creative_2016_25_1_85_92_001

Strong convergence results for nonlinear mappings in real Banach spaces

Mogbademu, Adesanmi Alao

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creative_2016_25_1_85_92

 

 

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Let X be a real Banach space, K be a nonempty closed convex subset of X, T: K \rightarrow K be a nearly uniformly L-Lipschitzian mapping with sequence \{a_n\}. Let {k_n}\subset [1,\infty) and \epsilon_n be sequences with \lim_{n\rightarrow\infty} k_n=1, \hspace{.05in}\lim_{n\rightarrow\infty} \epsilon_n=0 and F(T)=\{\rho\in K: T\rho=\rho\}\neq \emptyset. Let \{\alpha_n\}_{n\geq 0} be real sequence in [0,1] satisfying the following conditions: (i)\sum_{n\geq 0}\alpha_n=\infty (ii) \lim_{n\rightarrow\infty}\alpha_n=0. For arbitrary x_0\in K, let \{x_n\}_{n\geq 0} be iteratively defined by x_{n+1}= (1-\alpha_n)x_n + \alpha_nT^nx_n,\hspace{.1in}n\geq 0. If there exists a strictly increasing function \Phi:[0,\infty)\rightarrow [0,\infty) with \Phi(0)= 0 such that

    \[<T^nx - T^n\rho, j(x-\rho)> \leq k_n \|x -\rho\|^2 - \Phi(\|x -\rho\|)+\epsilon_n\]

for all x\in K, then, \{x_n\}_{n\geq 0} converges strongly to \rho\in F(T).

It is also proved that the sequence of iteration \{x_n\} defined by

    \[x_{n+1} =(1-b_n-d_n)x_n+b_nT^nx_n+d_nw_n, n\geq 0,\]

where \{w_n\}_{n\geq 0} is a bounded sequence in K and \{b_n\}_{n\geq 0},\hspace{0.02in}\{d_n\}_{n\geq 0} are sequences in [0,1] satisfying appropriate conditions, converges strongly to a fixed point of T.

 

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Mogbademu, Adesanmi Alao

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