Suppose that K is nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction and F : F T1 ∩ F T2 {x ∈ K : T1x T2x x}/ ∅. Let T1, T2 : K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences {k i n } ⊂ 1,∞ satisfying ∑∞ n 1 k i n − 1 < ∞ i 1, 2 , respectively. For any given x1 ∈ K, s...

Let E be a real reflexive Banach space which has uniformly Gâteaux differentiable norm. Let K be aclosed convex subset of E which is also a sunny nonexpansive retract of E, and T : K → E be nonexpansive mapping satisfying the weakly inward condition and F (T ) = {x ∈ K, Tx = x} 6= ∅, and f : K → K be a contractive mapping. Suppose that x0 ∈ K, {xn} is defined by { xn+1 = αnf(xn) + (1− αn)((1− δ...

and Applied Analysis 3 Very recently, Qin et al. 16 proposed the composite Halpern type iterative scheme in a uniformly smooth Banach space as follows: x0 x, u ∈ C, zn γnxn ( 1 − γn ) Txn, yn βnxn ( 1 − βn ) Tzn, xn 1 αnu 1 − αn yn, n ≥ 0, 1.6 and showed strong convergence of the sequence {xn} generated by 1.6 under the following control conditions: i ∑∞ n 0αn ∞; ii limn→∞αn 0, limn→∞βn 0 and 0...

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E∗, and K be a nonempty closed convex subset of E. Suppose that T, S : K → K are two nonexpansive mappings such that F := F (ST ) = F (T ) ∩ F (S) = ∅. For arbitrary initial value x0 ∈ K and fixed anchor u ∈ K, define iteratively a sequence {xn} as follows: { yn = βnxn + (1− βn)Txn xn+...

Suppose that K is a nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1, T2 : K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences {k n } ⊂ [1,∞) satisfying ∞n=1(k(i) n − 1) < ∞ (i = 1, 2) and F (T1) ∩ F (T2) = {x ∈ K : T1x = T2x = x} = ∅, respectively. For any giv...

Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2 : K → E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences {Kn},{ln} ⊂ [1,∞), limn→∞kn = 1, limn→∞ln = 1, F(T1)∩ F(T2) = {x ∈ K : T1x = T2x = x} =∅, respectively. Suppose that {xn} is a sequence in...

Let X be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from X to X∗, and C be a closed convex subset of X which is also a sunny nonexpansive retract of X, and T : C → X a non-expansive mapping satisfying the weakly inward condition and F (T ) = ∅, and f : C → C be a fixed contractive mapping. The sequence {xn} is given by xn+1 = P (αnf(xn) + (1− αn)...

and Applied Analysis 3 Lemma 2.2 cf., 4 . Let D be a nonempty subset of a reflexive, strictly convex, and smooth Banach space E. Let R be a retraction from E onto D. Then R is sunny and generalized nonexpansive if and only if 〈 x − Rx, JRx − Jy ≥ 0, 2.2 for all x ∈ E and y ∈ D. A generalized resolvent Jr of a maximal monotone operator B ⊂ E∗ × E is defined by Jr I rBJ −1 for any real number r >...

In this paper, we study nonlinear analytic methods for linear contractive operators in Banach spaces. Using these results, we obtain some new strong convergence theorems for commutative families of linear contractive operators in Banach spaces. In the results, the limit points are characterized by sunny generalized nonexpansive retractions.