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

Author

Listed:
  • Yongxiang Li

    (Department of Mathematics, Northwest Normal University, Lanzhou 730070, China)

  • Weifeng Ma

    (Department of Mathematics, Northwest Normal University, Lanzhou 730070, China)

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.

Suggested Citation

  • Yongxiang Li & Weifeng Ma, 2022. "Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem," Mathematics, MDPI, vol. 10(17), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:17:p:3063-:d:897326
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    References listed on IDEAS

    as
    1. C. De Coster & C. Fabry & F. Munyamarere, 1994. "Nonresonance conditions for fourth order nonlinear boundary value problems," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 17, pages 1-16, January.
    2. Mei Wei & Yongxiang Li, 2021. "Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-9, June.
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