SOME FIXED POINTS OF EXPANSION MAPPINGS

Wang et al. [11] proved some fixed point theorems on expansion mappings, which correspond some contractive mappings. Recently, several authors generalized their results by some war’s. In this paper, we give some fixed point theorems for expansion mappings, which improve the results of some authors.

2. THE MAIN THEOREMS.Throughout this paper, following Boyd and Wong [1], let " be the family of mappings such that for each E .T:, [0, o0) ---, [0, o0) is upper semi-continuous from the right and non-decreasing in each coordinate variable with (t) < for all > 0.
We also need the following Lemma due to Matkoski [6] in the proof of our main theorems.
THEOREM 2.1.Let S and T be rnappznga from a metric apace (X,d) into itaelf such that for each x, y in X, at leaat one of the following condtiona holds: (d(S2x, TSy)) >_ d(Sx, Sy).Then either S or T has a fixed point, or S and T have a common fixed point. (2.) (2.2) PROOF.Let x0 be an arbitrary point X.Since S(X) C_ S(X) and S(X) c_ TS(X), we have for 'x0 E X, there exists a point xl in X such that S2xi Sxo yo, say, and for this point xl, there exists a point x2 in X such that TSx2 Sx yl, say.Inductively, we can define a sequence {y,,} in S(X.) such that S2x,+ Sx2, yn and TSxn+2 It is easy to show that, for each of the inequalities (2.1).--(2.4),that we have (d(y,,, y,,+l)) > d(y,+, y,+2).Then one can show that (d(y,,+l, y,,+)) >_ d(y,+2, y,,+a), hence for arbitrary (d(yn, Yn+l)) d(yn+l,Yn+2).Now, if y, y2,+ for any n, one has that y, is a fixed point of S from the definition {y.}.It then follows that, also, Y,+a Y2+, which implies that {y,} is also a fixed point of T.
By Lemma, lim d(y,,, y,,,+a O. Now, using the technique of Kang [4], one would prove that {y,,} a Cauchy sequence and it converges to some point y in S(X).Consequently, the subsequences {y2,,}, {Y,,+I} and {y,,+2} converge to y.Let y Su and y TSv for some u and v in X., respectively.From inequalities (2.1),-(2.4), it follows that at least one of the following inequalities must be true for an infinite number of values of n: ,,,y)) >_ g[d(Sx,,+,,Sx2,,+,) + d(TSv, Sv) + d(S=:,,+,,Svl] Taking the limit as n c in each case yields y St,.A similar argument applies to proving that y Su.Therefore, y is a common fixed point of S and T. This completes the proof.THEOREM 2.2.Let S and T be continuous mappings from a metric space (X,d) into ttself such that S(X) C_ S2(X), S(X) C_ TS(X) and S(X.) is complete.Suppose that there exists such that (d(S2x, TSy)) > min{d(Sx, S2x),d(TSy, Sy),d(Sx, Sy)} (2.5) for all x, y in X.
Then S or T has a fixed point or S and T have a common fixed point.
PROOF.Define a sequence {y,} as in Theorem 2.1.If y, Y,,+I for any n, then S or T has a fixed point.
Hence, it follows that {y,} is a Cauchy sequence and it converges to some point y in S(X).
Thus, S and T have a common fixed point.
COROLLARY 2.3.(1) Let S and T be mappings from a metric space (X, d) into itself such that S(X) C_ S2(X), S(X) C_ TS(X) and S(X) is complete.Suppose that there exists real numbers h > 1 such that for each x, y in X, at least one of the following conditions holds: d(Sx, TSy) >_ hd(Sx, Sy). for dll x, y in X.
Then S or T has a fixed point or S and T have a common fixed point.
Then S and T have a common fixed point.
Therefbre, S and T have a common fixed point.

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 (d(Sx, TSy)) >_ [d(Sx, Sy) + d(TSy, Sy)]. ((Sz, TS)) > [d(Sz, S) + d(S,S)].(d(S2x, TSy)) >_ [[d(Sx, S2x) + d(TSy, Sy) + d(Sx, Sy)].

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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