THEOREMS FOR NON-SELF MAPS IN d-COMPLETE TOPOLOGICAL SPACES

Fixed point theorems are given for non-self maps and pairs of non-self maps defined on d-complete topological spaces.

THEOREM 2. Let X 1)c a d-conpletc Hausdorff topological space, C be a closed subset of X, T" C X with C C T(C).Suppose therc exists k'[0, o) [0, c) such that k(d(Wx, Wy)) _> d(x, y) for all x, y G C, k is non-decreasing, k(0) 0, and there exists x 0 C such that y kn(d(Tx0 x0) < cxz.If G(x)= d(Tx, x)is lower semi-continuous on C then T has a fixed point.
Thus Tp p.
In [5], Hicks gives several examples of functions k which satisfy the condition of theorem of that paper.These examples, with a slight modification, carry over to the non-self map case.The non-self map version of Example is givcn for completeness.The other examples carry over in a similar mazmer.

n=l n=l
Applying Theorem 1 we get a fixed point for T. (Note" d(x, y) _< A d(Tx, Ty) for 0 < , < 1 is equivalent to d(Tx, Ty) > a d(x, y) for o > 1.) The following examples show that conditions (2.2), (2.3) and (2.4) do not guarantee fixed points.
EXAMPLE 2. Let R denote the real numbers and CB(IR, I) denote the collection of all bounded and continuous functions which map 1 into 1.Let C {f CB(I, IR)-f(t) 0 for all < 0 and lira f(t) > 1}.
The following theorems were motivated by the work of Hicks and Rhoades [3].
THEOREM 3. Let C be a compact subset of a Hausdorff topological space (X, t) and d X x X [0, co) such that d(x, y) 0 if and only if x y.Suppose T C X with C C T(C), T and G(x)= d(x, Tx) are both continuous, and d(Tx, T2x)> d(x, Tx) for all x T-I(C) with x Tx.Then T has a fixed point in C.
PROOF.C is a compact subset of a Hausdorff space so it is closed.T is continuous so T-I(C) is closed and hence is compact since T-I(C)C C. G(x) is continuous so it attains its minimum on T-I(C), say at z. Now z C C T(C) so there exists y T-I(C) such that Ty z.If y z then d(z, Tz) d(Ty, T2y) > d(y, Ty), a contradiction.Thus y z Ty is a fixed point of T. THEOREM 4. Let C be a compact subset of a Hausdorff topological space (X, t) and d X X [0, co) such that d(x, y) 0 if and only if x y.Suppose T C X with C C T(C), T and G(x)= d(x, Tx) are both continuous, f: [0, co) [0, co) is continuous and f(t)> 0 for 0. If we know that d(Tx, W2x) _< f(d(x, Tx)) for all x T-(C) implies W has a fixed point where 0 < < 1, then d(Wx, T2x)< f(d(x, Wx)) for all x T-I(C) such that f(d(x, Tx)) 0 gives a fixed point.
PROOF.C is a compact subset of a Hausdorff space so it is closed.T is continuous gives that T-(C) is closed, and T-I(c)C C so T-(C) is compact.Suppose x Tx for all xW-(C).Then d(x, Wx)>0 so that f(d(x, Wx))>0 for all xW-l(c).Define P(x) on d(Tx T2x) T-I(C) by P(x)= f(d(x, Tx))" P is continuous since T, f and G(x) are continuous.Therefore P attains its maximum on T-I(C), say at z. P(x) < P(z) < so d(Wx, W2x) _< P(z)f(d(x, Ix)) and T must have a fixed point.THEOREM 5. Let C be a compact subset of a Hausdorff topological space (X, t) and d X x X [0, co) such that d(x, y) 0 if and only if x y.Suppose T C X with C C T(C), W and G(x)---d(x, Ix) are both continuous, f: [0, co) [0, co) is continuous and f(t)> 0 for 0. If we know that d(Wx, W2x) _> A f(d(x, Tx)) for all x T-(C) implies W has a fixed point wher A > 1, then d(Wx, T2x) > f(d(x, Tx)) for all x T-I(C) such that f(d(x, Tx)) 0 gives a fixed point.
C()R()LLARY 3. L('t A and B bc continuous mappings from C, a closed subset of X, into X sat,sfying C C A(C), C C B(C),tim d(Ax, By) h lnin{d(Ax, x), d(By, y), d(x, y)} for 11 x, yC with x#y where h> 1. Tlwn A or B has afixcd l)oint or A and B have a common fixed point.
Boyd and Wong [10] ('all the collection of all real functions :+ + vhich satisfy the follmvng onditaons : (C4) is ut)I)er-semicontinuous and non-decreasing, (C5) (t)<t for eacht>0.THEOREM 8. Let (X, t, d) be a d-complete symmetric Hausdorff topological space.If A and B are continuous mappings from C, a closed subset of X, into X such that C C A(C), C C B(C), and (d(Ax, By)) >_ ,nin{d(Ax, x), d(By, y), d(x, y)} for all x, y G C where , e and cn 0 l')n(t) < oo for each > 0, then either A or B has a fixed point or A and B have a comnon 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

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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