Abstract
Let
K be a closed convex subset of a Banach space
E and let
T
:
K
→
E be a continuous weakly inward pseudocontractive mapping. Then for
t
∈
(0,
1), there exists a sequence {
y
t
}
⊂
K satisfying
y
t
=
(1
−
t)
f(
y
t
)
+
tT(
y
t
), where
f
∈
Π
K
≔
{
f
:
K
→
K, a contraction with a suitable contractive constant}. Suppose further that
F(
T)
≠
∅ and
E is reflexive and strictly convex which has uniformly Gâteaux differentiable norm. Then it is proved that {
y
t
} converges strongly to a fixed point of
T which is also a solution of certain variational inequality. Moreover, an explicit iteration process which converges strongly to a fixed point of
T and hence to a solution of certain variational inequality is constructed provided that
T is Lipschitzian.