Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 RESEARCH Open Access Some results on zero points of m-accretive operators in reflexive Banach spaces Chang Qun Wu1 , Songtao Lv2* and Yunpeng Zhang3 * Correspondence: sqlvst@yeah.net School of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, Henan, China Full list of author information is available at the end of the article Abstract A modified proximal point algorithm is proposed for treating common zero points of a finite family of m-accretive operators A strong convergence theorem is established in a reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm Keywords: accretive operator; nonexpansive mapping; resolvent; fixed point; zero point Introduction and preliminaries Let E be a Banach space and let E∗ be the dual of E Let ·, · denote the pairing between ∗ E and E∗ The normalized duality mapping J : E → E is defined by J(x) = f ∈ E∗ : x, f = x = f , ∀x ∈ E A Banach space E is said to strictly convex if and only if x = y = ( – λ)x + λy for x, y ∈ E and < λ < implies that x = y Let UE = {x ∈ E : x = } The norm of E is said to be Gâteaux differentiable if the limit limt→ x+tyt – x exists for each x, y ∈ UE In this case, E is said to be smooth The norm of E is said to be uniformly Gâteaux differentiable if for each y ∈ UE , the limit is attained uniformly for all x ∈ UE The norm of E is said to be Fréchet differentiable if for each x ∈ UE , the limit is attained uniformly for all y ∈ UE The norm of E is said to be uniformly Fréchet differentiable if the limit is attained uniformly for all x, y ∈ UE It is well known that (uniform) Fréchet differentiability of the norm of E implies (uniform) Gâteaux differentiability of the norm of E Let ρE : [, ∞) → [, ∞) be the modulus of smoothness of E by ρE (t) = sup x+y – x–y – : x ∈ UE , y ≤ t A Banach space E is said to be uniformly smooth if ρEt(t) → as t → It is well known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single valued and uniformly norm to weak∗ continuous on each bounded subset of E Recall that a closed convex subset C of a Banach space E is said to have a normal structure if for each bounded closed convex subset K of C which contains at least two points, there exists an element x of K which is not a diametral point of K , i.e., sup{ x – y : y ∈ K} < d(K), where d(K) is the diameter of K ©2014 Wu et al.; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 Let D be a nonempty subset of a set C Let ProjD : C → D Q is said to be () sunny if for each x ∈ C and t ∈ (, ), we have ProjD (tx + ( – t)ProjD x) = ProjD x; () a contraction if ProjD = ProjD ; () a sunny nonexpansive retraction if ProjD is sunny, nonexpansive, and a contraction D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from C onto D The following result, which was established in [–], describes a characterization of sunny nonexpansive retractions on a smooth Banach space Let E be a smooth Banach space and let C be a nonempty subset of E Let ProjC : E → C be a retraction and Jϕ be the duality mapping on E Then the following are equivalent: () ProjC is sunny and nonexpansive; () x – ProjC x, Jϕ (y – ProjC x) ≤ , ∀x ∈ E, y ∈ C; () ProjC x – ProjC y ≤ x – y, Jϕ (ProjC x – ProjC y) , ∀x, y ∈ E It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction ProjC is coincident with the metric projection from E onto C Let C be a nonempty closed convex subset of a smooth Banach space E, let x ∈ E, and let x ∈ C Then we have from the above that x = ProjC x if and only if x – x , Jϕ (y – x ) ≤ for all y ∈ C, where ProjC is a sunny nonexpansive retraction from E onto C For more additional information on nonexpansive retracts, see [] and the references therein Let C be a nonempty closed convex subset of E Let T : C → C be a mapping In this paper, we use F(T) to denote the set of fixed points of T Recall that T is said to be an α-contractive mapping iff there exists a constant α ∈ [, ) such that Tx – Ty ≤ α x – y , ∀x, y ∈ C The Picard iterative process is an efficient method to study fixed points of α-contractive mappings It is well known that α-contractive mappings have a unique fixed point T is said to be nonexpansive iff Tx – Ty ≤ x – y , ∀x, y ∈ C It is well known that nonexpansive mappings have fixed points if the set C is closed and convex, and the space E is uniformly convex The Krasnoselski-Mann iterative process is an efficient method for studying fixed points of nonexpansive mappings The Krasnoselski-Mann iterative process generates a sequence {xn } in the following manner: x ∈ C, xn+ = αn Txn + ( – αn )xn , ∀n ≥ It is well known that the Krasnoselski-Mann iterative process only has weak convergence for nonexpansive mappings in infinite-dimensional Hilbert spaces; see [–] for more details and the references therein In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in infinite-dimensional spaces In such problems, strong convergence (norm convergence) is often much more desirable than weak convergence, for it translates the physically tangible property that the energy xn – x of the error between the iterate xn and the solution x eventually becomes arbitrarily small To improve the weak convergence of a Krasnoselski-Mann iterative process, so-called hybrid projections have been considered; see [–] for more details and the references therein The Halpern iterative process was initially introduced in []; see [] for more details and the references therein The Halpern iterative process generates a sequence {xn } in the following manner: x ∈ C, xn+ = αn u + ( – αn )Txn , ∀n ≥ , Page of 11 Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 where x is an initial and u is a fixed element in C Strong convergence of Halpern iterative process does not depend on metric projections The Halpern iterative process has recently been extensively studied for treating accretive operators; see [–] and the references therein Let I denote the identity operator on E An operator A ⊂ E × E with domain D(A) = {z ∈ E : Az = ∅} and range R(A) = {Az : z ∈ D(A)} is said to be accretive if for each xi ∈ D(A) and yi ∈ Axi , i = , , there exists j(x – x ) ∈ J(x – x ) such that y – y , j(x – x ) ≥ An accretive operator A is said to be m-accretive if R(I + rA) = E for all r > In this paper, we use A– () to denote the set of zero points of A For an accretive operator A, we can define a nonexpansive single valued mapping Jr : R(I + rA) → D(A) by Jr = (I + rA)– for each r > , which is called the resolvent of A Now, we are in a position to give the lemmas to prove main results Lemma . [] Let {an }, {bn }, {cn }, and {dn } be four nonnegative real sequences satisfying an+ ≤ ( – bn )an + bn cn + dn , ∀n ≥ n , where n is some positive integer, {bn } is a number sequence in (, ) such that ∞ n=n bn = ∞, {cn } is a number sequence such that lim supn→∞ cn ≤ , and {dn } is a positive number sequence such that ∞ n=n dn < ∞ Then limn→∞ an = Lemma . [] Let C be a closed convex subset of a strictly convex Banach space E Let N ≥ be some positive integer and let Ti : C → C be a nonexpansive mapping for each i ∈ {, , , N} Let {δi } be a real number sequence in (, ) with N i= δi = Suppose N F(T ) is nonempty Then the mapping T is defined to be nonexpansive with that N i i= i= i N N F( i= Ti ) = i= F(Ti ) Lemma . [] Let {xn } and {yn } be bounded sequences in a Banach space E and let βn be a sequence in [, ] with < lim infn→∞ βn ≤ lim supn→∞ βn < Suppose that xn+ = ( – βn )yn + βn xn for all n ≥ and lim sup yn+ – yn – xn+ – xn n→∞ ≤ Then limn→∞ yn – xn = Lemma . [] Let E be a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure, and let C be a nonempty closed convex subset of E Let f : C → C be α-contractive mapping and let T : C → C be a nonexpansive mapping with a fixed point Let {xt } be a sequence generated by the following: xt = tf (xt ) + ( – t)Txt , where t ∈ (, ) Then {xt } converges strongly as t → to a fixed point x∗ of T, which is the unique solution in F(T) to the following variational inequality: f (x∗ ) – x∗ , j(x∗ – p) ≥ , ∀p ∈ F(T) Main results Theorem . Let E be a real reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm Let N ≥ be some positive integer Let Am be an m-accretive operator in E for each m ∈ {, , , N} Assume that C := N m= D(Am ) is convex and has the normal structure Let f : C → C be an α-contractive mapping Let {αn }, {βn }, and {γn } be real number sequences in (, ) with the restriction αn + βn + γn = Let {δn,i } be a real number sequence in (, ) with the restriction δn, + δn, + · · · + δn,N = Let {rm } be a positive Page of 11 Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 Page of 11 real numbers sequence and {en,i } a sequence in E for each i ∈ {, , , N} Assume that N – i= Ai () is not empty Let {xn } be a sequence generated in the following manner: N x ∈ C, xn+ = αn f (xn ) + βn xn + γn ∀n ≥ , δn,i Jri (xn + en,i ), i= where Jri = (I + ri Ai )– Assume that the control sequences {αn }, {βn }, {γn }, and {δn,i } satisfy the following restrictions: (a) limn→∞ αn = , ∞ n= αn = ∞; (b) < lim infn→∞ βn ≤ lim supn→∞ βn < ; ∞ (c) n= en,m < ∞; (d) limn→∞ δn,i = δi ∈ (, ) Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following – variational inequality: f (¯x) – x¯ , J(p – x¯ ) ≤ , ∀p ∈ N i= Ai () N i= δn,i Jri (xn Proof Put yn = N – i= Ai (), + en,i ) Fixing p ∈ we have N yn – p ≤ δn,i Jri (xn + en,i ) – p i= N ≤ δn,i (xn + en,i ) – p i= N ≤ xn – p + en,i i= Hence, we have xn+ – p ≤ αn f (xn ) – p + βn xn – p + γn yn – p N ≤ αn α xn – p + αn f (p) – p + βn xn – p + γn xn – p + γn en,i i= ≤ – αn ( – α) xn – p + αn ( – α) f (p) – p + –α N en,i i= N ≤ max xn – p , f (p) – p + en,i i= ∞ ≤ max x – p , f (p) – p N + ej,i j= i= This proves that the sequence {xn } is bounded, and so is {yn } Since N yn – yn– = δn,i Jrm (xn + en,i ) – Jri (xn– + en–,i ) i= N + (δn,i – δn–,i )Jri (xn– + en–,i ), i= Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 Page of 11 we have N yn – yn– ≤ δn,i Jri (xn + en,i ) – Jri (xn– + en–,i ) i= N |δn,i – δn–,i | Jri (xn– + en–,i ) + i= N N ≤ xn – xn– + en,i + i= en–,i i= N |δn,i – δn–,i | Jri (xn– + en–,i ) + i= N N N ≤ xn – xn– + en,i + i= |δn,i – δn–,i |, en–,i + M i= i= where M is an appropriate constant such that M = max sup Jr (xn + en, ) , sup Jr (xn + en, ) , , sup JrN (xn + en,N ) n≥ n≥ Define a sequence {zn } by zn := xn+ –βn xn , –βn that is, xn+ = βn xn + ( – βn )zn It follows that αn αn– f (xn ) – yn + f (xn– ) – yn– + yn – yn– – βn – βn– αn αn– ≤ f (xn ) – yn + f (xn– ) – yn– + xn – xn– – βn – βn– yzn – zn– ≤ N |δn,i – δn–,i | Jri xn– + i= ≤ αn αn– f (xn ) – yn + f (xn– ) – yn– + xn – xn– – βn – βn– N N |δn,i – δi | + + M i= |δi – δn–,i | , i= where M is an appropriate constant such that M = max sup Jr xn , sup Jr xn , , sup JrN xn n≥ n≥ n≥ This implies that zn – zn– – xn – xn– ≤ αn– αn f (xn ) – yn + f (xn– ) – yn– – βn – βn– N N |δn,i – δi | + + M i= |δi – δn–,i | i= n≥ Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 Page of 11 From the restrictions (a), (b), (c), and (d), we find that lim sup zn – zn– – xn – xn– n→∞ ≤ Using Lemma ., we find that limn→∞ zn – xn = This further shows that lim supn→∞ xn+ – xn = Put T = N i= δi Jri It follows from Lemma . that T is nonexN N pansive with F(T) = i= F(Jri ) = i= A– i () Note that xn – Txn ≤ xn – xn+ + xn+ – Txn ≤ xn – xn+ + αn f (xn ) – Txn + βn xn – Txn + γn yn – Txn N |δn,i – δi | ≤ xn – xn+ + αn f (xn ) – Txn + βn xn – Txn + M i= This implies that N ( – βn ) xn – Txn ≤ xn – xn+ + αn f (xn ) – Txn + M |δn,i – δi | i= It follows from the restrictions (a), (b), and (d) that lim Txn – xn = n→∞ Now, we are in a position to prove that lim supn→∞ f (¯x) – x¯ , J(xn – x¯ ) ≤ , where x¯ = limt→ xt , and xt solves the fixed point equation xt = tf (xt ) + ( – t)Txt , ∀t ∈ (, ) It follows that xt – x n = t f (xt ) – xn , J(xt – xn ) + ( – t) Txt – xn , j(xt – xn ) = t f (xt ) – xt , J(xt – xn ) + t xt – xn , J(xt – xn ) + ( – t) Txt – Txn , J(xt – xn ) + ( – t) Txn – xn , J(xt – xn ) ≤ t f (xt ) – xt , J(xt – xn ) + xt – xn + Txn – xn xt – xn , ∀t ∈ (, ) This implies that xt – f (xt ), J(xt – xn ) ≤ Txn – xn t xt – xn , ∀t ∈ (, ) Since limn→∞ Txn – xn = , we find that lim supn→∞ xt – f (xt ), J(xt – xn ) ≤ Since J is strong to weak∗ uniformly continuous on bounded subsets of E, we find that f (¯x) – x¯ , J(xn – x¯ ) – xt – f (xt ), J(xt – xn ) ≤ f (¯x) – x¯ , J(xn – x¯ ) – f (¯x) – x¯ , J(xn – xt ) Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 Page of 11 + f (¯x) – x¯ , J(xn – xt ) – xt – f (xt ), J(xt – xn ) ≤ f (¯x) – x¯ , J(xn – x¯ ) – J(xn – xt ) + f (¯x) – x¯ + xt – f (xt ), J(xn – xt ) J(xn – x¯ ) – J(xn – xt ) + ( + α) x¯ – xt ≤ f (xt ) – x¯ xn – xt Since xt → x¯ , as t → , we have lim f (¯x) – x¯ , J(xn – x¯ ) – f (xt ) – xt , J(xn – xt ) = t→ For > , there exists δ > such that ∀t ∈ (, δ), we have f (¯x) – x¯ , J(xn – x¯ ) ≤ f (xt ) – xt , J(xn – xt ) + This implies that lim supn→∞ f (¯x) – x¯ , J(xn – x¯ ) ≤ Finally, we show that xn → x¯ as n → ∞ Since · is convex, we see that N yn – x¯ δn,i Jri (xn + en,i ) – x¯ = i= N δn,i Jri (xn + en,i ) – x¯ ≤ i= N ≤ xn – x¯ + en,i en,i + xn – x¯ i= It follows that xn+ – x¯ = αn f (xn ) – x¯ , J(xn+ – x¯ ) + βn xn – x¯ , J(xn+ – x¯ ) + γn yn – x¯ , J(xn+ – x¯ ) ≤ αn α xn – x¯ xn+ – x¯ + αn f (¯x) – x¯ , J(xn+ – x¯ ) + βn xn – x¯ xn+ – x¯ + γn yn – x¯ xn+ – x¯ αn α xn – x¯ + xn+ – x¯ + αn f (¯x) – x¯ , J(xn+ – x¯ ) ≤ βn γn + xn – x¯ + xn+ – x¯ + xn – x¯ N + en,i en,i + xn – x¯ + i= γn xn+ – x¯ Hence, we have xn+ – x¯ ≤ – αn ( – α) xn – x¯ + αn f (¯x) – x¯ , J(xn+ – x¯ ) N + en,i en,i + xn – x¯ i= Using Lemma ., we find xn → x¯ as n → ∞ This completes the proof Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 Page of 11 Remark . There are many spaces satisfying the restriction in Theorem ., for example Lp , where p > Corollary . Let E be a Hilbert space and let N ≥ be some positive integer Let Am be a maximal monotone operator in E for each m ∈ {, , , N} Assume that C := N m= D(Am ) is convex and has the normal structure Let f : C → C be an α-contractive mapping Let {αn }, {βn }, and {γn } be real number sequences in (, ) with the restriction αn + βn + γn = Let {δn,i } be a real number sequence in (, ) with the restriction δn, + δn, + · · · + δn,N = Let {rm } be a positive real numbers sequence and {en,i } a sequence in E for each i ∈ {, , , N} – Assume that N i= Ai () is not empty Let {xn } be a sequence generated in the following manner: N x ∈ C, xn+ = αn f (xn ) + βn xn + γn δn,i Jri (xn + en,i ), ∀n ≥ , i= where Jri = (I + ri Ai )– Assume that the control sequences {αn }, {βn }, {γn }, and {δn,i } satisfy the following restrictions: (a) limn→∞ αn = , ∞ n= αn = ∞; (b) < lim infn→∞ βn ≤ lim supn→∞ βn < ; ∞ (c) n= en,m < ∞; (d) limn→∞ δn,i = δi ∈ (, ) Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following – variational inequality: f (¯x) – x¯ , p – x¯ ≤ , ∀p ∈ N i= Ai () Applications In this section, we consider a variational inequality problem Let A : C → E∗ be a single valued monotone operator which is hemicontinuous; that is, continuous along each line segment in C with respect to the weak∗ topology of E∗ Consider the following variational inequality: find x ∈ C such that y – x, Ax ≥ , ∀y ∈ C The solution set of the variational inequality is denoted by VI(C, A) Recall that the normal cone NC (x) for C at a point x ∈ C is defined by NC (x) = x∗ ∈ E∗ : y – x, x∗ ≤ , ∀y ∈ C Now, we are in a position to give the convergence theorem Theorem . Let E be a real reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm Let N ≥ be some positive integer and let C be nonempty closed and convex subset of E Let Ai : C → E∗ a single valued, monotone and hemicontinuous operator Assume that N i= VI(C, Ai ) is not empty and C has the normal structure Let f : C → C be an α-contractive mapping Let {αn }, {βn }, and {γn } be real number sequences in (, ) with the restriction αn + βn + γn = Let {δn,i } be a real number sequence in (, ) with the restriction δn, + δn, + · · · + δn,N = Let {rm } be a positive real numbers sequence Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 Page of 11 and {en,i } a sequence in E for each i ∈ {, , , N} Let {xn } be a sequence generated in the following manner: N x ∈ C, xn+ = αn f (xn ) + βn xn + γn δn,i VI C, Ai + i= (I – xn ) , ri ∀n ≥ Assume that the control sequences {αn }, {βn }, {γn }, and {δn,i } satisfy the following restrictions: (a) limn→∞ αn = , ∞ n= αn = ∞; (b) < lim infn→∞ βn ≤ lim supn→∞ βn < ; ∞ (c) n= en,m < ∞; (d) limn→∞ δn,i = δi ∈ (, ) Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following variational inequality: f (¯x) – x¯ , J(p – x¯ ) ≤ , ∀p ∈ N i= VI(C, Ai ) Proof Define a mapping Ti ⊂ E × E∗ by ⎧ ⎨A x + N x, x ∈ C, i C Ti x := ⎩∅, x ∈/ C From Rockafellar [], we find that Ti is maximal monotone with Ti– () = VI(C, Ai ) For each ri > , and xn ∈ E, we see that there exists a unique xri ∈ D(Ti ) such that xn ∈ xri + ri Ti (xri ), where xri = (I + ri Ti )– xn Notice that yn,i = VI C, Ai + (I – xn ) , ri which is equivalent to y – yn,i , Ai yn,i + (yn,i – xn ) ≥ , ri ∀y ∈ C, that is, –Ai yn,i + ri (xn – yn,i ) ∈ NC (yn,i ) This implies that yn,i = (I + ri Ti )– xn Using Theorem ., we find the desired conclusion immediately From Theorem ., the following result is not hard to derive Corollary . Let E be a real reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm Let C be nonempty closed and convex subset of E Let A : C → E∗ a single valued, monotone and hemicontinuous operator with VI(C, A) Assume that C has the normal structure Let f : C → C be an α-contractive mapping Let {αn }, {βn }, and {γn } be real number sequences in (, ) with the restriction αn + βn + γn = Let {xn } be a sequence generated in the following manner: x ∈ C, xn+ = αn f (xn ) + βn xn + γn VI C, A + (I – xn ) , r ∀n ≥ Assume that the control sequences {αn }, {βn }, and {γn } satisfy the following restrictions: Wu et al Fixed Point Theory and Applications 2014, 2014:118 http://www.fixedpointtheoryandapplications.com/content/2014/1/118 (a) limn→∞ αn = , ∞ n= αn = ∞; (b) < lim infn→∞ βn ≤ lim supn→∞ βn < Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following variational inequality: f (¯x) – x¯ , J(p – x¯ ) ≤ , ∀p ∈ VI(C, Ai ) Competing interests The authors declare that they have no competing interests Authors’ contributions All authors contributed equally to this manuscript All authors read and approved the final manuscript Author details School of Business and Administration, Henan University, Kaifeng, Henan, China School of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, Henan, China Vietnam National University, Hanoi, Vietnam Acknowledgements The authors are grateful to the editor and the reviewers for useful suggestions which improved the contents of the article Received: 16 January 2014 Accepted: 30 April 2014 Published: 14 May 2014 References Bruck, RE: Nonexpansive projections on subsets of Banach spaces Pac J Math 47, 341-355 (1973) Reich, S: Asymptotic behavior of contractions in Banach spaces J Math Anal Appl 44, 57-70 (1973) Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings Dekker, New York (1984) Kopecká, EE, Reich, S: Nonexpansive 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2014:118 Page 11 of 11 ... algorithm for zero points of accretive operators Fixed Point Theory Appl 2013, Article ID 341 (2013) 30 Qing, Y, Cho, SY: Proximal point algorithms for zero points of nonlinear operators Fixed Point Theory... projection methods for zeros of monotone operators J Fixed Point Theory 2013, Article ID (2013) 20 Qin, X, Su, Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space Nonlinear... this article as: Wu et al.: Some results on zero points of m-accretive operators in reflexive Banach spaces Fixed Point Theory and Applications 2014, 2014:118 Page 11 of 11