solutions for nonlinear boundary value problems. In addition, the proposed method is capable of calculating analytical approximations for all branches of solutions. This method is a combination of reproducing kernel Hilbert space method and a shooting-like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems. The formulation and implementation of this iterative method is discussed for nonlinear second order with two and three-point boundary value problems. Also, the convergence of the proposed method is proved. To demonstrate the computational efficiency, the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear Bratu equation and nonlinear reaction–diffusion equation. It is also applied successfully to two nonlinear three-point boundary value problems with unknown exact solutions. In the last example a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method may be successful in cases where purely analytic methods are not.

Positive Solution of Nonlinear Two-Order Three-Point Boundary Value Problem for Difference Equation with Change of Sign

In this paper we investigate the existence of positive solution of the following discrete two-order three-point boundary value problemWherandis sign-changing on . By using the fixed-point index theory, the existence of positive solutions for the above boundary value problem is obtained.

Existence of Positive Solutions for a Kind of Fractional Boundary Value Problems

We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.

Nonlinear Sum Operator Equations with a Parameter and Application to Second-Order Three-Point BVPs

A class of nonlinear sum operator equations with a parameter on order Banach spaces were considered. The existence and uniqueness of positive solutions for this kind of operator equations and the dependence of solutions on the parameter have been obtained by using the properties of cone and nonlinear analysis methods. The critical value of the parameter was estimated. Further, the application to some nonlinear three-point boundary value problems was given to show the significance of the discussion.

On Higher-Order Sequential Fractional Differential Inclusions with Nonlocal Three-Point Boundary Conditions

We study a nonlinear three-point boundary value problem of sequential fractional differential inclusions of orderξ+1withn-1<ξ≤n,n≥2. Some new existence results for convex as well as nonconvex multivalued maps are obtained by using standard fixed point theorems. The paper concludes with an example.

Sequential fractional differential equations with three-point boundary conditions

This paper studies a nonlinear three-point boundary value problem of sequential fractional differential equations of order α+1 with 1<α≤2. The expression for Green’s function of the associated problem involving the classical gamma function and the generalized incomplete gamma function is obtained. Some existence results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented. Existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results.

A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space

In this paper, a new numerical algorithm is provided to solve nonlinear three-point boundary value problems in a very favorable reproducing kernel space which satisfies all boundary conditions. Its reproducing kernel function is discussed in detail. We also prove that the approximate solution and its first and second order derivatives all converge uniformly. The numerical experiments show that the algorithm is quite accurate and efficient for solving nonlinear second order three-point boundary value problems.

EXISTENCE OF POSITIVE SOLUTION FOR A NONLINEAR THREE-POINT BOUNDARY VALUE PROBLEM

The existence of positive solution is obtained for the following nonlinearthree-point boundary value problem\begin{equation*}\left\{\begin{array}{l}u^{\prime \prime }(t)+a(t)f(u(t))=0,\quad t\in (0,1) \\[12pt]\beta u(0)-\gamma u^{\prime }(0)=0,\quad u(1)=\alpha u(\eta ),%\end{array}%\right.\end{equation*}%where $$\beta ,\gamma \geq 0,0 0.$$