Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem

Yongxiang Li () and Weifeng Ma
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Yongxiang Li: Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Weifeng Ma: Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Mathematics, 2022, vol. 10, issue 17, 1-19

Abstract: This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem u ( 4 ) = f ( t , u , u ′ , u ″ , u ‴ ) on [ 0 , 1 ] with the boundary condition u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , which models a statically bending elastic beam whose two ends are simply supported, where f : [ 0 , 1 ] × R + × R × R − × R → R + is continuous. Some precise inequality conditions on f guaranteeing the existence of positive solutions are presented. The inequality conditions allow that f ( t , u , v , w , z ) may be asymptotically linear, superlinear, or sublinear on u , v , w , and z as | ( u , v , w , z ) | → 0 and | ( u , v , w , z ) | → ∞ . Our discussion is based on the fixed point index theory in cones.

Keywords: fully fourth-order boundary value problem; simply supported beam equation; positive solution; cone; fixed point index (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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