REMARKS ON CERTAIN SELECTED FIXED POINT THEOREMS

Theorem 1. Let A, S, I, and J be self-mappings of a complete metric space (X,d) such that the pairs (A,I) and (S,J) are commuting and A(X)⊂ J(X) and S(X)⊂ I(X) such [ 1+pd(Ax,Sy)]d(Ix,Jy) ≤ pmax{d(Ix,Ax)·d(Sy,Jy),d(Ix,Sy)·d(Jy,Ax)} +φ(d(Ax,Sy),d(Ix,Ax),d(Sy,Jy),d(Ix,Sy),d(Jy,Ax)), (1) for all x,y ∈ X where p ≥ 0 and φ ∈ Ψ . Then A, S, I, and J have a unique common fixed point provided one of these four functions is continuous.

Let R + denote the set of nonnegative reals and let ψ be the family of mappings φ from (R + ) 5 into R + such that (i) φ is nondecreasing, (ii) φ is upper semi-continuous in each coordinate variable, (iii) γ(t) = φ(t, t, a 1 t, a 2 t, t) < t, where γ : R + → R + is a mapping with γ(0) = 0 and a 1 + a 2 = 2. Theorem 3.2 of Lal et al. [11] for commuting mappings can be stated as follows.
for all x, y ∈ X where p ≥ 0 and φ ∈ Ψ . Then A, S, I, and J have a unique common fixed point provided one of these four functions is continuous. Remark 2. Theorem 1 was originally proved for "weakly compatible mappings of type (A)" (cf. [11]) but for a more natural setting we have adopted it for commuting mappings.
In this paper, as an application of Theorem 1, we derive a common fixed point theorem for six mappings which runs as follows.
for all x, y ∈ X where p ≥ 0 and φ ∈ Ψ . Then A, B, S, T , I, and J have a unique common fixed point provided one of these four mappings AB, ST , I, and J is continuous.

Proof.
We begin by observing that continuity of AB (resp., ST ) does not demand the continuities of the component maps A or B or both (resp., S or T or both). Since the pairs ( which shows that Az and Bz are other fixed points of the pair (AB, I) yielding thereby   [14], Husain and Sehgal [5], Khan and Imdad [10], Jungck [6],Ćirić [1], S. L. Singh and S. P. Singh [13], Fisher [3,4], Das and Naik [2], Kannan [9], Rhoades [12], and several others. Also setting p = 0 and choosing A, B, S, T , I, J, and φ suitably one can deduce the results proved in the above cited references and many others.
Next we wish to indicate a similar result in compact metric spaces. For this purpose one can adopt a general fixed point theorem for commuting mappings in compact metric spaces due to Jungck [8], which was originally proved for compatible mappings (a notion due to Jungck [7]).

Theorem 5 (see [8]). Let A, S, I, and J be self-mappings of a compact metric space (X, d) with A(X) ⊂ J(X) and S(X) ⊂ I(X). If the pairs (A, I) and (S, J) are commuting and d(Ax,Sy) < M(x,y),
with M(x, y) > 0, then A, S, I, and J have a unique common fixed point provided all four mappings A, S, I, and J are continuous.
As an application of Theorem 5 one can derive the following theorem in compact metric spaces involving six mappings. Proof. The proof is essentially the same as that of Theorem 3, hence we omit the proof.

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