In real Hilbert space H, from an arbitrary initial point x0∈H, an explicit iteration scheme is defined as follows: xn+1=αnxn+(1?αn)Tλn+1xn,n≥0, where Tλn+1xn=Txn?λn+1μF(Txn), T:H→H is a nonexpansive mapping such that F(T)={x∈K:Tx=x} is nonempty, F:H→H is a η-strongly monotone and k-Lipschitzian mapping, {αn}?(0,1), and {λn}?0,1). Under some suitable conditions, the sequence {xn} is shown to converge strongly to a fixed point of T and the necessary and sufficient conditions that {xn} converges strongly to a fixed point of T are obtained.
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