PROPERTIES OF SOLUTIONS TO A SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION

In this note, we show when all solutions to the nonlinear differential equation x" + c(t)f(x)g(x')x' + a(t,x) 0 are bounded. Furthermore, the solutions are either oscillatory or monotonic and asymptotically stable.


INTRODUCTION.
In thls note, we shall discuss under what conditions all solut{ons to the second order nonlinear differential equation, x" + c(t)f(x)g(x')x' + a(t,x) 0 (I.I) are bounded and are either oscillatory or monotonic and approach 0 along with their derivatives as t =.
By oscillatory, we shall mean a solution equaling 0 for arbitrarily large values of t.The above equation is a generalization of the well- known Lidnard equation (see, for example, Struble [I, p. 164-166] and Utz [2] for some classical results and Kroopnick [3] and Heldel [4,5] for some more recent though less general results to this type of equation).We now state and prove our main result.
THEOREM 2. I. Given the differential equation (l.I).Suppose C(.) C[O, ==) and is positive on [0, (R))with c > c(t) > 0 for some positive constant c and a(t,x) continuous on [0, =) x R with xa(t,x) > ao(t)Ixl p for x 0 (p I), ao(t) positive and continuous on [0, =) and ao(t)dt + (R).Furthermore, suppose x a(t,x) 0, o a(t,x) dx + uniformly in t and left [(.), g(.) be non-negatlve continuous x o functions on R. Then all solutions to (I.I) are bounded and are either oscillatory or monotonic and approach 0 along with their derivatives as t -. PROOF.
From standard existence theory, we know that a solution exists on the interval [O,t) for some t > O.All we need to show is that the solution stays bounded and, therefore, it may be extended for all t > 0. Now, multiply (I.I) by x' and integrate from 0 to t by parts in order to obtain the following, (2.1) (just differentiate (2.1) using Leibnlz's rule to get back to (I.I)).Should x and x' become unbounded then all terms on the LHS of (2.1) become positive and, furthermore, the LHS of (2.1) becomes unbounded which is impossible since the RHS of (2.1) is constant.A standard argument now completes the proof that x(t) may be extended as a bounded solution for all t > 0.
Next, suppose x(t) does not oscillate, then x(t) must eventually be of fixed sign for t > > 0. Assume x(t) > 0 for t > (a similar argument works for x(t) < 0).Also, x'(t) must be of fixed sign.Otherwise, x(t) will have an infinite number of consecutive relative maxima which Is impossible if x'(t) 0 infinitely often.In order to see this,-suppose x'(t I) and x'(t 2) are consecutive zeros of x'(t).Then x(t I) and x(t 2) are consecutive relative maxima of x(t) since x"(t) < 0 at these points from (I.I).Also, the sign of x' does not change on [tl,t2].Furthermore, we have x"(t o) 0 where t < t o < t 2. Should x'(t o) > 0 then from (I.I) we have that x"(t < 0 which is impossible.If x'(t < 0 then x' is negative on (t!,t2) since o o x' does not change sign on [tl,t2].However, if this is the case, then x(t) does not possess a relative maximum at t t 2.
This contradiction establishes that x' must eventually be of fixed slgn.
Since x(t) is positive, nonosclllatory and bounded, x'(t) must approach 0 as t .Should x(t) itself not approach 0 as t , then we have x'(t) as t .This follows from (I.I) when we Integrate (I.I) from t to t, i.e., t t x'(t) x'()c(s)f(x(s))g(x'(s))x'(s) ds a(s,x(s)) ds (2.2)   which diverges when x(t) approaches a positive limit.Note that the middle term of the LHS of (2.2) is bounded by 2KMc where M is a bound for f(x(t))g(x'(t)) and K is a bound for x(t) for t c [0,+) so it cannot cancel out the third term of the RHS of (2.2) which diverges as t / (R).
We have essentially shown that the bounded solutions to (I.I) behave very much like those of x" + cx' + ax 0 where a and c are positive constants, an observation similar to one made by Utz [2].x" + cx' + ax 0 (m and n are positive integers). (2.3) We readily see that all solutions to the above are bounded, oscillate or approach 0 from our theorem.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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:

First
Round of Reviews March 1, 2009