Let E be a real reflexive Banach space with a uniformly GÃ?¢teaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let ?? : ?? ? ?? a contractive mapping and ?? : ?? ? ?? be a uniformly continuous pseudocontractive mapping with ?? ( ?? ) ? Ã?Ë?. Let { ????} ? ( 0 , 1 / 2 ) be a sequence satisfying the following conditions: (i) l i m?? ? 8????= 0\r\n; (ii) ?8 ?? = 0????= 8. Define the sequence { ????} in K by??0? ??, ???? 1= ??\r\n???? ( ????) ( 1 - 2 ????) ???? ?????? ????, for all?? = 0. Under some ppropriate assumptions, we prove that the sequence { ????} converges strongly to a fixed point \r\n?? ? ?? ( ?? ) which is the unique solution of the following variational inequality: \r\n? ?? ( ?? ) - ?? , ?? ( ?? - ?? ) ? = 0, for all ?? ? ?? ( ?? ).
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