Random fixed point theorems for condensing 1-set contraction selfmaps are known. But no random fixed point theorem for more general asymptotic 1-set contraction selfmaps is yet available. The purpose of this paper is to prove random fixed point theorems for such maps. 1. Introduction. Random fixed point theorems are stochastic versions of (classical or deterministic) fixed point theorems and ar...

and Applied Analysis 3 Theorem 1.3. Let E be a real q-uniformly smooth Banach space which is also uniformly convex, let C be a nonempty closed convex subset of E, let T : C → C be a (λ, {kn})-strictly asymptotically pseudocontractive mapping such that ∑∞ n 1 k 2 n − 1 < ∞, and let F T / ∅. Let {αn} ⊂ 0, 1 be a real sequence satisfying the following condition: 0 < a ≤ αn ≤ b < q 1 − k 2cq 1 L − ...

and Applied Analysis 3 Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping A : C → H is called α-inverse strongly monotone if there exists a positive real number α such that 〈Ax − Ay, x − y〉 ≥ α‖Ax −Ay‖, for all x, y ∈ C. It is clear that any αinverse strongly monotone mapping is monotone and 1/α-Lipschitz continuous. Let f : C → H be a ρ-contraction; that ...

and Applied Analysis 3 for all x, y ∈ C, where cn max{0, supx,y∈C ‖Tnx − Tny‖ − ‖x − y‖ } so that limn→∞cn 0. Hence, T is a generalized asymptotically nonexpansive mapping. The mapping T : C → C is said to be demiclosed at 0 if for each sequence {xn} in C converging weakly to x and {Txn} converging strongly to 0, we have Tx 0. A Banach space E is said to satisfy Opial’s property, see 4 , if for...

and Applied Analysis 3 Moreover, PC is characterized by the following properties: 〈 x − PCx, y − PCx 〉 ≤ 0, ∥ ∥x − y∥∥2 ≥ ‖x − PCx‖ ∥ ∥y − PCx ∥ ∥ 2 , 2.4 for all x ∈ H and y ∈ C. We also need other sorts of nonlinear operators which are introduced below. Let T : H → H be a nonlinear operator. a T is nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ H. b T is firmly nonexpansive if 2T − I is n...

and Applied Analysis 3 In 2007, Marino and Xu 11 proved the following strong convergence theorem by using the hybrid projection method for a strict pseudocontraction. They defined a sequence as follows: x0 ∈ C chosen arbitrarily, yn αnxn 1 − αn Txn, Cn { z ∈ C : ∥∥yn − z ∥ ∥ 2 ‖xn − z‖ 1 − αn k − αn ‖xn − Txn‖ } , Qn {z ∈ C : 〈xn − z, x0 − xn〉 0}, xn 1 PCn∩Qn x0 , n 0, 1, 2, . . . , 1.5 They pr...

and Applied Analysis 3 Let {xn} be a bounded sequence in a CAT 0 space X. For x ∈ X, one sets r x, {xn} lim sup n→∞ d x, xn . 2.5 The asymptotic radius r {xn} of {xn} is given by r {xn} inf x∈X {r x, {xn} }, 2.6 the asymptotic radius rC {xn} of {xn}with respect to C ⊂ X is given by rC {xn} inf x∈C {r x, {xn} }, 2.7 the asymptotic center A {xn} of {xn} is the set A {xn} {x ∈ X : r x, {xn} r {xn}...

and Applied Analysis 3 for all x, y ∈ C. A mapping A : C → H is said to be α-inverse strongly g-monotone if and only if 〈 Ax −Ay, g x − gy ≥ α∥Ax −Ay∥2, 2.2 for some α > 0 and for all x, y ∈ C. A mapping g : C → C is said to be strongly monotone if there exists a constant γ > 0 such that 〈 g x − gy, x − y ≥ γ∥x − y∥2, 2.3 for all x, y ∈ C. Let B be a mapping ofH into 2 . The effective domain of...

and Applied Analysis 3 where {an}, {bn}, {αn} are appropriate sequences in 0, 1 . The theory of three-step iterative scheme is very rich, and this scheme, in the context of one or more mappings, has been extensively studied e.g., see Khan et al. 6 , Plubtieng and Wangkeeree 7 , Fukhar-ud-din and Khan 5 , Petrot 13 , and Suantai 14 . It has been shown in 15 that three-step method performs better...