Halpern type proximal point algorithm of accretive operators

The main objectives of this paper are to employ a new proof technique to prove the strong convergence of {xn}{xn} and {yn}{yn}, defined respectively by xn+1=αnu+βnxn+(1−αn−βn)JrnAxnand yn+1=βnyn+(1−βn)JrnA(αnu+(1−αn)yn), to some zero of accretive operator AA in a uniformly convex Banach space EE with a uniformly Gâteaux differentiable norm (or with a weakly continuous duality mapping JφJφ) whenever {αn}{αn} and {βn}{βn} are sequences in (0,1)(0,1) and {rn}⊂(0,+∞){rn}⊂(0,+∞) satisfying the conditions: limn→∞αn=0limn→∞αn=0, ∑n=1+∞αn=+∞, lim supn→∞βn<1lim supn→∞βn<1,lim infn→∞rn>0lim infn→∞rn>0.