Abstract: This paper shows the existence results of a resonance problem with derivative terms{u″(t)=f(t,u(t),u'(t)), t∈［0,1］,u(0)=εu'(0), u(1)=αu(η).under the condition of α(η+ε)=1+ε at resonance, where ε∈［0,+∞), α∈(0,∞), η∈(0,1)are given constants, and αη<21. f:［0,1］×R²→R is continuous and satisfies the Nagumo condition. The proof of the main results is based on the method of upper and lower solutions and the connectivity theory of the solution set.

Key words: existence, resonance, Nagumo condition, connectivity, disordered lower and upper solutions