APPROXIMATING FIXED POINTS OF NONEXPANSIVE MAPPINGS

We consider a mapping S of the form S =α0I+α1T1+α2T2+···+αkTk, where αi ≥ 0, α0 > 0, α1 > 0 and ∑k i=0αi = 1. We show that the Picard iterates of S converge to a common fixed point of Ti (i = 1,2, . . . ,k) in a Banach space when Ti (i = 1,2, . . . ,k) are nonexpansive.


Introduction.
Let X be a Banach space and C a convex subset of X.A mapping T : C → C is said to be nonexpansive if T x − T y ≤ x − y for all x, y in C.
Specifically, the iterative process studied by Kirk is given by where α i ≥ 0, α 1 > 0 and k i=0 α i = 1.Kirk [1] has investigated an iterative process for approximating fixed points of nonexpansive mapping on convex subset of a uniformly convex Banach space.Recently, Maiti and Saha [2] improved the result of Kirk.Let T i : C → C (i = 1, 2,...,k) be nonexpansive mappings, and let where α i ≥ 0, α 0 > 0, α 0 > 0 and k i=0 α i = 1.In this paper, we show that the Picard iterates (1.2) of S converge to a common fixed point of T i (i = 1, 2,...,k) in a Banach space, without any assumption on convexity of Banach space.This result generalizes the corresponding result of Kirk [1], Maiti and Saha [2], Senter and Dotson [4].

Main results
Lemma 2.1.Let X be a normed space and {a n } and {b n } be two sequences in X satisfying Proof.Suppose that d > 0 and it follows from (ii) that n+m−1 i=n b i is bounded for all n and m.Let Choose a number N such that We can choose a positive ε such that By 0 < t < 1, there exists a natural k such that Since lim n→∞ a n = d, lim sup n→∞ b n ≤ d and ε independent of n, without loss of generality we may assume that, for all n ≥ 1, Setting s = 1 − t from (iii), we obtain by induction Then it is clear that x, y ∈ B and a k+1 = s k x + (1 − s k )y.Therefore, (2.8) Hence, we have (2.9) On the other hand, we have arriving at a contradiction.This completes the proof.
Lemma 2.2.Let C be a subset of a normed space X and T n : C → C be a nonexpansive mapping for all n = 1, 2,...,k.If for an arbitrary x 0 ∈ C and {x n } is defined by (1.2), then for all n ≥ 1 and p ∈ F(T ), where F(T ) denotes the common fixed point set of T i (i = 1, 2,...,k).
Proof.Since p = Sp for all p ∈ F(T ) and T i (i = 1, 2,...,k) is nonexpansive, we have 12) for all n ≥ 1 and all p ∈ F(T ).This completes the proof.
Theorem 2.3.Let C be a nonempty closed convex and bounded subset of a Banach space X and T i : C → C (i = 1, 2,...,k) be nonexpansive mappings.If for an arbitrary Proof.By (1.2) and T i is nonexpansive mapping, we have (2.13) Finally, we have (2.16) Then n j=1 b j is bounded.Setting t = 1 − α 0 , then a n+1 = (1 − t)a n + tb n and 0 < t < 1.It follows from Lemma 2.1 that d = 0, this completes the proof.
Recall that a Banach space X is said to satisfy Opial's condition [3] if the condition x n → x 0 weakly implies lim sup for all y ≠ x 0 .
Theorem 2.4.Let X be a Banach space which satisfies Opial's condition, C be weakly compact and convex, and let T i (i = 1, 2,...,k) and {x n } be as in Theorem 2.3.Then {x n } converges weakly to a fixed point of S.
Proof.Due to weak compactness of C, there exists {x n j } of {x n } which converges weakly to a p ∈ C. With standard proof we show that p = Sp.We suppose that {x n } does not converge weakly to p; then there are {x n l } and q ≠ p such that x n l → q weakly and q = Sq.By Theorem 2.3 and Opial's condition of X, we have a contradiction.This completes the proof.
Theorem 2.5.Let X, C, and {x n } be as in Theorem 2.3.Let T i : C → X (i = 1, 2,...,k) be nonexpansive mappings with a nonempty common fixed points set F(T ) in C. If T i satisfies condition A, then {x n } converges to a member of F(T ).We can thus choose a subsequence {x n j } of {x n } such that

Proof. By condition A, we have
for all integers j ≥ 1 and some sequence {p j } in F(T ).Again by Lemma 2.2, we have x n j +1 − p j ≤ x n j − p j < 2 −j , and hence Theorem 2.7.Let C be a closed convex subset of a Banach space X, and T i (i = 1, 2,...,k) be nonexpansive mappings from C into a compact subset of X.If {x n } is as in Theorem 2.3, then {x n } converges to a fixed point of S.
Proof.By Theorem 2.3 and the precompactness of S(C), we see that {x n } admits a strongly convergent subsequence {x n j } whose limit we denote by z.Then, again by Theorem 2.3, we have z = Sz; namely, z is a fixed point of S. Since x n − z is decreasing by Lemma 2.1, z is actually the strong limit of the sequence {x n } itself.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning .22) which show that {p j } is Cauchy and therefore converges strongly to a point p in F(T ) since F(T ) is closed.Now it is readily seen that {x n j } and hence {x n } itself, by Lemma 2.2, converges strongly to p.

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation