Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 RESEARCH Open Access A general composite iterative method for strictly pseudocontractive mappings in Hilbert spaces Jong Soo Jung* * Correspondence: jungjs@dau.ac.kr; jungjs@mail.donga.ac.kr Department of Mathematics, Dong-A University, Busan, 604-714, Korea Abstract In this paper, we introduce a new general composite iterative method for finding a fixed point of a strictly pseudocontractive mapping in Hilbert spaces We establish the strong convergence of the sequence generated by the proposed iterative method to a fixed point of the mapping, which is the unique solution of a certain variational inequality In particular, we utilize weaker control conditions than previous ones in order to show strong convergence Our results complement, develop, and improve upon the corresponding ones given by some authors recently in this area MSC: 47H09; 47H05; 47H10; 47J25; 49M05; 47J05 Keywords: composite iterative method; k-strictly pseudocontractive mapping; nonexpansive mapping; fixed points; Lipschitzian; weakly asymptotically regular; ρ -Lipschitzian and η-strongly monotone operator; strongly positive bounded linear operator; Hilbert space; variational inequality Introduction Let H be a real Hilbert space with inner product ·, · and induced norm · Let C be a nonempty closed convex subset of H and let T : C → C be a self-mapping on C We denote by Fix(T) the set of fixed points of T We recall that a mapping T : C → H is said to be k-strictly pseudocontractive if there exists a constant k ∈ [, ) such that Tx – Ty ≤ x–y + k (I – T)x – (I – T)y , ∀x, y ∈ C The mapping T is pseudocontractive if and only if Tx – Ty, x – y ≤ x – y , ∀x, y ∈ C T is strongly pseudocontractive if and only if there exists a constant λ ∈ (, ) such that Tx – Ty, x – y ≤ ( – λ) x – y , ∀x, y ∈ C Note that the class of k-strictly pseudocontractive mappings includes the class of nonexpansive mappings T on C (i.e., Tx – Ty ≤ x – y , ∀x, y ∈ C) as a subclass That is, T is nonexpansive if and only if T is -strictly pseudocontractive The mapping T is also said © 2014 Jung; 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 Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page of 21 to be pseudocontractive if k = and T is said to be strongly pseudocontractive if there exists a positive constant λ ∈ (, ) such that T – λI is pseudocontractive Clearly, the class of k-strictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings Also we remark that the class of strongly pseudocontractive mappings is independent of the class of k-strictly pseudocontractive mappings (see [–]) The class of pseudocontractive mappings is one of the most important classes of mappings among nonlinear mappings Recently, many authors have been devoting the studies on the problems of finding fixed points for pseudocontractive mappings; see, for example, [–] and the references therein Let A be a strongly positive bounded linear operator on H That is, there is a constant γ > with the property Ax, x ≥ γ x , ∀x ∈ H It is well known that iterative methods for nonexpansive mappings can be used to solve a convex minimization problem: see, e.g., [–] and the references therein A typical problem is that of minimizing a quadratic function over the set of fixed points of a nonexpansive mapping on a real Hilbert space H: x∈C Ax, x – x, b , (.) where C is the fixed point set of a nonexpansive mapping S on H and b is a given point in H In [], Xu proved that the sequence {xn } generated by the iterative method for a nonexpansive mapping S presented below with the initial guess x ∈ H chosen arbitrary: xn+ = αn b + (I – αn A)Sxn , ∀n ≥ , (.) converges strongly to the unique solution of the minimization problem (.) provided the sequence {αn } satisfies certain conditions In [], combining the Moudafi viscosity approximation method [] with Xu’s method (.), Marino and Xu [] considered the following general iterative method for a nonexpansive mapping S: xn+ = αn γ fxn + (I – αn A)Sxn , ∀n ≥ , (.) where f is a contractive mapping on H with a constant α ∈ (, ) (i.e., there exists a constant α ∈ (, ) such that f (x) – f (y) ≤ α x – y , ∀x, y ∈ H) They proved that if the sequence {αn } of control parameters satisfies appropriate conditions, then the sequence {xn } generated by (.) converges strongly to the unique solution of the variational inequality (γ f – A)x, x – x ≤ , ∀x ∈ Fix(S), which is the optimality condition for the minimization problem x∈Fix(S) Ax, x – h(x), where h is a potential function for γ f (i.e., h = γ f ) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page of 21 On the other hand, Yamada [] introduced the following hybrid steepest-descent method for a nonexpansive mapping S for solving the variational inequality: xn+ = (I – ξn μF)Sxn , ∀n ≥ , (.) where S : H → H is a nonexpansive mapping with Fix(S) = ∅; F : H → H is a ρLipschitzian and η-strongly monotone operator with constants ρ > and η > (i.e., Fx – Fy ≤ ρ x – y and Fx – Fy, x – y ≥ η x – y , x, y ∈ H, respectively), and < μ < ρη , and then proved that if {ξn } satisfies appropriate conditions, the sequence {xn } generated by (.) converges strongly to the unique solution of the variational inequality: Fx, x – x ≥ , ∀x ∈ Fix(S) In , by combining Yamada’s hybrid steepest-descent method (.) with Marino with Xu’s method (.), Tian [] introduced the following general iterative method for a nonexpansive mapping S: xn+ = αn γ fxn + (I – αn μF)Sxn , ∀n ≥ , (.) where f is a contractive mapping on H with a constant α ∈ (, ) His results improved and complemented the corresponding results of Marino and Xu [] In [], Tian also considered the following general iterative method for a nonexpansive mapping S: xn+ = αn γ Vxn + (I – αn μF)Sxn , ∀n ≥ , (.) where V : H → H is a Lipschitzian mapping with a constant l ≥ In particular, the results in [] extended the results of Tian [] from the case of the contractive mapping f to the case of a Lipschitzian mapping V In , Ceng et al [] also introduced the following iterative method for the nonexpansive mapping S: xn+ = PC αn γ Vxn + (I – αn μF)Sxn , ∀n ≥ , (.) where F : C → H is a ρ-Lipschitzian and η-strongly monotone operator with constants ρ > and η > , V : C → H is an l-Lipschitzian mapping with a constant l ≥ and < μ < ρη In particular, by using appropriate control conditions on {αn }, they proved that the sequence {xn } generated by (.) converges strongly to a fixed point x of S, which is the unique solution of the following variational inequality related to the operator F: μFx – γ V x, x – p ≤ , ∀p ∈ Fix(S) Their results also improved the results of Tian [] from the case of the contractive mapping f to the case of a Lipschitzian mapping V In , Ceng et al [] introduced the following general composite iterative method for a nonexpansive mapping S: ⎧ ⎨y = (I – α μF)Sx + α γ fx , n n n n n ⎩xn+ = (I – βn A)Sxn + βn yn , ∀n ≥ , (.) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page of 21 which combines Xu’s method (.) with Tian’s method (.) Under appropriate control conditions on {αn } and {βn }, they proved that the sequence {xn } generated by (.) converges strongly to a fixed point x of S, which is the unique solution of the following variational inequality related to the operator A: (A – I)x, x – p ≤ , ∀p ∈ Fix(S) Their results supplemented and developed the corresponding ones of Marino and Xu [], Yamada [] and Tian [] On the another hand, in , by combining Yamada’s hybrid steepest-descent method (.) with Marino and Xu’s method (.), Jung [] considered the following explicit iterative scheme for finding fixed points of a k-strictly pseudocontractive mapping T for some ≤ k < : xn+ = αn γ f (xn ) + βn xn + ( – βn )I – αn μF PC Sxn , ∀n ≥ , (.) where S : C → H is a mapping defined by Sx = kx + ( – k)Tx; PC is the metric projection of H onto C; f : C → C is a contractive mapping with a constant α ∈ (, ); F : C → C is a ρ-Lipschitzian and η-strongly monotone operator with constants ρ > and η > ; and < μ < ρη Under suitable control conditions on {αn } and {βn }, he proved that the sequence {xn } generated by (.) converges strongly to a fixed point x of T, which is the unique solution of the following variational inequality related to the operator F: μFx – γ f x, x – p ≤ , ∀p ∈ Fix(T) His result also improved and complemented the corresponding results of Cho et al [], Jung [], Marino and Xu [] and Tian [] In this paper, motivated and inspired by the above-mentioned results, we will combine Xu’s method (.) with Tian’s method (.) for a k-strictly pseudocontractive mapping T for some ≤ k < and consider the following new general composite iterative method for finding an element of Fix(T): ⎧ ⎨y = α γ Vx + (I – α μF)T x , n n n n n n ⎩xn+ = (I – βn A)Tn xn + βn yn , ∀n ≥ , (.) where Tn : H → H is a mapping defined by Tn x = λn x + ( – λn )Tx for ≤ k ≤ λn ≤ λ < and limn→∞ λn = λ; A is a strongly positive bounded linear operator on H with a constant γ ∈ (, ); {αn } ⊂ [, ] and {βn } ⊂ (, ] satisfy appropriate conditions; V : H → H is a Lipschitzian mapping with a constant l ≥ ; F : H → H is a ρ-Lipschitzian and η-strongly monotone operator with constants ρ > and η > ; and < μ < ρη By using weaker control conditions than previous ones, we establish the strong convergence of the sequence generated by the proposed iterative method (.) to a point x in Fix(T), which is the unique solution of the variational inequality related to A: (A – I)x, x – p ≤ , ∀p ∈ Fix(T) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page of 21 Our results complement, develop, and improve upon the corresponding ones given by Cho et al [] and Jung [–] for the strictly pseudocontractive mapping as well as Yamada [], Marino and Xu [], Tian [] and Ceng et al [] and Ceng et al [] for the nonexpansive mapping Preliminaries and lemmas Throughout this paper, when {xn } is a sequence in H, xn → x (resp., xn x) will denote strong (resp., weak) convergence of the sequence {xn } to x For every point x ∈ H, there exists a unique nearest point in C, denoted by PC (x), such that x – PC (x) ≤ x – y , ∀y ∈ C PC is called the metric projection of H to C It is well known that PC is nonexpansive and that, for x ∈ H, z = PC x ⇐⇒ x – z, y – z ≤ , ∀y ∈ C (.) In a Hilbert space H, we have x–y = x + y – x, y , ∀x, y ∈ H (.) Lemma . In a real Hilbert space H, the following inequality holds: x+y ≤ x + y, x + y , ∀x, y ∈ H Let LIM be a Banach limit According to time and circumstances, we use LIMn (an ) instead of LIM(a) for every a = {an } ∈ ∞ The following properties are well known: (i) for all n ≥ , an ≤ cn implies LIMn (an ) ≤ LIMn (cn ), (ii) LIMn (an+N ) = LIMn (an ) for any fixed positive integer N , (iii) lim infn→∞ an ≤ LIMn (an ) ≤ lim supn→∞ an for all {an } ∈ l∞ The following lemma was given in [, Proposition ] Lemma . Let a ∈ R be a real number and let a sequence {an } ∈ l∞ satisfy the condition LIMn (an ) ≤ a for all Banach limit LIM If lim supn→∞ (an+ – an ) ≤ , then lim supn→∞ an ≤ a We also need the following lemmas for the proof of our main results Lemma . ([, ]) Let {sn } be a sequence of non-negative real numbers satisfying sn+ ≤ ( – ωn )sn + ωn δn + rn , ∀n ≥ , where {ωn }, {δn }, and {rn } satisfy the following conditions: (i) {ωn } ⊂ [, ] and ∞ n= ωn = ∞, (ii) lim supn→∞ δn ≤ or ∞ n= ωn |δn | < ∞, (iii) rn ≥ (n ≥ ), ∞ r < n= n ∞ Then limn→∞ sn = Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page of 21 Lemma . ([] Demiclosedness principle) Let C be a nonempty closed convex subset of a real Hilbert space H, and let S : C → C be a nonexpansive mapping Then the mapping x∗ and (I – S)xn → y, I – S is demiclosed That is, if {xn } is a sequence in C such that xn ∗ then (I – S)x = y Lemma . ([]) Let H be a real Hilbert space and let C be a closed convex subset of H Let T : C → H be a k-strictly pseudocontractive mapping on C Then the following hold: (i) The fixed point set Fix(T) is closed convex, so that the projection PFix(T) is well defined (ii) Fix(PC T) = Fix(T) (iii) If we define a mapping S : C → H by Sx = λx + ( – λ)Tx for all x ∈ C then, as λ ∈ [k, ), S is a nonexpansive mapping such that Fix(T) = Fix(S) The following lemma can easily be proven (see also []) Lemma . Let H be a real Hilbert space H Let F : H → H be a ρ-Lipschitzian and ηstrongly monotone operator with constants ρ > and η > Let < μ < ρη and < t < ξ ≤ Then G := ξ I – tμF : H → H is a contractive mapping with constant ξ – tτ , where τ = – – μ(η – μρ ) Lemma . ([]) Assume that A is a strongly positive bounded linear operator on H with a coefficient γ > and < ζ ≤ A – Then I – ζ A ≤ – ζ γ Finally, we recall that the sequence {xn } in H is said to be weakly asymptotically regular if w- lim (xn+ – xn ) = , n→∞ that is, xn+ – xn and asymptotically regular if lim xn+ – xn = , n→∞ respectively The main results Throughout the rest of this paper, we always assume the following: • H is a real Hilbert space; • T : H → H is a k-strictly pseudocontractive mapping with Fix(T) = ∅ for some ≤ k < ; • F : H → H is a ρ-Lipschitzian and η-strongly monotone operator with constants ρ > and η > ; • A : H → H is a strongly positive linear bounded operator on H with a constant γ ∈ (, ); • V : H → H is an l-Lipschitzian mapping with a constant l ≥ ; • < μ < ρη and ≤ γ l < τ , where τ = – – μ(η – μρ ); • Tt : H → H is a mapping defined by Tt x = λt x + ( – λt )Tx, t ∈ (, ), for ≤ k ≤ λt ≤ λ < and limt→ λt = λ; • Tn : H → H is a mapping defined by Tn x = λn x + ( – λn )Tx for ≤ k ≤ λn ≤ λ < and limn→∞ λn = λ; • PFix(T) is a metric projection of H onto Fix(T) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page of 21 By Lemma .(iii), we note that Tt and Tn are nonexpansive and Fix(T) = Fix(Tt ) = Fix(Tn ) In this section, we introduce the following general composite scheme that generates a net {xt }t∈(,min{, –γ }) in an implicit way: τ –γ l xt = (I – θt A)Tt xt + θt tγ Vxt + (I – tμF)Tt xt (.) We prove strong convergence of {xt } as t → to a fixed point x of T which is a solution of the following variational inequality: (A – I)x, x – p ≤ , ∀p ∈ Fix(T) (.) We also propose the following general composite explicit scheme, which generates a sequence in an explicit way: ⎧ ⎨y = α γ Vx + (I – α μF)T x , n n n n n n ⎩xn+ = (I – βn A)Tn xn + βn yn , ∀n ≥ , (.) where {αn } ∈ [, ], {βn } ⊂ (, ] and x ∈ H is an arbitrary initial guess, and establish strong convergence of this sequence to a fixed point x of T, which is also the unique solution of the variational inequality (.) Now, for t ∈ (, min{, τ–γ }) and θt ∈ (, A – ], consider a mapping Qt : H → H defined –γ l by Qt x = (I – θt A)Tt x + θt tγ Vx + (I – tμF)Tt x , ∀x ∈ H It is easy to see that Qt is a contractive mapping with constant – θt (γ – + t(τ – γ l)) Indeed, by Lemma . and Lemma ., we have Qt x – Qt y ≤ (I – θt A)Tt x – (I – θt A)Tt y + θt tγ Vx + (I – tμF)Tt x – tγ Vy + (I – tμF)Tt y ≤ ( – θt γ ) x – y + θt tγ Vx – Vy + (I – tμF)Tt x – (I – tμF)Tt y ≤ ( – θt γ ) x – y + θt – t(τ – γ l) x – y = – θt γ – + t(τ – γ l) Since γ ∈ (, ), τ – γ l > , and < t < , –γ –γ , ≤ τ –γl τ –γl it follows that < γ – + t(τ – γ l) < , x–y Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 which along with < θt ≤ A – Page of 21 < yields < – θt γ – + t(τ – γ l) < Hence Qt is a contractive mapping By the Banach contraction principle, Qt has a unique fixed point, denoted xt , which uniquely solves the fixed point equation (.) We summary the basic properties of {xt }, which can be proved by the same method in [] We include its proof for the sake of completeness Proposition . Let {xt } be defined via (.) Then }); (i) {xt } is bounded for t ∈ (, min{, τ–γ –γ l (ii) limt→ xt – Tt xt = provided limt→ θt = ; }) → H is locally Lipschitzian provided θt : (, min{, τ–γ }) → (iii) xt : (, min{, τ–γ –γ l –γ l – (, A ] is locally Lipschitzian, and λt : (, min{, τ–γ }) → [k, λ] is locally –γ l Lipschitzian; (iv) xt defines a continuous path from (, min{, τ–γ }) into H provided –γ l –γ – }) → [k, λ] is θt : (, min{, τ –γ l }) → (, A ] is continuous, and λt : (, min{, τ–γ –γ l continuous Proof () Let p ∈ Fix(T) Observing Fix(T) = Fix(Tt ) by Lemma .(iii), we have xt – p = (I – θt A)Tt xt + θt tγ Vxt + (I – tμF)Tt xt – p = (I – θt A)Tt xt – (I – θt A)Tt p + θt tγ Vxt + (I – tμF)Tt xt – p + θt (I – A)p ≤ (I – θt A)Tt xt – (I – θt A)Tt p + θt tγ Vxt + (I – tμF)Tt xt – p + θt (I – A)p = (I – θt A)Tt xt – (I – θt A)Tt p + θt (I – tμF)Tt xt – (I – tμF)Tt p + t(γ Vxt – μFp) + θt I – A p ≤ ( – θt γ ) xt – p + θt ( – tτ ) xt – p + t γ l xt – p + γ Vp – μFp + θt I – A p So, it follows that xt – p ≤ ≤ I – A p + t γ Vp – μFp I – A p + t γ Vp – μFp ≤ γ – + t(τ – γ l) γ – I – A p + γ Vp – μFp γ – Hence {xt } is bounded and so are {Vxt }, {Txt }, {Tt xt }, and {FTt xt } (ii) By the definition of {xt }, we have xt – Tt xt = θt (I – A)Tt xt + t(γ Vxt – μFTt xt ) = θt (I – A)Tt xt + t(γ Vxt – μFTt xt ) ≤ θt I – A Tt xt + t γ Vxt – μFTt xt → by the boundedness of {Vxt } and {FTt xt } in (i) as t → , Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page of 21 (iii) Let t, t ∈ (, min{, τ–γ }), Noting that –γ l Tt xt – Tt xt ≤ Tt xt – Tt xt + Tt xt – Tt xt ≤ xt – xt + |λt – λt | xt – Txt , we calculate x t – x t = (I – θt A)Tt xt + θt tγ Vxt + (I – tμF)Tt xt – (I – θt A)Tt xt – θt t γ Vxt + (I – t μF)Tt xt ≤ (I – θt A)Tt xt – (I – θt A)Tt xt + (I – θt A)Tt xt – (I – θt A)Tt xt + |θt – θt | tγ Vxt + (I – tμF)Tt xt + θ t tγ Vxt + (I – tμF)Tt xt – t γ Vxt + (I – t μF)Tt xt ≤ |θt – θt | A Tt xt + ( – θt γ ) Tt xt – Tt xt + |θt – θt | tγ Vxt + (I – tμF)Tt xt + θt (t – t )γ Vxt + t γ (Vxt – Vxt ) – (t – t )μFTt xt + (I – t μF)Tt xt – (I – t μF)Tt xt ≤ |θt – θt | A Tt xt + ( – θt γ ) xt – xt + |λt – λt | xt – Txt + |θt – θt | Tt xt + t γ Vxt + μ FTt xt + θt γ Vxt + μ FTt xt |t – t | + t γ l xt – xt + ( – t τ ) Tt xt – Tt xt ≤ |θt – θt | A Tt xt + ( – θt γ ) xt – xt + |λt – λt | xt – Txt + |θt – θt | Tt xt + γ Vxt + μ FTt xt + θt γ Vxt + μ FTt xt |t – t | + θt t γ l xt – xt + θt ( – t τ ) xt – xt + |λt – λt | xt – Txt This implies that x t – x t ≤ A Tt xt + Tt xt + γ Vxt + μ FTt xt |θt – θt | θt (γ – + t (τ – γ l)) + γ Vxt + μ FTt xt |t – t | γ – + t (τ – γ l) + [ – θt (γ – + t τ )] xt – Txt |λt – λt | θt (γ – + t (τ – γ l)) }) → (, A – ] is locally Lipschitzian, and λt : (, min{, τ–γ }) → Since θt : (, min{, τ–γ –γ l –γ l [k, λ] is locally Lipschitzian, xt is also locally Lipschitzian (iv) From the last inequality in (iii), the result follows immediately We prove the following theorem for strong convergence of the net {xt } as t → , which guarantees the existence of solutions of the variational inequality (.) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 10 of 21 Theorem . Let the net {xt } be defined via (.) If limt→ θt = , then xt converges strongly to a fixed point x of T as t → , which solves the variational inequality (.) Equivalently, we have PFix(T) (I – A)x = x Proof We first show the uniqueness of a solution of the variational inequality (.), which is indeed a consequence of the strong monotonicity of A – I In fact, since A is a strongly positive bounded linear operator with a coefficient γ ∈ (, ), we know that A – I is strongly monotone with a coefficient γ – ∈ (, ) Suppose that x ∈ Fix(T) and x ∈ Fix(T) both are solutions to (.) Then we have (A – I)x, x – x ≤ (.) (A – I)x, x – x ≤ (.) and Adding up (.) and (.) yields (A – I)x – (A – I)x, x – x ≤ The strong monotonicity of A – I implies that x = x and the uniqueness is proved Next, we prove that xt → x as t → Observing Fix(T) = Fix(Tt ) by Lemma .(iii), from (.), we write, for given p ∈ Fix(T), xt – p = (I – θt A)Tt xt – (I – θt A)Tt p + θt tγ Vxt + (I – tμF)Tt xt – p + θt (I – A)p = (I – θt A)(Tt xt – Tt p) + θt t(γ Vxt – μFp) + (I – tμF)Tt xt – (I – tμF)p + θt (I – A)p, to derive xt – p = (I – θt A)(Tt xt – Tt p), xt – p + θt t γ Vxt – μFp, xt – p + (I – tμF)Tt xt – (I – tμF)p, xt – p + θt (I – A)p, xt – p ≤ ( – θt γ ) xt – p + θt ( – tτ ) xt – p + tγ l xt – p + t γ Vp – μFp, xt – p + θt (I – A)p, xt – p = – θt γ – + t(τ – γ l) xt – p + θt t γ Vp – μFp, xt – p + (I – A)p, xt – p Therefore, xt – p ≤ t γ Vp – μFp, xt – p + (I – A)p, xt – p γ – + t(τ – γ l) (.) Since {xt } is bounded as t → (by Proposition .(i)), we see that if {tn } is a subsequence }) such that tn → and xtn x∗ , then from (.), we obtain xtn → x∗ We in (, min{, τ–γ –γ l Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 11 of 21 show that x∗ ∈ Fix(T) To this end, define S : H → H by Sx = λx + ( – λ)Tx, ∀x ∈ H, for ≤ k ≤ λ < Then S is nonexpansive with Fix(S) = Fix(T) by Lemma .(iii) Noticing that Sxtn – xtn ≤ Sxtn – Ttn xtn + Ttn xtn – xtn = (λ – λtn ) xtn – Txtn + Ttn xtn – xtn = λ – λ tn xt – Ttn xtn + Ttn xtn – xtn – λ tn n = + λ – λtn xtn – Ttn xtn , – λ tn by Proposition .(ii) and λtn → λ as tn → , we have limn→∞ (I – S)xtn = Thus it follows from Lemma . that x∗ ∈ Fix(S) By Lemma .(iii), we get x∗ ∈ Fix(T) Finally, we prove that x∗ is a solution of the variational inequality (.) Since xt = (I – θt A)Tt xt + θt tγ Vxt + (I – tμF)Tt xt , we have xt – Tt xt = θt (I – A)Tt xt + θt t(γ Vxt – μFTt xt ) Since Tt is nonexpansive, I – Tt is monotone So, from the monotonicity of I – Tt , it follows that, for p ∈ Fix(T) = Fix(Tt ), ≤ (I – Tt )xt – (I – Tt )p, xt – p = (I – Tt )xt , xt – p = θt (I – A)Tt xt , xt – p + θt t γ Vxt – μFTt xt , xt – p = θt (I – A)xt , xt – p + θt (I – A)(Tt – I)xt , xt – p + θt t γ Vxt – μFTt xt , xt – p This implies that (A – I)xt , xt – p ≤ (I – A)(Tt – I)xt , xt – p + t γ Vxt – μFTt xt , xt – p (.) Now, replacing t in (.) with tn and letting n → ∞, noticing the boundedness of {γ Vxtn – μFTtn xtn } and the fact that (I – A)(Ttn – I)xtn → as n → ∞ by Proposition .(ii), we obtain (A – I)x∗ , x∗ – p ≤ That is, x∗ ∈ Fix(T) is a solution of the variational inequality (.); hence x∗ = x by uniqueness In summary, we have shown that each cluster point of {xt } (at t → ) equals x Therefore xt → x as t → The variational inequality (.) can be rewritten as (I – A)x – x, x – p ≥ , ∀p ∈ Fix(T) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 12 of 21 Recalling Lemma .(i) and (.), this is equivalent to the fixed point equation PFix(T) (I – A)x) = x Taking F = I, μ = and γ = in Theorem ., we get Corollary . Let {xt } be defined by xt = (I – θt A)Tt xt + θt tVxt + ( – t)Tt xt If limt→ θt = , then {xt } converges strongly as t → to a fixed point x of T, which is the unique solution of variational inequality (.) First, we prove the following result in order to establish strong convergence of the sequence {xn } generated by the general composite explicit scheme (.) Theorem . Let {xn } be the sequence generated by the explicit scheme (.), where {αn } and {βn } satisfy the following condition: (C) {αn } ⊂ [, ] and {βn } ⊂ (, ], αn → and βn → as n → ∞ Let LIM be a Banach limit Then LIMn (A – I)x, x – xn ≤ , where x = limt→+ xt with xt being defined by xt = (I – θt A)Sxt + θt tγ Vxt + (I – tμF)Sxt , (.) where S : H → H is defined by Sx = λx + ( – λ)Tx for ≤ k ≤ λ < Proof First, note that from the condition (C), without loss of generality, we assume that < βn ≤ A – for all n ≥ Let {xt } be the net generated by (.) Since S is a nonexpansive mapping on H, by Theorem . with Tt = S and Lemma ., there exists limt→ xt ∈ Fix(S) = Fix(T) Denote it by x Moreover, x is the unique solution of the variational inequality (.) From Proposition .(i) with Tt = S, we know that {xt } is bounded, so are {Vxt } and {FSxt } First of all, let us show that {xn } is bounded To this end, take p ∈ Fix(T) = Fix(Tn ), Then it follows that yn – p = αn γ Vxn + (I – αn μF)Tn xn – p = αn (γ Vxn – μFp) + (I – αn μF)Tn xn – (I – αn μF)Tn p ≤ – αn (τ – γ l) xn – p + αn γ Vp – μFp , and hence xn+ – p = (I – βn A)Tn xn + βn yn – p = (I – βn A)Tn xn – (I – βn A)Tn p + βn (yn – p) + βn (I – A)p Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 13 of 21 ≤ (I – βn A)Tn xn – (I – βn A)Tn p + βn yn – p + βn I – A p ≤ ( – βn γ ) xn – p + βn – αn (τ – γ l) xn – p + αn γ Vp – μFp + βn I – A p ≤ – βn (γ – ) xn – p + βn γ Vp – μFp + I – A p = – βn (γ – ) xn – p + βn (γ – ) ≤ max xn – p , γ Vp – μFp + I – A p γ – γ Vp – μFp + I – A p γ – By induction xn – p ≤ max x – p , γ Vp – μFp + I – A p γ – , ∀n ≥ This implies that {xn } is bounded and so are {Txn }, {Tn xn }, {FTn xn }, {Vxn }, and {yn } As a consequence, with the control condition (C), we get xn+ – Tn xn = βn yn – ATn xn → (n → ∞), and Sxt – xn+ ≤ Sxt – Sxn + Sxn – Tn xn + Tn xn – xn+ ≤ xt – xn + |λ – λn | xn – Txn + Tn xn – xn+ = xt – xn + en , (.) where en = |λ – λn | xn – Txn + xn+ – Tn xn → as n → ∞ Also observing that A is strongly positive, we have Axt – Axn , xt – xn = A(xt – xn ), xt – xn ≥ γ xt – xn (.) Now, by (.), we have xt – xn+ = (I – θt A)Sxt + θt tγ Vxt + (I – tμF)Sxt – xn+ = (I – θn A)Sxt – (I – θt A)xn+ + θt tγ Vxt + (I – tμF)Sxt – Axn+ Applying Lemma ., we have xt – xn+ ≤ (I – θt A)Sxt – (I – θt A)xn+ + θt Sxt – t(μFSxt – γ Vxt ) – Axn+ , xt – xn+ ≤ ( – θt γ ) Sxt – xn+ + θt Sxt – xt , xt – xn+ – θt t μFSxt – γ Vxt , xt – xn+ + θt xt – Axn+ , xt – xn+ (.) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 14 of 21 Using (.) and (.) in (.), we obtain xt – xn+ ≤ ( – θt γ ) Sxt – xn+ + θt Sxt – xt , xt – xn+ + θt t γ Vxt – μFSxt , xt – xn+ + θt xt – Axn+ , xt – xn+ ≤ ( – θt γ ) xt – xn + en + θt t γ Vxt – μFSxt = θt γ – θt γ xt – xn + θt Sxt – xt + θt Sxt – xt xt – xn+ xt – xn+ + θt xt – Axn+ , xt – xn+ + xt – xn + ( – θt γ ) xt – xn en + en xt – xn+ + θt t γ Vxt – μFSxt xt – xn+ + θt xt – Axn+ , xt – xn+ ≤ θt γ – θt Axt – Axn , xt – xn + xt – xn + θt Sxt – xt + ( – θt γ ) xt – xn en + en xt – xn+ + θt t γ Vxt – μFSxt xt – xn+ + θt xt – Axn+ , xt – xn+ = θt γ Axt – Axn , xt – xn + xt – xn + θt Sxt – xt + ( – θt γ ) xt – xn en + en xt – xn+ + θt t γ Vxt – μF(Sxt ) xt – xn+ + θt xt – Axn+ , xt – xn+ – Axt – Axn , xt – xn = θt γ A(xt – xn ), xt – xn + xt – xn + θt Sxt – xt + ( – θt γ ) xt – xn en + en xt – xn+ + θt t γ Vxt – μFSxt xt – xn+ + θt (I – A)xt , xt – xn+ + A(xt – xn+ ), xt – xn+ – A(xt – xn ), xt – xn (.) Applying the Banach limit LIM to (.), together with limn→∞ en = , we have LIMn xt – xn+ ≤ θt γ LIMn A(xt – xn ), xt – xn + LIMn xt – xn + θt Sxt – xt LIMn xt – xn+ + θt t γ Vxt – μFSxt LIMn xt – xn+ + θt LIMn (I – A)xt , xt – xn+ + LIMn A(xt – xn+ ), xt – xn+ – LIMn A(xt – xn ), xt – xn Using the property LIMn (an ) = LIMn (an+ ) of the Banach limit in (.), we obtain LIMn (A – I)xt , xt – xn = LIMn (A – I)xt , xt – xn+ ≤ θt γ LIMn A(xt – xn ), xt – xn (.) Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 + LIMn xt – xn θt – LIMn xt – xn+ + Sxt – xt LIMn xt – xn Page 15 of 21 + t γ Vxt – μFSxt LIMn xt – xn + LIMn A(xt – xn+ ), xt – xn+ – LIMn A(xt – xn ), xt – xn = θt γ LIMn A(xt – xn ), xt – xn + Sxt – xt LIMn xt – xn + t γ Vxt – μFSxt LIMn xt – xn (.) Since θt A(xt – xn ), xt – xn ≤ θt A xt – xn ≤ θt K → (as t → ), where A xt – xn Sxt – xt → , (.) ≤ K, and t γ Vxt – μFSxt → (as t → ), (.) we conclude from (.)-(.) that LIMn (A – I)x, x – xn ≤ lim sup LIMn (A – I)xt , xt – xn t→ ≤ lim sup t→ θt γ LIMn A(xt – xn ), xt – xn + lim sup Sxt – xt LIMn xt – xn t→ + lim sup t γ Vxt – μFSxt LIMn xt – xn t→ = This completes the proof Now, using Theorem ., we establish strong convergence of the sequence {xn } generated by the general composite explicit scheme (.) to a fixed point x of T, which is also the unique solution of the variational inequality (.) Theorem . Let {xn } be the sequence generated by the explicit scheme (.), where {αn } and {βn } satisfy the following conditions: (C) {αn } ⊂ [, ] and {βn } ⊂ (, ], αn → and βn → as n → ∞; ∞ (C) n= βn = ∞ If {xn } is weakly asymptotically regular, then {xn } converges strongly to x ∈ Fix(T), which is the unique solution of the variational inequality (.) Proof First, note that from the condition (C), without loss of generality, we assume that n (γ –) < for all n ≥ αn τ < and β–β n Let xt be defined by (.), that is, xt = (I – θt A)Sxt + θt Sxt – t μF(Sxt ) – γ Vxt Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 16 of 21 for t ∈ (, min{, τ–γ }), where Sx = λx + ( – λ)Tx for ≤ k ≤ λ < , and limt→ xt := x ∈ –γ l Fix(S) = Fix(T) (by using Theorem . and Lemma .(iii)) Then x is the unique solution of the variational inequality (.) We divide the proof into several steps as follows Step We see that xn – p ≤ max x – p , γ Vp – μFp + I – A p γ – , ∀n ≥ , for all p ∈ Fix(T) as in the proof of Theorem . Hence {xn } is bounded and so are {Txn }, {Tn xn }, {FTn xn }, {Vxn }, and {yn } Step We show that lim supn→∞ (I – A)x, xn – x ≤ To this end, put an := (A – I)x, x – xn , ∀n ≥ Then Theorem . implies that LIMn (an ) ≤ for any Banach limit LIM Since {xn } is bounded, there exists a subsequence {xnj } of {xn } such that lim sup(an+ – an ) = lim (anj + – anj ) j→∞ n→∞ and xnj v ∈ H This implies that xnj + Therefore, we have v since {xn } is weakly asymptotically regular w – lim (x – xnj + ) = w – lim (x – xnj ) = (x – v), j→∞ j→∞ and so lim sup(an+ – an ) = lim (A – I)x, (x – xnj + ) – (x – xnj ) = j→∞ n→∞ Then Lemma . implies that lim supn→∞ an ≤ , that is, lim sup (I – A)x, xn – x = lim sup (A – I)x, x – xn ≤ n→∞ n→∞ Step We show that limn→∞ xn – x = By using (.) and Tn x = x, we have yn – x = (I – αn μF)Tn xn – (I – αn μF)Tn x + αn (γ Vxn – μFx), and xn+ – x = (I – βn A)(Tn xn – Tn x) + βn (yn – x) + βn (I – A)x Applying Lemma ., Lemma . and Lemma ., we obtain yn – x = (I – μαn F)Tn xn – (I – μαn F)Tn x + αn (γ Vxn – μFx) ≤ (I – μαn F)Tn xn – (I – μαn F)Tn x + αn γ Vxn – μFx, yn – x Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 ≤ ( – αn τ ) xn – x ≤ xn – x Page 17 of 21 + αn γ Vxn – μFx yn – x + αn γ Vxn – μFx yn – x , and hence xn+ – x = (I – βn A)(Tn xn – Tn x) + βn (yn – x) + βn (I – A)x ≤ (I – βn A)(Tn xn – Tn x) + βn yn – x, xn+ – x + βn (I – A)x, xn+ – x ≤ ( – βn γ ) xn – x + βn yn – x xn+ – x + βn (I – A)x, xn+ – x ≤ ( – βn γ ) xn – x + βn yn – x + xn+ – x + βn (I – A)x, xn+ – x ≤ ( – βn γ ) xn – x + βn x n – x + βn xn+ – x + αn γ Vxn – μFx yn – x + βn (I – A)x, xn+ – x = ( – βn γ ) + βn xn – x + βn xn+ – x + αn βn γ Vxn – μFx yn – x + βn (I – A)x, xn+ – x (.) It then follows from (.) that xn+ – x ≤ ( – βn γ ) + βn xn – x – βn + = βn αn γ Vxn – μFx yn – x + (I – A)x, xn+ – x – βn – + βn (γ – ) – βn xn – x βn (γ – ) αn γ Vxn – μFx yn – x · – βn (γ – ) + βn γ x n – x + (I – A)x, xn+ – x = ( – ωn ) xn – x + ωn δn , where ωn = βn (γ – ) – βn δn = αn γ Vxn – μFx yn – x + βn γ xn – x (γ – ) and + (I – A)x, xn+ – x It can easily be seen from Step and conditions (C) and (C) that ωn → , ∞ n= ωn = ∞ and lim supn→∞ δn ≤ From Lemma . with rn = , we conclude that limn→∞ xn – x = This completes the proof Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 18 of 21 Corollary . Let {xn } be the sequence generated by the explicit scheme (.) Assume that the sequence {αn } and {βn } satisfy the conditions (C) and (C) in Theorem . If {xn } is asymptotically regular, then {xn } converges strongly to x ∈ Fix(T), which is the unique solution of the variational inequality (.) Putting μ = , F = I and γ = in Theorem ., we obtain the following Corollary . Let {xn } be generated by the following iterative scheme: ⎧ ⎨y = α Vx + ( – α )T x , n n n n n n ⎩xn+ = (I – βn A)Tn xn + βn yn , ∀n ≥ Assume that the sequence {αn } and {βn } satisfy the conditions (C) and (C) in Theorem . If {xn } is weakly asymptotically regular, then {xn } converges strongly to x ∈ Fix(T), which is the unique solution of the variational inequality (.) Putting αn = , ∀n ≥ in Corollary ., we get the following Corollary . Let {xn } be generated by the following iterative scheme: xn+ = (I – βn A)Tn xn + βn Tn xn , ∀n ≥ Assume that the sequence {βn } satisfies the conditions (C) and (C) in Theorem . with αn = , ∀n ≥ If {xn } is weakly asymptotically regular, then {xn } converges strongly to x ∈ Fix(T), which is the unique solution of the variational inequality (.) Remark . If {αn }, {βn } in Corollary . and {λn } in Tn satisfy conditions (C) and ∞ ∞ (C) n= |αn+ – αn | < ∞ and n= |βn+ – βn | < ∞; or ∞ βn αn –αn+ (C) n= |αn+ – αn | < ∞ and limn→∞ βn+ = or, equivalently, limn→∞ αn+ = and n+ limn→∞ βnβ–β = ; or, n+ ∞ ∞ (C) |α – α n | < ∞ and |βn+ – βn | ≤ o(βn+ ) + σn , n= n+ n= σn < ∞ (the perturbed control condition); (C) n= |λn+ – λn | < ∞, then the sequence {xn } generated by (.) is asymptotically regular Now we give only the proof in the case when {αn }, {βn }, and {λn } satisfy the conditions (C), (C), and (C) By Step in the proof of Theorem ., there exists a constant M > such that, for all n ≥ , xn – Txn ≤ M, μ FTn xn + γ Vxn ≤ M, and A Tn xn + yn ≤ M Next, we notice that Tn xn – Tn– xn– ≤ Tn xn – Tn xn– + Tn xn– – Tn– xn– ≤ xn – xn– + |λn – λn– | xn– – Txn– ≤ xn – xn– + |λn – λn– |M Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 Page 19 of 21 So we obtain, for all n ≥ , yn – yn– = αn γ (Vxn – Vxn– ) + γ (αn – αn– )Vxn– + (I – αn μF)Tn xn – (I – αn μF)Tn– xn– + μ(αn – αn– )FTn– xn– ≤ – αn (τ – γ l) Tn xn – Tn– xn– + |αn – αn– | γ Vxn– + μ FTn– xn– ≤ – αn (τ – γ l) xn – xn– + |λn – λn– |M + |αn – αn– |M, and hence xn+ – xn = (I – βn A)Tn xn + βn yn – (I – βn– A)Tn– xn– – βn– yn– ≤ (I – βn A)(Tn xn – Tn– xn– ) + |βn – βn– | A Tn– xn– + βn yn – yn– + |βn – βn– | yn– ≤ ( – βn γ ) Tn xn – Tn– xn– + βn – αn (τ – γ l) xn – xn– + |λn – λn– |M + βn |αn – αn– |M + |βn – βn– | A Tn– xn– + yn– ≤ ( – βn γ ) xn – xn– + |λn – λn– |M + βn – αn (τ – γ l) xn – xn– + |λn – λn– |M + |αn – αn– |M + |βn – βn– |M ≤ ( – βn γ ) xn – xn– + |λn – λn– |M + βn xn – xn– + |λn – λn– |M + |αn – αn– |M + |βn – βn– |M = – βn (γ – ) xn – xn– + |βn – βn– |M + |λn – λn– |M + |αn – αn– |M ≤ – βn (γ – ) xn – xn– + o(βn ) + σn– M + |αn – αn– |M + |λn – λn– |M (.) By taking sn+ = xn+ – xn , ωn = βn (γ – ), ωn δn = Mo(βn ) and rn = (|αn – αn– | + σn– + |λn – λn– |)M, from (.) we have sn+ ≤ ( – ωn )sn + ωn δn + rn Hence, by the conditions (C), (C), (C), and Lemma ., we obtain lim xn+ – xn = n→∞ In view of this observation, we have the following Corollary . Let {xn } be the sequence generated by the explicit scheme (.), where the sequences {αn }, {βn }, and {λn } satisfy the conditions (C), (C), (C), and (C) (or the conditions (C), (C), (C) and (C), or the conditions (C), (C), (C), and (C)) Then {xn } Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 converges strongly to x ∈ Fix(T), which is the unique solution of the variational inequality (.) Remark . () Our results improve and extend the corresponding results of Ceng et al [] in the following respects: (a) The nonexpansive mapping S : H → H in [] is extended to the case of a k-strictly pseudocontractive mapping T : H → H (b) The contractive mapping f in [] with constant α ∈ (, ) is extended to the case of a Lipschitzian mapping V with constant l ≥ (c) The range < γ α < τ = μ(η – μρ ) in [] is extended to the case of range < γ l < τ = – – μ(η – μρ ) (For this fact, see Remark . of [].) ∞ () We point out that the condition (C) ∞ n= |αn+ – αn | < ∞ and n= |βn+ – βn | < ∞ in [, Theorem .] is relaxed to the case of the weak asymptotic regularity on {xn } in Theorem . () The condition (C) on {βn } in Corollary . is independent of condition (C) or (C) in Remark ., which was imposed in Theorem . of Ceng et al [] For this fact, see [, ] () Our results also complement and develop the corresponding ones given by Cho et al [] and Jung [–] for the strictly pseudocontractive mapping as well as Yamada [], Marino and Xu [], Tian [] and Ceng et al [] for the nonexpansive mapping () For several iterative schemes based on hybrid steepest-descent method for generalized mixed equilibrium problems, variational inequality problems, and fixed point problems for strictly pseudocontractive mappings, we can also refer to [–] and the references therein Competing interests The author declares to have no competing interests Acknowledgements The author would like to thank the anonymous referees for their careful reading and valuable comments along with providing some recent related papers, which improved the presentation of this manuscript This study was supported by research funds from Dong-A University Received: May 2014 Accepted: 21 July 2014 Published: 18 August 2014 References Browder, FE: Fixed point theorems for noncompact mappings Proc Natl Acad Sci USA 53, 1272-1276 (1965) Browder, FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces Arch Ration Mech Anal 24, 82-90 (1967) Browder, FE, Petryshn, WV: Construction of fixed points of nonlinear mappings Hilbert space J Math Anal Appl 20, 197-228 (1967) Acedo, GL, Xu, HK: Iterative methods for strictly pseudo-contractions in Hilbert space Nonlinear Anal 67, 2258-2271 (2007) Cho, YJ, Kang, SM, Qin, X: Some results on k-strictly pseudo-contractive mappings in Hilbert spaces Nonlinear Anal 70, 1956-1964 (2009) Jung, JS: Strong convergence of iterative methods for k-strictly pseudo-contractive mappings in Hilbert spaces Appl Math Comput 215, 3746-3753 (2010) Jung, JS: Some results on a general iterative method for k-strictly pseudo-contractive mappings Fixed Point Theory Appl 2011, 24 (2011) doi:10.1186/1687-1812-2011-24 Jung, JS: A general iterative method with some control conditions for k-strictly pseudo-contractive mappings J Comput Anal Appl 14, 1165-1177 (2012) Morales, CH, Jung, JS: Convergence of paths for pseudo-contractive mappings in Banach spaces Proc Am Math Soc 128, 4311-4319 (2000) 10 Deutch, F, Yamada, I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings Numer Funct Anal Optim 19, 33-56 (1998) 11 Xu, HK: An iterative approach to quadratic optimization J Optim Theory Appl 116, 659-678 (2003) Page 20 of 21 Jung Fixed Point Theory and Applications 2014, 2014:173 http://www.fixedpointtheoryandapplications.com/content/2014/1/173 12 Yamada, I: The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings In: Butnariu, D, Censor, Y, Reich, S (eds.) 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