On the Sixth-Order Beam Equation of Small Deflection with Variable Parameters

• Published in: Mathematics (Basel) Vol. 13; no. 5; p. 727
• Main Authors: Khanfer, Ammar, Bougoffa, Lazhar, Alhelali, Nawal
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.03.2025
• Subjects: Boundary conditions; Boundary value problems; Differential equations; Existence theorems; ordinary differential equations; Parameters; Uniqueness theorems; Upper bounds
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math13050727

This paper establishes an existence and uniqueness theorem for the nonlocal sixth-order nonlinear beam differential equations with four parameters of the form u(6)+A(x)u(4)+B(x)u″+C(x)u=λf(x,u,u″,u(4)),0<x<1, subject to the integral boundary conditions: u(0)=u(1)=∫01p(x)u(x)dx, u″(0)=u″(1)=∫01q(x)u″(x)dx and u(4)(0)=u(4)(1)=∫01s(x)u(4)(x)dx such that 1−∫01p2(x)dx=α>0,1−∫01q2(x)dx=β>0,1−∫01s2(x)dx=γ>0, under some growth condition on f, and provided that an upper bound exists for the flexural rigidity λ to guarantee that no large deflections will occur.