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

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.

Existence and uniqueness of solutions to the nonlinear boundary value problem for fourth-order differential equations with all derivatives

“…For the nonlinear boundary, there are few studies at present, and the theories need to be further studied. For the linear boundary, by contrast to [1], the condition of Theorem 3.4 is not optimal. The results in [1] offer an advancement over traditional approaches based on the Rus fixed point theorem and two metrics.…”

“…For the linear boundary, by contrast to [1], the condition of Theorem 3.4 is not optimal. The results in [1] offer an advancement over traditional approaches based on the Rus fixed point theorem and two metrics. It is notable that the method used in [1] seems to be invalid for (1.3), (1.4) because the nonlinear boundary conditions are dominated by the function of u( 1).…”

“…In [29], the authors obtained new conclusion on the uniqueness of solutions to problem (1.1) with the order reduction method and the linear operator theory. In [1], Almuthaybiri and Tisdell considered the following nonlinear fourth-order differential equation with linear boundary conditions (LBC) for continuous f : [0, 1] × R 4 → R: ⎧ ⎨ ⎩ u (4) (t) = f (t, u(t), u (t), u (t), u (t)), t ∈ [0, 1],…”

Laminar flow in channels with porous walls: advancing the existence, uniqueness and approximation of solutions via fixed point approaches

“…This is in contrast to third order approaches where there are constants of integration in the equation and a fourth "hanging" boundary condition to consider. In addition, the mathematical theory regarding solutions to fourth order BVPs has recently been advanced in directions [1] that potentially can shine new light on (1.1), (1.2) and so we feel that this presents a timely opportunity to directly work with the form of the fourth order BVP (1.1), (1.2).…”

Solutions to Differential Equations via Fixed Point Approaches: New Mathematical Foundations and Applications

“…Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc. S. S. G. Almuthaybiri [2] are researching the theoretical aspects of differential equations, and those who are interested in better understanding their applications. Some of the research from the thesis has been published in [1][2][3][4][5][6].…”