Johnson, Jacob D.

Kong, Lingju

Ruddy, Michael G.

Ruys de Perez, Alexander M.

English

Johnson Jacob D.

Kong Lingju

Ruddy Michael G.

Ruys de Perez Alexander M.

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2014, No. 3, 1–8; http://www.math.u-szeged.hu/ejqtde/Existence of positive periodic solutions for higherorder singular functional difference equationsJacob D. Johnson1, Lingju KongB2, Michael G. Ruddy3 andAlexander M. Ruys de Perez41Department of Mathematics, University of Tennessee at Knoxville, Knoxville, TN 37916, USA2Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA3Department of Mathematics, University of Tennessee at Martin, Martin, TN 38237, USA4Department of Mathematics,Vanderbilt University, Nashville, TN 37235, USAReceived 14 January 2014, appeared 13 March 2014Communicated by Paul EloeAbstract. We study a higher order singular functional difference equation on Z. Suffi-cient conditions are obtained for the existence of at least one positive periodic solutionof the equation. Our proof utilizes the nonlinear alternative of Leray–Schauder.Keywords: positive periodic solutions, higher order, functional difference equations,singular, nonlinear alternative of Leray–Schauder.2010 Mathematics Subject Classification: 39A23, 39A10.1 IntroductionNonlinear difference equations have numerous applications in modeling processes in biology,physics, statistics, and many other areas. For this reason, the existence of positive solutionsto these equations is of great interest to many researchers. We refer the reader to [6–12, 15–18]for some recent work on this subject. In this paper, we are concerned with a higher orderfunctional difference equation. To introduce our equation, we let a 6= 1, b 6= 1 be any fixedpositive numbers, and m, k, ω be any fixed positive integers, and for any u : Z→ R, defineLu(n) = u(n + m + k)− au(n + m)− bu(n + k) + abu(n).Here, we study the existence of positive periodic solutions of the higher order functional dif-ference equationLu(n) = f (n, u(n− τ(n))) + r(n), n ∈ Z, (1.1)where f : Z × (0, ∞) → R, τ : Z → Z, and r : Z → R are ω-periodic on Z, and f (n, x) iscontinuous in x.Equation (1.1) with r(n) ≡ 0, i.e., the equationLu(n) = f (n, u(n− τ(n))), n ∈ Z,BCorresponding author. Email: Lingju-Kong@utc.edu2 J. Johnson, L. Kong, M. Ruddy and A. Ruys de Perezhas been recently studied by Wang and Chen in [18] using Krasnosel’skii’s fixed point theorem.When f (n, x) is nonsingular at x = 0, sufficient conditions were found there for the existenceof positive periodic solutions. In this paper, we will establish a new existence criterion forequation (1.1). The nonlinear term f (n, x) is allowed to be singular at x = 0 in (1.1). The proofwill employ a nonlinear alternative of Leray–Schauder. Our approach involves examining aone-parameter family of nonsingular problems constructed from a sequence of nonsingularperturbations of f . For each of these nonsingular problems, we will apply the nonlinear al-ternative of Leray–Schauder to obtain the existence of at least one positive periodic solution.From this sequence of solutions, we will extract a subsequence that converges to a positiveperiodic solution of (1.1). This type of technique has been successfully used in obtaining pos-itive solutions for several classes of singular problems, see, for example, [1, 2, 6, 12, 13]. Ourproofs are partly motivated by these works. Other results on singular problems can be foundin [3–5, 7, 14].As a simple application of our general existence theorem, we also derive some sufficientconditions for the existence of at least one positive periodic solution of the functional differenceequationLu = c(n)(u(n− τ(n)))−α + µd(n)(u(n− τ(n)))β + r(n), n ∈ Z, (1.2)where α ≥ 0 and β ≥ 0 are constants, c, d, and r are ω-periodic functions on Z with c(n) > 0and d(n) ≥ 0 on Z, and µ > 0 is a parameter.The remainder of this paper is laid out as follows. In Section 2, we present our assumptionsand main results. Some preliminary lemmas as well as the proofs are given in Section 3.2 Main resultsIn this paper, for any c, d ∈ Z with c ≤ d, let [c, d]Z denote the discrete interval {c, . . . , d}. Forthe function r(n) given in equation (1.1), define a function γ : Z→ R byγ(n) =ω∑i=1ω∑j=1G(i, j)r(nij), (2.1)whereG(i, j) =aωbωa−ib−j(1− aω)(1− bω) , i, j ∈ [1, ω]Z, (2.2)andnij = n + (i− 1)k + (j− 1)m. (2.3)Letδ1 =aωbω|(1− aω)(1− bω)| min{a−1, a−ω} ·min{b−1, b−ω}andδ2 =aωbω|(1− aω)(1− bω)| max{a−1, a−ω} ·max{b−1, b−ω}.Then, (2.2) implies that{δ1 ≤ G(i, j) ≤ δ2 for i, j ∈ [1, ω]Z if (a− 1)(b− 1) > 0,δ1 ≤ −G(i, j) ≤ δ2 for i, j ∈ [1, ω]Z if (a− 1)(b− 1) < 0.(2.4)We make the following assumptions.Positive periodic solution 3(H1) (k, ω) = (m, ω) = 1, where (x, y) denotes the greatest common divisor of x and y;(H2) either(a) (a− 1)(b− 1) > 0, γ(n) ≥ 0 on Z, and there exist continuous, nonnegative functionsg(x), h(x), and φ(n) such that g(x) > 0 is nonincreasing on (0, ∞), h(x)/g(x) isnondecreasing on (0, ∞), and0 ≤ f (n, x + γ(n− τ(n))) ≤ φ(n)(g(x) + h(x)) for (n, x) ∈ Z× (0, ∞),or(b) (a− 1)(b− 1) < 0, γ(n) ≥ 0 on Z, and there exist continuous, nonpositive functionsg(x), h(x), and nonnegative φ(n) such that g(x) < 0 is nondecreasing on (0, ∞),h(x)/g(x) is nondecreasing on (0, ∞), and0 ≥ f (n, x + γ(n− τ(n))) ≥ φ(n)(g(x) + h(x)) for (n, x) ∈ Z× (0, ∞);(H3) for each q > 0, there exists a continuous function ψq(n) such that either(i) ψq(n) is nonnegative, ψq(n) > 0 for some n ∈ [1, ω]Z, andf (n, x + γ) ≥ ψq(n) for (n, x) ∈ [1, ω]Z × (0, q],if (H2)(a) holds, or(ii) ψq(n) is nonpositive, ψq(n) < 0 for some n ∈ [1, ω]Z, andf (n, x + γ) ≤ ψq(n) for (n, x) ∈ [1, ω]Z × (0, q],if (H2)(b) holds;(H4) there exists R > 0 such thatR > δ2ωg(δR)(1 +h(R)g(R)) ω∑i=1φ(i),where δ = δ1/δ2 ∈ (0, 1].Now, we state our main results.Theorem 2.1. Assume that (H1)–(H4) hold. Then equation (1.1) has at least one positive periodicsolution y(n) satisfying y(n) > γ(n) on [1, ω]Z and 0 < maxn∈[1,ω]Z |y(n)− γ(n)| < R.As a consequence of Theorem 2.1, we have the following corollary.Corollary 2.2. Assume that (H1) holds, (a− 1)(b− 1) > 0, and γ(n) ≥ 0 on Z. Then, we have(i) if β < 1, then equation (1.2) has at least one positive periodic solution for each µ ∈ (0, ∞);(ii) if β ≥ 1, then there exists µ̄ > 0 such that equation (1.2) has at least one positive periodic solutionfor each µ ∈ (0, µ̄).4 J. Johnson, L. Kong, M. Ruddy and A. Ruys de Perez3 Proofs of the main resultsThroughout this section, let X be the set of all real ω-periodic functions on Z. Then, equippedwith the maximum norm ‖u‖ = maxn∈[1,ω]Z |u(n)|, X is a Banach space.Lemma 3.1 below can be proved using [18, Lemma 2.1].Lemma 3.1. Assume (H1) holds. Then for any h ∈ X, u(n) is a periodic solution of the equationLu(n) = h(n), n ∈ Z,if and only if u(n) is a solution of the summation equationu(n) =ω∑i=1ω∑j=1G(i, j)h(nij),where G(i, j) and nij are given by (2.2) and (2.3) respectively.We refer the reader to [1, Theorem 1.2.3] for the following version of the well known non-linear alternative of Leray–Schauder.Lemma 3.2. Let K be a convex subset of a normed linear space X, and let Ω be a bounded open subsetwith p̃ ∈ Ω. Then every compact map N : Ω→ K has at least one of the following properties:(i) N has at least one fixed point in Ω;(ii) there is u ∈ ∂Ω and λ ∈ (0, 1) such that u = (1− λ) p̃ + λNu.For any h ∈ X, it is easy to see thatω∑i=1ω∑j=1h(nij) = ωω∑i=1h(i).We will use this identity in the proof of Theorem 2.1.Now, we are ready to prove our results.Proof of Theorem 2.1. We show the case where (H2)(a) holds. The proof for (H2)(b) is similar.Let Ω = {u ∈ X : ‖u‖ < R}, where R is given in (H4). We first observe that, to prove thetheorem, it suffices to show that the equationLu(n) = f (n, u(n− τ(n)) + γ(n− τ(n))), n ∈ Z. (3.1)has a positive periodic solution u ∈ Ω satisfying 0 < ‖u‖ < R. If fact, if this is true, we lety(n) = u(n) + γ(n). Then, y(n) > γ(n) on [1, ω]Z, 0 < ‖y− γ‖ < R, andLy(n) = Lu(n) + Lγ(n)= f (n, u(n− τ(n)) + γ(n− τ(n))) + r(n)= f (n, y(n− τ(n))) + r(n), n ∈ Z.Thus, y(n) is a positive periodic solution of (1.1) with the required properties.From (H4), there exists k0 ∈N such thatR >1k0+ δ2ωg(δR)(1 +h(R)g(R)) ω∑i=1φ(i). (3.2)Positive periodic solution 5Let K0 = {k0, k0 + 1, . . .}. For any fixed k ∈ K0, consider the family of equationsu(n) =1k+ λω∑i=1ω∑j=1G(i, j) fk(nij, u(nτij) + γ(nτij)) =1k+ λTku(n), n ∈ Z, (3.3)where λ ∈ (0, 1), fk(n, x) = f (n, max{x, 1/k}) for (n, x) ∈ Z×R, nτij = nij − τ(nij), andTku(n) =ω∑i=1ω∑j=1G(i, j) fk(nij, u(nτij) + γ(nτij)).We now prove two claims.Claim 1. For any λ ∈ (0, 1), any possible solution u(n) of (3.3) satisfies u(n) ≥ δ||u|| for n ∈ Z,where δ = δ1/δ2 ∈ (0, 1] is given in (H4).In fact, from (2.4), we haveu(n) ≤ 1k+ δ2ω∑i=1ω∑j=1f (nij, u(nτij) + γ(nτij))andu(n) ≥ 1k+ δ1ω∑i=1ω∑j=1f (nij, u(nτij) + γ(nτij))≥ δ(1k+ δ2ω∑i=1ω∑j=1f (nij, u(nτij) + γ(nτij)))for n ∈ Z. Thus, u(n) ≥ δ||u|| on Z, i.e., Claim 1 holds.Claim 2. For any λ ∈ (0, 1), any possible solution of (3.3) satisfies ‖u‖ 6= R.Suppose the claim is not true and assume that u(n) is a solution of (3.3) for some λ ∈ (0, 1)with ‖u‖ = R. Since λTku(n) ≥ 0 on Z, we have u(n) ≥ 1/k, which implies that u(n) + γ(n) ≥u(n) ≥ 1/k on Z. Then, (3.3) becomesu(n) =1k+ λω∑i=1ω∑j=1G(i, j) f (nij, u(nτij) + γ(nτij)). (3.4)From (2.4), (H2), and Claim 1, it follows thatR = ‖u‖ ≤ 1k0+ λω∑i=1ω∑j=1G(i, j) f (nij, u(nτij) + γ(nτij))≤ 1k0+ δ2ωω∑i=1f (i, u(i− τ(i)) + γ(i− τ(i)))≤ 1k0+ δ2ωω∑i=1φ(i)g(u(i− τ(i)))(1 +h(u(i− τ(i)))g(u(i− τ(i))))≤ 1k0+ δ2ωg(δ‖u‖)(1 +h(‖u‖)g(‖u‖)) ω∑i=1φ(i)=1k0+ δ2ωg(δR)(1 +h(R)g(R)) ω∑i=1φ(i),which contradicts (3.2). Hence, Claim 2 holds.6 J. Johnson, L. Kong, M. Ruddy and A. Ruys de PerezNote that 1/k ≤ 1/k0 < R and (3.3) can be rewritten asu(n) = (1− λ)1k+ λNku(n),where Nku(n) = Tku(n) + 1/k. Clearly, Nk is compact, and as in Claim 1, we haveNku(n) ≥ δ||Nku|| for n ∈ Z.Then, Nk maps Ω into K, where K = {u ∈ X : u(n) ≥ δ‖u‖ on Z}. Therefore, by Lemma 3.2,Nk has at least one fixed point uk ∈ Ω. Thus, for any k ∈ K0, we have proven that the equationuk(n) =1k+ω∑i=1ω∑j=1G(i, j) fk(nij, u(nτij) + γ(nτij)), n ∈ Z,has a solution uk(n) with ‖uk‖ ≤ R. Since uk(n) + γ(n) ≥ uk(n) ≥ 1/k, we see that uk(n) isactually a solution of the equationuk(n) =1k+ω∑i=1ω∑j=1G(i, j) f (nij, u(nτij) + γ(nτij)), n ∈ Z. (3.5)This, together with (H3), implies thatuk(n) >ω∑i=1ω∑j=1G(i, j) f (nij, u(nτij) + γ(nτij))≥ ωδ1ω∑i=1ψR(i) > 0 on Z. (3.6)Since ‖uk‖ ≤ R for all k ∈ K0, we know that the sequence {uk(n)}k∈K0 has a subsequence,which converges uniformly to a function u ∈ X. For simplicity, we still denote this subsequenceby {uk(n)}k∈K0 . If we let k→ ∞ in (3.5) and (3.6), we obtainu(n) =ω∑i=1ω∑j=1G(i, j) f (nij, u(nτij) + γ(nτij)) (3.7)andu(n) ≥ ωδ1ω∑i=1ψR(i) > 0for n ∈ Z. Thus, by Lemma 3.1, u(n) is a positive solution of (3.1). Since ‖u(n)‖ ≤ R andu = limk→∞uk, we have ‖u‖ ≤ R. By an argument similar to the one used to show Claim 2, wehave ‖u‖ < R. Hence, 0 < ‖u‖ < R. This completes the proof of the theorem.Proof of Corollary 2.2. We will apply Theorem 2.1. To this end, let f (n, x) = c(n)x−α + µd(n)xβ,g(x) = x−α, h(x) = µxβ, and φ(n) = max{c(n), d(n)}. Then, (H1), (H2), and (H3) withψq(n) = q−αc(n) hold. Let A = δ2ω ∑ωi=1 φ(i). Then, 0 < A < ∞ and (H4) becomesµ <δαRα+1 − AARα+βfor some R > 0.Hence, equation (1.2) has at least one positive solution for0 < µ < µ̄ := supR>0δαRα+1 − AARα+β.Note that µ̄ = ∞ if β < 1 and µ̄ < ∞ if β ≥ 1. This completes the proof of the corollary.Positive periodic solution 7AcknowledgmentsThe research by Jacob D. Johnson, Michael G. Ruddy, and Alexander M. Ruys de Perez wasconducted as part of a 2013 Research Experience for Undergraduates at the University of Ten-nessee at Chattanooga that was supported by NSF Grant DMS-1261308.References[1] R. P. AGARWAL, D. O’REGAN, Singular Differential and Integral Equations with Appli-cations, Kluwer Academic Publishers, Boston, 2003. MR2011127[2] J. CHU, P. J. TORRES, M. ZHANG, Periodic solutions of second order non-autonomoussingular dynamical systems, J. 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Existence of positive periodic solutions for higher order singular functional difference equations

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Existence of positive periodic solutions for higher order singular functional difference equations

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