Positive Periodic Solutions in Neutral Delay Difference Equations

Transcription

1 Advances in Dynamical Systems and Applications ISSN , Volume 5, Number, pp (200) Positive Periodic Solutions in Neutral Delay Difference Equations Youssef N. Raffoul University of Dayton Department of Mathematics Dayton, OH , U.S.A. Ernest Yankson University of Cape Coast Department of Mathematics and Statistics Cape Coast, Ghana Abstract We use Krasnoselskii s fixed point theorem to obtain sufficient conditions for the existence of a positive periodic solution of the neutral delay difference equation x(n + ) = a(n)x(n) + c x(n τ) + g(n, x(n τ)). AMS Subject Classifications: 39A0, 39A2. Keywords: Krasnoselskii, neutral, positive periodic solutions. Introduction In this paper we use Krasnoselskii s fixed point theorem to study the existence of positive periodic solutions of a certain type of difference equation with delay which appear in ecological models. The existence of positive periodic solutions of functional differential equations has gained the attention of many researchers in recent times. We are mainly motivated by the work of the first author 5 and the references therein on neutral differential equations. Received December 7, 2009; Accepted May 20, 200 Communicated by Martin Bohner

2 24 Y. N. Raffoul and E. Yankson Let τ be a nonnegative integer and consider the neutral delay difference equation x(n + ) = a(n)x(n) + c x(n τ) + g(n, x(n τ)), (.) where g is continuous in x. The operator is defined as x(n) = x(n + ) x(n). In this paper, we denote by E the shift operator defined as Ex(n) = x(n + ). Also, the b product of x(n) from n = a to n = b is denoted by x(n). For more on the calculus of difference equation we refer the reader to and 2. In the continuous case, equations in the form of. have applications in food-limited populations, see biological 5 and the references therein. In 3, the first author considered a more complicated form of. and analyzed the existence of periodic solutions. On the other hand, the second author studied the boundedness of solutions and the stability of the zero solution. In 4, using cone theory, the first author obtained sufficient conditions that guaranteed the existence of multiple positive periodic solutions for the nonlinear delay difference equation n=a x(n + ) = a(n)x(n) ± λh(n)f(x(n τ(n))). 2 Preliminaries We begin this section by introducing some notations. Let P T be the set of all real T - periodic sequences, where T is an integer with T. Then P T is a Banach space when it is endowed with the maximum norm x = It is natural to ask for the periodicity condition max x(n). n 0,T a(n + T ) = a(t ), g(n + T, ) = g(n, ), (2.) to hold for all n Z. In addition to (2.), we assume that 0 < a(n) <. (2.2) Let s=u+ G(n, u) = a(s), u n, n + T. (2.3) s=n a(s) Note that the denominator in G(n, u) is not zero since 0 < a(n) < for n 0, T. Also, let m := min{g(n, u) : n 0, u T } = G(n, n) > 0, (2.4) M := max{g(n, u) : n 0, u T } = G(n, n + T ) = G(0, T ) > 0. (2.5)

3 Delay Difference Equations 25 Lemma 2.. Suppose (2.) and (2.2) hold. If x(n) P T, then x(n) is a solution of (.) if and only if x(n) = cx(n τ) + where G(n, u) is defined by (2.3). G(n, u) g(u, x(u τ)) c( a(u))x(u τ), (2.6) Proof. Rewrite (.) as x(n) a (s) = c x(n τ) + g(n, x(n τ)) Summing (2.7) from n to n + T, we obtain i.e., u x(u) a (s) = x(n + T ) a (s) x(n) a (s) = Since x(n + T ) = x(n), we obtain x(n) a (s) a (s) But c x(u τ) = cx(n τ) = cx(n τ) u = a (s) = c E n a (s). (2.7) c x(u τ) + g(u, x(u τ)) u a (s), c x(u τ) + g(u, x(u τ)) c x(u τ) + g(u, x(u τ)) u a (s) a (s) a (s) a (s) x(u τ)c a(u) u a (s). a (s) x(u τ) c x(u τ) u a (s). u a (s). u a (s)

4 26 Y. N. Raffoul and E. Yankson Thus (2.7) becomes x(n) a (s) a (s) = cx(n τ) x(u τ)c a(u) u a (s) + Dividing both sides of the above equation by proof. Now for n 0, Equation (.) is equivalent to x(n) 0 s= u a (s) a (s) a (s) g(u, x(u τ)). a (s) a (s) completes the a (s) = c x(n τ) + g(n, x(n τ)) 0 s=n+ a (s). Summing the above expression from n to n + T, we obtain (.) by a similar argument. We next state Krasnoselskii s theorem in the following lemma. Lemma 2.2 (Krasnoselskii). Let M be a closed convex nonempty subset of a Banach space (B, ). Suppose that C and B map M into B such that (i) x, y M implies Cx + By M; (ii) C is continuous and CM is contained in a compact set; (iii) B is a contraction mapping. Then there exists z M with z = Cz + Bz. 3 Main Results In this section we obtain the existence of positive periodic solution of (.). For some nonnegative constant L and a positive constant K we define the set M = {φ P T : L φ K}, (3.) which is a closed convex and bounded subset of the Banach space P T. In addition we assume that for all u Z and ρ M, ( c)l mt g(u, ρ) c a(u)ρ ( c)k, (3.2) MT

5 Delay Difference Equations 27 where m and M are defined by (2.4) and (2.5), respectively. We will treat separately the cases 0 c < and < c 0. Thus, for our first theorem we assume 0 c <. (3.3) To apply the theorem stated in Lemma 2.2, we will need to construct two mappings; one is contraction and the other is compact. In view of this we define the map B : M P T by (Bϕ)(n) = cx(n τ). In a similar way we define the map C : M P T by (Cϕ)(n) = G(n, u) g(u, x(u τ)) c( a(u))x(u τ). It is clear from condition (3.3) that B defines a contraction map under the supremum norm. Lemma 3.. If (2.), (2.2), (3.2) and (3.3) hold, then the operator C is completely continuous on M. Proof. For n 0, T and for ϕ M, we have by (3.2) that (Cϕ)(n) T M G(n, u)g(u, x(u τ)) c( a(u))x(u τ) ( c)k MT = ( c)k. From the estimation of Cϕ(n) it follows that Cϕ ( c)k K. This shows that C(M) is uniformly bounded. Due to the continuity of all terms, we have that C is continuous. Next, we show that A maps bounded subsets into compact sets. Let J be given, S = {ϕ P T : ϕ J} and Q = {(Cϕ)(t) : ϕ S}. Then S is a subset of R T which is closed and bounded and thus compact. As C is continuous in ϕ, it maps compact sets into compact sets. Therefore Q = C(S) is compact. This completes the proof. Theorem 3.2. Suppose that (2.), (2.2), (3.2) and (3.3) hold. Then equation (.) has a positive periodic solution z satisfying L z K.

6 28 Y. N. Raffoul and E. Yankson Proof. Let ϕ, ψ M. Then, by (3.2), we have that (Aϕ)(n) + (Cψ)(n) = cϕ(n τ) + ck + M ck + MT On the other hand, (Aϕ)(n) + (Cψ)(n) = cϕ(n τ) + cl + m cl + mt G(n, u) g(u, ψ(u τ)) c( a(u))ψ(u τ) g(u, ψ(u τ)) c( a(u))ψ(u τ) ( c)k MT ( c)l mt = K. G(n, u) g(u, ψ(u τ)) c( a(u))ψ(u τ) g(u, ψ(u τ)) c( a(u))ψ(u τ) = L. This shows that Aϕ + Cψ M. All the hypotheses of the theorem stated in Lemma 2.2 are satisfied and therefore equation (.) has a positive periodic solution, say z, residing in M. This completes the proof. For the next theorem we substitute conditions (3.2) and (3.3) with and for all u R and ρ M < c 0 (3.4) L ck mt g(u, ρ) c a(u)ρ K cl MT, (3.5) where M and m are defined by (2.4) and (2.5), respectively. Theorem 3.3. If (2.), (2.2), (3.2), (3.4) and (3.5) hold, then Equation (.) has a positive periodic solution z satisfying L z K. Proof. The proof follows along the lines of Theorem 3.2, and hence we omit it.

7 Delay Difference Equations 29 4 Example The neutral difference equation x(n + ) = 8 x(n) + 0 x(n 4) + x 2 (n 4) x(n 4) + 20 (4.) has a positive periodic solution x of period 4 satisfying we have g(u, ρ) = 8428 ρ ρ + 20, a(n) = 8, c =, and T = 4. 0 φ 2. To see this, A simple calculation yields M = and m = 8. Let K = 2, L = , and define the set { } M = φ P 4 : 8428 φ 2. Then for ρ 8428, 2 we have On the other hand, g(u, ρ) c a(u)ρ = g(u, ρ) c a(u)ρ = ρ ρ + 20 ρ K( c) = 0.26 <. MT ρ ρ + 20 ρ 0 8 > 7 80 ρ ρ > L( c) 2 >. 80 mt By Theorem 3.2, equation (4.) has a positive periodic solution x with period 4 such that x References S. Elaydi, An Introduction to Difference Equations, Second edition. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 999.

8 30 Y. N. Raffoul and E. Yankson 2 W. Kelley and A. Peterson, Difference Equations: An Introduction with Applications, Harcourt Academic Press, San Diego, Y. Raffoul, Periodic solutions for scaler and vector nonlinear difference equations, Panamer. J. Math. 9 (999), Y. Raffoul, Positive periodic solutions of nonlinear functional difference equations, Electr. J. Diff. Eq. 55 (2002), 8. 5 Y. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, Electr. J. Qual. Th. Diff. Eq. 6 (2007), 0.