Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations

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1 Positivity 1 (26), Birkhäuser Verlag Basel/Switzerland / , published online April 26, 26 DOI 1.17/s Positivity Positive Periodic Solutions of Systems of Second Order Ordinary Differential Equations DONAL O REGAN 1 and HAIYAN WANG 2 1 Department of Mathematics, National University of Ireland, Galway, Ireland. donal.oregan@nuigalway.ie 2 New College of Interdisciplinary Arts and Sciences, Arizona State University, PO Box 371, Phoenix, AZ 8569, USA. wangh@asu.edu Received 26 April 24; accepted 7 August 24 Abstract. We establish the existence, multiplicity and nonexistence of positive periodic solutions of systems of second order ordinary differential equations. Mathematics Subject Classification 2: Primary: 34B15 Key words: Periodic boundary value problem, existence, fixed point theorem, cone 1. Introduction In this paper we consider the existence, multiplicity and nonexistence of positive solutions for the periodic boundary value problem x + m 2 x = λg(t)f(x) t 2π, x() = x(2π), x () = x (2π), (1.1) where m (, 1/2) is a constant, x = [x 1,...,x n ] T, G(t) = diag[g 1 (t), g 2 (t),...,g n (t)], F(x) = [f 1 (x), f 2 (x),...,f n (x)] T, and λ> is a positive parameter. The existence of positive solutions for the periodic scalar boundary value problem have been studied in many papers, see Jiang-Chu-O Regan-Agarwal [3] and Torres [5] and the references therein. It was shown that the scalar periodic boundary value problem has one positive solution provided f is superlinear or sublinear at zero and infinity. Periodic boundary value problems are closely related to the Dirichlet/Neumann boundary value problems, which have been extensively investigated in the literature (see, e.g., Agarwal-O Regan-Wong [1] and Wang [7]). Our arguments as in Jiang- Chu-O Regan-Agarwal [3] and Torres [5] are based on the fixed point index in a cone. We are able to establish several criteria for the existence, multiplicity and nonexistence of positive solutions of (1.1). The same approach

2 286 D. O REGAN AND H. WANG Positivity also was used in Wang [6, 7] to prove analogous results for the existence, multiplicity and nonexistence of positive solutions of boundary value problems solutions for other boundary data. Let R = (, ), R + = [, ), R+ n = n R +, and for any u = (u 1,...,u n ) R+ n, u = n u i. Our assumptions for this paper are: (H1) f i : R+ n [, ) is continuous, i = 1,...,n. (H2) g i (t) : [, 2π] [, ) is continuous and g i (t)dt >,i = 1,...,n. (H3) f i (u) >for u >,i = 1,...,n. In order to state our results we introduce the notation f i = lim f i (u) u F = max,...,n {f i }, Our main results are: u, fi = lim u f i (u) u, u Rn +, i = 1,...,n F = max {f i }. (1.2),...,n THEOREM 1.1. Assume (H1) (H2) hold. (a). If F = and F =, then for all λ> (1.1) has a positive solution. (b). If F = and F =, then for all λ> (1.1) has a positive solution. THEOREM 1.2. Assume (H1) (H3) hold. (a). If F = orf =, then there exists a λ > such that (1.1) has a positive solution for λ>λ. (b). If F = or F =, then there exists a λ > such that (1.1) has a positive solution for <λ<λ. (c). If F = F =, then there exists a λ > such that (1.1) has two positive solutions for λ>λ. (d). If F = F =, then there exists a λ > such that (1.1) has two positive solutions for <λ<λ. (e). If F < and F <, then there exists a λ > such that for all <λ<λ (1.1) has no positive solution. (f). If F > and F >, then there exists a λ > such that for all λ>λ (1.1) has no positive solution. This paper is organized in the following ways. In Section 2, we transform (1.1) into a fixed point problem. Furthermore, we establish several inequalities. In Section 3, we apply the fixed point index to show the existence, multiplicity and nonexistence of positive solutions of (1.1) based on the inequalities.

3 Vol. 1 (26) POSITIVE PERIODIC SOLUTIONS OF SYSTEMS Preliminaries We recall some concepts and conclusions on the fixed point index in a cone in Guo-Lakshmikantham [2] and Krasnoselskii [4]. Let E be a Banach space and K be a closed, nonempty subset of E. K is said to be a cone if (i) αu + βv K for all u, v K and all α, β and (ii) u, u K imply u =. Assume is a bounded open subset in E with the boundary, and let T : K K be completely continuous such that Tx x for x K, then the fixed point index i(t,k,k) is defined. If i(t,k,k), then T has a fixed point in K. The following wellknown result of the fixed point index is crucial in our arguments. LEMMA 2.1. (Guo-Lakshmikantham [2] and Krasnoselskii [4]). Let E be a Banach space and K a cone in E. Forr >, define K r ={v K : x < r}. Assume that T : K r K is completely continuous such that Tx x for x K r ={v K : x =r}. (i) If Tx x for x K r, then i(t,k r,k)=. (ii) If Tx x for x K r, then i(t,k r,k)= 1. Following Jiang-Chu-O Regan-Agarwal [3], we consider the function G(t, s) = { sin m(t s)+sin m(2π t+s) 2m(1 cos 2mπ) s t 2π, sin m(s t)+sin m(2π s+t) 2m(1 cos 2mπ) t s 2π. (2.3) Let Ĝ(x) = (sin(mx) + sin m(2π x))/(2m(1 cos 2mπ)), x [, 2π]. It is easy to check that Ĝ is increasing on [,π] and decreasing on [π, 2π], and G(t, s) = Ĝ( t s ). Thus sin 2mπ = Ĝ() G(t, s) Ĝ(π) = 2m(1 cos 2mπ) sin mπ m(1 cos 2mπ) for s, t [, 2π]. Let X be the Banach space C[, 2π] C[, 2π] }{{} n (u 1,...,u n ) X, and for u = u = sup t [,2π] u i (t).

4 288 D. O REGAN AND H. WANG Positivity For u X or R+ n, u denotes the norm of u in X or Rn +, respectively. Let K be the cone given by K ={u = (u 1,...,u n ) X : u i (t), and min u i (t) σ u }, t 2π t [, 2π], i = 1,...,n, where σ = cos mπ >. Also, define, for r a positive number, r by r ={u K : u <r}. Note that r ={u K : u =r}. Let T λ : K X be a map with components (T 1 λ,...,tn λ ). We define T i λ, i = 1,...,n,by Tλ i u(t) = λg(t, s)g i (s)f i (u(s))ds, t 2π. (2.4) LEMMA 2.2. Assume (H1) (H2) hold. Then T λ (K) K and T λ : K K is compact and continuous. Proof. Let u K, then, for i = 1,...,n min T λ i u(t) Ĝ()λ t 2π = σ Ĝ(π)λ σ sup t 2π g i (s)f i (u(s))ds T i λ u(t). g i (s)f i (u(s))ds Thus, T λ (K) K (note min n [,2π] T λ iu(t) n min [,2π] Tλ i u(t) ). It is easy to verify that T λ is compact and continuous. Notice that u K is a positive fixed point of T λ if only if u is a positive solution of (1.1). Let Ɣ = Ĝ()σ min g i (s)ds >.,...,n

5 Vol. 1 (26) POSITIVE PERIODIC SOLUTIONS OF SYSTEMS 289 LEMMA 2.3. Assume (H1) (H2) hold. Let u = (u 1 (t),...,u n (t)) K and η>. If there exists a component f i of f such that then f i (u(t)) η u i (t) for t [, 2π] T λ u λɣη u. Proof. From the definition of T λ u it follows that T λ u max T λ i u(t) t 2π λĝ() λĝ() λĝ()σ = λɣη u. For each i = 1,...,n, let Let g i (s)f i (u(s))ds g i (s)η u i (s)ds g i (s)dsη u fˆ i (t) = max{f i (u) : u R+ n and u t}. ˆ f i = lim t ˆ f i (t): R + R + be the function given by ˆ f i (t) t and ˆ f i = lim t ˆ f i (t) t. LEMMA 2.4. (Wang [7]). Assume (H1) holds. Then ˆ f i = f i and ˆ f i = f i, i = 1,...,n. LEMMA 2.5. Assume (H1) (H2) hold and let r>. If there exits an ε> such that ˆ f i (r) εr, i = 1,...,n, then T λ u λĉε u for u r, where the constant Ĉ = Ĝ(π) n g i (s)ds.

6 29 D. O REGAN AND H. WANG Positivity Proof. From the definition of T λ,wehaveforu r, T λ u = max T λ i u(t) t 2π λĝ(π) g i (s)f i (u(s))ds λĝ(π) g i (s) fˆ i (r)ds λĝ(π) g i (s)dsε u = λĉε u. The following two lemmas are weak forms of Lemmas 2.3 and 2.5. LEMMA 2.6. Assume (H1) (H3) hold. If u r, r>, then T λ u λ ˆm r Ɣ 1 σ, where ˆm r = min{f i (u) : u R+ n and σ r u r,,...,n} >. Proof. Since f i (u(t)) ˆm r for t [, 2π], i = 1,...,n (we just note that r = u sup [,2π] n u i(t) inf [,2π] n u i(t) σr if u r ), a slight modification of the proof in Lemma 2.3 yields the result. LEMMA 2.7. Assume (H1) (H3) hold. If u r, r>, then T λ u λ ˆM r Ĉ, where ˆM r = max{f i (u) : u R+ n and u r, i = 1,...,n} > and Ĉ is the positive constant defined in Lemma 2.5. Proof. Since f i (u(t)) ˆM r for t [, 2π], i = 1,...,n, a slight modification of the proof in Lemma 2.5 guarantees the result.

7 Vol. 1 (26) POSITIVE PERIODIC SOLUTIONS OF SYSTEMS Proof of Theorem 1.1 Proof. Part (a). F = implies that f i =, i = 1,...,n. It follows from Lemma 2.4 that fˆ i =, i = 1,...,n. Therefore, we can choose r 1 > so that fˆ i (r 1 ) εr 1,i= 1,...,n, where the constant ε> satisfies λεĉ <1, and Ĉ is the positive constant defined in Lemma 2.5. We have by Lemma 2.5 that T λ u λεĉ u < u for u r1. Now, since F =, there exists a component f i of F such that f i =. Therefore, there is an H> ˆ such that f i (u) η u for u = (u 1,...,u n ) R n + and u ˆ H, where η> is chosen so that λɣη > 1. Let r 2 = max{2r 1, 1 H ˆ }. Ifu = (u σ 1,...,u n ) r2, then min t 2π u i (t) σ u =σr 2 which implies that ˆ H, f i (u(t)) η u i (t) for t [, 2π]. It follows from Lemma 2.3 that T λ u λɣη u > u for u r2. By Lemma 2.1, i(t λ, r1,k)= 1 and i(t λ, r2,k)=.

8 292 D. O REGAN AND H. WANG Positivity It follows from the additivity of the fixed point index that i(t λ, r2 \ r1,k)= 1. Thus, i(t λ, r2 \ r1,k), which implies T λ has a fixed point u r2 \ r1 by the existence property of the fixed point index. The fixed point u r2 \ r1 is the desired positive solution of (1.1). Part (b). If F =, there exists a component f i such that f i =. Therefore, there is an r 1 > such that f i (u) η u for u = (u 1,...,u n ) R n + and u r 1, where η> is chosen so that λɣη > 1. If u = (u 1,...,u n ) r1, then f i (u(t)) η u i (t) for t [, 1]. Lemma 2.3 implies that T λ u λɣη u > u for u r1. We now determine r2. F = implies that f i =, i = 1,...,n. It follows from Lemma 2.4 that fˆ i =, i = 1,...,n. Therefore there is an r 2 > 2r 1 such that ˆ f i (r 2 ) εr 2, i = 1,...,n, where the constant ε> satisfies λεĉ <1, and Ĉ is the positive constant defined in Lemma 2.5. Thus, we have by Lemma 2.5 that T λ u λεĉ u < u for u r2. By Lemma 2.1, i(t λ, r1,k)= and i(t λ, r2,k)= 1.

9 Vol. 1 (26) POSITIVE PERIODIC SOLUTIONS OF SYSTEMS 293 It follows from the additivity of the fixed point index that i(t λ, r2 \ r1,k) = 1. Thus, T λ has a fixed point in r2 \ r1, which is the desired positive solution of (1.1). 4. Proof of Theorem 1.2 Proof. Part (a). Fix a number r 1 >. Lemma 2.6 implies that there exists a λ > such that T λ u > u for u r1, λ>λ. If F =, then f i =, i = 1,...,n. It follows from Lemma 2.4 that ˆ f i =, i = 1,...,n. Therefore, we can choose <r 2 <r 1 so that ˆ f i (r 2 ) εr 2, i = 1,...,n, where the constant ε> satisfies λεĉ <1, and Ĉ is the positive constant defined in Lemma 2.5. We have by Lemma 2.5 that T λ u λεĉ u < u for u r2. If F =, then f i =, i = 1,...,n. It follows from Lemma 2.4 that fˆ i =, i = 1,...,n. Therefore there is an r 3 > 2r 1 such that ˆ f i (r 3 ) εr 3, i = 1,...,n, where the constant ε> satisfies λεĉ <1, and Ĉ is the positive constant defined in Lemma 2.5. Thus, we have by Lemma 2.5 that T λ u λεĉ u < u for u r3.

10 294 D. O REGAN AND H. WANG Positivity It follows from Lemma 2.1 that i(t λ, r1,k)=, i(t λ, r2,k)= 1 and i(t λ, r3,k)= 1. Thus i(t λ, r1 \ r2,k) = 1and i(t λ, r3 \ r1,k) = 1. Hence, T λ has a fixed point in r1 \ r2 or r3 \ r1 according to F = orf =, respectively. Consequently, (1.1) has a positive solution for λ>λ. Part (b). Fix a number r 1 >. Lemma 2.7 implies that there exists a λ > such that T λ u < u for u r1, <λ<λ. If F =, there exists a component f i of F such that f i there is a positive number r 2 <r 1 such that =. Therefore, f i (u) η u for u = (u 1,...,u n ) R+ n and u r 2, where η> is chosen so that λɣη > 1. Then f i (u(t)) η u i (t), for u = (u 1,...,u n ) r2, t [, 1]. Lemma 2.3 implies that T λ u λɣη u > u for u r2. =. There- If F =, there exists a component f i of F such that f i fore, there is an H> ˆ such that f i (u) η u for u = (u 1,...,u n ) R n + and u ˆ H, where η> is chosen so that λɣη > 1. Let r 3 = max{2r 1, ˆ H σ }.Ifu = (u 1,...,u n ) r3, then min t 2π u i (t) σ u =σr 3 ˆ H,

11 Vol. 1 (26) POSITIVE PERIODIC SOLUTIONS OF SYSTEMS 295 which implies that f i (u(t)) η u i (t) for t [, 2π]. It follows from Lemma 2.3 that T λ u λɣη u > u for u r3. It follows from Lemma 2.1 that i(t λ, r1,k)= 1, i(t λ, r2,k)= and i(t λ, r3,k)=, and hence, i(t λ, r1 \ r2,k) = 1 and i(t λ, r3 \ r1,k) = 1. Thus, T λ has a fixed point in r1 \ r2 or r3 \ r1 according to f = or f =, respectively. Consequently, (1.1) has a positive solution for <λ<λ. Part (c). Fix two numbers <r 3 <r 4. Lemma 2.6 implies that there exists a λ > such that we have for λ>λ, T λ u > u for u ri (i = 3, 4). Since F = and F =, it follows from the proof of Theorem that we can choose <r 1 <r 3 /2 and r 2 > 2r 4 such that 1.2 (a) T λ u < u for u ri (i = 1, 2). It follows from Lemma 2.1 that i(t λ, r1,k)= 1, i(t λ, r2,k)= 1, and i(t λ, r3,k)=, i(t λ, r4,k)= and hence, i(t λ, r3 \ r1,k)= 1 and i(t λ, r2 \ r4,k)= 1. Thus, T λ has two fixed points u 1 (t) and u 2 (t) such that u 1 (t) r3 \ r1 and u 2 (t) r2 \ r4, which are the desired distinct positive solutions of (1.1) for λ>λ satisfying r 1 < u 1 <r 3 <r 4 < u 2 <r 2.

12 296 D. O REGAN AND H. WANG Positivity Part (d). Fix two numbers <r 3 <r 4. Lemma 2.7 implies that there exists a λ > such that we have, for <λ<λ, T λ u < u for u ri (i = 3, 4). Since F = and F =, it follows from the proof of Theorem that we can choose <r 1 <r 3 /2 and r 2 > 2r 4 such that 1.2 (b) T λ u > u for u ri (i = 1, 2). It follows from Lemma 2.1 that i(t λ, r1,k)=, i(t λ, r2,k)=, and i(t λ, r3,k)= 1, i(t λ, r4,k)= 1 and hence, i(t λ, r3 \ r1,k) = 1 and i(t λ, r2 \ r4,k) = 1. Thus, T λ has two fixed points u 1 (t) and u 2 (t) such that u 1 (t) r3 \ r1 and u 2 (t) r2 \ r4, which are the desired distinct positive solutions of (1.1) for λ<λ satisfying r 1 < u 1 <r 3 <r 4 < u 2 <r 2. Part (e). Since F < and F <, then f i < and f i <, i = 1,...,n. It is easy to show (see Wang [7]) that there exists an ε> such that f i (u) ε u for u R n + (i = 1,...,n). Assume v(t) is a positive solution of (1.1). We will show that this leads to a contradiction for <λ<λ, where λ = 1 n Ĝ(π) g i (s)dsε.

13 Vol. 1 (26) POSITIVE PERIODIC SOLUTIONS OF SYSTEMS 297 In fact, for <λ<λ, since T λ v(t) = v(t) for t [, 1], we find v = T λ v = max T λ i v(t) t 2π λĝ(π) g i (s)f i (v(s))ds λĝ(π) g i (s)dsε u < v, which is a contradiction. Part (f). Since F > and F >, there exist two components f i and f j of F such that f i > and f j >. It is easy to show (see Wang [7]) that there exist positive numbers η, r 1 such that and f i (u) η u for u R n +, u r 1 (4.5) f j (u) η u for u R n +, u σr 1. (4.6) Assume v(t) = (v 1,...,v n ) is a positive solution of (1.1). We will show that this leads to a contradiction for λ>λ = 1 Ɣη. In fact, if v r 1, (4.5) implies that f i (v(t)) η v i (t) for t [, 1]. On the other hand, if v >r 1, then min t 2π v i (t) σ v >σr 1, which, together with (4.6), implies that f j (v(t)) η v i (t) for t [, 2π].

14 298 D. O REGAN AND H. WANG Positivity Since T λ v(t) = v(t) for t [, 1], it follows from Lemma λ> λ, 2.3 that, for v = T λ v λɣη v > v, which is a contradiction. References 1. Agarwal, R. O Regan, D. and Wong, P.: Positive Solution of Differential, Difference and Integral Equations, Kluwer, Dordrecht, Guo, D. and Lakshmikantham, V.: Nonlinear Problems in Abstract Cones, Academic Press, Orlando, FL, Jiang, D. Chu, J. O Regan D. and Agarwal, R.: Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces, J. Math. Anal. Appl. 286 (23), Krasnoselskii, M.: Positive Solutions of Operator Equations, Noordhoff, Groningen, Torres, P.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Diff. Eqs. 19 (23), Wang, H.: On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Differential Equations 19 (1994), Wang, H.: On the number of positive solutions of nonlinear systems. J. Math. Anal. Appl. 281 (23),