Introduction to Functional Analysis.

Measures of noncompactness in banach spaces

In this paper we discuss the existence of positive solutions of the fully fourth-order boundary value problem { u ( 4 ) = f ( t , u , u ′ , u ″ , u ‴ ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ′ ( 0 ) = u ″ ( 1 ) = u ‴ ( 1 ) = 0 , which models a statically elastic beam fixed at the left and freed at the right, and it is called cantilever beam in mechanics, where f : [ 0 , 1 ] × R + 3 × R − → R + is continuous. Some inequality conditions on f guaranteeing the existence of positive solutions are presented. Our conditions allow that f ( t , x 0 , x 1 , x 2 , x 3 ) is superlinear or sublinear growth on x 0 , x 1 , x 2 , x 3 . For the superlinear case, a Nagumo-type condition is presented to restrict the growth of f on x 2 and x 3 . Our discussion is based on the fixed point index theory in cones.

Existence of positive solutions for the cantilever beam equations with fully nonlinear terms

We prove new results on the existence of positive solutions for some cantilever equation subject to nonlocal and nonlinear boundary conditions. Our main ingredient is the classical fixed point index.

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Yongxiang Li

Papers

On the existence of positive solutions for the bending elastic beam equations

A monotone iterative technique for solving the bending elastic beam equations

Monotone iterative technique for second order delayed periodic problem in Banach spaces