Journal of the
Korean Mathematical Society JKMS

Let $X$ be a real reflexive Banach space with a uniformly G\^ateaux differentiable norm, $C$ a nonempty closed convex subset of $X, T:C\to X$ a continuous pseudocontractive mapping, and $A:C\to C$ 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 $x_t=tA x_t+(1-t) (2I-T)x_t$ to a fixed point of firmly pseudocontractive mapping $T$.

Keywords: pseudocontractive mapping, strongly pseudocontractive mapping, firmly pseudocontractive mapping, nonexpansive mapping, fixed points, uniformly G\^ateaux differentiable norm, variational inequality

MSC numbers: Primary 47H10, 47J20