Non-Oscillating solutions of a generalized system of ODEs with derivative terms


 We consider a system of ODEs of mixed order with derivative terms appearing in the non-linear function and show the existence of a solution which does not oscillate for such system. We applied the fixed point technique to show that under certain conditions there exists at least one solution to the system which is not only non-oscillating, but also asymptotically constant.


Introduction
Systems of di erential equations arise while modelling many situations. We will now look at few systems of coupled ODEs which arise in nature. In [1], the authors discuss some qualitative properties of elliptic systems of the type (1. 3) The following system of ODEs appears in Non-linear Optics [11]: where a, b are positive constants and w, v are functions de ned for all x ∈ R. This system arises from a system of χ −SHG equations which govern phenomena in non-linear optics. The χ −SHG equations arise while studying the Parametric interactions of intense light signals in materials with second-order non-linearities. It is very important to know the asymptotic behaviour of solutions of such systems. In fact, Lopes in [11] talks about solutions of (1.4) with nite energy (equivalently tending to zero as the independent variable say x → ∞). While considering the steady ow of an electrically conducting uid between two horizontal parallel plates where the uid and the plates rotate in unison about an axis normal to the plates with an angular velocity, the following system of ODEs arise.
where A is a constant, R is the viscosity parameter, M is the magnetic parameter and K is the rotational parameter. The functions f and g are de ned on the real line. These equations are found in [12]. These examples lead us to consider the more generalized forms of (1.3). So, we consider the following generalized version of coupled system of non-linear ordinary di erential equations.
Lot of work is done in providing conditions to nd non oscillating solutions of systems and equations. We refer to articles like [2-4, 6, 7, 9, 14? ]. In most of these article authors proved conditions for non oscillation or positive solutions for either lower order systems which are fully coupled or higher order systems which are weakly coupled. For example, Graef et al [13] considered the following second order system which is fully coupled.
Whereas Henderson et al [14] considered the following higher order system.
To the best of our knowledge more general systems of the type (1.5) are not considered in existing literature, which makes this problem worth attempting. In this article we presented di erent conditions under which the existence of a non-oscillating solution for system (1.5) is guaranteed. At the end we apply the theory developed on a theoretical example.

Mathematical Preliminaries
We present here some mathematical preliminaries required to prove the main results presented in the next section.
We will now state Shauder's theorem Let E be a Banach space and X any nonempty convex and closed subset of E. If S is a continuous mapping of X into itself and SX is relatively compact, then the mapping S has at least one xed point (i.e. there exists an x ∈ X with x = Sx). Consider B ([ , ∞)) to be the Banach space of all continuous and bounded real valued functions on the interval [ , ∞], endowed with the sup-norm . : ) be the Cartesian product of (B) N ([ , ∞)) with itself with the product topology. This product space is endowed with the norm . N×N de ned as We will now need a criterion to show the compactness of subsets of the space (B) N ([ , ∞]). In that direction, we will revisit few de nitions of concepts related to sets of functions taking real values.

De nition 2.3. A set of real-valued functions de ned on the interval [ , ∞), U is called uniformly bounded if there exists a positive constant M such that, for all functions
De nition 2.4. U is said to be equicontinuous if, for each ϵ > , there exists a δ ≡ δ(ϵ) > such that, for all De nition 2.5. U is called equiconvergent at ∞ if all functions in U are convergent in R at the point ∞ and, for each ϵ > , there exists a T > such that, for all u ∈ U We will use the following lemma from [5] which is a generalization of the above mentioned compactness criterion.

Main Results
The following is our main theorem.

4)
where Θ l , Θ l are de ned as Clearly X is a non-empty closed convex subset of E. Now pick x and x , two arbitrary functions in the set X. Then

From (3.3) and (3.4) for every t ≥ T we get
for every t ≥ T and j = , , , .., N − . As this is true for any pair x , x , we now de ne mapping S = (S (x , x ), S (x , x )) on X × X as It is not di cult to notice here that the map S, is a self map from Y = X × X to itself and is well de ned because S (X × X) ⊆ X and S (X × X) ⊆ X . We apply the Shauder's theorem and show that S has a xed point pair.
We rst show that S Y is relatively compact. We do this for S and an analogous proof follows for S also to show that S Y is relatively compact, which we exclude. After applying the compactness criterion Lemma 2.6 given earlier, it su ces to show that each one of the sets (S Y) (k) , k = , , .., n − , are uniformly bounded, equicontinuous and equiconvergent at ∞. Since S Y ⊂ Y, we obviously have (S (x , x )) (k) < K for k = , , .., n − for all (x , x ) ∈ Y. So (S X) (k) for k = , , .., n − are uniformly bounded. Moreover for some t ≥ T ., n − , we see that the right hand side of the inequality can be made close to zero, therefore we conclude that S Y is equiconvergent at ∞. Now by using (3.5) for any (x , x ) ∈ Y, for every t , t with T ′ ≤ t < t and for k < n − , we see that Also for k = n − we have (n − )! r (s)ds. Now, by choosing |t − t | < ϵ A(N +N )max{Θ l } we conclude that S Y is equicontinuous. Therefore, S Y is relatively compact. A similar proof can be given to show that S Y is relatively compact. Therefore, SY is relatively compact. To apply Schauder's theorem, the mapping S has to be continuous. Consider (x v , x v ) to be a random sequence in Y, converging to (x , x ) under the norm de ned before. From (3.5) we have for every t ≥ T and for all v ∈ N. Now, due to the Lebesgue's dominated convergence theorem we have This proves the pointwise convergence i.e Now, consider any random subsequence (uµ , uµ ) of S(x v , x v ). The relatively compactness of SY guarantees the existence of a subsequence (η λ , η λ ) of (uµ , uµ ) and a pair of functions (η , η ) in E such that (v λ , v λ ) converges uniformly to (v , v ). So for all t ≥ T under the sup-norm. Therefore S is continuous. Thus S satis es all the assumptions of Schauder's theorem, therefore S has a xed point pair (x , x ) ∈ Y such that S (x , x ) = x and S (x , x ) = x . That implies