In this paper, by means of the computation of fixed point index, we obtain the existence of positive solutions for four-point boundary value problem involving the $$p(t)$$ p ( t ) -Laplacian $$\begin{aligned}&(\phi (t,u'(t)))'+ f(t,u(t))=0,\quad t\in (0,1), \\&u'(0)=\alpha u'(\xi ),\quad u(1)=\beta u(\eta ), \end{aligned}$$ ( ϕ ( t , u ′ ( t ) ) ) ′ + f ( t , u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ′ ( 0 ) = α u ′ ( ξ ) , u ( 1 ) = β u ( η ) , where $$\phi (t,x)=|x|^{p(t)-2} \ x,\ p\in C([0,1],(1,+\infty )),\ \alpha ,\beta ,\xi ,\eta \in (0,1),\ \xi >\eta ,\ p(0)=p(\xi ),\ f\in C([0,1]\times [0,+\infty ),[0,+\infty ))$$ ϕ ( t , x ) = | x | p ( t ) - 2 x , p ∈ C ( [ 0 , 1 ] , ( 1 , + ∞ ) ) , α , β , ξ , η ∈ ( 0 , 1 ) , ξ > η , p ( 0 ) = p ( ξ ) , f ∈ C ( [ 0 , 1 ] × [ 0 , + ∞ ) , [ 0 , + ∞ ) ) .