Oscillation of Noncanonical Nonlinear Second-Order Neutral Difference Equations via One Condition

1. Examples

In this section, four examples are provided to illustrate the significance of the main results. First example may be used in population biology to model species growth and interaction. Second example may be used in signal processing in communication systems to model how signals decay with distance but still oscillate due to interference. Third example may be used in control systems and delayed feed back to model feed back loops in mechanical systems where oscillation persist. The last example serves as a counterexample. In all cases, we present at least one explicit solution.

Example 1. 

Consider the half-linear neutral difference equation

$$\Delta \left({\upsilon }^5{(\upsilon +1)}^5{(\Delta (y\left(\upsilon \right)+2y(\upsilon -1)))}^5\right)+q\left(\upsilon \right)y^5(\upsilon -2)=0,\upsilon \ge 2,$$

(59)

where $q\left(\upsilon \right)=32{(\upsilon +1)}^5({\upsilon }^5+{(\upsilon +2)}^5).$

Here $a\left(\upsilon \right)={\upsilon }^5{(\upsilon +1)}^5,$$p\left(\upsilon \right)=2,$$q\left(\upsilon \right)=32{(\upsilon +1)}^5({\upsilon }^5+{(\upsilon +2)}^5),$$\alpha =\beta =5,$$\tau =1$ and $\sigma =2.$ It is easy to see that the hypotheses $\left(H_1\right)$-$\left(H_3\right)$ are clearly satisfied. Further a simple calculation shows that $B\left(\upsilon \right)={\displaystyle \frac{1}{\upsilon }},$$m=2$, $k={\displaystyle \frac{{(m-1)}^{\beta }}{m^{\beta }}}={\displaystyle \frac{1}{32}},$${\displaystyle \frac{\alpha }{k}}=160$ and $Q\left(\upsilon \right)={(\upsilon +1)}^5({\upsilon }^5+{(\upsilon +2)}^5).$ The condition (43) becomes

$$lim\underset{\upsilon \to \infty }{sup}\sum _{s=\upsilon +\tau -\sigma }^{\upsilon -1}{B}^{\alpha }(s+1)Q\left(s\right)=\underset{\upsilon \to \infty }{lim}({(\upsilon -1)}^5+{(\upsilon +1)}^5)=\infty >160,$$

By Theorem 2, the equation (59) is oscillatory and in fact $\left\{y\left(\upsilon \right)\right\}=\left\{{(-1)}^{\upsilon }\right\}$ is one such oscillatory solution of equation (59).

Example 2. 

Consider the second-order nonlinear neutral difference equation

$$\Delta \left(2^{3\upsilon }{(\Delta (y\left(\upsilon \right)+2^{\upsilon }y(\upsilon -2)))}^3\right)+q\left(\upsilon \right)y(\upsilon -3)=0,\upsilon \ge 1,$$

(60)

where $q\left(\upsilon \right)=2^{4\upsilon +6}[{(2+{\displaystyle \frac{3}{2^{\upsilon +4}}})}^3+{(1+{\displaystyle \frac{3}{2^{\upsilon +4}}})}^3].$

Here $a\left(\upsilon \right)=2^{3\upsilon },$$\tau =2,$$\sigma =3,$$\alpha =3,$$\beta =1,$ and $p\left(\upsilon \right)=2^{\upsilon }.$ It is easy to see that the hypotheses $\left(H_1\right)$-$\left(H_3\right)$ are clearly hold. By a simple calculation, we see that $B\left(\upsilon \right)={\displaystyle \frac{1}{2^{\upsilon }}},$$m=2,$$k={\displaystyle \frac{1}{2}},$${\displaystyle \frac{\alpha }{k}}=6$ and $Q\left(\upsilon \right)\approx 9\left(2^{3\upsilon +7}\right).$The condition (32) becomes

$$lim\underset{\upsilon \to \infty }{sup}\Omega \left(\upsilon \right)=2592>6.$$

Now, by applying Theorem 1, the equation (60) is oscillatory and in fact $\left\{y\left(\upsilon \right)\right\}=\left\{{(-1/2)}^{\upsilon }\right\}$ is one such oscillatory solution of equation (60).

Example 3. 

Consider the nonlinear second-order neutral delay difference equation

$$\Delta \left({\upsilon }^3{(\upsilon +1)}^3{(\Delta (y\left(\upsilon \right)+2\upsilon y(\upsilon -1)))}^3\right)+q\left(\upsilon \right)y^5(\upsilon -2)=0,\upsilon \ge 2,$$

(61)

where $q\left(\upsilon \right)=2{(\upsilon +1)}^3({\upsilon }^3+{(\upsilon +2)}^3).$

Here $a\left(\upsilon \right)={\upsilon }^3{(\upsilon +1)}^3,$$\tau =1,$$\sigma =2,$$\alpha =3,$$\beta =5,$$p\left(\upsilon \right)=2$ and $q\left(\upsilon \right)=2{(\upsilon +1)}^3({\upsilon }^3+{(\upsilon +2)}^3).$ It is easy to see that the hypotheses $\left(H_1\right)$-$\left(H_3\right)$ are clearly hold. By a simple calculation, we find that $B\left(\upsilon \right)={\displaystyle \frac{1}{\upsilon }},$$m=2$ and $k={\displaystyle \frac{1}{32}}.$ The condition (56) becomes

$$lim\underset{\upsilon \to \infty }{sup}\sum _{s=\upsilon -1}^{\upsilon -1}2(s-1)({\left({\displaystyle \frac{s}{s+1}}\right)}^3+{\left({\displaystyle \frac{s+2}{s+1}}\right)}^3)=\infty >{\displaystyle \frac{1}{32}},$$

that is, condition (56) holds. Hence, by Theorem 3, equation (61) is oscillatory and in fact $\left\{y\left(\upsilon \right)\right\}=\left\{{(-1)}^{\upsilon }\right\}$ is one such oscillatory solution of equation (61).

Next, we present a counterexample showing that if one condition is not satisfied, then the equation may fail to be oscillatory and instead admit a nonoscillatory solution.

Example 4. 

Consider the nonlinear second-order neutral delay difference equation

$$\Delta \left(\upsilon (\upsilon +1)(\upsilon +2){(\Delta (y\left(\upsilon \right)+2\upsilon y(\upsilon -1)))}^3\right)+q\left(\upsilon \right)y^5(\upsilon -2)=0,\upsilon \ge 2,$$

(62)

where $q\left(\upsilon \right)={\displaystyle \frac{3\upsilon }{\upsilon +1}}.$

Here $a\left(\upsilon \right)=\upsilon (\upsilon +1)(\upsilon +2),$$\tau =1,$$\sigma =2,$$\alpha =1,$$\beta =1,$$p\left(\upsilon \right)=2$ and $q\left(\upsilon \right)={\displaystyle \frac{3\upsilon }{\upsilon +1}}.$ The hypotheses $\left(H_1\right)$-$\left(H_3\right)$ are clearly satisfied. A simple computation shows that $B\left(\upsilon \right)={\displaystyle \frac{1}{2\upsilon (\upsilon +1)}},$$m=2,$$k=1/2,$${\displaystyle \frac{\alpha }{k}}=2$ and $Q\left(\upsilon \right)={\displaystyle \frac{3\upsilon }{2(\upsilon +1)}}.$ The condition (43) becomes

$$lim\underset{\upsilon \to \infty }{sup}\sum _{s=\upsilon +\tau -\sigma }^{\upsilon -1}B(s+1)Q\left(s\right)=\underset{\upsilon \to \infty }{lim}{\displaystyle \frac{3(\upsilon -1)}{4{\upsilon }^2(\upsilon +1)}}=0>2,$$

that is, condition (43) is not satisfied. Hence, by Theorem2, equation(62) is not oscillatory and in fact equation(62) has a nonoscillatory solution $\left\{y\left(\upsilon \right)\right\}=\left\{{\displaystyle \frac{\upsilon +2}{\upsilon +1}}\right\}.$

Remark 3. 

From the four examples, we see that:

(i) If conditions are satisfied then oscillatory solutions appear (persistent cycle);

(ii) If one condition fails (as in Example 3.4) nonoscillatory solutions emerge, showing stable, monotone behavior;

(iii)This demonstrates how parameters $\alpha ,$$\beta ,$$\tau ,$ and $\sigma ,$ changes directly control whether a system oscillates or stabilizes.

2. Conclusion

In this paper, we have investigated the oscillatory properties of second-order noncanonical difference equations with bounded and unbounded neutral terms. The obtained oscillation criteria require only a single condition, in contrast to the two conditions used in [22,23]. This result is achieved through the application of the linearization method combined with the summation averaging technique. Four examples are presented to illustrate the importance and novelty of the main findings. Furthermore, our results are applicable to linear, half-linear, and nonlinear types of equations.

Next we present directions for future study:

(1) In the case of $0<\alpha <1,$ the same method cannot be used since the linearization technique fails to hold. So it is interesting to extend the results of this paper when $0<\alpha <1$ using other methods.

(2) Integrating the linearization techniques developed in this paper with fuzzy set theory, or extending them to higher-dimensional systems, may lead to more realistic and flexible models in fields such as engineering, economics, and biology. This represents a challenging yet promising direction for future investigation.

(3) Another interesting problem is to apply the method developed in this paper to study the oscillatory behavior of fractional-order difference equations.