Let E be a real reflexive Banach space with a uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed convex subset of E, and every nonempty closed convex bounded subset of K has the fixed point property for non-expansive self-mappings. Let f : K → K a contractive mapping and T : K → K be a uniformly continuous pseudocontractive mapping with F T / ∅. Let {λn} ⊂ 0, 1/2 be a seq...

Throughout this paper, we always assume thatH is a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ‖ · ‖. The symbols → and ⇀ are denoted by strong convergence and weak convergence, respectively. ωw xn {x : ∃xni ⇀ x} denotes the weak w-limit set of {xn}. Let C be a nonempty closed and convex subset of H and T : C → C a mapping. In this paper, we denote the fixed point...

Abstract The aim of this work is to establish fixed point results in ordered Hilbert spaces for monotone operators with a pseudocontractive property. We state versions Theorem 12 [F. E. Browder, W. V. Petryshyn, Construction points nonlinear mappings space, J. Math. Anal. Appl. 20 (1967), 197–228] and 2.1 [Berinde, Vasile. Weak strong convergence theorems the Krasnoselskij iterative algorithm c...

and Applied Analysis 3 Lemma 1 (see [1, 2]). Let E be a Banach space and let J be the normalized duality mapping on E. Then for any x, y ∈ E, the following inequality holds: 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2⟨y, j (x + y)⟩, ∀j (x + y) ∈ J (x + y) . (14) Lemma 2 (see [20]). Let {s n } be a sequence of nonnegative real numbers satisfying s n+1 ≤ (1 − λ n ) s n + λ n δ n , ∀n ≥ 0, (15) where {λ n } and ...

Let D be an open subset of a real uniformly smooth Banach space E. Suppose T : D̄ → E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where D̄ denotes the closure ofD. Then, it is proved that (i) D̄ ⊆ (I + r(I −T)) for every r > 0; (ii) for a given y0 ∈ D, there exists a unique path t → yt ∈ D̄, t ∈ [0,1], satisfying yt := tT yt + (1− t)y0. Moreover, if F(T) = ∅ o...

and Applied Analysis 3 limn→∞tn 1, ∑∞ n 1 tn 1 − tn ∞, and limn→∞ kn − 1 / kn − tn 0, where ξn min{ 1 − α kn/ kn − α , 1/kn}. For an arbitrary z0 ∈ K let the sequence {zn} be iteratively defined by zn 1 ( 1 − tn kn ) f zn tn kn Tzn, n ∈ N. 1.7 Then i for each integer n ≥ 0, there is a unique xn ∈ K such that xn ( 1 − tn kn ) f xn tn kn Txn; 1.8

where E∗ denotes the dual space of E and 〈·, ·〉 denotes the generalized duality pairing. In the sequel, we denote a single-valued normalized duality mapping by j. Throughout this paper, we use F T to denote the set of fixed points of the mapping T . ⇀ and → denote weak and strong convergence, respectively. Let K be a nonempty subset of E. For a given sequence {xn} ⊂ K, let ωω xn denote the weak...

and Applied Analysis 3

Throughout this paper, we always assume thatH is a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ‖ · ‖. The symbols → and ⇀ are denoted by strong convergence and weak convergence, respectively. ωw xn {x : ∃xni ⇀ x} denotes the weak w-limit set of {xn}. Let C be a nonempty closed and convex subset of H and T : C → C a mapping. In this paper, we denote the fixed point...