Abstract
The singular fourth-order nonlocal boundary value problem { u ⁗ ( t ) = h ( t ) f ( t , u ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = ∫ 0 1 p ( s ) u ( s ) d s , u ″ ( 0 ) = u ″ ( 1 ) = ∫ 0 1 q ( s ) u ( s ) d s is considered under some suitable conditions concerning the first eigenvalue of the corresponding linear operator, where p , q ∈ L 1 [ 0 , 1 ] , h : ( 0 , 1 ) → [ 0 , + ∞ ) is continuous, symmetric on (0, 1) and may be singular at t = 0 and t = 1 , f : [ 0 , 1 ] × [ 0 , + ∞ ) → [ 0 , + ∞ ) is continuous and f ( ⋅ , x ) is symmetric on [0,1] for all x ∈ [ 0 , + ∞ ) . The existence of at least one symmetric positive solution is obtained by the application of the fixed point index in cones.
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