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

Ammar Khanfer (), Lazhar Bougoffa and Nawal Alhelali
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Ammar Khanfer: Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
Lazhar Bougoffa: Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh 11623, Saudi Arabia
Nawal Alhelali: Department of Mathematics, Faculty of Science, King Saud University, Riyadh 11451, Saudi Arabia

Mathematics, 2025, vol. 13, issue 5, 1-15

Abstract: 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 = λ f ( x , u , u ″ , u ( 4 ) ) , 0 < x < 1 , subject to the integral boundary conditions: u ( 0 ) = u ( 1 ) = ∫ 0 1 p ( x ) u ( x ) d x , u ″ ( 0 ) = u ″ ( 1 ) = ∫ 0 1 q ( x ) u ″ ( x ) d x and u ( 4 ) ( 0 ) = u ( 4 ) ( 1 ) = ∫ 0 1 s ( x ) u ( 4 ) ( x ) d x such that 1 − ∫ 0 1 p 2 ( x ) d x = α > 0 , 1 − ∫ 0 1 q 2 ( x ) d x = β > 0 , 1 − ∫ 0 1 s 2 ( x ) d x = 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.

Keywords: boundary value problems; ordinary differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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