Positive periodic solutions of nonlinear functional. difference equation

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1 Electronic Journal of Differential Equations, Vol. 2002(2002), No. 55, pp. 8. ISSN: URL: or ftp ejde.math.swt.edu (login: ftp) Positive periodic solutions of nonlinear functional difference equations Youssef N. Raffoul Abstract In this paper, we apply a cone theoretic fied point theorem to obtain sufficient conditions for the eistence of multiple positive periodic solutions to the nonlinear functional difference equations (n + ) = a(n)(n) ± λh(n)f((n τ(n))). Introduction Let R denote the real numbers, Z the integers and R + the positive real numbers. Given a < b in Z, let [a, b] = {a, a +,..., b}. In this paper, we investigate the eistence of multiple positive periodic solutions for the nonlinear delay functional difference equation (n + ) = a(n)(n) + λh(n)f((n τ(n))) (.) where a(n), h(n) and τ(n) are T -periodic for T is an integer with T. We assume that λ, a(n), and h(n) are nonnegative with 0 < a(n) < for all n [0, T ]. The eistence of multiple positive periodic solutions of nonlinear functional differential equations have been studied etensively in recent years. We cite some appropriate references here [2] and []. We are particularly motivated by the work of Cheng and Zhang [2] on functional differential equations and the work of Eloe, Raffoul and others [5] on a boundary value problem involving functional difference equation. It is customary when working with boundary value problems, whether in differential or difference equations, to display the desired solution in terms of a suitable Green function and then apply cone theory [, 3, 4, 5, 6, 7, 9]. Since our equation (.) is not of the type of boundary value we obtain a variation of parameters formula and then try to find a lower and upper estimates for the kernel inside the summation. Once those estimates are found we use Krasnoselskii s fied point theorem to show the eistence of multiple positive periodic solutions. In [0], the author studied the eistence of Mathematics Subject Classifications: 39A0, 39A2. Key words: Cone theory, positive, periodic, functional difference equations. c 2002 Southwest Teas State University. Submitted December 22, 200. Published June 3, 2002.

2 2 Positive periodic solutions EJDE 2002/55 periodic solutions of an equation similar to equation (.) using Schauder s Second fied point theorem. Throughout this paper, we denote the product of y(n) from n = a to n = b by b n=a y(n) with the understanding that b n=a y(n) = for all a > b, 2 Positive periodic solutions We now state Krasnosel skii fied point theorem [8]. Theorem 2. (Krasnosel skii) Let B be a Banach space, and let P be a cone in B. Suppose Ω and Ω 2 are open subsets of B such that 0 Ω Ω Ω 2 and suppose that T : P (Ω 2 \Ω ) P is a completely continuous operator such that (i) T u u, u P Ω, and T u u, u P Ω 2 ; or (ii) T u u, u P Ω, and T u u, u P Ω 2. Then T has a fied point in P (Ω 2 \Ω ). Let X be the set of all real T -periodic sequences. This set endowed with the maimum norm = ma n [0,T ] (n), X is a Banach space. The net Lemma is essential in obtaining our results. Lemma 2.2 (n) X is a solution of equation (.) if and only if where (n) = λ G(n, u)h(u)f((u τ(u))) (2.) G(n, u) = s=u+ a(s), u [n, n + T ]. (2.2) s=n a(s) Note that the denominator in G(n, u) is not zero since 0 < a(n) < for n [0, T ]. The proof of Lemma 2. is easily obtained by noting that (.) is equivalent to ( n s= ) a (s)(n) = λh(n)f((n τ(n)) n s= a (s). By summing the above equation from u = n to u = n + T we obtain (2.). Note that since 0 < a(n) < for all n [0, T ], we have N G(n, n) G(n, u) G(n, n + T ) = G(0, T ) M

3 EJDE 2002/55 Youssef N. Raffoul 3 for n u n + T and For each X, define a cone by G(n, u) G(n, n + T ) G(n, n) G(n, n + T ) = N M > 0. P = { y X : y(n) 0, n Z and y(n) η y }, where η = N/M. Clearly, η (0, ). Define a mapping T : X X by (T )(n) = λ G(n, u)h(u)f((u τ(u)) where G(n, u) is given by (2.2). By the nonnegativity of λ, f, a, h, and G, T (n) 0 on [0, T ]. It is clear that (T )(n + T ) = (T )(n) and T is completely continuous on bounded subset of P. Also, for any P we have Thus, Therefore, (T )(n) = λ G(n, u)h(u)f((u τ(u))) G(0, T )h(u)f((u τ(u)). T λ T T = ma T (n) λ G(0, T )h(u)f((u τ(u))). n [0,T ] T (n) = λ G(n, u)h(u)f((u τ(u))) λn h(u)f((u τ(u))) = λn η T. G(0, T ) h(u)f((u τ(u))) M That is, T P is contained in P. In this paper we shall make the following assumptions. (A) the function f : R + R + is continuous (A2) h(n) > 0 for n Z

4 4 Positive periodic solutions EJDE 2002/55 (L) lim 0 (L2) lim = = (L3) lim 0 = 0 (L4) lim = 0 (L5) lim 0 = l with 0 < l < (L6) lim = L with 0 < L <. For the net theorem we let and A = B = ma 0 n T min 0 n T G(n, u)h(u) (2.3) G(n, u)h(u). (2.4) Theorem 2.3 Assume that (A), (A2), (L5), and (L6) hold. Then, for each λ satisfying ηbl < λ < (2.5) Al or ηbl < λ < AL equation (.) has at least one positive periodic solution. (2.6) Proof Suppose (2.5) hold. We construct the sets Ω and Ω 2 in order to apply Theorem 2.. Let ɛ > 0 be such that ηb(l ɛ) λ A(l + ɛ). By condition (L5), there eists H > 0 such that f(y) (l +ɛ)y for 0 < y H. Define Ω = { P : < H } Then, if P Ω, (T )(n) λ(l + ɛ) G(n, u)h(u)(u τ(u))) λ(l + ɛ) G(n, u)h(u) u=o λa(l + ɛ).

5 EJDE 2002/55 Youssef N. Raffoul 5 In particular, T, for all P Ω. Net we construct the set Ω 2. Apply condition (L6) and find H such that f(y) (L ɛ)y, for all y H. Let H 2 = ma{2h, ηh}. Define Then, if P Ω 2, Ω 2 = { P : < H 2 } (T )(n) λ(l ɛ) G(n, u)h(u)(u τ(u))) λ(l ɛ)η G(n, u)h(u) u=o λ(l ɛ)ηb. In particular, T, for all P Ω 2. Apply condition (i) of Theorem 2., and this completes the proof. When condition (2.6) holds, the proof can be similarly obtained by invoking condition (ii) of Theorem 2.. Theorem 2.4 Assume that (A) and (A2) hold. Also, if either (L) and (L4) hold, or, (L2) and (L3) hold, then (.) has at least one positive periodic solution for any λ > 0. Proof: Apply (L) and choose H > 0 such that if 0 < y < H, then f(y) y ληb. Define Ω = { P : < H }. If P Ω, then (T )(n) λ G(n, u)h(u)(u τ(u))) ληb T ληb λ G(n, u)h(u). In particular, T, for all P Ω. In order to construct Ω 2, we consider two cases, f bounded and f unbounded. The case where f is bounded is straight forward. If f(y) is bounded by Q > 0, set Then if P and = H 2, u=o H 2 = ma{2h, λqa}. T (n) λn G(n, u)h(u) λqa H 2.

6 6 Positive periodic solutions EJDE 2002/55 Now assume f is unbounded. Apply condition (L4) and set ɛ > 0 such that if > ɛ, then f(y) < y λa. Set H 2 = ma{2h, ɛ } and define Ω 2 = { P : < H 2 }. If P Ω 2, then (T )(n) λ G(n, u)h(u)(u τ(u)) λa T λa λh 2 G(n, u)h(u) u=o H 2 =. In particular, T, for all P Ω 2. Apply condition (ii) of Theorem 2., and this completes the proof. The proof of the other part, follows similarly by invoking condition (ii) of Theorem 2.. The net two corollaries are consequence of the previous two theorems. Corollary 2.5 Assume that (A) and (A2) hold. Also, if either (L) and (L6) hold, or, (L2) and (L5) hold, then (.) has at least one positive periodic solution if λ satisfies either 0 < λ < /(AL), or, 0 < λ < /(Al). Corollary 2.6 Assume that (A) and (A2) hold. Also, if either (L3) and (L6) hold, or, (L4) and (L5) hold, then (.) has at least one positive periodic solution if λ satisfies either /(ηbl) < λ <, or, /(ηbl) < λ <. Net we turn our attention to the equation (n + ) = a(n)(n) λh(n)f((n τ(n))) (2.7) where λ, a(n), and h(n) satisfy the same assumptions stated for (.) ecept that a(n) > for all n [0, T ]. In view of (2.7) we have that (n) = λ K(n, u)h(u)f((u τ(u))) (2.8) where s=u+ K(n, u) = a(s), u [n, n + T ]. (2.9) s=n a(s) Note that the denominator in G(n, u) is not zero since a(n) > for n [0, T ]. Also, it is easily seen that since a(n) > for all n [0, T ], we have M K(n, n) K(n, u) K(n, n + T ) = K(0, T ) N for n u n + T and K(n, u) K(n, n + T ) = N K(n, n) K(n, n) M > 0.

7 EJDE 2002/55 Youssef N. Raffoul 7 Finally, by defining and A = B = ma 0 n T min 0 n T K(n, u)h(u) K(n, u)h(u) similar theorems and corollaries can be easily stated and proven regarding equation (2.7). We conclude this paper with the following open problems. Assume that (A) and (A2) hold. In view of this paper, what can be said about equations (.) and (2.7) when:. The conditions (L) and (L2) hold? 2. The conditions (L3) and (L4) hold? 3. 0 < a(n) < in (2.7) and a(n) > in (.) for all n [0, T ]? References [] A. Datta and J. Henderson, Differences and smoothness of solutions for functional difference equations, Proceedings Difference Equations (995), [2] S. Cheng and G. Zhang, Eistence of positive periodic solutions for nonautonomous functional differential equations, Electronis Journal of Differential Equations 59 (200), -8. [3] J. Henderson and A. Peterson, Properties of delay variation in solutions of delay difference equations, Journal of Differential Equations (995), [4] R.P. Agarwal and P.J.Y. Wong, On the eistence of positive solutions of higher order difference equations, Topological Methods in Nonlinear Analysis 0 (997) 2, [5] P.W. Eloe, Y. Raffoul, D. Reid and K. Yin, Positive solutions of nonlinear Functional Difference Equations, Computers and Mathematics With applications 42 (200), [6] J. Henderson and S. Lauer, Eistence of a positive solution for an nth order boundary value problem for nonlinear difference equations, Applied and Abstract Analysis, in press. [7] J. Henderson and W. N. Hudson, Eigenvalue problems for nonlinear differential equations, Communications on Applied Nonlinear Analysis 3 (996), 5-58.

8 8 Positive periodic solutions EJDE 2002/55 [8] M. A. Krasnosel skii, Positive solutions of operator Equations Noordhoff, Groningen, (964). [9] F. Merdivenci, Two positive solutions of a boundary value problem for difference equations, Journal of Difference Equations and Application (995), [0] Y. Raffoul, Periodic solutions for scalar and vector nonlinear difference equations, Pan-American Journal of Mathematics 9 (999), 97-. [] W. Yin, Eigenvalue problems for functional differential equations, Journal of Nonlinear Differential Equations 3 (997), Youssef N. Raffoul Department of Mathematics, University of Dayton Dayton, OH USA youssef.raffoul@notes.udayton.edu