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

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