LIMINF AND LIMSUP CONTRACTIONS

We give some theorems related to the contraction mapping principle of BanachCaccioppoli and Edelstein. The contractive conditions we consider involve the quantities liminfξ→·d(ξ,fξ) and limsupξ→·d(ξ,fξ) instead ofd(·,f ·). Some examples are provided to show the difference between our results and the classical ones. 2000 Mathematics Subject Classification. 54H25, 47H10.


Introduction.
One of the simplest and most useful results in the fixed point theory is the Banach-Caccioppoli contraction mapping principle (see [1,2]), which in the general setting of complete metric spaces reads as follows.
Theorem 1.1.Let (X, d) be a complete metric space, f : X → X a mapping and c ∈ [0, 1[ such that d(f x, f y) ≤ cd(x, y), ∀x, y ∈ X; (1.1) then (i) there exists a point a ∈ X such that for each x ∈ X, lim n→+∞ f n x = a; (ii) a is the unique fixed point for f ; (iii) for each x ∈ X, d(f n x, a) ≤ c n /(1 − c)d(x, f x).
In 1962, in the case of compact metric spaces, Edelstein in [3] has proved the following generalization of the contraction mapping principle.Theorem 1.2.Let (X, d) be a compact metric space and f : X → X be a mapping such that d(f x, f y) < d(x, y), ∀x, y ∈ X, x ≠ y; (1.2) then there exists a unique fixed point for f .
Before stating our theorems we need some notations and definitions: by Z, Z + , R, and R + we denote, respectively, the sets of integers, nonnegative integers, real numbers and nonnegative real numbers; let now X be the cluster set of X and ϕ : X → R + be a real-valued mapping, then ϕ is called (weak) lower semicontinuous at x ∈ X if and only if if this happens for all x ∈ X then we simply say that ϕ is a (weak) lower semicontinuous mapping.Finally, taking a mapping f : X → X then ϕ is said to be f -orbitally (weak) lower semicontinuous at a ∈ X if and only if for each x ∈ X and for each sequence 2. Main results.We are now ready to state and prove our results; the first two are related to Theorem 1.1, while the third one is related to Theorem 1.2.
Theorem 2.1.Let (X, d) be a complete metric space such that X ≠ ∅ and let f : X → X be a mapping such that f (X ) ⊆ X .Suppose that there exists a point x ∈ X such that and that the mapping ϕ(•) then for all x ∈ X satisfying (2.1), (i) there exists a point a ∈ X such that f n x → a as n → +∞; Proof.Let x ∈ X be such that (2.1) holds and consider the sequence (f n x) n∈Z + , thus for n ∈ Z + we have lim inf further, by the lower semicontinuity of ϕ, one has this implies that (f n x) n∈Z + is a Cauchy sequence so that, for the completeness of (X, d), there exists lim n→+∞ f n x = a ∈ X so that (i) is proved.Further, again by the lower semicontinuity of ϕ, one has thus f a = a and (ii) is proved.Finally, to see the validity of (iii), we note that so that, letting m → +∞, one has which proves item (iii).
Theorem 2.2.Let (X, d) be a complete metric space such that X ≠ ∅ and let f : X → X be a mapping such that f (X ) ⊆ X .Suppose that there exists a point x ∈ X such that and that the mapping ϕ( then for all x ∈ X satisfying (2.9) (i) there exists a point a ∈ X such that f n x → a as n → +∞; (ii) f a = a if and only if ϕ is f -orbitally weak lower semicontinuous at a; Proof.We start with a point x ∈ X such that (2.9) holds and consider the sequence (f n x) n∈Z + , thus as in the previous proof we have lim sup (2.11) using the weak lower semicontinuity of ϕ one now has hence (f n x) n∈Z + is a Cauchy sequence in the complete metric space (X, d), so that there exists a = lim n→+∞ f n x, thus (i) is proved.Now let a = f a, then which is true for each subsequence (f n k y) k∈Z + of (f n y) n∈Z + converging to a as k → +∞, that is, ϕ is f -orbitally weak lower semicontinuous at a. Conversely, suppose the f -orbitally weak lower semicontinuity of ϕ at a, then that is, f a = a, which guarantees (ii).Finally, as in the proof of Theorem 2.1, we have and thus, as m → +∞, one has which proves item (iii).
Theorem 2.3.Let (X, d) be a compact metric space such that X ≠ ∅ and let f : and that the mapping ϕ(•) := d(•,f •) is lower semicontinuous; then f has a fixed point.
Proof.We define φ : X → [0, +∞] by thus we can observe that such φ is lower semicontinuous, in fact for each a ∈ (X ) one has Further, φ is defined on the compact set X , in fact it is a closed subset of the compact set X, thus φ has a minimum on X ; we call it a, that is,

.21)
We now claim that φ(a) = 0, in fact suppose by contradiction that this is false, then by the hypotheses we have f a ∈ X and Mappings satisfying (2.2), (2.10), and (2.18) will be called in the next section, respectively, liminf contractions, limsup contractions, and weak liminf contractions.

Some remarks and examples.
In this section, we give some remarks and examples concerning liminf, limsup, and weak liminf contractions which are not classical ones; we make use of the following notations: 1/2 ∞ := 0 (for ∞ we mean +∞), P := {2k | k ∈ Z}, and Remark 3.1.All the contractive conditions we have considered have a local character in the sense that they do not involve two generic points of the underlying space, but a single points and its orbit; under this aspects the results in this paper are related more to [4, Hicks-Rhoades theorem] rather than to Banach-Caccioppoli principle.
Remark 3.2.It is obvious that a Banach-Caccioppoli or an Edelstein contraction f is also, respectively, a liminf and limsup or weak liminf contraction if it satisfies the additional hypothesis f (X ) ⊆ X ≠ ∅, so that our results are sometimes generalizations of the classical ones; the opposite is not true as we will see in the sequel.
Example 3.3.We consider the metric space (X, d) where and for each This metric is equivalent to the Euclidean one in R 2 (restricted to X), and X is a closed subset of R 2 so that (X, d) is actually a complete metric space.Now consider the mapping f : X → X defined by (3.9) In short for each x ∈ X one has and, by (3.6) and (3.9), the mapping y d(y, f y) is lower semicontinuous, so that all the hypotheses of Theorem 2.1 are satisfied (in fact f has (0, 0) as fixed point), but f is not a contraction in the sense of Hicks and Rhoades (see [4]), in fact for The mapping of this example is actually both a liminf and a limsup contraction, but starting from it we give two other examples: in the first one (Example 3.4) the mapping we give is a liminf but not limsup contraction, while the opposite is true in Example 3.5 (all the details are left to the interested reader).
Example 3.4.Let (X, d) be as above, we consider the following mapping: (3.12) It is easy to see that f is a liminf but not limsup contraction, in fact for (3.13) Example 3.5.Let (X, d) be as in Examples 3.4 and 3.5 and let f be the following mapping: if m ∈ D, n ∈ P .(3.15) The final example shows that a weak liminf contraction may not be an Edelstein one (even in this case all the details are left to the interested reader).
Example 3.6.Let X be the following set: and let d : X 2 → R + be a mapping such that for each (x, y), (u, v) ∈ X one has with the definitions above it is easy to see that (X, d) becomes a compact metric space and that X = {(1/m, 0) | m ∈ (Z + − {0}) ∪ {∞}}.We define the mapping f : X → X in the following manner (in this example D and P denote, respectively, the set of positive

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) but this contradicts the minimality of a ∈ X , thus φ(a) = 0. Now for the lower semicontinuity of ϕ one has d(a, f a) ≤ lim inf ξ→a d(ξ, f ξ) = φ(a) = 0, (2.23)thus f a = a, and the theorem is proved.

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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