STABILITY AND BOUNDEDNESS PROPERTIES OF A RATIONAL EXPONENTIAL DIFFERENCE EQUATION

This article aims to discuss, the stability and boundedness character of the solutions of the rational equation of the form (0.1) yt+1 = νρ−yt + δρ−yt−1 μ+ νyt + δyt−1 , t ∈ N(0). Here, ρ > 1, ν, δ, μ ∈ (0,∞) and y0, y1 are taken as arbitrary non-negative reals and N(a) = {a, a + 1, a + 2, · · · }. Relevant examples are provided to validate our results. The exactness is tested using MATLAB.


Introduction
Difference equations involving geometrical and exponential functions have many applications in biology.Growth of a perennial grass, generally rely on the parameters 1 biomass, like litter mass and soil nitrogen, was described by the difference equations + µsN e ν−δLt 1 + e ν−δLt , s ∈ (0, 1). (2) Here, the parameters B, L and N denote biomass, litter mass, soil nitrogen respectively, ν, δ, µ, d > 0 are fixed.Oscillatory behaviour and chaotic climate of (2) was discussed in [22].
The boundedness, stability and periodic character of the solution obtained from exponential rational equation was obtained by El-Metwally et all [13].The population growth rate β and immigration rate α are positive reals with initial conditions y 0 and y 1 .
Global asymptotic behavior and Boundedness behavior of the difference equations and have been developed by Ozturk et all [19,20], where α > 0, β > 0 and s ∈ N (1) and the y −j for j = 0, 1, 2, • • • , s can be taken as reals.
For given reals δ, µ, d, s and 0 < a < 1, the entity of periodical solution is given by the equation Authors in [3] discussed stability behaviour of the equation Here, the initial conditions are taken as arbitrary reals and α, β are positive numbers.
Difference equations are normally discrete version of differential equations which preserve symmetries.The role of difference equations are well established in the study of Lie theory.One can refer [10]- [12] for a detailed study on this aspect.
In this paper, we extend the theory to (3) and establish new conditions for stability and other behaviors of the equations (1) for > 1. MATLAB is used to test the exactness of the behavior of the solutions.
(ii) If at least one of the roots of ( 6) is in the region | | < 1, then the equilibrium ȳ of ( 1) is unstable.
(iii) The two roots of (6) will lie in the open region | | < 1 if and only if This locally asymptotically stable equilibrium point ȳ is called a sink.
(iv) The magnitude of one of the two roots of ( 6) is more than unity if and only if This equilibrium point ȳ is called a repeller.
(v) The absolute value of one of the roots of (6) is more than unity and the other has absolute value less than unity if and only if and this unstable equilibrium point ȳ is called a saddle point.
(vi) If a root of ( 6) has absolute value unity, then |p| = |1 − q| or q = −1 and |p| ≤ 2. Conversely, if |p| = |1 − q| or q = −1 and |p| ≤ 2 then we get one root whose absolute value is equal to unity and hence we get the equilibrium point ȳ, which is non hyperbolic.

Main Results
Here, we discuss the existence, uniqueness and stability of the equation (1).This gives us that (1) has equilibrium ȳ.
which implies that F is decreasing.Hence, the equilibrium ȳ is unique.
(i) Every positive solution of equation ( 1) is bounded.
(ii) The unique equilibrium point ȳ > 0 of the equation (1) is bounded.
Proof.From Definition 2.2, we get the linearized equation and the characteristic equation associated with (1) about ȳ is and respectively.From Theorem 2.3 we obtain We derive and Substituting (16), we get Again substituting in (14), we get Example 3.5.For ν = 3, δ = 5, µ = 2, = 3 and condition (10) of the Theorem 3.4 does not hold, then every positive equilibrium solution of ( 1) is not locally asymptotically stable.(ii).Equilibrium solution ȳ is not a saddle point.
Thus the equilibrium solution ȳ is nonrepeller.

Conclusion
In this paper, we discuss the different characters like periodic, stability and boundedness of the solutions of the rational exponential difference equation (1).
Earlier results exist for similar type of difference equation when the independent variable is raised as a power of e.Here we have generalized the results when the independent variable is raised to any > 1.Earlier results are available only on the study of the stability of the solutions but, we have analyzed more characters like boundedness and the asymptotic behavior of solutions of the equation ( 1

1 . [ 8 ]
Let f : I × I → I, I ∈ R, be a continuous function and y 0 , y −1 ∈ I be given values.Then, for