On solvability of second-order Sturm-Liouville boundary value problems at resonance

Abstract

In this paper, based on of the concept q 0 ∈ H 0 ( p , ( 0 , 1 ) , α , β ) q_0\in H_0(p,(0,1),\alpha ,\beta ) , which is a generalized form of the first resonant point π 2 \pi ^2 to the Picard problem x + λ x = 0 x+\lambda x=0 , x ( 0 ) = x ( 1 ) = 0 x(0)=x(1)=0 , we study the solvability of second-order Sturm-Liouville boundary value problems at resonance ( p ( t ) x ′ ) ′ + q 0 ( t ) x + g ( t , x ) = h ( t ) (p(t)x’)’+q_0(t)x+g(t,x)=h(t) , x ( 0 ) cos ⁡ α − p ( 0 ) x ′ ( 0 ) sin ⁡ α = 0 x(0){\cos \alpha }-p(0)x’(0)\sin \alpha =0 , x ( 1 ) cos ⁡ β − p ( 1 ) x ′ ( 1 ) sin ⁡ β = 0 x(1)\cos \beta -p(1)x’(1)\sin \beta =0 , and improve the previous results about problems x + π 2 x + g ( t , x ) = h ( t ) , x ( 0 ) = x ( 1 ) = 0 x+\pi ^2x+g(t,x)=h(t),x(0)=x(1)=0 derived by Chaitan P. Gupta, R. Iannacci and M. N. Nkashama, and Ma Ruyun, respectively.