SOME RESULTS ON (δ-PRE, s)-CONTINUOUS FUNCTIONS

One of the important and basic topics in general topology and several branches of mathematics which have been researched by many authors is the continuity of functions. In this paper, we study (δ-pre, s)-continuous functions as a new weaker form of continuity. In 1996, Dontchev [3] introduced contra-continuous functions, and Jafari and Noiri [10] introduced contra-precontinuous functions (in 2002). Ekici [8] studied the notion of almost contra-precontinuous functions. Recently, Ekici [7] introduced and studied the notion of (δ-pre, s)-continuous functions as a new weaker form of almost contraprecontinuous functions. The aim of this paper is to study some properties of (δ-pre, s)-continuous functions and modification of the results due to Ekici [7]. Basic characterizations concerning (δ-pre, s)-continuous functions are investigated and some results are obtained. Moreover, we obtain some properties in general cases concerning composition of functions under specific conditions, where the composition would yield a (δ-pre, s)-continuous function. Finally, if given a composition of functions, which are (δ-pre, s)-continuous, we obtain the first function in the composition, which will be (δ-pre, s)continuous.


Introduction
One of the important and basic topics in general topology and several branches of mathematics which have been researched by many authors is the continuity of functions. In this paper, we study (δ-pre, s)-continuous functions as a new weaker form of continuity. In 1996, Dontchev [3] introduced contra-continuous functions, and Jafari and Noiri [10] introduced contra-precontinuous functions (in 2002). Ekici [8] studied the notion of almost contra-precontinuous functions. Recently, Ekici [7] introduced and studied the notion of (δ-pre, s)-continuous functions as a new weaker form of almost contraprecontinuous functions. The aim of this paper is to study some properties of (δ-pre, s)-continuous functions and modification of the results due to Ekici [7]. Basic characterizations concerning (δ-pre, s)-continuous functions are investigated and some results are obtained. Moreover, we obtain some properties in general cases concerning composition of functions under specific conditions, where the composition would yield a (δ-pre, s)-continuous function. Finally, if given a composition of functions, which are (δ-pre, s)-continuous, we obtain the first function in the composition, which will be (δ-pre, s)continuous.

Preliminaries
Throughout this paper, all spaces X and Y (or (X,τ) and (Y ,υ)) are always mean topological spaces. Let A be a subset of a space X. For a subset A of (X,τ), Cl(A) and Int(A) represent the closure and interior of A with respect to τ, respectively.
A subset A of a space X is said to be regular open (resp., regular closed) if A = Int(Cl(A)) (resp., A = Cl(Int(A))). The family of all regular open (resp., regular closed) sets of X is denoted by RO(X) (resp., RC(X)). We put RO(X,x) = {U ∈ RO(X) : x ∈ U} and RC(X,x) = {F ∈ RC(X) : x ∈ F}.
The δ-interior [17] of a subset A of X is the union of all regular open sets of X contained in A and is denoted by δ − Int(A).

Some results
In this section, the modification of results due to Ekici [7] is investigated. Basic characterizations and some properties of (δ-pre, s)-continuous functions are also investigated.
Proof. For each α ∈ Δ, since A α is δ-preopen in X, we have A α ⊆ Int(δ − Cl(A α )). Then The following theorem is obtained by modification and extending the results from [7, Theorem 1].
Theorem 3.2. The following are equivalent for a function f : X → Y : (1) f is (δ-pre, s)-continuous; Proof. (1)⇔(2): let F be any semiclosed set in Y not containing f (x). Then Y \ F is a semiopen set in Y containing f (x). By (1), there exists a δ-preopen set U in X containing The converse can be shown similarly.
, then This shows that f is (δ-pre, s)-continuous.
The converse can be obtained similarly.
This shows that f is (δ-pre, s)-continuous.
Conversely, let x ∈ X and let F be a closed subset of Y containing f (x). Since Cl(Int(F)) is regular closed, then it is semiopen in Y containing f (x). By (1), there exists a δ-preopen set U x in X containing x such that (3.7) Hence, x ∈ U x ⊆ f −1 (Cl(Int(F))) and f −1 (Cl(Int(F))) = x∈ f −1 (Cl(Int(F))) U x . This shows that f −1 (Cl(Int(F))) ∈ δPO(X) by Lemma 3.1. The following example shows that (δ-pre, s)-continuous function does not imply almost contraprecontinuous function. Recall that for a function f : X → Y , the subset {(x, f (x)) : x ∈ X} ⊆ X × Y is called the graph of f . The following theorems are obtained in [7] and proved by using [7, Theorem 1(3)]. We prove here by using different technique, that is, by using Theorem 3.2(5) in this paper.
Theorem 3.5. Let f : X → Y be a function and let g : Thus, f is (δ-pre, s)-continuous by Theorem 3.2.

Theorem 3.8. If f : X → Y is a (δ-pre, s)-continuous function and A is any δ-open subset of X, then the restriction f |
by Lemma 3.1. This gives that f is a (δ-pre, s)-continuous function.
, it follows from Lemma 3.7 that V ∈ δPO(X,x). Since x ∈ X is arbitrary, this shows that f is (δ-pre, s)-continuous function.
Definition 3.11. A function f : X → Y is said to be (1) θ-irresolute [11] if for each x ∈ X and each V ∈ SO(Y , f (x)), there exists U ∈ SO(X,x) such that f (Cl(U)) ⊆ Cl(V ), (2) δ-preirresolute [7] if for each x ∈ X and each V ∈ δPO(Y , f (x)), there exists a δ-preopen set U in X containing x such that f (U) ⊆ V .
In [7, Theorem 10], Ekici has proved that composition of two functions with specific condition would yield the (δ-pre, s)-continuous function. For the composition of three functions, we have the following results. (1) If f and g are δ-preirresolute, and h is (δ -pre, s) f is (δ-pre, s)-continuous, and g and h are θ- f is δ-preirresolute, g is (δ-pre, s)-continuous, and h is θ- Next, we obtained Corollaries 3.13 and 3.14 as general cases, obvious from [7, Theorem 10] and Propositions 3.12(1) and 3.12(2), by repeating application of δ-preirresolute and θ-irresolute functions, respectively.
Observe that, in Corollary 3.13, the (δ-pre, s)-continuous function lies at the beginning of the composition function, while in Corollary 3.14, the (δ-pre, s)-continuous function lies at the end. How about, if the (δ-pre, s)-continuous function lies inside of the composition function? We have the following results. (1) If f and g are δ-preirresolute, h is (δ-pre, s)-continuous, and p is θ-irresolute, then Since h is (δ-pre, s)-continuous, there exists a δ-preopen set N in Z containing (g • f )(x) such that h(N) ⊆ Cl(F). Since g is δ-preirresolute, there exists a δ-preopen set M in Y containing f (x) such that g(M) ⊆ N. Since f is δ-preirresolute, there exists a δ-preopen set U in X containing x such that Clearly, from Propositions 3.12(3) and 3.15, we obtain the following corollary.
In [7,Theorem 11], Ekici has also proved that, given a composition of two functions with specific conditions where the (δ-pre, s)-continuous function would be yield, the first function in the composition is (δ-pre, s)-continuous. For the composition of three functions, we give the following proposition. Proof. Suppose that x, y, and z are three points in X, Y , and Z, respectively, such that f (x) = y and g(y Since g is also δ-preopen, g( f (U)) is a δ-preopen set in Z containing z such that h(g( f (U))) ⊆ Cl(V ). This implies that h is (δ-pre, s)-continuous.
As in [7, Corollary 1], we obtained the following corollary.  Proof. It can be obtained from Propositions 3.12(1) and 3.18.
The following corollaries are considered as general cases obtained from the above discussions.
The proof of Corollary 3.20 is obvious from [7,Theorem 11] and Proposition 3.18.
The proof of Corollary 3.21 can be obtained from Corollaries 3.13 and 3.20.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009