and

Our main theorem establishes the uniqueness of the common fixed point of two set-valued mappings and of two single-valued mappings defined on a complete metric space, 
under a contractive condition and a weak commutativity concept. This improves a theorem 
of the second author.


6(A,C) + 6(C,B)
for all A B C in B(X).
We say that a subset A of X is the limit of a sequence {A of nonempty subn sets of X if each point a in A is the limit of a convergent sequence {a }, where to the bounded sets A and B respectively, then the sequence {6(An,Bn)} verge converges to (A,B).
This lemma was proved in [2].
Now let F be a mapping of X into B(X) We say that F is continuous at the point in X if whenever {x is a sequence of points in X converging to x n the sequence {Fx in B(X) converges to Fx in B(X).If F is continuous at n each point x in X, we say that F is a continuous mapping of X into B(X).A point z in X is said to be a fixed point of F if z is in Fz.
For a selfmap I of (X,d), the authors of [3], extending the results of [2] and [4], defined F and I to be weakly commuting on X if (Flx,IFx) !max{(Ix,Fx), diam IFx} (I.I) for all x in X.Two commuting mappings F and I clearly commute, but two weakly )] [0, x/(ax + ah+l)]--IFx but for any x in X we have 6(Flx, IFx) x/(x + ah+l) x/a (Ix, Fx) Note that if F is a single-valued mapping, then the set {IFx} consists of a single point and therefore diam {IFx} 0 for all x in X. Condition (I.I) therefore reduces to the condition given in [5], i.e. d(Flx, IFx) !d(Ix, Fx) (1.2) for all x in X.
In this paper we consider the family F of functions f from [0, ) into [0, ) The proof of this lemma is obvious but see also [15].
2. RESULTS IN COMPLETE METRIC SPACES.
Let F, G be two set-valued mappings of X into B(X) and let I J be two elfmaps of X such that F(X) !l(X), G(X) i J(X) (2.1) l.et Xo (Resp.yo be an arbitrary point in X and define inductively a sequence {i resp.{yn }) such that, having defined the point Xn_l (resp.yn_l), choose a point :n (resp.yn with IXn (resp.JYn) in FXn_l (resp.GYn_ 1) for n |,2, 'I'his can be done since the range of I (resp.J) contains the range of F (resp.G).
We consider the following conditions: ( (y2) F continuous and IFx !Fix for all x in X ([) J continuous, (2) G continuous and JGx !GJx for all x in X Modifying the proof of theorem of [I] we are now able to prove the following: THEOREM I. Let F, G be two set-valued mappings of X into B(X) and let I, J be two selfmaps of X satisfying (2.1) and As in the proof of theorem of [24], we have on using inequality (2.3) p times and property (a): 6(FXm,GYn) fP(max{6(FXr'Gyq) m p _< r _< m n-p!q!n}) for m n > p.Thus (Fx m,Fxn) _< 6(FXm,GYs) + 6(GYs,FXr) 2 for m n > p The sequence {z is therefore a Cauchy sequence in the complete metric space X and so has a limit Z in X where z is independent of the particu- lar choice of each z It follows in particular that the sequence {Ix converges n n to z and the sequence of sets {Fx converges to the set {z} n Similarly, it can be proved that the sequence {Jyn converges to a point w and the sequence of sets {Gy n} converges to the set {w} Using (2.3) we have 6(Fx n,Gyn) _< f(max{d(IXn,JYn), 6 (Ixn'Gyn), 6(JYn,FXn   Letting n tend to infinity and using lemma and properties (aa) and (aaa) it is seen that w z.
Now suppose that (yl) holds.Then the sequence {12Xn and {IFXn converge to Iz and {Iz} respectively.Let w be an arbitrary point in Fix for n n n 1,2, Then since I weakly commutes with F we have on using (I.I) Letting n tend to infinity and using lemma we see that the sequence {w converges n to Iz But Iz is independent of the particular choice of w in Fix and this n n means that the sequence of sets {Fix converges to the set {Iz} n Using inequality (2.3) we have (FlXn'GYn) -< f(max{d(12Xn'JYn ), 6(12Xn,GYn), 6(Jy n,Flxn)}) Letting n tend to infinity and using lemma and property (aa), we have d(Iz,z) _< f(d(Iz,z)) which implies Iz z by (aaa).
Letting n tend to infinity and using lemma and property (a), we obtain the ine- quality (Fu,z) !f(max{d(lu,z), 6(z,Fu)}) f((z,Fu)) Thus Fu {z} by (aaa) and since F and I weakly commute, we have z Fz Flu IFu Iz}.
It follows that Iz z.
We have therefore shown that if the conditions (yi) and (j), with i, I, 2, hold then Iz Jz z and Fz Gz {z}.
That z is the unique common fixed point of F and I and of G and J follows easily.This completes the proof of the theorem.
COROLLARY I. Let F, G be two set-valued mappings of X into B(X) and let I, J be two selfmaps of X satisfying (2.1) and (Fx,Fy) c.max{d(Ix,Jy),(Ix,Gy),(Jy,Fx)} ( for all x, y in X, where 0 c < I. Further, let F and G commute with I and J respectively.If F or I and G or J are continuous, then F, G, I and J have a unique common fixed point z.Further, Fz Gz {z and z is the unique common fixed point of F and I and of G and J. PROOF.As in the proof of theorem of [I], it is proved that (2.2) holds for any x0' Y0 in X.Since F and G commute with I and J respectively, we have Fix IFx and GJx JGx for all x in X.The thesis then follows from theorem if we assume that f(t) ct for all t O.
The result of this corollary was given in [I].
We now give an example in which theorem holds but corollary is not applicable.Gx [0,x/(x + 8)], Ix Jx x for all x in X.
By example i, F and G weakly commute with I. Further, we have GJx for all x in X. + 2y- + 2y' y and so I/(I + 2y) _< c which as y tends to zero, gives c _> I, a contradiction.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

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: and if for arbitrary O, there exists an integer n n N such that A ! for n N where A is the union of all open spheres with n centres in A and radius LEMM_A I.If {A and {B are sequences of bounded subsets of (X,d) which con- n n