The contraction-proximal point algorithm with square-summable errors
© Tian and Wang; licensee Springer 2013
Received: 7 September 2012
Accepted: 26 March 2013
Published: 11 April 2013
In this paper, we study the contraction-proximal point algorithm for approximating a zero of a maximal monotone mapping. The norm convergence of such an algorithm has been established under two new conditions. This extends a recent result obtained by Ceng, Wu and Yao to a more general case.
MSC:47J20, 49J40, 65J15, 90C25.
that is, we only need to assume the sequence is bounded below away from zero. The paper is organized as follows. In Section 2, we prove two useful lemmas that are very useful for proving the boundedness of the iteration. In Section 3, we establish norm convergence of the CPPA under two different conditions. As a result, we extend the corresponding result obtained in .
2 Some lemmas
is not properly contained in the graph of any other monotone operator.
the resolvent of A, where and I is the identity operator. The zero set of A is denoted by . The resolvent operator has the following properties (see ).
is single-valued and firmly nonexpansive;
, where denotes the fixed point set of ;
its graph is weak-to-strong closed in .
for all and all . In what follows, we present two lemmas that are very useful for proving the boundedness of the iterative sequence.
where the last inequality follows from the basic inequality: for all . □
where and are real sequences. If , then is bounded; more precisely, .
By induction, we can show the result as desired. □
If , , and , then .
3 Convergence analysis
In what follows, we assume that A is a maximal monotone mapping and its zero set S is nonempty. To establish the convergence, we need the following lemma, which is indeed proved in . We present here a different proof that is mainly based on property of firmly nonexpansive mappings.
Consequently, the desired inequality (7) follows from the fact . □
We now are ready to prove our main results.
Applying Lemma 2 to the last inequality, we conclude that is bounded.
It is obvious that .
We next consider two possible cases on the sequence .
By using Lemma 5, we conclude that .
Since and implies , this together with the fact immediately yields . □
to ensure the convergence of the CPPA. In the following theorem, we shall present a similar condition under the accuracy criterion (II).
where satisfying , so we assume without loss of generality that . Applying Lemma 3, we conclude that is bounded.
where we define .
To show , we consider two possible cases for .
We note that by our hypothesis goes to zero, and thus apply Lemma 5 to the previous inequality to conclude that .
Consequently, follows from (6) immediately. □
The authors thank the referees for their useful comments and suggestions. This work is supported by the National Natural Science Foundation of China, Tianyuan Foundation (11226227), the Basic Science and Technological Frontier Project of Henan (122300410268, 122300410375).
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