On the Sixth-Order Beam Equation of Small Deflection with Variable Parameters
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)=∫01...
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| Published in | Mathematics (Basel) Vol. 13; no. 5; p. 727 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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| ISSN | 2227-7390 2227-7390 |
| DOI | 10.3390/math13050727 |
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| 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=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. |
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| AbstractList | 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. 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. 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. This paper establishes an existence and uniqueness theorem for the nonlocal sixth-order nonlinear beam differential equations with four parameters of the form u[sup.(6)]+A(x)u[sup.(4)]+B(x)u[sup.″]+C(x)u=λf(x,u,u[sup.″],u[sup.(4)]), 0<x<1, subject to the integral boundary conditions: u(0)=u(1)=∫[sub.0] [sup.1]p(x)u(x)dx, u[sup.″](0)=u[sup.″](1)=∫[sub.0] [sup.1]q(x)u[sup.″](x)dx and u[sup.(4)](0)=u[sup.(4)](1)=∫[sub.0] [sup.1]s(x)u[sup.(4)](x)dx such that 1−∫[sub.0] [sup.1]p[sup.2](x)dx=α>0, 1−∫[sub.0] [sup.1]q[sup.2](x)dx=β>0, 1−∫[sub.0] [sup.1]s[sup.2](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. |
| Audience | Academic |
| Author | Bougoffa, Lazhar Alhelali, Nawal Khanfer, Ammar |
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| Cites_doi | 10.1016/j.compstruct.2013.02.011 10.1016/j.cnsns.2010.01.028 10.1080/00207169208804130 10.3934/math.2021575 10.3934/math.2021664 10.1086/154522 10.1016/j.cnsns.2008.10.013 10.30538/oms2020.0088 10.1016/j.chaos.2005.08.180 10.1016/j.camwa.2007.11.026 10.1080/00207169608804493 10.1186/1687-2770-2012-22 10.1090/S0025-5718-03-01587-4 10.1090/S0002-9939-1988-0958062-3 10.1186/s13662-019-2385-9 10.1155/2010/878131 |
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| SubjectTerms | Boundary conditions Boundary value problems Differential equations Existence theorems ordinary differential equations Parameters Uniqueness theorems Upper bounds |
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| Title | On the Sixth-Order Beam Equation of Small Deflection with Variable Parameters |
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