We shall study the existence of solutions for a ( k , n − k ) $(k,n-k)$ conjugate boundary-value problem at resonance with dim ker L = 2 in this paper. The boundary-value problem is shown as follows: ( − 1 ) n − k φ ( n ) ( x ) = f ( x , φ ( x ) , φ ′ ( x ) , … , φ ( n − 1 ) ( x ) ) , x ∈ [ 0 , 1 ] , φ ( i ) ( 0 ) = φ ( j ) ( 1 ) = 0 , 1 ≤ i ≤ k − 1 , 1 ≤ j ≤ n − k − 1 , φ ( 0 ) = ∫ 0 1 φ ( x ) d A ( x ) , φ ( 1 ) = ∫ 0 1 φ ( x ) d B ( x ) . $$\begin{aligned}& (-1)^{n-k}\varphi^{(n)}(x)=f \bigl(x,\varphi(x), \varphi'(x),\ldots,\varphi^{(n-1)}(x) \bigr), \quad x\in[0,1], \\ & \varphi^{(i)}(0)=\varphi^{(j)}(1)=0, \quad1\leq i\leq k-1, 1\leq j \leq n-k-1, \\ & \varphi(0)= \int^{1}_{0}\varphi(x)\,dA(x), \quad\quad \varphi(1)= \int^{1}_{0}\varphi(x)\,dB(x). \end{aligned}$$ We can obtain that this boundary-value problem has at least one solution under the conditions we provided through Mawhin’s continuation theorem, and an example is also provided for our new results.