RESEARC H Open Access On relaxed and contraction-proximal point algorithms in hilbert spaces Shuyu Wang * and Fenghui Wang * Correspondence: shyuwang@163. com Department Of Mathematics, Luoyang Normal University, Luoyang 471022, China Abstract We consider the relaxed and contraction-proximal point algorithms in Hilbert spaces. Some conditions on the parameters for gu aranteeing the convergence of the algorithm are relaxed or removed. As a result, we extend some recent results of Ceng-Wu-Yao and Noor-Yao. Keywords: maximal monotone operator, proximal point algorithm, firmly nonexpan- sive operator 1. Introduction Throughout, H denotes a real Hilbert s pace and A a multi-valued operator with domain D(A). We know that A is called monotone if 〈u - v, x - y〉 ≥ 0, for any u Î Ax, v Î Ay; maximal monotone if its graph G(A)={(x,y): x Î D(A), y Î Ax} is not prop- erly contained in the graph of any other monotone operator. Denote by S:={x Î D (A): 0 Î Ax}thezerosetandbyJ c :=(I + cA) -1 the resolvent of A.Itiswellknown that J c is single valued and D(J c )=H for any c >0. A fundamental problem of monotone operators is that of finding an element x so that 0 Î Ax. This problem is essential because it includes many concrete examples, such as convex pr ogrammi ng and monotone variational inequaliti es. A successfu l and powerful algorithm for solving this problem is the well-known proximal point algo- rithm (PPA), which generates, for any initial guess, x 0 Î H, an iterative sequence as x n+1 = J c n (x n + e n ) , (1:1) where (c n ) is a positive real sequence and (e n ) is the error sequence (see [1]). To guarantee the convergence of PPA, there are two kinds of accuracy criterion posed on the error sequence: (I)  e n  ≤ ε n , ∞  n = 0 ε n < ∞ o r (II)  e n  ≤ η n   ˜ x n − x n   , ∞  n = 0 η n < ∞ , where ˜ x n = J c n (x n + e n ) . In 2001, Han and He [2] proved that in finite dimensional Hilbert space criterion (II) can be replaced by Wang and Wang Journal of Inequalities and Applications 2011, 2011:41 http://www.journalofinequalitiesandapplications.com/content/2011/1/41 © 2011 Wang and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecomm ons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductio n in any medium, provided the original work is prop erly cited. (II’)  e n  ≤ η n   ˜ x n − x n   , ∞  n = 0 η 2 n < ∞ . The infinite version was obtained by Marino and Xu [3]. There are various generations or modifications on the PPA. Among them Eckstein and Bertsekas [4] proposed the relaxed proximal point algorithm (RPPA): x n+1 =(1− ρ n )x n + ρ n J c n (x n )+e n , (1:2) where (r n ) ⊂ (0, 2) is a relaxation factor. The weak convergence of (1.2) is guaran- teed provided that (e n ) satisfies criterion (I), c n ≥ ¯ c > 0, 0 <δ≤ ρ n ≤ 2 − δ . (1:3) On the other hand, since the PPA does not necessarily converge strongly (see [5]), many authors have conducted worthwhile studies on modifying the PPA so that the strong convergence is guaranteed (see, for instance, [6-8]). In particular, Marino and Xu [3] proposed the contraction-proximal point algorithm (CPPA): x n+1 = λ n u +(1− λ n )J c n (x n )+e n , (1:4) where the parameters above satisfy (i) lim n l n =0,Σ n l n = ∞; (ii) either Σ n |l n +1- l n |<∞; or lim n l n /l n +1 = 1; (iii) 0 < c ≤ c n ≤ ¯ c < ∞,  n |c n+1 − c n | < ∞ ; (iv) Σ n ||e n || < ∞ . Under these assumptions, the CPPA converges strongly to P S (u), the projection of u onto S. In this article, we shall focus on the RPPA and CPPA. We note that the resolvent is in fact the arithmetic mean of the identity and a nonexpansive operator. By using this fact, we relax or remove some sufficient conditions to guarantee the convergence of the algorithms. As a result, we extend and improve some recent results on the PPA. 2. Some lemmas We know that an operator T : H ® H is called (i) nonexpansive if ||Tx - Ty|| ≤ || x - y|| ∀x,y Î H; and (ii) firmly nonexpansive if 〈Tx - Ty, x - y〉 ≥ ||Tx - Ty|| 2 ∀x ,y Î H. Denote by Fix(T)={x Î H : x = Tx}thefixedpointsetofT.Itiswellknownthat firmly nonexpansive operators have the following properties. Lemma 1 (Goebel-Kirk [9]). Let T be firmly nonexpansive. Then (1) 2T - I is nonexpan- sive; (2) 〈Tx - x, Tx - z〉 ≤ 0 for all x Î H and for all z Î H Fix(T). It is well known that J c is firmly nonexpansive and consequently nonexpansive; moreover, S =Fix(J c ). Since the fixed point set of nonexpansive operators is closed convex, the projection P s onto the solution set S is well defined whenever S ≠ ∅. Here- aft er, we assume that S is nonempty. The following lemmas play an important role in our convergence analysis. Lemma 2 (resolvent identity [3]). Let c, t >0. Then for any x Î H, J c x = J t  t c x +  1 − t c  J c x  . Lemma 3 ([10]). Let (r n ) be real sequence satisfying 0 < lim inf n→∞ ρ n ≤ lim sup n →∞ ρ n < 1 . Wang and Wang Journal of Inequalities and Applications 2011, 2011:41 http://www.journalofinequalitiesandapplications.com/content/2011/1/41 Page 2 of 8 Assume that (x n ) and (y n ) are bounded sequences in H satisfying x n +1 = (1 - r n )x n + r n y n . If lim sup n → ∞ (   y n+1 − y n   −  x n+1 − x n  ) ≤ 0 , then lim n ®∞||x n -y n || = 0. Lemma 4 For r, s,>0,let T r =2J r - I. Then for any x Î H,  T s x − T r x  ≤    1 − s r     x − T r x  . (2:1) Proof. Using the resolvent identity, we have  T s x − T r x  =2    J s x − J s  s r x +  1 − s r  J r x     ≤ 2    x −  s r x +  1 − s r  J r x     =2    1 − s r     x − J r x  =    1 − s r     x − T r x  , where the inequality uses the nonexpansive property of the resolvent. Lemma 5 ([11]). Let (ε n ) and (s n ) be positive real sequences. Assume that Σ n ε n < ∞. If either (i) s n+ 1≤ (1+ε n )s n , or (ii) s n+ 1≤ ε n , then the limit of (s n ) exists. 3. The relaxed proximal point algorithm Under criterion (II’), Ceng et al. [12] considered another type, RPPA:  ˜ x n = J c n (x n + e n ), x n+1 =(1− ρ n )x n + ρ n ˜ x n , (3:1) and proved the weak convergence of (3.1) under the assumptions: c n ≥ ¯ c > 0, 0 <δ≤ ρ n ≤ 1 . We note that the choice of (r n ) excludes the case whenever r n Î (1,2), the overre- laxation. The overrelaxation, however, may indeed speed up the convergence of the algorithm (see [13]). Below, we shall improve their conditions on the relaxation factor from 0 <δ ≤ r n ≤ 1to0<δ ≤ r n ≤ 2-δ. Theorem 6 . Assume that the following conditions hold: (a) c n ≥ ¯ c > 0 ; (b) 0<δ ≤ r n ≤ 2-δ; (c)  n  e n  ≤ η n   ˜ x n − x n   ,  n η 2 n < ∞ . Then the sequence generated by (3.1) converges weakly to a point in S. Proof. The key point of our proof is to show lim n s n =0,where s n =   x n − J c n (x n )   . To see this, let z Î S be fixed. Since J c n is firmly nonexpan sive and z ∈ Fix(J c n ) , apply- ing Lemma 1 yields  ˜ x n − z, ˜ x n − x n − e n  ≤ 0 . This together with (3.1) enables us to get  x n+1 − z  2 −  x n − z  2 =   (x n − z)+ρ n ( ˜ x n − x n )   2 −  x n − z  2 =2ρ n x n − z, ˜ x n − x n  + ρ 2 n   ˜ x n − x n   2 =2ρ n  ˜ x n − z, ˜ x n − x n −ρ n (2 − ρ n )   ˜ x n − x n   2 ≤ 2ρ n  ˜ x n − z, e n −ρ n (2 − ρ n )   ˜ x n − x n   2 =2ρ n  ˜ x n − x n , e n  +2ρ n x n − z, e n −ρ n (2 − ρ n )   ˜ x n − x n   2 ≤ 2ρ n  e n    ˜ x n − x n   +2ρ n  e n  x n − z  − ρ n (2 − ρ n )   ˜ x n − x n   2 ≤ 2ρ n η n   ˜ x n − x n   2 +2ρ n η n   ˜ x n − x n    x n − z  − ρ n (2 − ρ n )   ˜ x n − x n   2 . Wang and Wang Journal of Inequalities and Applications 2011, 2011:41 http://www.journalofinequalitiesandapplications.com/content/2011/1/41 Page 3 of 8 Using the basic inequality 2ab ≤ a 2 / ε + εb 2 (a,b Î ℝ, ε > 0), we arrive at 2ρ n η n  x n − z    ˜ x n − x n   ≤ 2ρ n 2 − ρ n ( η n  x n − z  ) 2 + 2 − ρ n 2ρ n  ρ n   ˜ x n − x n    2 = 2ρ n η 2 n 2 − ρ n  x n − z  2 + ρ n (2 − ρ n ) 2   ˜ x n − x n   2 ≤ 2(2 − δ)η 2 n δ  x n − z  2 + ρ n (2 − ρ n ) 2   ˜ x n − x n   2 = ε n  x n − z  2 + ρ n (2 − ρ n ) 2   ˜ x n − x n   2 , where ε n =2(2− δ)η 2 n / δ is a summable sequence. Substituting this into above yields  x n+1 − z  2 ≤ (1 + ε n )  x n − z  2 − ρ n (2 − ρ n − 4η n ) 2   ˜ x n − x n   2 . Since by Lemma 5 the limit of ||x n - z || 2 exists and lim inf n r n (2 - r n -4h n ) ≥ δ (2 - δ), this implies that   ˜ x n − x n   → 0 . On the other hand, we note that for all n Î N s n ≤ (1 + η n )   x n − ˜ x n   → 0 ; therefore, lim n s n = 0. The rest proof is similar to that of [12, Theorem 3.1]. We now turn to the RPPA (1.2). Under the criterion (I), the assumptions on relaxa- tion factors can be relaxed to Σr n (2 - r n )=∞ (see [3, Theorem 3.3]). Since the proof there is very technical, we wang to restate this result with a simple proof. Theorem 7 . Assume that the following conditions hold: (a) Σ n ||e n || < ∞; (b) Σ n r n (2 - r n )=∞; (c) 0 < ¯ c ≤ c n ≤ ˜ c < ∞ ; (d) Σ n |c n +1- c n |<∞. Then the sequence generated by (1.2) converges weakly to a point in S. Proof. The key step is to show lim n s n =0,where s n =   x n − J c n (x n )   . It has been shown that Σ n r n (2 - r n )s n < ∞ (see [3, Lemma 3.2]). Therefore, it remains to show that lim n s n exists. By letting T n =2J n - I, we rewrite (2) as x n+1 =  1 − ρ n 2  x n + ρ n 2 T n x n + e n . In view of Lemma 4 and condition (c),  T n+1 x n+1 − T n x n  ≤  T n+1 x n+1 − T n+1 x n  +  T n+1 x n − T n x n  ≤x n − x n+1  + T n+1 x n − T n x n  ≤x n − x n+1  +     1 − c n+1 c n     T n x n − x n  ≤x n − x n+1  + |c n+1 − c n | ¯ c T n x n − x n  ≤  x n − x n+1  + M | c n+1 − c n | , Wang and Wang Journal of Inequalities and Applications 2011, 2011:41 http://www.journalofinequalitiesandapplications.com/content/2011/1/41 Page 4 of 8 where M > 0 is a suitable number. Consequently,  x n+1 − T n+1 x n+1  =     1 − ρ n 2  x n + ρ n 2 T n x n + e n − T n+1 x n+1    =     1 − ρ n 2  (x n − T n x n )+(T n x n − T n+1 x n+1 )+e n    ≤  1 − ρ n 2   x n − T n x n  +  T n x n − T n+1 x n+1  +  e n  ≤  1 − ρ n 2   x n − T n x n  +  x n − x n+1  + M|c n+1 − c n | +  e n  =  1 − ρ n 2   x n − T n x n  +    ρ n 2 (x n − T n x n )+e n    + M|c n+1 − c n | +  e n  ≤  x n − T n x n  + M|c n+1 − c n | +2  e n  . Using s n =||x n - T n x n ||/2, we therefore arrive at s n +1 ≤ s n + σ n , where s n =2M |c n +1- c n |+4||e n || satisfying Σ n s n < ∞ (due to (a) and (d)). By Lemma 5, we finally conclude that lim n s n =0. 4. The contraction-proximal point algorithm Recently, Yao and Noor [14] extended the CPPA to the following form: x n+1 = λ n u + r n x n + δ n J c n (x n )+e n , (4:1) where (l n ),(r n ),(δ n )⊆ (0,1) and l n + r n + δ n = 1. They proved the strong convergence of the algorithm provided that (i) c n ≥ ¯ c > 0, lim n | c n+1 − c n | =0 ; (ii) 0 < lim inf n r n ≤ lim sup n r n < 1; and (iii) Σ n ||e n || < ∞. Also, they claimed that their algorithm includes the CPPA as a special case. This is, however, not the case, because condition (ii) excludes the special case r n ≡ 0. To overcome this drawback, we shall show the same result by replacing condition (ii) with the weak condition: lim sup n →∞ r n < 1 ⇔ lim inf n→∞ δ n > 0 . In this situation, the CPPA is evidently a speci al case of algorithm (4.1). The idea of the following proof is followed by the second author [15]. Theorem 8. Let be (l n ), (r n ) and (δ n ) be parameters in (4.1). Assume that the foll ow- ing conditions hold: (a) lim n l n =0,Σ n l n = ∞; (b) lim sup n r n <1⇔ lim inf n δ n >0; (c) c n ≥ ¯ c > 0, | c n+1 − c n | → 0 ; (d) Σ n ||e n || < ∞. Then the sequence generated by (4.1) converges strongly to P S (u). Proof. All we need to do is to prove ||x n +1- x n || ® 0, since the rest proof is similar to that of [14, Theorem 3.3]. To this end, set J n = J c n and T n =2J n - I.Itthenfollows from (4.1) that x n+1 = λ n u + r n x n + δ n 2 (I + T n )x n + e n =  r n + δ n 2  x n + λ n u + δ n 2 T n x n + e n . Wang and Wang Journal of Inequalities and Applications 2011, 2011:41 http://www.journalofinequalitiesandapplications.com/content/2011/1/41 Page 5 of 8 Let r n = l n +(δ n /2). Then the algorithm has the form: x n+1 = ( 1 − ρ n ) x n + ρ n y n , (4:2) where y n =(2l n u + δ n T n x n +2e n )/2r n. Using nonex pansiveness of T n and Lemma 4, we have  T n+1 x n+1 − T n x n  ≤  T n+1 x n+1 − T n+1 x n  +  T n+1 x n − T n x n  ≤  x n+1 − x n  +     1 − c n+1 c n      T n x n − x n  ≤  x n+1 − x n  + |c n − c n+1 | ¯ c  T n x n − x n  . (4:3) On the other hand, it follows from the definition of y n that   y n+1 − y n   =     1 2ρ n+1 (2λ n+1 u + δ n+1 T n+1 x n+1 +2e n+1 ) − 1 2ρ n (2λ n u + δ n T n x n +2e n )     ≤     λ n+1 ρ n+1 − λ n ρ n      u  +  e n+1  ρ n+1 +  e n  ρ n +     δ n+1 2ρ n+1 T n+1 x n+1 − δ n 2ρ n T n x n     ≤     λ n+1 ρ n+1 − λ n ρ n      u  +  e n+1  ρ n+1 +  e n  ρ n +     δ n+1 2ρ n+1 − δ n 2ρ n      T n+1 x n+1  + δ n 2ρ n  T n+1 x n+1 − T n x n  . (4:4) Since (x n ) is bounded and T n is nonexpansive, we can find M > 0 so that (||T n x n || + ||x n || + ||u||) ≤ M for all n Î N Adding (4.3) and (4.4) and noting δ n ≤ 2r n yield   y n+1 − y n   ≤     λ n+1 ρ n+1 − λ n ρ n      u  +  e n+1  ρ n+1 +  e n  ρ n +     δ n+1 2ρ n+1 − δ n 2ρ n      T n+1 x n+1  +  x n+1 − x n  + |c n − c n+1 | ¯ c  T n x n − x n  ≤  x n+1 − x n  + M      λ n+1 ρ n+1 − λ n ρ n     +  e n+1  ρ n+1 +  e n  ρ n +     δ n+1 2ρ n+1 − δ n 2ρ n     + |c n − c n+1 | ¯ c  . With the knowledge that ||e n ||® 0 and λ n ρ n = 2λ n 2λ n + δ n → 0, δ n 2 ρ n = δ n 2λ n + δ n → 1 , Wang and Wang Journal of Inequalities and Applications 2011, 2011:41 http://www.journalofinequalitiesandapplications.com/content/2011/1/41 Page 6 of 8 we therefore deduce from (b) and (c) that lim sup n→∞ (   y n+1 − y n   −  x n+1 − x n  ) ≤ lim sup n→∞ M      λ n+1 ρ n+1 − λ n ρ n     +  e n+1  ρ n+1 +  e n  ρ n +     δ n+1 2ρ n+1 − δ n 2ρ n     + |c n − c n+1 | ¯ c  → 0. Note that lim inf n r n = lim inf n (δ n /2)> 0 and lim sup n r n = lim sup n (δ n /2) ≤ 1/2 < 1. On the other hand, it is easy to check that (x n ) is bounded and so is (y n ) We therefore apply Lemma 3 to yield lim n ||x n - y n || = 0. By means of (4.2), we finally have  x n+1 − x n  = ρ n   x n − y n   → , and thus the required result at once follows. As a corollary, we improve [3, Theorem 4.1] as follows. Theorem 9 . Assume that the following conditions hold: (a) lim n l n =0,Σ n l n = ∞; (b) c n ≥ ¯ c > 0, | c n+1 − c n | → 0 ; (c) Σ n ||e n || < ∞. Then the sequence generated by (1.4) converges strongly to P S (u). Abbreviations CPPA: contraction-proximal point algorithm; PPA: proximal point algorithm; RPPA: relaxed proximal point algorithm. Acknowledgements The authors would like to express thier sincere thanks to the referees for their valuable suggestions. This study is supported by the Natural Science Foundation of Department of Education, Henan(2011B110023). Authors’ contributions Both authors contributed equally to this work. All authors read and approved the final manuscript. Competing interests The authors declare that they have no competing interests. Received: 13 March 2011 Accepted: 25 August 2011 Published: 25 August 2011 References 1. Rockafellar, RT: Monotone operators and the proximal point algorithm. SIAM J Control Optim. 14, 877–898 (1976). doi:10.1137/0314056 2. 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Eckstein, J, Ferris, MC: Operator-splitting methods for monotone affine variational inequalities, with a parallel application to optimal control. INFORMS J Comput. 10, 218–235 (1998). doi:10.1287/ijoc.10.2.218 14. Yao, Y, Noor, MA: On convergence criteria of generalized proximal point algorithms. J Comput Appl Math. 217,46–55 (2008). doi:10.1016/j.cam.2007.06.013 15. Wang, F: A note on the regularized proximal point algorithm. J Global Optim. 50, 531–535 (2011). doi:10.1007/s10898- 010-9611-z doi:10.1186/1029-242X-2011-41 Cite this article as: Wang and Wang: On relaxed and contraction-proximal point algorithms in hilbert spaces. Journal of Inequalities and Applications 2011 2011:41. Submit your manuscript to a journal and benefi t from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the fi eld 7 Retaining the copyright to your article Submit your next manuscript at 7 springeropen.com Wang and Wang Journal of Inequalities and Applications 2011, 2011:41 http://www.journalofinequalitiesandapplications.com/content/2011/1/41 Page 8 of 8 . China Abstract We consider the relaxed and contraction-proximal point algorithms in Hilbert spaces. Some conditions on the parameters for gu aranteeing the convergence of the algorithm are relaxed. as: Wang and Wang: On relaxed and contraction-proximal point algorithms in hilbert spaces. Journal of Inequalities and Applications 2011 2011:41. Submit your manuscript to a journal and benefi. ∞ . The infinite version was obtained by Marino and Xu [3]. There are various generations or modifications on the PPA. Among them Eckstein and Bertsekas [4] proposed the relaxed proximal point algorithm