BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF A FORCED DIFFERENCE EQUATION

The authors consider the nonlinear difference equation where Ay, Y,+I Y,, {P,}, {q,}, and {rn} are real sequences, and uf(u) > 0 for u # 0. Sufficient conditions for boundedness and convergence to zero of certain types of solutions axe given. Examples illustrating the results are also included.

numbers, and f" ]R ]R is continuous with uf(u) > 0 for u 0. A solution of (E) is a sequence {yn} defined for n > No max{h, k}, No > 0, which satisfies (E) for n > No.We will classify solutions of (E) by borrowing some terminology introduced in [5] for the solutions of differential equations.A nontrivial solution {y, of (E) is said to be oscillatory if for every positive integer N] > No there exists n > Na such that g,,y,,+ < 0; it will be called nonoscillatory if there exists a positive integer N2 such that y, has fixed sign for all n > N2; and will be called a Z-type solution if there exists a positive integer N3 such that y, does not change sign for n > N3 but y,, 0 for arbitrarily large values of n.
Our interest here is in obtaining results on the convergence to zero of all the nonoscillatory solutions of (E).The Z-type solutions, even though they have arbitrarily large zeros, behave in many respects like the nonoscillatory solutions.As a consequence, our results include that type of solution as well.Very few results of this type are known for nonlinear difference equations, and essentially no such results are known for equations with a forcing term.Most of the results known to this point in time arc sufficient conditions for the oscillation of solutions of unforced equations.
Recent contributions in this direction can be found, for example, in [2 4, 6, 8, and 10] and in the references contained therein.For a discussion of basic notions on difference equations, we refer the reader to Kelley and Peterson [7] and Mickens [11]; for more advanced topics we refer to the monographs by Agarwal [1] and Lakshmikanth d Trigiante [9].

MAIN RESULTS.
In the remainder of this paper, we will let w, y, + P,Y,-h.Our first theorem gives sufficient conditions to ensure that certain types of solutions of (E) are bounded, and our final result gives conditions that imply these types of solutions tend on to zero as n .
Then, as argued above for the case L < 0, eventually p, < 0 and {y, is not only bounded but satisfies The proof for y, < 0 is similar and will be omitted.THEOREM 2. Suppose that, in addition to (1) ( 2), E q, cx3, and f(u) is bounded away from zero when u is bounded away from zero.( p, ---0 as n c, ( then any nonoscillatory or Z-type solution {y,,} of (E) satisfies y, 0 as n -cx.
PROOF.Let {y,,} be a nonoscillatory or Z-type solution of (E), say y, > 0 for n > N-h-k where N1 > No is a positive integer.Observe that (9)implies that (3) eventually holds, so {y,,} is bounded.From the proof of Theorem 1, either (6) holds, or (5) holds and w, L as n c for some constant L. Furthermore, it was also shown in the proof of Theorem 1 that y, 0 as n o if either (6) holds or (5) holds with L1 < 0. Hence, we only need consider the case when (5) holds and L1 > 0.

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.