CUBO, A Mathematical Journal Vol. 23, no. 01, pp. 109–119, April 2021 DOI: 10.4067/S0719-06462021000100109 Inequalities and sufficient conditions for exponen- tial stability and instability for nonlinear Volterra difference equations with variable delay Ernest Yankson Department of Mathematics, University of Cape Coast, Ghana. ernestoyank@gmail.com ABSTRACT Inequalities and sufficient conditions that lead to exponential stability of the zero solution of the variable delay nonlinear Volterra difference equation x(n + 1) = a(n)h(x(n)) + n−1∑ s=n−g(n) b(n, s)h(x(s)) are obtained. Lyapunov functionals are constructed and em- ployed in obtaining the main results. A criterion for the in- stability of the zero solution is also provided. The results generalizes some results in the literature. RESUMEN Se obtienen desigualdades y condiciones suficientes que im- plican la estabilidad exponencial de la solución cero de la ecuación en diferencias no lineal de Volterra con retardo va- riable x(n + 1) = a(n)h(x(n)) + n−1∑ s=n−g(n) b(n, s)h(x(s)). Se construyen funcionales de Lyapunov y se utilizan para obtener los resultados principales. Se entrega también un cri- terio para la inestabilidad de la solución cero. Los resultados generalizan algunos resultados en la literatura. Keywords and Phrases: Exponential stability, Lyapunov functional, Instability. 2020 AMS Mathematics Subject Classification: 34D20, 34D40, 34K20. Accepted: 01 February, 2021 Received: 02 June, 2020 ©2021 E. Yankson. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000100109 https://orcid.org/0000-0002-5621-1048 110 Ernest Yankson CUBO 23, 1 (2021) 1 Introduction Let R and Z+ denote the set of real numbers and the set of positive integers respectively. In recent times, research into the stability properties of solutions of difference equations have gained the attention of many Mathematicians, see [1], [2], [4], [6], [7], [8] and the references cited therein. We are mainly motivated by the work of Kublik and Raffoul in [6] in which the authors obtained inequalities that lead to the exponential stability of the zero solution of the linear Volterra difference equation with finite delay x(n + 1) = a(n)x(n) + n−1∑ s=n−r b(n,s)x(s), (1.1) for some positive constant r. In this paper we consider the scalar nonlinear Volterra difference equation with variable delay x(n + 1) = a(n)h(x(n)) + n−1∑ s=n−g(n) b(n,s)h(x(s)), (1.2) where a : Z+ → R, b : Z+ × [−g0,∞) → R, h : R → R and 0 < g(n) ≤ g0, for all n ∈ Z+ for some positive constant g0. We will obtain some inequalities regarding the solutions of (1.2) by employing Lyapunov functionals. These inequalities can be used to deduce exponential stability of the zero solution. Also, by means of a Lyapunov functional an instability criterion of the zero solution of equation (1.2) will be provided. Let ψ : [−g0, 0] → (−∞,∞) be a given bounded initial function with ||ψ|| = max −g0≤s≤0 |ψ(s)|. We further denote the norm of a function ϕ : [−g0,∞) → (−∞,∞) by ||ϕ|| = sup −g0≤s≤∞ |ϕ(s)|. Throughout this paper we let h(x) = xh1(x). The notation xn means that xn(τ) = x(n + τ),τ ∈ [−g0, 0] as long as x(n + τ) is defined. Thus, xn is a function mapping an interval [−g0, 0] into R. We say that x(n) ≡ x(n,n0,ψ) is a solution of (1.2) if x(n) satisfies (1.2) for n ≥ n0 and xn0 = x(n0 + s) = ψ(s), s ∈ [−g0, 0]. In this paper we use the convention that ∑b s=a h(s) = 0 if a > b. The following notation is introduced. Let A(n,s) = γ∑ u=n−s b(u + s,s), where 0 < γ ≤ g(n− 1) for all n ∈ Z+. (1.3) CUBO 23, 1 (2021) Inequalities and sufficient conditions for exponential stability . . . 111 It follows from (1.3) that A(n,n−g(n− 1) − 1) = 0. (1.4) We assume throughout the paper that ∆nA 2(n,z) ≤ 0, for all n + s + 1 ≤ z ≤ n− 1. (1.5) Due to (1.3) we can express (1.2) in the equivalent form ∆x(n) = ( a(n)h1(x(n)) + A(n + 1,n)h1(x(n)) − 1 ) x(n) − ∆n n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s)). (1.6) Definition 1.1. The zero solution of (1.2) is said to be exponentially stable if any solution x(n,n0,ψ) of (1.2) satisfies |x(n,n0,ψ)| ≤ C(||ψ||,n0)ζγ(n−n0), for all n ≥ n0, where ζ is a constant with 0 < ζ < 1,C : R+ × Z+ → R+, and γ is a positive constant. The zero solution of (1.2) is said to be uniformly exponentially stable if C is independent of n0. We end this section by stating a fact which will be used in the proof of Lemma 2.1, that is, if u(n) is a sequence, then ∆u2(n) = u(n + 1)∆u(n) + u(n)∆u(n). For more on the calculus of difference equations we refer to [3] and [5]. 2 Exponential Stability In this section we obtain inequalities that can be used to deduce the exponential stability of (1.2). To simplify notation we let Q(n,x) = ( a(n) + A(n + 1,n) ) h1(x(n)) − 1, and Q1(n) = ( a(n) + A(n + 1,n) ) − 1. Lemma 2.1. Suppose that (1.3), (1.5) and for δ > 0, − δ δg0 + g(n) ≤ Q(n,x) ≤−δg0A2(n + 1,n)h21(x(n)) −Q 2(n,x), (2.1) 112 Ernest Yankson CUBO 23, 1 (2021) holds. If 1 ≤ h1(x), and V (n) =  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))  2 + δ −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)), (2.2) then based on the solutions of (1.2) we have ∆V (n) ≤ Q1(n)V (n). (2.3) Proof. Let x(n,n0,ψ) be a solution of (1.2) and let V (n) be defined by (2.2). It must also be noted that in view of condition (2.1), Q(n,x) < 0 for all n ≥ 0. This together with the fact that 1 ≤ h1(x) also implies that Q(n,x) ≤ Q1(n) < 0. Then based on the solutions of (1.2) we have ∆V (n) =  x(n + 1) + n∑ s=n−g(n) A(n + 1,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))   +  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))   + δ∆n −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)). (2.4) But x(n + 1) + n∑ s=n−g(n) A(n + 1,s)h(x(s)) = ( Q(n,x) + 1 ) x(n) − ∆n n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s)) + n∑ s=n−g(n) A(n + 1,s)h(x(s)) = ( Q(n,x) + 1 ) x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s)) = ( Q(n,x) + 1 ) x(n) + n−1∑ s=n−g(n) A(n,s)h(x(s)) (2.5) CUBO 23, 1 (2021) Inequalities and sufficient conditions for exponential stability . . . 113 where we have used the fact that A(n,n−g(n− 1) − 1) = 0. Using (2.5) in (2.4) we obtain ∆V (n) =  (Q(n,x) + 1) x(n) + n−1∑ s=n−g(n) A(n,s)h(x(s))  Q(n,x)x(n) +  x(n) + n−1∑ s=n−g(n) A(n,s)h(x(s))  Q(n,x)x(n) + δ∆n −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) = Q(n,x)V (n) + (Q2(n,x) + Q(n,x))x2(n) + δ∆n −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) − Q(n,x)   n−1∑ s=n−g(n) A(n,s)h(x(s))  2 − δQ(n,x) −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) (2.6) Considering the third term on the right hand side of (2.6) we obtain ∆n −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) = −1∑ s=−g0 n∑ z=n+s+1 A2(n + 1,z)h2(x(z)) − −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) = −1∑ s=−g0 [ A2(n + 1,n)h(x2(n)) + n−1∑ z=n+s+1 A2(n + 1,z)h2(x(z)) − n−1∑ z=n+s+1 A2(n,z)h2(x(z)) −A2(n,n + s)h2(x(n + s)) ] = −1∑ s=−g0 ( A2(n + 1,n)h21(x(n))x 2(n) −A2(n,n + s)h2(x(n + s)) ) + −2∑ s=−g0 n−1∑ z=n+s+1 ∆nA 2(n,z)h2(x(z)) = g0A 2(n + 1,n)h21(x(n))x 2(n) − −1∑ s=−g0 A2(n,n + s)h2(x(n + s)) + −2∑ s=−g0 n−1∑ z=n+s+1 ∆nA 2(n,z)h2(x(z)) ≤ g0A2(n + 1,n)h21(x(n))x 2(n) − −1∑ s=−g0 A2(n,n + s)h2(x(n + s)). = g0A 2(n + 1,n)h21(x(n))x 2(n) − n−1∑ z=n−g0 A2(n,z)h2(x(z)) (2.7) 114 Ernest Yankson CUBO 23, 1 (2021) Applying the Holder’s inequality to the squared term in the fourth term on the right hand side of (2.6) gives   n−1∑ s=n−g(n) A(n,s)h(x(s))  2 ≤ g(n) n−1∑ s=n−g(n) A2(n,s)h2(x(s)) ≤ g(n) n−1∑ s=n−g0 A2(n,s)h2(x(s)). (2.8) Considering the last term on the right hand side of (2.6) we obtain −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) ≤ g0 n−1∑ s=n−g0 A2(n,s)h2(x(s)) (2.9) Substituting (2.7), (2.8) and (2.9) in (2.6) we obtain ∆V (n) ≤ Q(n,x)V (n) + (Q2(n,x) + Q(n,x) + δg0A2(n + 1,n)h21(x(n)))x 2(n) + [−(g(n) + δg0)Q(n,x) − δ] n−1∑ s=n−g0 A2(n,s)h2(x(s)) ≤ Q(n,x)V (n) + (Q2(n,x) + Q(n,x) + δg0A2(n + 1,n))x2(n) + [−(g(n) + δg0)Q(n,x) −δ] n−1∑ s=n−g0 A2(n,s)h2(x(s)) ≤ Q(n,x)V (n) ≤ Q1(n)V (n). Theorem 2.2. Suppose the hypothesis of Lemma 2.1 hold. Then any solution x(n) = x(n,n0,ψ) of (1.2) satisfies the exponential inequality |x(n)| ≤ √√√√g0 + δ δ V (n0) n−1∏ s=n0 ( a(n) + A(n + 1,n) ) (2.10) for n ≥ n0. Proof. Let V (n) be defined by (2.2). Changing the order of summation in the second term on the right hand side of (2.2) we obtain δ −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) = δ n−1∑ z=n−g0 z−n∑ s=−g0 A2(n,z)h2(x(z)) = δ n−1∑ z=n−g0 A2(n,z)h2(x(z))(z −n + g0 + 1) ≥ δ n−1∑ z=n−g0 A2(n,z)h2(x(z)) ≥ δ n−1∑ z=n−g(n) A2(n,z)h2(x(z)), CUBO 23, 1 (2021) Inequalities and sufficient conditions for exponential stability . . . 115 where we have used the fact that if n − g0 ≤ z ≤ n − 1 then 1 ≤ z − n + g0 + 1 ≤ g0 and n−g0 ≤ n−g(n). Also, we note that   n−1∑ z=n−g(n) A(n,z)h(x(z))  2 ≤ g0 n−1∑ z=n−g(n) A2(n,z)h2(x(z)). Hence, δ −1∑ s=−g0 n−1∑ z=n+s A2(n,z)h2(x(z)) ≥ δ g0   n−1∑ z=n−g(n) A(n,z)h(x(z))  2 Thus, V (n) ≥  x(n) + n−1∑ s=n−g(n) A2(n,z)h2(x(z))  2 + δ g0   n−1∑ z=n−g(n) A(n,z)h(x(z))  2 = δ g0 + δ x2(n) +  √ g0 g0 + δ x(n) + √ g0 + δ g0 n−1∑ z=n−g(n) A(n,z)h(x(z))  2 ≥ δ g0 + δ x2(n). But V (n) ≤ V (n0) n−1∏ s=n0 ( (a(n) + A(n + 1,n) ) This implies that δ g0 + δ x2(n) ≤ V (n0) n−1∏ s=n0 ( (a(n) + A(n + 1,n) ) Hence, |x(n)| ≤ √√√√g0 + δ δ V (n0) n−1∏ s=n0 ( a(n) + A(n + 1,n) ) . (2.11) This completes the proof. Corollary 2.3. Suppose that the hypotheses of Theorem 3.2 hold. Suppose that there exists a positive number α < 1 such that 0 < a(n) + A(n + 1,n) ≤ α. Then the zero solution of (1.2) is exponentially stable. 116 Ernest Yankson CUBO 23, 1 (2021) Proof. It follows from (2.10) that |x(n)| ≤ √√√√g0 + δ δ V (n0) n−1∏ s=n0 (a(n) + A(n + 1,n)) ≤ √ g0 + δ δ V (n0)αn−n0 for n ≥ n0. Since α ∈ (0, 1) the proof is complete. 3 Instability Criteria In this section we consider the problem of finding a criteria for instability of the zero solution of (1.2). A suitable Lyapunov functional will be used to obtain the instability criteria. Theorem 3.1. Assume that (1.3), (1.5) hold and let ρ > g0 be a constant. Assume that Q1(n) > 0 and Q(n,x) > 0 such that Q2(n,x) + Q(n,x) −ρA2(n + 1,n)h21(x(n)) ≥ 0. (3.1) If 1 ≤ h1(x) and V (n) =  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))  2 −ρ n−1∑ s=n−g(n−1)−1 A2(n,s)h2(x(s)) (3.2) then, based on the solutions of (1.2) we have ∆V (n) ≥ Q1(n)V (n). Proof. Let x(n,n0,ψ) be a solution of (1.2) and let V (n) be defined by (3.2). Then based on the solutions of (1.2) we have ∆V (n) =  x(n + 1) + n−1∑ s=n−g(n) A(n,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))   +  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))   × ∆  x(n) + n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))   − ρ  A2(n + 1,n)h2(x(n)) + n−1∑ s=n−g(n) ∆nA 2(n,s)h2(x(s))   CUBO 23, 1 (2021) Inequalities and sufficient conditions for exponential stability . . . 117 ≥  (Q(n,x) + 1)x(n) + n−1∑ s=n−g(n) A(n,s)h(x(s))  Q(n,x)x(n) +  x(n) + n−1∑ s=n−g(n) A(n,s)h(x(s))  Q(n,x)x(n) − ρA2(n + 1,n)h2(x(n)) = Q(n,x)V (n) + (Q2(n,x) + Q(n,x) −ρA2(n + 1,n)h21(x(n)))x 2(n) − Q(n,x)   n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))  2 + Q(n,x)ρ n−1∑ s=n−g(n−1)−1 A2(n,s)h2(x(s)) ≥ Q(n,x)V (n) + (Q2(n,x) + Q(n,x) −ρA2(n + 1,n)h21(x(n)))x 2(n) + Q(n,x)(ρ−g0) n−1∑ s=n−g(n−1)−1 A2(n,s)h2(x(s)) ≥ Q(n,x)V (n) ≥ Q1(n)V (n). This completes the proof. Theorem 3.2. Suppose the hypothesis of Theorem 3.1 hold. Then the zero solution of (1.2) is unstable, provided that ∞∏ s=0 (a(n) + A(n + 1,n)) = ∞. Proof. We have from Theorem 3.1 that ∆V (n) ≥ Q1(n)V (n), which implies that V (n) ≥ V (n0) ∞∏ s=n0 (a(s) + A(s + 1,s)). (3.3) Using the definition of V (n) in (3.2) we have that V (n) = x2(n) + 2x(n) n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s)) +   n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))  2 −ρ n−1∑ s=n−g(n−1)−1 A2(n,s)h2(x(s)) (3.4) Now let β = ρ−g0, then from (√g0√ β a− √ β √ g0 b )2 ≥ 0, 118 Ernest Yankson CUBO 23, 1 (2021) we have 2ab ≤ g0 β a2 + β g0 b2. It follows from this inequality that 2x(n) n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s)) ≤ 2|x(n)| ∣∣∣∣∣∣ n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s)) ∣∣∣∣∣∣ ≤ g0 β x2(n) + β g0   n−1∑ s=n−g(n−1)−1 A(n,s)h(x(s))  2 ≤ g0 β x2(n) + β g0 g0 n−1∑ s=n−g(n−1)−1 A2(n,s)h2(x(s)). (3.5) Substituting (3.5) into (3.4) we obtain V (n) ≤ x2(n) + g0 β x2(n) + (β + g0 −ρ) n−1∑ s=n−g(n−1)−1 A2(n,s)h2(x(s)) = β + g0 β x2(n) ≤ ρ ρ−g0 x2(n). Using the last inequality and (3.3) we obtain |x(n)|2 ≥ ρ−g0 ρ V (n) = ρ−g0 ρ V (n0) ∞∏ s=n0 [a(n) + A(n + 1,n)]. This completes the proof. CUBO 23, 1 (2021) Inequalities and sufficient conditions for exponential stability . . . 119 References [1] I. Berezansky, and E. Braverman, “Exponential stability of difference equations with several delays: Recursive approach”, Adv. Difference Edu., article ID 104310, pp. 13, 2009. [2] El-Morshedy, “New explicit global asymptotic stability criteria for higher order difference equations”, vol. 336, no. 1, pp. 262–276, 2007. [3] S. Elaydi, An introduction to Difference Equations, Springer Verlage, New York, 3rd Edition, 2005. [4] M. Islam, and E. Yankson, “Boundedness and stability in nonlinear delay difference equations employing fixed point theory”, Electron. J. Qual. Theory Differ. Equ., vol. 26, 2005. [5] W. Kelley, and A. Peterson, Difference Equations: An Introduction with Applications, Second Edition, Academic Press, New York, 2001. [6] C. Kublik, and Y. Raffoul, “Lyapunov functionals that lead to exponential stability and instability in finite delay Volterra difference equations”, Acta Mathematica Vietnamica, vol. 41, pp. 77–89, 2016. [7] Y. Raffoul, “Stability and periodicity in discrete delay equations”, J. Math. Anal. Appl., vol. 324, pp. 1356–1362, 2006. [8] Y. Raffoul, “Inequalities that lead to exponential stability and instability in delay difference equations”, J. Inequal. Pure Appl. Math, vol. 10, no. 3, article 70, pp. 9, 2009. Introduction Exponential Stability Instability Criteria