Said R Grace*
Received: November 28, 2022; Published: December 09, 2022
*Corresponding author: Said R Grace, Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt
DOI: 10.26717/BJSTR.2022.47.007522
This paper deals with the oscillation of second order nonlinear difference equations with mixed nonlinear neutral terms. The purpose of the present paper is the linearization of the considered equation in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature. The results are illustrated by some examples.
Keywords: Second Order; Nonlinear; Neutral; Mixed Type; Oscillation AMS (MOS) Classification 34N05, 39A10
This paper is concerned with oscillatory behavior of all solutions of the nonlinear second order difference equations with mixed neutral terms of the form
where
We shall assume that
α,β,γ,μ” and “ δ are the ratios of positive odd integers,α≥1
{p_1 (n)},{p_2 (n)}, {q (n)} and {p (n)} are sequences of positive real numbers.
k, m, m*are positive real numbers with h (n) = n– m + k + 1 and h* (n) = n + m* + k.
We let
for
. By a solution of equation (1.1), we mean a real sequence {x(t)} defined for allt≥t_0-θ and satisfies equation (1.1) for all t≥t_0. A solution of equation (1.1) is called oscillatory if its terms are neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. If all solutions of the equation are oscillatory then the equation itself called oscillatory. In recent years, there has been a great interest in establishing criteria for the oscillation and asymptotic behavior of solutions of various classes of second-order difference equations, see [1-15] and the references cited therein. However, to the best of our knowledge, there are no results for second-order difference equations with mixed neutral terms of type (1.1). More exactly, existing literature does not provide any criteria which ensure oscillation of all solutions of equations (1.1). The aim of the present paper is the linearization of equation (1.1) in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria (taking the linear form of equation (1.1) into account) via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature for equation (1.1).In this section we study some oscillation criteria for equation (1.1) when β<1” and “ δ>1. We start with the following fundamental result. See [10, Lemma 1], and for the proof of (I), see [15, Lemma 2.2].
Lemma 2.1. Let {q (n)} be a sequence of positive real numbers, m and m*are positive real number and f: R→ R is a continuous nondecreasing function, and x f(x) > 0 for x≠0,
The first order delay differential inequality
y(n) + q(n) f ( y(n −m+1)) ≤ 0 has an eventually positive solution, so does the delay equation
y(n) + q(t) f ( y(t −m+)) ≤ 0
The first order advanced differential inequality
y(t) − q(n) f ( y(n + m*)) ≥ 0 has an eventually positive solution, so does the delay equation y(n) − q(n) f ( y(n + m*)) = 0 Lemma 2.2. [13]. If X and Y are nonnegative, then
Xλ + (λ −1)Yλ −λ XYλ −1 ≥ 0 forλ >1, (2.1)
Xλ − (1−λ )Yλ −λ XYλ −1 ≤ 0 for0 < λ <1 (2.2)
where equality is held if and only if X = Y.
In what follows, we let
"for some n"≥n_0," where " {b (n)} is a sequence of positive real numbers
Now, we present the following oscillation result. Theorem 2.1. Let β<1” and “ δ>1, conditions (i) – (iv) and (1.3) hold. Assume that there exist positive sequences (b (t)} and positive real numbers k1and k2 such that k1 < m-k1 and k2 < m* + k +1 such that
(2.3)If the first order advanced equation
is oscillatory, where ρ(n) = n +m*+k-k2>t, and assume that there exists a number θ∈(0,1) such that both the delay equations
for some n1≥n0 and
where, ξ(n)=n-m+k1+1<t
Proof. Let {w (n)} be a nonoscillatory solution of equation (1.1), say w (n) > 0, w(n-k)>0 ,w(n-m+1)>0 , and w(n+ m*+1)>0 for n≥n1 for some n1≥n. It follows from equation (1.1) that
Hence
is nonincreasing and of one sign. That is, there exists a such that Now, we see that Taking the difference of the above inequality, we getFrom equation (1.1) one can easily see that
From (2.9) one can easily get
We shall distinguish the following four cases:
First, we consider Case (I): Since
By condition (1.2), we conclude that a contradiction to the fact that y (n) is eventually positive. Next, we consider Case (II). Now, from the definition of y (n), we getOr
If we apply (2.1) with
we have
Thus, we see that
Since y (t) in nondecreasing, there exists a constant C > 0 such that y(n)≥C , and so, we have
Now, there exists a constant c1∈(0,1) such that
Using (2.11) in (2.10), we have
Clearly, we see that
Using this inequality in (2.12), we find
(2.13) It is easy to see that the function (2.13) is a nonincreasing and so, we get (2.14) We let and so, we see that Using this inequality in (2.14), we haveIt follows from Lemma 2.1. (I) that the corresponding differential equation (2.5) also has a positive solution, a contradiction. Next, we consider the cases when y (n) < 0 fort≥t2.
Let
Or
From (2.16) one can easily find that
Using (2.19) in (2.17), we have
which finally takes the form
where W (n) =
. The rest of the proof is similar to that of Case (I) and hence is omitted. Next, we consider Case (IV), i.e., z(n)>0 and Δz(n)>0. Now,We note that the results of this paper can be extended easily to the more general equations of the for
where the coefficients are the same as in equation (1,1) with γ and μ are the ratio of positive odd integers. The details are left to the reader. For the special case when α= δ, i.e. ,the equation &
(2.20) and when α=1,i.e.,the equation(2.21) we have the following interesting results Corollary 2.1. Let the hypotheses of Theorem 2.1 hold with equations (2.4) and (2.6) are replaced (respectively) by:
(2.23) Then equation (2.20) is oscillatory.
We also have the following result from corollary 2.1 for equation (2.21): Corollary 2.2. Let the hypotheses of Theorem 2.1 hold with equations (2.4) and (2.6) are replaced (respectively) by
Then equation (2.21) is oscillatory.
The following corollary is to employ some integral conditions rather than the oscillatory behavior of first order equations involved. Corollary 2.3. Let the hypotheses of Theorem 2.1 hold. If
And
then equation (2.20) is oscillatory. The following example is illustrative: Example 2.1. Consider the mixed neutral second order differential equations
Here we have
are are positive sequences of real numbers ,a (n)= n3,A(n,n1 )= p_1 (n)=1/n→0 as n→∞ and p_2 (n)=1=b(n), α =5/3= δ and β=1/3, k, m,m*are positive real numbers with h (n) = n – m + k + 1 and h* (n)=n+m*+k and positive real numbers k1 and k2 such that k1 < m-k1 and k2 < m* + k +1 with ρ(n) = n +m*+k-k2>n and ξ(n)=n-m+k1+1<n. It is easy to see for appropriate function p and q and the numbers k1 and k2 that all conditions of Corollary 2.3 are satisfied and hence every solution x (t) of equation (2.29) (respectively (2.30)) is oscillatory.The paper is presented in a form which is essentially and of high degree of generality. It will be of interest to study these results for the higher order of the
The author declare that they have no competing interests.
Not Applicable.