ON THE SOLUTION SETS OF FOUR-POINT BOUNDARY VALUE PROBLEMS FOR NONCONVEX DIFFERENTIAL INCLUSIONS

2011 ◽  
Vol 08 (01) ◽  
pp. 23-37 ◽  
Author(s):  
ADEL MAHMOUD GOMAA
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Differential Inclusions ◽  
Boundary Value ◽  
Point Boundary ◽  
Boundary Problems ◽  
Solution Sets ◽  
Nonempty Compact ◽  
Nonconvex Differential Inclusions

We consider the multivalued problem [Formula: see text] under four boundary conditions u(0) = x0, u(η) = u(θ) = u(T) where 0 < η < θ < T and for F is a multifunctions from [0, T] × ℝn × ℝn to the nonempty compact subsets of ℝn not necessary convex. We give a lemma which is useful in the study of four boundary problems for the differential equations and the differential inclusions. Further we have results that improve earlier theorems.

scholarly journals Iterative numerical method for fractional order two-point boundary value problems

2021 ◽  
Vol 38 (1) ◽  
pp. 47-55
Author(s):  
ALEXANDRU MIHAI BICA ◽  
Keyword(s):  
Differential Equations ◽  
Error Estimate ◽  
Boundary Value Problems ◽  
Numerical Method ◽  
Fractional Order ◽  
Boundary Value ◽  
Point Boundary ◽  
Boundary Problems ◽  
Numerical Example

In this paper we develop an iterative numerical method based on Bernstein splines for solving two-point boundary problems associated to differential equations of fractional order $\alpha\in\left( 0,1\right) $. The convergence of the method is proved by providing the error estimate and it is tested on a numerical example.

scholarly journals POSITIVE SOLUTIONS FOR SINGULAR SYSTEMS OF HIGHER-ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS

2013 ◽  
Vol 18 (3) ◽  
pp. 309-324 ◽  
Author(s):  
Johnny Henderson ◽  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Singular Systems ◽  
Boundary Value ◽  
Higher Order ◽  
Point Boundary ◽  
Existence Of Positive Solutions ◽  
Higher Order Differential Equations

We investigate the existence of positive solutions for systems of singular nonlinear higher-order differential equations subject to multi-point boundary conditions.

scholarly journals Boundary Value Problems for ψ-Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1001
Author(s):  
Surang Sitho ◽  
Sotiris K. Ntouyas ◽  
Ayub Samadi ◽  
Jessada Tariboon
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Point Theorem ◽  
Existence Result ◽  
Boundary Value ◽  
Point Boundary ◽  
Differential Equations And Inclusions

In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ-Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from fixed point theory. Thus, in the single-valued case, the existence of a unique solution is proved by using the classical Banach fixed point theorem while an existence result is established via Krasnosel’skiĭ’s fixed point theorem. The Leray–Schauder nonlinear alternative for multi-valued maps is the basic tool to prove an existence result in the multi-valued case. Finally, our results are well illustrated by numerical examples.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

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

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ji Lin ◽  
Yuhui Zhang ◽  
Chein-Shan Liu
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Initial Conditions ◽  
Boundary Value ◽  
Accurate Solution ◽  
Point Boundary ◽  
Third Order ◽  
Boundary Shape ◽  
Free Function ◽  
New Algorithms

AbstractFor 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.

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