Using the new analysis techniques, the existence and iterative approximation problem of a solution for a class of nonlinear
variational inclusions with accretive type mappings are discussed in arbitrary Banach spaces. The results extend and improve some recent
results

G. Feng

Using the new analysis techniques, the existence and iterative approximation problem of a solution for a class of nonlinear

variational inclusions with accretive type mappings are discussed in arbitrary Banach spaces. The results extend and improve some recent

results

Feng, G.

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Mathematical Communications 8(2003), 1-8 1On the Ishikawa iterative approximation withmixed errors for solutions to variational inclusionswith accretive type mappings in Banach spaces∗Gu Feng†Abstract. Using the new analysis techniques, the existence anditerative approximation problem of a solution for a class of nonlinearvariational inclusions with accretive type mappings are discussed in ar-bitrary Banach spaces. The results extend and improve some recentresults.Key words: variational inclusion; accretive mapping; Ishikawa it-erative sequence with mixed errors; Mann iterative sequence with mixederrors.AMS subject classifications: 47H06, 49J40, 47J15Received October 30, 2002 Accepted November 29, 20021. IntroductionThroughout this paper we suppose that X is a real Banach space, X∗ is its dualspace, 〈·, ·〉 is the pairing of X and X∗. D(T ) and R(T ) denote the domain and therange of T , respectively.Let T, A : X → X, g : X → X∗ be three mappings and ϕ : X∗ → R ∪ {+∞}be a proper convex lower semicontinuous function.In 1999, Chang [1] introduced and studied the existence and approximationproblem of solutions for a class of nonlinear variational inclusions with accretivemappings in uniformly smooth Banach space as follows:For any given f ∈ X , to find an u ∈ X such that{g(u) ∈ D(∂ϕ),〈Tu−Au − f, v − g(u)〉 ≥ ϕ(g(u))− ϕ(v), ∀v ∈ X∗, (1)where ∂ϕ denotes the subdifferential of ϕ.∗Project Supported by the Key Teacher Creating Capacity Fund of Heilongjiang General Col-lege (1053G015), the Natural Science Foundation of Heilongjiang Province (A0211) and the Foun-dation of Heilongjiang Education Committee of China (10511132)†Department of Mathematics, Qiqihar University, Qiqihar, Heilongjiang 161006, People’s Re-public of China, e-mail: gufeng99@sohu.com2 GuFengThe purpose of this paper is to study further the existence and uniqueness ofsolutions and the convergence problem of Ishikawa and Mann iterative processeswith mixed errors for a class of accretive type variational inclusion in arbitraryBanach spaces. The results presented in this paper not only extend and improvethe main results in Chang [1], but also extend and improve the correspondingresults in Chang [2,3], Chang, Cho, Lee et al [4], Ding [5,6], Hassouni and Moudafi[7], Huang [8-10], Kazmi [12], Noor [15,16], Siddiqi and Ansari [17], Siddiqi, Ansariand Kazmi [18] and Zeng [19].2. PreliminariesA mapping J : X → 2X∗ is said to be a normalized duality mapping, if it is definedbyJ(x) ={f ∈ X∗ : 〈x, f〉 = ‖f‖2 = ‖x‖2} , ∀x ∈ X.Definition 1. A mapping T : D(T ) ⊂ X → X is said to be accretive, if for anyx, y ∈ D(T ), there exists j(x − y) ∈ J(x− y) such that〈Tx− Ty, j(x− y)〉 ≥ 0.If T is accretive and R(I + rT ) = X for all r > 0, then T is called m-accretive.In the sequel we shall use the following Proposition and Lemmas.Proposition 1 [14]. Let X be a real Banach space, T : D(T ) ⊂ X → X isaccretive and continuous, and D(T ) = X. Then T is m-accretive.Lemma 1 [13]. Let {an}, {bn} and {cn} be three nonnegative real sequencessatisfying the inequality.:an+1 ≤ (1− tn)an + bntn + cn, ∀n ≥ 0,where {tn} ⊂ [0, 1],∑∞n=0 tn = ∞, limn→∞ bn = 0 and∑∞n=0 cn < ∞. Thenlimn→∞ an = 0.Lemma 2 [20]. Let X be a real Banach space, T : D(T ) ⊂ X → X an m-accretive mapping. Then the equation x + Tx = f has a unique solution in D(T )for any f ∈ X.Lemma 3 [2]. Let X be an arbitrary real Banach space, ∂ϕ ◦ g : X → 2X amapping, then the following conclusions are equivalent to each other:(i) x∗ ∈ X is a solution of variational inclusion problem (1);(ii) x∗ ∈ X is a fixed point of the mapping S : X → 2X :S(x) = f − (Tx−Ax+ ∂ϕ(g(x))) + x;(iii) x∗ ∈ X is a solution of the equation f ∈ Tx−Ax+ ∂ϕ(g(x)).On the Ishikawa iterative approximation 33. Main resultsTheorem 1. Let X be an arbitrary real Banach space, T,A : X → X, g : X → X∗three mappings, and ϕ : X∗ → R ∪ {+∞} a function with a continuous Ga¨teauxdifferential ∂ϕ. For any given f ∈ X, define a mapping S : X → X bySx = f − (Tx−Ax+ ∂ϕ(g(x))) + x.Let x0 ∈ X be any given point and {xn} the Ishikawa iterative sequence with mixederrors defined by{xn+1 = (1− αn)xn + αnSyn + un,yn = (1− βn)xn + βnSxn + vn, ∀n ≥ 0, (2)where {αn}, {βn} are two real sequences in [0, 1], and {un}, {vn} are two sequencesin X such that un = u′n + u′′n for any sequences {u′n} and {u′′n} in X satisfying thefollowing conditions(i) T −A+ ∂ϕ ◦ g − I : X → X is accretive,(ii) T −A+ ∂ϕ ◦ g : X → X is a Lipschitz operator with constant L,(iii) Kn = (1 + L∗)(1 + L2∗)αn + L∗(1 + L∗)βn ≤ 1− r and∑∞n=0 αn =∞,(iv)∑∞n=0 ‖u′n‖ <∞, ‖u′′n‖ = γnαn and ‖vn‖ → 0 (n→∞),where L∗ = 1+L, r ∈ (0, 1) is a constant and γn → 0 (n→∞). Then the followingconclusions hold:(1) The nonlinear variational inclusion problem (1) has a unique solution x∗ ∈ X,(2) The Ishikawa iterative sequence {xn} with mixed errors converges strongly tothe unique solution x∗ ∈ X of the variational inclusion problem (1).Proof. (1) First we prove that the variational inclusion problem (1) has a uniquesolution x∗ ∈ X .From conditions (i) and (ii), the mapping T − A + ∂ϕ ◦ g − I : X → X iscontinuous and accretive. By Proposition 1 we know that T − A + ∂ϕ ◦ g − I ism-accretive. Therefore, by Lemma2, for any given f ∈ X , the equationf = x+ (T −A+ ∂ϕ ◦ g − I)(x)has a unique solution x∗ ∈ X . Hence, by Lemma 3, we know that x∗ is a uniquesolution of the variational inclusion problem (1), and it is also a fixed point of S,i.e., Sx∗ = x∗.(2) Next we prove that the Ishikawa iterative sequence {xn} with mixed errorsconverges strongly to x∗.By condition (i), for any x, y ∈ X , there exists j(x − y) ∈ J(x− y) such that〈Sx−Sy, j(x−y)〉=−〈(T−A+∂ϕ ◦ g−I)x−(T−A+∂ϕ ◦ g−I)y, j(x−y)〉 ≤ 0.4 GuFengIt follows from Lemma 1.1 of Kato [11] that‖x− y‖ ≤ ‖x− y − t(Sx− Sy)‖ (3)for all x, y ∈ X and t > 0. Using (2), we easily conclude that for all n ≥ 0,xn = xn+1 + αnxn − αnSyn − un= (1 + αn)xn+1 − αnSxn+1 + αn2(xn − Syn)+ αn(Sxn+1 − Syn)− (1 + αn)un. (4)Note thatx∗ = (1 + αn)x∗ − αnSx∗ (5)for all n ≥ 0. It follows from (3), (4) and (5) that‖xn − x∗‖ ≥ (1 + αn)∥∥∥∥xn+1 − x∗ − αn1 + αn (Sxn+1 − Sx∗)∥∥∥∥− α2n‖xn − Syn‖ − αn‖Sxn+1 − Syn‖ − (1 + αn)‖un‖≥ (1 + αn)‖xn+1 − x∗‖ − αn2‖xn − Syn‖− αn‖Sxn+1 − Syn‖ − (1 + αn)‖un‖,which implies that‖xn+1−x∗‖≤ 11+αn ‖xn−x∗‖+αn2‖xn−Syn‖+αn‖Sxn+1−Syn‖+‖un‖. (6)Since T − A + ∂ϕ ◦ g is a Lipschitz mapping with the constant L, it is easy toverify that S is also Lipschitz with the constant L∗ = 1+L. Furthermore, we havethe following estimates:‖xn − Syn‖ ≤ ‖xn − x∗‖+ ‖Syn − Sx∗‖≤ ‖xn − x∗‖+ L∗‖yn − x∗‖≤ ‖xn−x∗‖+L∗[(1−βn)‖xn−x∗‖+βnL∗‖xn−x∗‖+‖vn‖]≤ (1 + L2∗)‖xn − x∗‖+ L∗‖vn‖ (7)and‖Sxn+1−Syn‖≤L∗‖xn+1−yn‖=L∗‖αn(Syn−xn)+βn(xn−Sxn)+un−vn‖≤ L∗αn‖xn− Syn‖+L∗βn(‖xn−x∗‖+‖Sxn−x∗‖)+L∗‖un‖+L∗‖vn‖≤ [L∗(1+L2∗)αn+L∗(1+L∗)βn]‖xn−x∗‖+L∗(1+L∗αn)‖vn‖+L∗‖un‖. (8)Substituting (7) and (8) into (6), and by conditions (iii) and (iv), we infer that‖xn+1−x∗‖ ≤ 11+αn{1+[(1+L∗)(1+L2∗)αn+L∗(1+L∗)βn]αn} ‖xn−x∗‖+11+αnL∗[1+(1+L∗)αn]αn‖vn‖+ 11+αn [1+(1+L∗)αn]‖un‖≤(1− 1−Kn1+αnαn)‖xn−x∗‖+L∗(2+L∗)αn‖vn‖+(2+L∗)(‖u′n‖+‖u′′n‖)≤(1− r2αn)‖xn−x∗‖+(L∗‖vn‖+γn)(2+L∗)αn+(2+L∗)‖u′n‖. (9)On the Ishikawa iterative approximation 5Setan = ‖xn−x∗‖, tn = r2αn, bn =2r(L∗‖vn‖+γn)(2+L∗), and cn = (2+L∗)‖u′n‖Then (9) is equivalent to the following inequality:an+1 ≤ (1− tn)an + bntn + cn, ∀n ≥ 0.In view of Lemma1, conditions (iii) and (iv), we know that an → 0 (n→∞), thatis, xn → x∗ (n→∞). This completes the proof. ✷Remark 1. Theorem1 improves and extends the corresponding results of [1] inits four aspects:(1) It abolishes the condition that X is uniformly smooth,(2) The Ishikawa iterative process is replaced by the more general Ishikawa itera-tive process with mixed errors,(3) It abolishes the condition that the range R(S) of S is bounded,(4) Sequences {αn} and {βn} need not converge to zero.Remark 2. Theorem1 extends and improves the main results of [2] in thefollowing ways:(1) Sequences {αn} and {βn} need not converge to zero,(2) It abolishes the condition that the {Sxn} and {Syn} are bounded,(3) The Ishikawa and Mann iterative process with errors is replaced by the moregeneral Ishikawa iterative process with mixed errors.Remark 3. Theorem1 also extends and improves the corresponding results ofChang [3], Chang, Cho and Lee et al [4], Ding [5,6], Hassouni and Moudafi [7],Huang [8-10], Kazmi [12], Noor [15,16], Siddiqi and Ansari [17], Siddiqi, Ansariand Kazmi [18] and Zeng [19].In Theorem 1, if βn ≡ 0, vn ≡ 0, ∀n ≥ 0, then yn = xn, hence we have thefollowing result.Theorem 2. Let X be an arbitrary real Banach space, T,A : X → X, g :X → X∗ be three mappings, and ϕ : X∗ → R∪{+∞} a function with a continuousGa¨teaux differential ∂ϕ. For any given f ∈ X, define a mapping S : X → X bySx = f − (Tx−Ax+ ∂ϕ(g(x))) + x.Let x0 ∈ X be any given point and {xn} the Mann iterative sequence with mixederrors defined byxn+1 = (1− αn)xn + αnSxn + un, ∀n ≥ 0, (10)where {αn} is a real sequence in [0, 1], and {un} is a sequence in X such that un =u′n + u′′n for any sequences {u′n} and {u′′n} in X satisfying the following conditions:6 GuFeng(i) T −A+ ∂ϕ ◦ g − I : X → X is accretive,(ii) T −A+ ∂ϕ ◦ g : X → X is a Lipschitz operator with the constant L,(iii) αn ≤ 1−r(1+L∗)(1+L2∗) and∑∞n=0 αn =∞,(iv)∑∞n=0 ‖u′n‖ <∞ and ‖u′′n‖ = γnαn, where L∗ = 1+L, r ∈ (0, 1) is a constantand γn → 0 (n→∞).Then the following conclusions hold:(1) The nonlinear variational inclusion problem (1) has a unique solution x∗ ∈ X,(2) The Mann iterative sequence {xn} with mixed errors converges strongly to theunique solution x∗ ∈ X of the variational inclusion problem (1).If ϕ ≡ 0 in Theorem 1, we have the following result.Theorem 3. Let X be an arbitrary real Banach space and let T,A : X → X,g : X → X∗ be three mappings. For any given f ∈ X, define a mapping S : X → XbySx = f − (Tx− Ax) + x.Let x0 ∈ X be any given point and {xn} the Ishikawa iterative sequence with mixederrors defined by {xn+1 = (1 − αn)xn + αnSyn + un,yn = (1− βn)xn + βnSxn + vn, ∀n ≥ 0, (11)where {αn}, {βn} are two real sequences in [0, 1], and {un}, {vn} are two sequencesin X such that un = u′n + u′′n for any sequences {u′n} and {u′′n} in X satisfying thefollowing conditions:(i) T −A− I : X → X is accretive,(ii) T −A : X → X is a Lipschitz operator with the constant L,(iii) Kn = (1 + L∗)(1 + L2∗)αn + L∗(1 + L∗)βn ≤ 1− r and∑∞n=0 αn =∞,(iv)∑∞n=0 ‖u′n‖ < ∞, ‖u′′n‖ = γnαn and ‖vn‖ → 0 (n → ∞), where L∗ = 1 + L,r ∈ (0, 1) is a constant and γn → 0 (n→∞).Then the following conclusions hold:(1) The variational inequality〈Tx−Ax− f, v − g(x)〉 ≥ 0, ∀v ∈ X∗ (12)has a unique solution x∗ ∈ X,(2) The Ishikawa iterative sequence {xn} with mixed errors converges strongly tothe unique solution x∗ ∈ X of the variational inequality (12).On the Ishikawa iterative approximation 7In Theorem 3, if βn ≡ 0, vn ≡ 0, ∀n ≥ 0, then yn = xn, hence we have thefollowing result.Theorem 4. Let X be an arbitrary real Banach space, and let T,A : X → X,g : X → X∗ be three mappings. For any given f ∈ X, define a mapping S : X → XbySx = f − (Tx−Ax) + x.Let x0 ∈ X be any given point and {xn} the Mann iterative sequence with mixederrors defined byxn+1 = (1− αn)xn + αnSxn + un, ∀n ≥ 0, (13)where {αn} is a real sequence in [0, 1], and {un} is a sequence in X such that un =u′n + u′′n for any sequences {u′n} and {u′′n} in X satisfying the following conditions:(i) T −A− I : X → X is accretive,(ii) T −A : X → X is a Lipschitz operator with the constant L,(iii) αn ≤ 1−r(1+L∗)(1+L2∗) and∑∞n=0 αn =∞,(iv)∑∞n=0 ‖u′n‖ <∞ and ‖u′′n‖ = γnαn, where L∗ = 1+L, r ∈ (0, 1) is a constantand γn → 0 (n→∞).Then the following conclusions hold:(1) The variational inequality (12) has a unique solution x∗ ∈ X,(2) The Mann iterative sequence {xn} with mixed errors converges strongly to theunique solution x∗ ∈ X of the variational inequality (12).Remark 4. The following example reveals that Theorem1 extends properlyTheorem 3.1 of Chang [1] and Theorem 2.1 of Chang [2].Example 1. Let X,T,A, g, f, S, ϕ be as in Theorem1 andαn =1− r2(1 + L∗)(1 + L2∗), βn =1− r2L∗(1 + L∗),‖u′n‖ =1(n+ 1)2, ‖u′′n‖ =1(n+1)1− r2(1+L∗)(1+L2∗), ‖vn‖ = 1n+1for all n ≥ 0. Then the conditions of Theorem1 are satisfied. But Theorem 3.1 in[1] and Theorem 2.1 in [2] are not applicable since {αn} and {βn} do not convergeto 0.References[1] S. S. Chang,On the Mann and Ishikawa iterative approximation of solutionsto variational inclusions with accretive type mappings, Computers Math. Appl.37(1999), 17–24.8 GuFeng[2] S. S. Chang, Existence and approximation of solutions to variational inclu-sions with accretive mappings in Banach spaces, Appl. Math. Mechanics.22(2001), 898–904.[3] S. S. Chang, Set-valued variational inclusions in Banach spaces, J. Math.Anal. Appl. 248(2000), 438–454.[4] S. S. Chang, Y. J. Cho, B. S. 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On the Ishikawa iterative approximation with mixed errors for solutions to variational inclusions with accretive type mappings in Banach spaces

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