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Let X be a real reflexive Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of X, T : C → X a continuous pseudocontractive mapping, and A : C → C a continuous strongly pseudocontractive mapping. We show the existence of a path {xt} satisfying xt = tAxt +(1− t)Txt, t ∈ (0, 1) and prove that {xt} 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 {xt} defined by xt = tAxt + (1− t)(2I −T )xt to a fixed point of firmly pseudocontractive mapping T .

CONVERGENCE OF APPROXIMATING PATHS TO SOLUTIONS OF VARIATIONAL INEQUALITIES INVOLVING NON-LIPSCHITZIAN MAPPINGS