Oscillation Criteria for Third Order Neutral Generalized Difference Equations with Distributed Delay

: This paper aims to investigate the criteria of behavior of a certain type of third order neutral generalized difference equations with distributed delay. With the technique of generalized Riccati transformation and Philos-type method, we obtain criteria to ensure convergence and oscillatory solutions and suitable examples are provided to illustrate the main results.


Introduction
Difference equations and functional equations usually occur due to certain phenomena over time and play essential roles in the field of discrete dynamical systems [1].Difference equations and their associated operators play a vital role as direct mathematical models of physical phenomena but also provide powerful tools in numerical methods.Given a differential equation with symmetries, we can construct a difference scheme, which appropriates it while preserving the symmetries.Difference equations play an important role in Lie theory.An important and significant observation is that difference equations either appear in themselves and they can be used in Lie theory to get classes of exact solutions, or they can be obtained by discretizing the continuum equation in such a way to preserve the symmetries.That is, we can create sets of discrete equations which provide numerical schemes approximating the continuum equations.For a detailed study on this aspect one can refer to [2][3][4][5].
Here, we obtain conditions for the existence of convergent oscillatory solutions of Equation (1) with the help of the generalized Riccati transformation.In fact, by choosing an appropriate function, we shall present several oscillation criteria easily.The technique adopted in the present paper are different from technique used in the references cited earlier, and the results are the generalization of the existing results.
This paper is structured as follows: A few standard definitions and preliminaries are discussed in Section 2. Section 3 deals with new oscillation results for (1), and in Section 4 we provide suitable examples to demonstrate the main findings.

Preliminaries
In this section, some basic definitions and preliminary results are presented, which will be useful for further discussion.
We denote the polynomial factorial λ (m) by the expression 1) .

Main Results
In this section, we obtain new oscillation criteria for the Equation (1) by using the generalized Riccati transformation and Philos type technique.For Philos type technique, we define functions q, Q : N × N → R such that Theorem 1.Consider the Condition (7) and {ρ(λ)} satisfies m 1 (λ) where and then, every solution of Equation ( 1) is either oscillatory or converges to zero.
then for every m ≥ 1, each solution of ( 1) is oscillatory or converges to 0.
then, for every m ≥ 1, each solution of ( 1) is oscillatory or converges to 0.
Proof.Let y(λ) be a non oscillatory solution of Equation ( 1).As in Therorem 2, when x(λ) satisfies property (i), from (26) and by rearranging the terms we obtain and then lim inf Then, The above inequality can be expressed as where .
We shall prove that Suppose to the contrary that from Equation (32), we have inf Then, we can find a positive constant M 3 > 0 with the condition Thus for λ ≥ λ 3 and using Equation (45), we obtain Theorem 4. Assume that all hypotheses of Theorem 3 are satisfied except condition (33).Also let and then, every solution of ( 1) is convergent to zero or oscillatory.
Proof.The proof is similar to that of Theorem 3 and hence the details are omitted.
If there is a sequence {Φ(λ)} satisfying (34) and then, every solution of Equation ( 1) is oscillatory or converges to zero.
Corollary 4. Let m ≥ 1 be a constant, and If there is a sequence {Φ(λ)} satisfying (34) and then every solution of Equation (1) will either oscillate or converge to 0.

Conclusions
In this paper, we present new oscillation criteria for the generalized difference equation with distributed delay, which is new in the literature.Similar results are available in the literature for difference equations with delay involving the conventional difference operator ∆.The results we obtained in this paper for difference equations involving the generalized difference operator ∆ with distributed delay are rare and new in the literature.Also, the techniques we adopted are different from the techniques adopted by other researchers.Our results generalize the results on the oscillatory behavior of the continuous third order dynamical systems discussed in [5].In addition, our results generalize the results presented in [18].Other researchers considered only the oscillation criteria, but we obtained conditions for the solutions to be convergent apart form obtaining oscillation results.