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 Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis

2020 ◽  
Vol 18 (1) ◽  
pp. 1006-1024
Author(s):  
Saleh S. Almuthaybiri ◽  
Christopher C. Tisdell
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Boundary Value ◽  
Elastic Beam ◽  
Point Theory ◽  
Uniqueness Of Solutions ◽  
Contraction Mapping Theorem ◽  
Uniqueness Results

Abstract We examine the existence and uniqueness of solutions to two-point boundary value problems involving fourth-order, ordinary differential equations. Such problems have interesting applications to modelling the deflections of beams. We sharpen traditional results by showing that a larger class of problems admit a unique solution. We achieve this by drawing on fixed-point theory in an interesting and alternative way via an application of Rus’s contraction mapping theorem. The idea is to utilize two metrics on a metric space, where one pair is complete. Our theoretical results are applied to the area of elastic beam deflections when the beam is subjected to a loading force and the ends of the beam are either both clamped or one end is clamped and the other end is free. The existence and uniqueness of solutions to the models are guaranteed for certain classes of linear and nonlinear loading forces.

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 Solutions to Third-Order Boundary Value Problems

2021 ◽  
Vol 22 (2) ◽  
pp. 221-240
Author(s):  
S. S. Almuthaybiri ◽  
J. M. Jonnalagadda ◽  
C. C. Tisdell
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Fixed Point Methods ◽  
Bounded Sets

The purpose of this research is to connect fixed point methods with certain third-order boundary value problems in new and interesting ways. Our strategy involves an analysis of the problem under consideration within closed and bounded sets. We develop sufficient conditions under which the associated mappings will be contractive and invariant in these sets, which generates new advances concerning the existence, uniqueness and approximation of solutions.

scholarly journals Notes on Local and Nonlocal Intuitionistic Fuzzy Fractional Boundary Value Problems with Caputo Fractional Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ali El Mfadel ◽  
Said Melliani ◽  
M’hamed Elomari
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Caputo Fractional Derivatives ◽  
Existence And Uniqueness Results ◽  
Intuitionistic Fuzzy ◽  
Uniqueness Results ◽  
Fractional Boundary Value Problems

In this paper, we investigate the existence and uniqueness results of intuitionistic fuzzy local and nonlocal fractional boundary value problems by employing intuitionistic fuzzy fractional calculus and some fixed-point theorems. As an application, we conclude this manuscript by giving an example to illustrate the obtained results.

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