Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption

In the present paper, the following result is shown: Let X X be a real Banach space with a uniformly convex dual X ∗ X^* , and let K K be a nonempty closed convex and bounded subset of X X . Assume that T : K → K T:\,K\rightarrow K is a continuous strong pseudocontraction. Let { α n } n = 1 ∞ \{\alpha _n\}^{\infty }_{n=1} and { β n } n = 1 ∞ \{\beta _n\}^{\infty }_{n=1} be two real sequences satisfying (i) 0 > α n , β n > 1 0>\alpha _n,\,\beta _n>1 for all n ≥ 1 n\ge 1 ; (ii) ∑ n = 1 ∞ α n = ∞ \sum _{n=1}^{\infty }\alpha _n=\infty ; and (iii) α n → 0 , β n → 0 \alpha _n \rightarrow 0,\, \beta _n \rightarrow 0 as n → ∞ . n\rightarrow \infty . Then the Ishikawa iterative sequence { x n } n = 1 ∞ \{x_n\}_{n=1}^{\infty } generated by ( I ) { x 1 ∈ K , x n + 1 = ( 1 − α n ) x n + α n T y n , y n = ( 1 − β n ) x n + β n T x n , n ≥ 1 , \begin{equation*} \mathrm {(I)} \quad \left \{ \begin {array}{l} x_1\in K,\\ x_{n+1}=(1-\alpha _n)x_n+\alpha _nTy_n,\\ y_n=(1-\beta _n)x_n+\beta _nTx_n,\,n\geq 1, \end{array} \right . \end{equation*} converges strongly to the unique fixed point of T T .