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By means of the Schechter's Linking method, we study the existence of 2T-periodic solutions of the non-autonomous fourth-order ordinary differential equation $$ u''''-Au''-Bu-V_u(t,u)=0 $$ where $A>0$, $B>0$, $V(t,u)in mathbb{C}^1(mathbb{R}imesmathbb{R}, mathbb{R})$ is 2T-periodic in t and satisfies either $02$, or $u V_u(t,u)-2V(t,u)geq d_3|u|^r$ with $rgeq 1$.