Let X be a real reflexive Banach space with a uniformly $Ghat{a}teaux$ differentiable norm, C a nonempty closed convex subset of X, T : C $ ightarrow$ X a continuous pseudocontractive mapping, and A : C $ ightarrow$ C a continuous strongly pseudocontractive mapping. We show the existence of a path ${x_t}$ satisfying $x_t=tAx_t+(1- t)Tx_t$, t $in$ (0,1) and prove that ${x_t}$ converges strongly to a fixed point of T, which solves the variational inequality involving the mapping A. As an application, we give strong convergence of the path ${x_t}$ defined by $x_t=tAx_t+(1-t)(2I-T)x_t$ to a fixed point of firmly pseudocontractive mapping T.