A MINIMIZATION THEOREM IN QUASI-METRIC SPACES AND ITS APPLICATIONS

We prove a new minimization theorem in quasi-metric spaces, which improves the results of Takahashi (1993). Further, this theorem 
is used to generalize Caristi's fixed point theorem and Ekeland's 
ϵ-variational principle.

Definition 2.4.A sequence {x n } in X is said to be a left k-Cauchy sequence if for each k ∈ N there is an N k such that A quasi-metric space is left k-sequentially complete if each left k-Cauchy sequence is convergent.
Theorem 2.5.Let (X, d) be left k-sequentially complete quasi metric space such that for each x ∈ X the mapping u → d(x, u) is a lower semicontinuous on X.Let f : X → (−∞, ∞] be a proper weak lower semicontinuous function bounded from below such that for any u ∈ X with inf x∈X f (x) < f (u), there exists v ∈ X with v ≠ u and f (v)+ d(u, v) ≤ f (u).Then there exists x 0 ∈ X such that inf x∈X f (x) = f (x 0 ).
Proof.Suppose that inf x∈X f (x) < f (y) for every y ∈ X.For each y ∈ X, we define S(y) by 3) and hypotheses of the theorem we have the following: ( * ) For each y ∈ X, there exists v ∈ X with v ≠ y such that v ∈ S(y), and for each z ∈ S(y), S(z) ⊆ S(y).
For each y ∈ X, we define A(y) by Choose u ∈ X with f (u) < ∞.Then we choose a sequence {u n } in S(u) as follows: when u = u 1 ,u 2 ,...,u n have been chosen, choose u n+1 ∈ S(u n ) such that (2.5) Thus, we obtain a sequence {u n } such that (2.7) By (2.6), {f (u n )} is a nonincreasing sequence of reals and so it converges.Therefore, by (2.7) there is some α in R such that Let k ∈ N be arbitrary.By (2.8) there exists some N k such that f (u n ) < α+ 1/k for all n ≥ N k .Thus, by monotony of {f (u n )}, for m ≥ n ≥ N k , we have and hence (2.10) From the triangle inequality, (2.6) and (2.10), we get for all m > n ≥ N k .Therefore, {u n } is a left k-Cauchy sequence in X.By completeness, there exists z ∈ X such that u n → z.Since f is a weak lower semicontinuous; by (2.8), we have (2.12) From (2.11), we obtain Since f is a weak lower semicontinuous on X and u → d(x, u) on X is a lower semicontinuous, we have From (2.3) and (2.15), we obtain that z ∈ S(u n ) for every n ∈ N and hence (2.16) Taking the limit when n tends to infinity, we have (2.17) From (2.8), (2.12), and (2.17), we have Taking the limit when n tends to infinity, we get This is in contradiction with f (v 1 ) < α.Hence S(z) = {z}.But, by (2.3) and hypothesis of a function f in theorem there exists y ∈ X such that y ≠ z and {y,z} ⊆ S(z).This is a contradiction.This completes the proof.
Theorem 2.7.Let (X, d) be left k-sequentially complete quasi-metric space such that for each x ∈ X, the mapping u → d(x, u) is a lower semicontinuous on X.Let f : X → (−∞, ∞] be a proper weak lower semicontinuous function bounded from below.Assume that there exists a selfmapping T of X such that (2.21) Then T has a fixed point in X.
Then, since f is weak lower semicontinuous, Z is closed.So Z is left k-sequentially complete.Let x ∈ Z.Then, Since So Z is invariant under T .Assume that T x ≠ x for every x ∈ Z. Then by Theorem 2.5, there exists This is a contradiction.Therefore T has a fixed point u in Z.This completes the proof.
Theorem 2.9.Let (X, d) be left k-sequentially complete quasi-metric space such that for each x ∈ X the mapping u → d(x, u) is a lower semicontinuous on X.Let f : X → (−∞, ∞] be a proper weak lower semicontinuous function bounded from below.Then, (1) Then Y is nonempty and complete.We prove that there exists v ∈ Y such that f (w) > f (v)−d(v, w) for every w ∈ X with w ≠ v.If not, for every x ∈ Y , there exists w ∈ X such that w ≠ x and f (w)+d(x, w) ≤ f (x).Since f (w) ≤ f (x) ≤ f (u), w ∈ X is an element of Y .By Theorem 2.5, there exists x 0 ∈ Y such that f (x 0 ) = inf x∈Y f (x).For this x 0 ∈ Y , there exists x 1 ∈ Y such that x 0 ≠ x 1 and f (x 1 ) + d(x 0 ,x 1 ) ≤ f (x 0 ).Thus we have f (x 0 ) = f (x 1 ) and d(x 0 ,x 1 ) = 0. Hence x 0 = x 1 .This is a contradiction.Therefore (1) holds. ( Then Z is nonempty and complete.Since εd(u, x) is a quasi metric, as in the proof of (1), we have that there exists v ∈ Z such that f (w This completes the proof of (2).

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