Behavior of Second Order Nonlinear Differential Equations

Qualitative behavior of second order nonlinear differential equations with variable coefficients is studied. It includes properties such as posltlvlty, number of zeroes, oscillatory behavior, boundedness and monotoniclty of the solutions.

RINA LING are obtained.It would be assumed that the coefficients and their derivatives are continuous real-valued functions on the interval of interest.The work is divided into four parts; the first part deals with the case of p(t) < 0 and q(t) < 0, the second part with the case of p(t) < 0 and q(t) > O, the third part with p(t) > 0 and q(t) < 0 and the fourth part with p(t) > 0 and q(t) > O.
Nonllnear differential equations have been investigated in Chen  [i0] and Chen, Yeh and Yu [11], the former is on oscillatory behavior of bounded solutlons and the latter on asymptotic behavior of solutions.The results here are of a different nature and are independent of theirs.
(2.1) Let z(t) be real-valued and satisfy the linear equation + p(t)z 0.
(2.2) By a theorem in Hartman [12, p. 346-347], z(t) has no zero.Since n i is even and q(t) < 0, we have p(t) + q(t)y n-1 < p(t) and by Sturm First Comparison Theorem, z(t) has at least a zero on (0,=), if y has a zero on (0,), a contradiction.
The concavity of the graph follows from (I.i) written in the form n y -p(t)yq(t)y The case of n being even is considered in the following theorem.
If n is odd, the following two theorems on the number of zeroes of the solution can be obtained.THEOREM 4.1.If (i) p(t) > 0, a < t < b, (2) q(t) < 0, a < t < b and (3) n is odd, then a necessary condition for y to have two zeroes on (a PROOF.Equation (i.i) can be written in the form + (p(t) + q(t)yn-l)y 0.
Let z(t) be real-valued and satisfy the linear equation + p(t)z 0.
Since q(t) < 0 and (ni) is even, p(t) + q(t)y n-I < p(t) and by Hrtman [12], z(t) has at least two zeroes on (a, b) if y(t) has two zeroes on (a, b].By Lyapunov Theorem in Hartman [12, p. 346], a necessary condition for z(t n is odd, (4) y(t) has N zeroes on (0,T], then T N < (T p(t)dt) + i.
PROOF.As in the proof of Theorem 4.1, it can be shown that if z(t) has M zeroes on (0,T), then N <_ M.But by Hartman [12, p. 346-347 ], T M < (T p(t)dt + i 0 and the conclusion follows.
For p(t) > 0, q(t) > 0 and n odd, the following theorems on the oscillatory behavior and boundedness of the solutions can be obtained.
Let z(t) be a real-valued solution to E + p(t)z 0 which has been widely discussed.Since q(t) > 0 and (n i) is even, p(t) < p(t) + q(t)y n-1 and the conclusion follows from comparison theorems in Hartman and Sanchez [12,14].
PROOF.(P0 A sin e + q0 sin n e) sine de, 0 where A is a constant.
Since P0 > 0, q0 > 0 and n is odd, the integrand in s e)de 0 is positive, so a > 0.
By Theorem 5.4, y is bounded.The fact that ldt=m 0 and y is bounded imply that y is oscillatory, see Hartman [12, p. 354].
In the next two theorems, n is assumed to be even.
PROOF.The equation is + p(t)y + q(t)y n 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.