A new approximation algorithm for fixed points of nonexpansive mappings

Abstract The aim of this paper is to establish a new approximation algorithm for fixed points of nonexpansive mappings in general Banach spaces and to illustrate some numerical results. The approximation algorithm we shall discuss is xt,n = (tT) n x0, where x0 ∈ D(T) is arbitrary, n is a natural number, and t ∈ (0,1). We shall also provide some numerical error estimates.


Introduction and Preliminaries
From the applications point of view the construction of fixed points for nonexpansive mappings is an important topic in the theory of nonlinear analysis. For nonexpansive mappings classical fixed point theorems have been established in [1,4,10], whereas Mann [5], Ishikawa [3], and Halpern [2] have analyzed some iterative schemes. Various improvements and extensions of the iterative schemes presented in [2,3,5] have been offered in recent years in [6,7,8,9,11,13]. However, although these improvements and extensions advance the theory, they have only limited applicability in applied problems. The purpose of this paper is to study a new approximation algorithm of fixed points for nonexpansive mappings in general Banach spaces and to illustrate its applications in numerical computation. Our approximation scheme is defined by x t,n = (tT ) n x 0 , where x 0 ∈ D(T ) is arbitrary, n is a natural number, and t ∈ (0, 1).
In order to study our new approximation algorithm, the following notation will be useful. A mapping T : K → K, in a Banach space, is said to be almost invariant, if the range R(T ) is bounded and there exists some constant t 0 ∈ (0, 1) such that tR(T ) ⊂ K for all t ∈ [t 0 , 1]. Example 1. Let E be a Banach space and T be a retraction from K = {x : Then for all α ∈ [1/2, 1], we have αR(T ) ⊂ K, and hence T is almost invariant.
Example 2. Let E be a Banach space, S(E) denote the unit ball of E, and let T : S(E) → S(E) be a mapping, then T is almost invariant.

New approximation algorithm
Let E be a Banach space, K be a nonempty closed subset of E and T : K → K be an almost invariant nonexpansive mapping. For an arbitrary x 0 ∈ K we define the sequence {x t,n } by where t ∈ (0, 1) and n is a natural number.
Recall that a Banach space E is said to satisfy Opial's condition, if whenever {x n } is a sequence in E which converges weakly to x, then We also recall that a mapping T in a Banach space is said to be demi-closed at zero if for any sequence {x n } which converges weakly to x * and {T x n } converges strongly to zero, T x * = 0. We shall need the following lemma. Lemma 1. [12] Let E be a real reflexive Banach space which satisfy Opial's condition. Let K be a nonempty closed convex subset of E, and T : K → K be a continuous pseudocontractive mapping. Then (I − T ) is demi-closed at zero.
Since any nonexpansive mapping is continuous pseudocontractive mapping, the above demi-closed principle also holds for nonexpansive mappings.

Error Estimation for Approximate Fixed Points
Let E be a real Banach space, K be a closed subset of E, T : K → K be an almost invariant nonexpansive mapping and x t,n be the path defined by (1.1). Assume that t → 1, then there exists some neighbourhood [t 0 , 1) such that, for any t ∈ [t 0 , 1), the contraction tT : K → K has a unique fixed point y t ∈ K, i.e., y t = tT y t . Further, for any given t ∈ [t 0 , 1), by the Banach fixed point theorem, the approximation sequence {x t,n } defined by (1.1) converges strongly to the y t , as n → ∞ (the Picard iterative process). Now from the error estimation formula for the Picard iterative process, we have In addition, from y t = tT y t , we find Combining (2.1) and (2.2), we obtain Hence, we have established an approximate fixed point path {x t,n } = (tT ) n x 0 and the error estimation (2.3). For any given acceptable error ε > 0, in the inequality (2.3) we take t ∈ (0, 1) and a natural number n such that x t,n − T x t,n ≤ ε, where x t,n = (tT ) n x 0 . Clearly, to reduce the number of iterations, we must take the least natural numbers n so that the above inequality is satisfied. For this, we consider the following nonlinear optimal problem min n = n(t), x t,n = (tT ) n x 0 , x t,n − T x t,n ≤ ε, t ∈ (0, 1).
Next, in the inequality (2.3), we let so it can be written as Now for any given acceptable error ε > 0, we let which is equivalent to Thus, it follows that Next, without any loss of generality, we can assume that so that n > 0. Clearly, for any given ε > 0, if we take t ∈ (0, 1) and a nature number n such that (2.4) holds, then x t,n − T x t,n ≤ ε, where x t,n = (tT ) n x 0 .
Thus, we can take x t,n = (0.573433T ) 7 x 0 , as an approximate fixed point of T . Clearly, it follows that x t,n − T x t,n ≤ 0.5.
We compute the functional values of f 0.001,1,1,1 (t) for some 0.999 < t < 1 in the following: Thus, we can take x t,n = (0.999053156T ) 17741 x 0 as an approximate fixed point of T . Clearly, it follows that x t,n − T x t,n ≤ 0.001.

Some Approximation Fixed Point Sequences and Convergence
In particular, let t = α n → 1, as n → ∞, be a real sequence in (0, 1) such that lim n→∞ α n n 1−αn = 0, then the approximation algorithm (1.1) takes the following special form x n = (α n T ) n x 0 . Example 3. Let α n = √ n/( √ n + 1). Then, it is easy to show that as n → ∞. In this case, the approximation algorithm (1.1) simply takes the following form Theorem 1. Let E be a real reflexive Banach space which satisfies Opial's condition, K be a closed convex subset of E, and T : K → K be an almost invariant nonexpansive mapping. Then the fixed points set F (T ) is nonempty. Moreover let lim n→∞ α n n 1−αn = 0. Then for any initial x 0 ∈ K, w w (x n ) ⊂ F (T ), where w w (x n ) denote the weak limit set of {x n } defined by (3.1).
Proof. From (3.2) we have x n − T x n → 0 as n → ∞. This together with the reflexivity of Banach space E, and Lemma 1, implies that, for any subsequence {x n k } of {x n }, there must exist a subsequence {x n k i } of {x n k } which converges weakly to a fixed point x * of T . This completes the proof. ⊓ ⊔ Example 4. From Example 3, it follows that the sequence {x n } defined by is also an approximation fixed point sequence. The numerical error estimates of (3.3) and (3.4) can be computed as in Section 2.
Theorem 2. Let E be a real reflexive Banach space which satisfy Opial's condition, K be a closed convex subset of E, and T : K → K be an almost invariant nonexpansive mapping. Then the fixed points set F (T ) is nonempty and for any initial x 0 ∈ K, w w (x n ) ⊂ F (T ), where w w (x n ) denote the weak limit set of {x n } defined by (3.4).