scholarly journals GREEN’S FUNCTION AND EXISTENCE OF SOLUTIONS FOR A THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

2019 ◽  
Vol 24 (2) ◽  
pp. 171-178 ◽  
Author(s):  
Sergey Smirnov
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Point Boundary ◽  
Nonlinear Boundary Value Problems ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results

The solutions of third-order three-point boundary value problemx′′′+f(t,x) = 0, t∈[a,b],x(a) =x′(a) = 0, x(b) =kx(η),whereη∈(a,b),k∈R,f∈C([a,b]×R,R) andf(t,0)6= 0, are the subject of thisinvestigation. In order to establish existence and uniqueness results for the solutions,attention is focused on applications of the corresponding Green’s function. As anapplication, also one example is given to illustrate the result.Keywords:Green’s function, nonlinear boundary value problems, three-point boundaryconditions, existence and uniqueness of solutions.

scholarly journals Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Peiguo Zhang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Nonlinear Boundary Value Problems ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Equation Boundary

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.

scholarly journals SHARPER EXISTENCE AND UNIQUENESS RESULTS FOR SOLUTIONS TO THIRD-ORDER BOUNDARY VALUE PROBLEMS

2020 ◽  
Vol 25 (3) ◽  
pp. 409-420 ◽  
Author(s):  
Saleh S. Almuthaybiri ◽  
Christopher C. Tisdell
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Contraction Mapping Theorem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach’S Fixed Point Theorem

The purpose of this note is to sharpen Smirnov’s recent work on existence and uniqueness of solutions to third-order ordinary differential equations that are subjected to two- and three-point boundary conditions. The advancement is achieved in the following ways. Firstly, we provide sharp and sharpened estimates for integrals regarding various Green’s functions. Secondly, we apply these sharper estimates to problems in conjunction with Banach’s fixed point theorem. Thirdly, we apply Rus’s contraction mapping theorem in a metric space, where two metrics are employed. Our new results improve those of Smirnov by showing that a larger class of boundary value problems admit a unique solution.

scholarly journals Existence and uniqueness results for Φ-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Thabet Abdeljawad ◽  
Fahd Jarad ◽  
Piyachat Borisut ◽  
...  
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Uniqueness Of Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Special Cases ◽  
Generalized Fractional Derivative

Abstract The present paper describes the implicit fractional pantograph differential equation in the context of generalized fractional derivative and anti-periodic conditions. We formulated the Green’s function of the proposed problems. With the aid of a Green’s function, we obtain an analogous integral equation of the proposed problems and demonstrate the existence and uniqueness of solutions using the techniques of the Schaefer and Banach fixed point theorems. Besides, some special cases that show the proposed problems extend the current ones in the literature are presented. Finally, two examples were given as an application to illustrate the results obtained.

scholarly journals EXISTENCE AND UNIQUENESS RESULTS FOR A COUPLED SYSTEM OF HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS WITH MULTI-POINT BOUNDARY CONDITIONS

2020 ◽  
pp. 843
Author(s):  
Mohamed Houas ◽  
Khellaf Ould Melha
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Coupled System ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential

In this paper, we have studied existence and uniqueness of solutions for a coupled system of multi-point boundary value problems for Hadamard fractional differential equations. By applying principle contraction and Shaefer's fixed point theorem new existence results have been obtained.

scholarly journals A Study of Caputo-Hadamard-Type Fractional Differential Equations with Nonlocal Boundary Conditions

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Wafa Shammakh
Keyword(s):  
Differential Equations ◽  
Green’S Function ◽  
Green's Function ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Nonlinear Boundary Value Problems ◽  
Uniqueness Results ◽  
Fractional Differential

Existence and uniqueness results of positive solutions to nonlinear boundary value problems for Caputo-Hadamard fractional differential equations by using some fixed point theorems are presented. The related Green’s function for the boundary value problem is given, and some useful properties of Green’s function are obtained. Example is presented to illustrate the main results.

scholarly journals Iteration for a Third-Order Three-Point BVP with Sign-Changing Green's Function

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Ya-Hong Zhao ◽  
Xing-Long Li
Keyword(s):  
Iterative Method ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solution ◽  
Green's Function ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Monotone Positive Solution

We are concerned with the following third-order three-point boundary value problem:u‴(t)=f(t,u(t)),t∈[0,1],u′(0)=u(1)=0,u″(η)+αu(0)=0, whereα∈[0,2)andη∈[2/3,1). Although corresponding Green's function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions onfby applying iterative method. An example is also included to illustrate the main results obtained.

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