Stability in delay Volterra difference equations of neutral type

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1 Proyecciones Journal of Mathematics Vol. 34, N o 3, pp , September Universidad Católica del Norte Antofagasta - Chile Stability in delay Volterra difference equations of neutral type Ernest Yankson Emmanuel K. Essel University of Cape Coast, Ghana Received : March Accepted : June 2015 Abstract Sufficient conditions for the zero solution of a certain class of neutral Volterra difference equations with variable delays to be asymptotically stable are obtained. The Banach s fixed point theorem is employed in proving our results Mathematics Subject Classification: 349A30, 39A70. Keywords : Banach s Fixed point theorem, Volterra difference equation, asymptotic stability.

2 230 Ernest Yankson Emmanuel K. Essel 1. Introduction The study of the stability of the zero solution of difference equations has gained the attention of many mathematicians lately, see [1], [2], [3], [5], [7], [9], [11] [12]. In this paper we consider the nonlinear difference equation with variable delays (1.1) x(n) = a j (n)x(n τ j (n)) Q j (n, x(n τ j (n))) k j (n, s)f j (s, x(s)), s=n τ j (n) with the initial condition x(n) =ψ(n) forn [m(n 0 ),n 0 ] Z, where ψ :[m(n 0 ),n 0 ] Z R is a bounded sequence for n 0 0, m j (n 0 )=inf{n τ j (n), n n 0 },m(n 0 )=min{m j (n 0 ), 1 j N}. Here denotes the forward difference operator. That is, x(n) = x(n 1) x(n) for any sequence {x(n) : n Z }. We assume throughout this paper that a j : Z R, k j : Z ([m j (n 0 ), ) Z) R, f j : Z R R, Q j : Z R R τ j : Z Z, for j =1,...,N. Special cases of (1.1) have been considered by a number of researchers in recent times. For instance, Raffoul in [7] considered the equation (1.2) x(n) = a(n)x(n τ), where τ is a positive constant. The first author in [11], extended the results obtained in [7] to the equation (1.3) x(n) = a j (n)x(n τ j (n)).

3 Stability in delay Volterra difference equations of neutral type 231 The first author also in [12] considered the the following nonlinear Volterra difference equation (1.4) x(n 1) = a(n)x(n)c(n) x(n τ(n)) s=n τ(n) k(n, s)q(x(s)). Ardjouni Djoudi in [1] considered the nonlinear Volterra difference equation with variable delays x(n 1) = a(n)x(n τ 1 (n)) c(n) x(n τ 2 (n)) (1.5) s=n τ 2 (n) k(n, s)q(x(s)). Moreover, Ardjouni Djoudi in [2] considered the difference equations with variable delays (1.6) x(n) = a j (n)x(n τ j (n)) c j (n) x(n τ j (n)). Motivated by the above mentioned papers, we obtain in this paper sufficient conditions for the zero solution of (1.1) to be asymptotically stable. 2. Stability Let n 0 Z [0, ), be fixed. We let D(n 0 ) be the set of bounded sequences ψ :[m(n 0 ),n 0 ] Z R, with the norm ψ 0 =max{ ψ(n) : n [m(n 0 ),n 0 ] Z}. Also, let (B,. ) be the Banach space of bounded sequences x :[m(n 0 ), ) Z R with the maximum norm.. In this paper we assume that for,...,n, (2.1) Q j (n, x) Q j (n, y) L 1 x y, (2.2) f j (n, x) f j (n, y) L 2 x y for some positive constants L 1 L 2. Also, for,...,n,

4 232 Ernest Yankson Emmanuel K. Essel (2.3) f j (n, 0) = 0, Q j (n, 0) = 0, (2.4) n τ j (n) as n. Lemma 2.1 Let h j :[m(n 0 ), ) Z R be an arbitrary sequence, for j = 1,...,N. Suppose that H(n) =1 P N h j (n) 6= 0, for all n [n o, ) Z. Then x is a solution of equation (1.1) if only if x(n) = (2.5) nx 0 1 x(n 0 ) Q j (n 0,x(n 0 τ j (n 0 ))) h j (s)x(s) s=n 0 τ j (n 0 ) H(u) Q j (n, x(n τ j (n))) h j (s)x(s) s=n τ j (n) {h j (n τ j (n)) a j (n)}x(n τ j (n)) s=n 0 [1 H(s)] Q j (s, x(s τ j (s))) [1 H(s)] h j (r)x(r) r=s τ j (s) k j (s, u)f j (u, x(u)) H(u). u=s τ j (s) Proof. Rewrite (1.1) as x(n) = h j (n)x(n) n h j (s)x(s) s=n τ j (n)

5 Stability in delay Volterra difference equations of neutral type 233 {h j (n τ j (n)) a j (n)}x(n τ j (n)) Q j (n, x(n τ j (n))) k j (n, s)f j (s, x(s)), s=n τ j (n) where n denotes the difference taken with respect to n. The above equation is equivalent to (2.6) x(n 1) = H(n)x(n) n N X s=n τ j (n) h j (s)x(s) {h j (n τ j (n)) a j (n)}x(n τ j (n)) Q j (n, x(n τ j (n))) k j (n, s)f j (s, x(s)). s=n τ j (n) Rewrite equation (2.6) as n H(u) 1 x(u) (2.7) = n N X h j (s)x(s) s=n τ j (n) {h j (n τ j (n)) a j (n)}x(n τ j (n)) Q j (n, x(n τ j (n))) s=n τ j (n) k j (n, s)f j (s, x(s)) ny H(u) 1.

6 234 Ernest Yankson Emmanuel K. Essel Summing (2.7) from n 0 to n 1weobtain s=n 0 s s 1 Y H(u) 1 x(s) Consequently, we have = s=n 0 s N X r=s τ j (s) h j (r)x(r) {h j (n τ j (n)) a j (n)}x(n τ j (n)) Q j (n, x(n τ j (n))) sy k j (s, u)f j (u, x(u)) H(u) 1. u=s τ j (s) (2.8) s=n 0 n 0 1 Y H(u) 1 x(n) H(u) 1 x(n 0 ) = s h j (r)x(r) r=s τ j (s) {h j (n τ j (n)) a j (n)}x(n τ j (n)) Q j (n, x(n τ j (n))) sy k j (s, u)f j (u, x(u)) H(u) 1. u=s τ j (s) Dividing both sides of (2.8) by Q n 1 H(u) 1 we obtain x(n) = x(n 0 ) H(u) s=n 0 s N X r=s τ j (s) {h j (n τ j (n)) a j (n)}x(n τ j (n)) Q j (n, x(n τ j (n))) h j (r)x(r)

7 Stability in delay Volterra difference equations of neutral type 235 (2.9) u=s τ j (s) k j (s, u)f j (u, x(u)) H(u). Using the summation by parts formula, we obtain (2.10) H(u) Q j (n, x(n τ j (n))) s=n 0 = Q j (n, x(n τ j (n))) Q j (n 0,x(n 0 τ j (n 0 ))) H(u) Q j (s, x(s τ j (s)))[1 H(s)] H(u), s=n 0 (2.11) s=n 0 H(u) s r=s τ j (s) h j (r)x(r) nx 0 1 = h j (s)x(s) H(u) h j (s)x(s) s=n τ j (n) s=n 0 τ j (n 0 ) [1 H(s)] H(u) h j (r)x(r). s=n 0 r=s τ j (s) Substituting (2.10) (2.11) into (2.9) gives the desired results. Theorem 2.1 Suppose (2.1), (2.2), (2.3) (2.4) hold let h j : [m(n 0 ), ) Z R be an arbitrary sequence, for j =1,...,N, such that H(n) =1 P N h j (n) 6= 0, for all n [n o, ) Z. Suppose further that there exist a constant α (0, 1) such that NL 1 s=n 0 s=n τ j (n) h j (s) h j (n τ j (n)) a j (n)

8 236 Ernest Yankson Emmanuel K. Essel (2.12) 1 H(s) NL 1 1 H(s) L 2 N X r=s τ j (s) k j (s, u) H(u) u=s τ j (s) α. h j (r) (2.13) Moreover, assume that there exist a positive constant G such that H(u) G, (2.14) H(u) 0asn. Then the zero solution of (1.1) is asymptotically stable. Proof. Let >0begiven.Chooseδ>0 such that δg[1 α] α. Let ψ D(n 0 ) such that ψ(n) δ define S = {ϕ B : ϕ(n) =ψ(n) if n [m(n 0 ),n 0 ] Z, ϕ ϕ(n) 0asn }. Then (S,. ) is a complete metric space where,. is the maximum norm. Define the mapping P : S S by (Pϕ)(n) =ψ(n) forn [m(n 0 ),n 0 ] Z, nx 0 1 (Pϕ)(n) = ψ(n 0 ) Q j (n 0,ψ(n 0 τ j (n 0 ))) h j (s)ψ(s) s=n 0 τ j (n 0 ) H(u)

9 Stability in delay Volterra difference equations of neutral type 237 Q j (n, ϕ(n τ j (n))) h j (s)ϕ(s) s=n τ j (n) {h j (n τ j (n)) a j (n)}ϕ(n τ j (n)) s=n 0 [1 H(s)] Q j (s, ϕ(s τ j (s))) [1 H(s)] h j (r)ϕ(r) r=s τ j (s) k j (s, u)f j (u, ϕ(u)) H(u), n n 0. u=s τ j (s) (2.15) Clearly, Pϕ is continuous. We first show that P : S S. Using (2.15) we obtain ½ (Pϕ)(n) δg[1 α] NL 1 h j (s) s=n τ j (n) h j (n τ j (n)) a j (n) s=n 0 1 H(s) NL 1 1 H(s) h j (r) r=s τ j (s) ¾ L 2 k j (s, u) H(u) ϕ u=s τ j (s) δg[1 α]α. We next show that (Pϕ)(n) 0asn. The first term on the right h side of (2.15) goes to zero in view of condition (2.14). Since ϕ(n) 0 n τ j (n) as n, we have that Q j (n, ϕ(n τ j (n))) Q j (n, 0) = 0 as n for j =1,...,N. Thus showing that the second term on the right h side of (2.15) goes to zero as n.

10 238 Ernest Yankson Emmanuel K. Essel Let ϕ S be fixed. Thefactthatϕ(n) 0n τ j (n) as n, implies that, given 1 > 0 there exists N 1 >n τ j (n) for j =1,...,N such that ϕ(s) 1 for s N 1. Thus X h j (s)ϕ(s) N 1 h j (s) s=n τ j (n) s=n τ j (n) α 1 < 1. Thus showing that the third term on the right h side of (2.15) goes to zero as n. We next show that the last term on the right h side of (2.15) goes to zero as n. Since ϕ(n) 0n τ j (n) as n,foreach 2 > 0, there exists N 2 >n 0 such that s N 2 implies ϕ(s τ j (s)) < 2 for j =1,...,N. Thus for n N 2, the last term on the right h side of (2.15) satisfies {h j (n τ j (n)) a j (n)}ϕ(n τ j (n)) s=n 0 [1 H(s)] Q j (s, ϕ(s τ j (s))) [1 H(s)] h j (r)ϕ(r) r=s τ j (s) k j (s, u)f j (u, ϕ(u)) H(u) u=s τ j (s) 2 1 h j (s τ j (s)) a j (s) ϕ(s τ j (s)) s=n 0 1 H(s) L 1 ϕ(s τ j (s)) 1 H(s) h j (r) ϕ(r) r=s τ j (s) L 2 k j (s, u) ϕ(u) H(u) u=s τ j (s) h j (s τ j (s)) a j (s) ϕ(s τ j (s)) s=n 2 1 H(s) L 1 ϕ(s τ j (s)) 1 H(s) h j (r) ϕ(r) r=s τ j (s)

11 Stability in delay Volterra difference equations of neutral type 239 L 2 N X u=s τ j (s) k j (s, u) ϕ(u) H(u) N2 1 max ϕ(σ) X h j (s τ j (s)) a j (s) σ m(n 0 ) s=n 0 1 H(s) L 1 N 1 H(s) L 2 N X 2 r=s τ j (s) k j (s, u) H(u) u=s τ j (s) s=n 2 h j (s τ j (s)) a j (s) 1 H(s) L 1 N 1 H(s) L 2 N X r=s τ j (s) k j (s, u) H(u) u=s τ j (s) 2 2 α<2 2. h j (r) h j (r) Thus showing that the last term on the right h side of (2.15) goes to zero as n. Therefore, (Pϕ) 0asn. It therefore follows that P maps S into S. We finally show that P is a contraction. Let ϕ, η S. Then (Pϕ)(n) (Pη)(n) ½ NL 1 h j (s) s=n τ j (n) h j (n τ j (n)) a j (n) s=n 0 1 H(s) NL 1 1 H(s) h j (r) r=s τ j (s) L 2 N X k j (s, u) u=s τ j (s) α ϕ η. ¾ H(u) ϕ η

12 240 Ernest Yankson Emmanuel K. Essel This shows that P is a contraction. Therefore, by the contraction mapping principle, P has a unique fixed point in S which solves (1.1) for any ϕ S, Pϕ. This shows that the zero solution of (1.1) is stable. Moreover, (Pϕ) 0asn, showing that the zero solution of (1.1) is asymptotically stable. This completes the proof. References [1] A. Ardjouni, A. Djoudi; Stability in nonlinear neutral Volterra difference equations with variable delays, Journal of Nonlinear Evolution Equations Applications, No. 7, pp , (2013). [2] A. Ardjouni, A. Djoudi; Stability in linear neutral difference equations with variable delays, Mathematica Bohemica, No. 3, pp , (2013). [3] S. Elaydi, Periodicity stability of linear Volterra difference systems, Journal of Mathematical Analysis Applications, 181, pp , (1994). [4] S. Elaydi, An Introduction to Difference Equations, Springer, New York, (1999). [5] M. Islam E. Yankson, Boundedness stability in nonlinear delay difference equations employing fixed point theory, Electronic Journal of Qualitative Theory of Differential Equations, No. 26, pp. 1-18, (2005). [6] W.G.KellyA.C.Peterson,Difference Equations : An Introduction with Applications, Academic Press, (2001). [7] Y. N. Raffoul, Stability periodicity in discrete delay equations, Journal of Mathematical Analysis Applications, 324, No. 2, pp , (2006). [8] Y. N. Raffoul, Periodicity in general delay nonlinear difference equations using fixed point theory, Journal of Difference Equations Applications, 10, No. 1315, pp , (2004). [9] Y. N. Raffoul, General theorems for stability boundedness for nonlinear functional discrete systems, Journal of Mathematical Analysis Applications, 279, pp , (2003).

13 Stability in delay Volterra difference equations of neutral type 241 [10] D. R. Smart, Fixed point theorems ; Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, (1974). [11] E. Yankson, Stability in discrete equations with variable delays, Electronic Journal of Qualitative Theory of Differential Equations, No. 8, pp. 1-7, (2009). [12] E. Yankson, Stability of Volterra difference delay equations, Electronic Journal of Qualitative Theory of Differential Equations, No. 20, pp. 1-14, (2006). Ernest Yankson Department of Mathematics Statistics University of Cape Coast Ghana ernestoyank@gmail.com Emmanuel K. Essel Department of Mathematics Statistics University of Cape Coast Ghana ekessel04@yahoo.co.uk