Parameter-dependent periodic problems for non-autonomous Duffing equations with sign-changing forcing term

Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 65, pp. 1–23. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.65 PARAMETER-DEPENDENT PERIODIC PROBLEMS FOR NON-AUTONOMOUS DUFFING EQUATIONS WITH SIGN-CHANGING FORCING TERM JIˇ R´ Iˇ SREMR Abstract. We study the existence, exact multiplicity, and structure of the set of positive solutions to the periodic problem u00 =p(t)u+h(t)|u|λsgn u+µf(t); u(0) = u(ω), u0(0) = u0(ω), where µ∈Ris a parameter. We assume that p, h, f ∈L([0, ω]), λ > 1, and the function his non-negative. The results obtained extend the results known in the existing literature. We do not require that the Green’s function of the corresponding linear problem be positive and we allow the forcing term fto change its sign. 1. Statement of the problem We consider the periodic problem u00 =p(t)u+h(t)|u|λsgn u+µf(t); u(0) = u(ω), u0(0) = u0(ω),(1.1) where p, h, f ∈L([0, ω]), h≥0 a.e. on [0, ω], λ > 1, and µ∈Ris a parameter. By a solution to problem (1.1), as usual, we understand a function u: [0, ω]→R which is absolutely continuous together with its first derivative, satisfies the given equation almost everywhere, and meets the periodic conditions. In [11], we considered problem (1.1) with µ= 0 and we showed, among other things, that for the existence of a positive solution it is necessary that p6∈ V−(ω)∪ V0(ω). Using a technique developed in [11], we provided in [15] effective conditions for the existence and exact multiplicity of positive solutions to the periodic problem for a non-autonomous Duffing equation with a sign-changing forcing term, i.e., problem (1.1) with µ= 1. In the present paper, we conclude our studies and show, in the case of p6∈ V−(ω)∪V0(ω), the existence/non-existence as well as the exact multiplicity of sign-constant solutions to problem (1.1) depending on the choice of the parameter µ. The results obtained are compared with the results known for the autonomous case and the results available in the existing literature. For the results covering the multiplicity and local/global bifurcations of periodic solutions to super-linear equations (and their systems), we refer the readers, for instance, to [1, 2, 3, 4, 6, 8, 12, 13] (see also the references therein). We studied 2020 Mathematics Subject Classification. 34B08, 34C23, 34C25, 34B18. Key words and phrases. Periodic solution; second-order differential equation; existence; Duffing equation; multiplicity; bifurcation; positive solution. ©2023. This work is licensed under a CC BY 4.0 license. Submitted November 22, 2022. Published October 5, 2023. 1 2 J. ˇ SREMR EJDE-2023/65 a bifurcation of positive solutions to problem (1.1), with the non-positive function h, in [14]. In [2], the authors study the parameter-dependent problem x00 +cx0+a(t)x−b(t)x3=λd(t); x(0) = x(T), x0(0) = x0(T),(1.2) where c > 0, λ∈Ris a parameter, and a, b, d: [0, T ]→Rare continuous functions such that a(t)≤π2 T2+c2 4for t∈[0, T],ZT 0 a(s) ds > 0,(1.3) and b(t)>0, d(t)>0 for t∈[0, T].(1.4) Theorem 1.1 ([2, Theorem 1.1]).Assume that (1.3) and (1.4) hold. Then, all solutions to (1.2) are of one sign and there is λ0>0such that (1) problem (1.2) has a unique solution which is negative (positive) and unstable for λ>λ0(λ < −λ0), (2) problem (1.2) has exactly three ordered solutions for |λ|<|λ0|. Moreover, the middle solution is asymptotically stable and the remaining two are unstable. When −λ0< λ < 0, the maximal solution is positive and the other two are negative. When λ= 0, problem (1.2) has one positive, one 0, and one negative solution. When 0< λ < λ0, the minimal solution is negative and the other two are positive. (3) problem (1.2) has exactly two one-signed solutions for λ=±λ0; both of them are unstable. Recently, Liang [8] proved the conclusion of Theorem 1.1 under the positivity of a, b, d and the hypothesis kakp≤(1 + c2)K(2p∗) with some p≥1. It seems from the proof of Theorem 1.1 that its conclusions, which concern the existence and multiplicity of solutions, remain true even in the case of c= 0. In Section 3, we extend the conclusions of Theorem 1.1 for the case of undamped Duffing equation (i.e., for c= 0). Moreover, we weaken hypotheses (1.3) and (1.4). In particular, (1.3) is replaced by a weaker assumption −a∈ V+(T) (see Definition 2.1), bmay be equal to zero on a set of positive measure, and dmay change its sign so that (−a, d)∈ U(T) (see Definition 2.7). Furthermore, we prove the existence/non-existence of solutions to problem (1.2), with c= 0, depending on the choice of the parameter λin the case of a(t)>π2 T2on a set of positive measure. At the end of this section, we show, as a motivation, what happens in the autonomous case of (1.1). If p(t) := −a, then p6∈ V−(ω)∪V0(ω) if and only if a > 0 (see Remark 2.4). Therefore, we consider the equation x00 =−ax +b|x|λsgn x+µ, (1.5) where a > 0 and b, µ ∈R. In this paper, we are interested in the equation in (1.1) with a non-negative hand, thus, we assume that b > 0 in (1.5). By direct calculation, the phase portraits of this equation can be elaborated depending on the choice of the parameter µand, thus, one can prove the following proposition concerning periodic solutions to equation (1.5). Proposition 1.2. Let λ > 1and a, b > 0. Then, the following conclusions hold: (i) If µ > (λ−1)a λa λb 1 λ−1, then equation (1.5) has a unique negative equilibrium (saddle) and no other periodic solutions occur. EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 3 (ii) If µ=(λ−1)a λa λb 1 λ−1, then equation (1.5) has a unique positive equilibrium (cusp), a unique negative equilibrium (saddle), and no other periodic solutions occur. (iii) If 0< µ < (λ−1)a λa λb 1 λ−1, then equation (1.5) possesses exactly two positive equilibria x1> x2(x1is a saddle and x2is a center), a unique negative equilibrium x3(saddle), and non-constant (both positive and sign-changing) periodic solutions with different periods. Moreover, all non-constant periodic solutions oscillate around x2between x3and x1. (iv) If µ= 0, then equation (1.5) possesses a unique positive equilibrium x0 (saddle), a trivial equilibrium (center), a unique negative equilibrium −x0, and non-constant sign-changing periodic solutions with different periods. Moreover, all non-constant periodic solutions oscillate around 0between −x0and x0. (v) If −(λ−1)a λa λb 1 λ−1< µ < 0, then equation (1.5) possesses exactly two negative equilibria x1< x2(x1is a saddle and x2is a center), a unique positive equilibrium x3(saddle), and non-constant (both positive and sign-changing) periodic solutions with different periods. Moreover, all non-constant periodic solutions oscillate around x2between x1and x3. (vi) If µ=−(λ−1)a λa λb 1 λ−1, then equation (1.5) has a unique negative equilibrium (cusp), a unique positive equilibrium (saddle), and no other periodic solutions occur. (vii) If µ < −(λ−1)a λa λb 1 λ−1, then equation (1.5) has a unique positive equilibrium (saddle) and no other periodic solutions occur. 2. Notation and definitions The following notation is used throughout this article: •Ris the set of real numbers. For x∈R, we put [x]+=1 2(|x|+x) and [x]−=1 2(|x|−x). •C(I) denotes the set of continuous real functions defined on the interval I⊆R. For u∈C([a, b]), we put kukC= max{|u(t)|:t∈[a, b]}. •AC1([a, b]) is the set of functions u: [a, b]→Rwhich are absolutely continuous together with their first derivatives. •AC`([a, b]) (resp. ACu([a, b])) is the set of absolutely continuous functions u: [a, b]→Rsuch that u0admits the representation u0(t) = γ(t) + σ(t) for a. e. t∈[a, b], where γ: [a, b]→Ris absolutely continuous and σ: [a, b]→R is a non-decreasing (resp. non-increasing) function whose derivative is equal to zero almost everywhere on [a, b]. •L([a, b]) is the Banach space of Lebesgue integrable functions p: [a, b]→R equipped with the norm kpkL=Rb a|p(s)|ds. The symbol Int Astands for the interior of the set A⊂L([a, b]). Definition 2.1 ([10, Definition 0.1]).We say that a function p∈L([0, ω]) belongs to the set V+(ω) (resp. V−(ω)) if, for any function u∈AC1([0, ω]) satisfying u00(t)≥p(t)u(t) for a.e. t∈[0, ω], u(0) = u(ω), u0(0) = u0(ω), the inequality u(t)≥0 for t∈[0, ω]resp. u(t)≤0 for t∈[0, ω] 4 J. ˇ SREMR EJDE-2023/65 holds. Remark 2.2. In an alternative terminology, p∈ V−(ω) (resp. p∈ V+(ω)) means that the maximum principle (resp. the anti-maximum principle) holds for the linear periodic problem u00 =p(t)u;u(0) = u(ω), u0(0) = u0(ω).(2.1) Definition 2.3 ([10, Definition 0.2]).We say that a function p∈L([0, ω]) belongs to the set V0(ω)if problem (2.1) has a positive solution. Remark 2.4. Let ω > 0. If p(t) := p0for t∈[0, ω], then one can show by direct calculation that: Bp∈ V−(ω)if and only if p0>0, Bp∈ V0(ω)if and only if p0= 0, Bp∈ V+(ω)if and only if p0∈−π2 ω2,0, Bp∈Int V+(ω)if and only if p0∈−π2 ω2,0. When the function p∈L([0, ω]) is not constant, efficient conditions for pto belong to each of the sets V+(ω)and V−(ω)are provided in [10] (see also [1, 16]). Remark 2.5. It is well known that, if the homogeneous problem (2.1) has only the trivial solution, then, for any f∈L([0, ω]), the problem u00 =p(t)u+f(t); u(0) = u(ω), u0(0) = u0(ω) (2.2) possesses a unique solution uand this solution satisfies |u(t)| ≤ ∆(p)Zω 0|f(s)|dsfor t∈[0, ω], where ∆(p), depending only on p, denotes a norm of the Green’s operator of problem (2.1). Clearly, ∆(p)>0. Assuming that p∈Int V+(ω), we extend the function pperiodically to the whole real axis denoting it by the same symbol. It is proved in [10, Section 6] that, for any a∈R, the problem u00 =p(t)u;u(a)=1, u(a+ω)=1 has a unique solution uaand ua(t)>0 for t∈[0, ω]. We put Γ(p) := sup kuakC:a∈[0, ω]eRω 0[p(s)]+ds.(2.3) It is clear that Γ(p)≥1. Remark 2.6. If p∈ V+(ω), then the number ∆(p) defined in Remark 2.5 can be estimated, for example, by the a maximal value of the Green’s function of problem (2.1) (see, e.g., [16]). On the other hand, assuming p∈Int V+(ω), some estimates of the number Γ(p) are provided in [10, Section 6]. For instance, if p(t) := p0for t∈[0, ω] and p0∈[−π2 ω2,0[ , resp. p0∈−π2 ω2,0, then ∆(p)≤2p|p0|sin ωp|p0| 2−1 ,resp. Γ(p) = cos ωp|p0| 2−1 . Definition 2.7 ([10, Definition 16.1]).Let p, f ∈L([0, ω]). We say that a pair (p, f) belongs to the set U(ω), if problem (2.1) has a unique solution which is positive. EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 5 3. Main results This section contains formulations of all the main results of the paper. Their proofs are presented in detail in Section 5. We start with the most general statement of the paper, which provides the existence/non-existence results in the case of p6∈ V−(ω)∪V0(ω). This condition is satisfied, for instance, if Zω 0 p(s) ds≤0, p(t)6≡ 0 (see Lemma 4.15). Note also that, for the Duffing equation with the constant coefficients x00 +ax −bx3=µf(t), the above-mentioned condition is satisfied if and only if a > 0. Theorem 3.1. Let λ > 1,p6∈ V−(ω)∪V0(ω),f(t)6≡ 0, and h(t)>0for a.e. t∈[0, ω].(3.1) Then, there exist −∞ ≤ µ∗<0and 0< µ∗≤+∞such that the following conclusions hold: (1) For any µ∈]µ∗, µ∗[, problem (1.1) has a positive solution u∗such that every solution uto problem (1.1) satisfies either u(t)< u∗(t)for t∈[0, ω],or u(t)≡u∗(t).(3.2) Moreover, for any couple of distinct positive solutions u1,u2to (1.1) satisfying u1(t)6≡ u∗(t), u2(t)6≡ u∗(t),(3.3) the conditions min{u1(t)−u2(t) : t∈[0, ω]}<0, max{u1(t)−u2(t) : t∈[0, ω]}>0(3.4) hold. (2) If µ∗<+∞, then (a) for µ>µ∗, problem (1.1) has no positive solution, (b) for µ=µ∗, problem (1.1) has a unique non-negative solution u∗and every solution uto (1.1) satisfies (3.2). (3) If µ∗>−∞, then (a) for µ<µ∗, problem (1.1) has no positive solution, (b) for µ=µ∗, problem (1.1) has a unique non-negative solution u∗and every solution uto (1.1) satisfies (3.2). (4) If Rω 0f(s) ds > 0(resp. Rω 0f(s) ds < 0), then µ∗<+∞(resp. µ∗>−∞). Corollary 3.2. Let λ > 1,p6∈ V−(ω)∪V0(ω),f(t)6≡ 0, and condition (3.1) hold. Then, there exists 0< µ0<+∞such that, for any µ∈]−µ0, µ0[, problem (1.1) has a negative solution u∗and a positive solution u∗such that every solution uto problem (1.1) different from u∗,u∗satisfies u∗(t)< u(t)< u∗(t)for t∈[0, ω].(3.5) 6 J. ˇ SREMR EJDE-2023/65 Remark 3.3. The conclusions of Theorem 3.1 and Corollary 3.2 extend the conclusions of Proposition 1.2 for non-autonomous Duffing equations with a sign-changing forcing term. Indeed, let ω > 0 and p(t) := −a, h(t) := b, f(t) := 1 for t∈[0, ω], where a, b > 0. Then, condition (3.1) holds and, by Remark 2.4, we obtain p6∈ V−(ω)∪V0(ω). We emphasize, in particular, the conclusion of Corollary 3.2 which claims that there exists 0 < µ0<+∞such that, for any µ∈]−µ0, µ0[ , equation (1.5) has a maximal (resp. a minimal) ω-periodic solution which is positive (resp. negative); compare it with conclusions (iii), (iv), (v) of Proposition 1.2. We now provide a lower (resp. an upper) estimate of the number µ∗(resp. µ∗) appearing in the conclusion of Theorem 3.1. Proposition 3.4. Let λ > 1,p6∈ V−(ω)∪V0(ω),f(t)6≡ 0,hsatisfy (3.1), and µ∗, µ∗be the numbers appearing in the conclusion of Theorem 3.1. If [f(t)]+6≡ 0, then µ∗≥1 Rω 0[f(s)]+dssup nr ∆p+rλ−1h:r > 0, p +rλ−1h∈ V+(ω)o,(3.6) and, if [f(t)]−6≡ 0, then µ∗≤ − 1 Rω 0[f(s)]−dssup nr ∆p+rλ−1h:r > 0, p +rλ−1h∈ V+(ω)o,(3.7) where ∆is defined in Remark 2.5. Remark 3.5. Let λ > 1, ω > 0, and p(t) := −a, h(t) := bfor t∈[0, ω],(3.8) where a, b > 0, and Φ(a, b, λ, ω) := (2ω π (λ−1)a λ(a λb )1 λ−1if a < λ λ−1π ω2, 2π ω[1 b(a−π2 ω2)] 1 λ−1if a≥λ λ−1(π ω)2. It follows from the proof of [15, Corollary 3.19] that, if [f(t)]+6≡ 0 and [f(t)]−6≡ 0, then µ∗≥Φ(a, b, λ, ω) Rω 0[f(s)]+ds, µ∗≤ − Φ(a, b, λ, ω) Rω 0[f(s)]−ds. If f(t)≥0 for t∈[0, ω], f(t)6≡ 0,(3.9) then it follows from [15, Theorem 3.15(3)] that, for any µ > 0, problem (1.1) has a unique negative solution. Therefore, the conclusions of Theorem 3.1 can be refined as follows. Theorem 3.6. Let λ > 1,p6∈ V−(ω)∪ V0(ω)and conditions (3.1) and (3.9) be fulfilled. Then, there exists 0< µ0<+∞such that the following conclusions hold: (1) For any µ>µ0, problem (1.1) has a unique negative solution u∗and no positive solution. Moreover, every solution uto (1.1) satisfies either u(t)> u∗(t)for t∈[0, ω],or u(t)≡u∗(t).(3.10) (2) For µ=µ0, problem (1.1) has a unique negative solution u∗and a unique non-negative solution u∗. Moreover, every solution uto problem (1.1) different from u∗,u∗satisfies (3.5). EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 7 (3) For µ∈]0, µ0[, problem (1.1) has a unique negative solution u∗and a positive solution u∗such that every solution uto problem (1.1) different from u∗,u∗satisfies (3.5). (4) For µ= 0, problem (1.1) has a unique positive solution u0, the trivial solution, and a unique negative solution −u0. Moreover, every solution u to problem (1.1) different from u∗,u∗changes its sign and satisfies (3.5). (5) For µ∈]−µ0,0[ , problem (1.1) has a unique negative solution u∗and a positive solution u∗such that every solution uto problem (1.1) different from u∗,u∗satisfies (3.5). (6) For µ=−µ0, problem (1.1) has a unique non-positive solution u∗and a unique positive solution u∗. Moreover, every solution uto problem (1.1) different from u∗,u∗satisfies (3.5). (7) For any µ < −µ0, problem (1.1) has a unique positive solution u∗an no negative solution. Moreover, every solution uto (1.1) satisfies (3.2). Remark 3.7. It follows from Theorem 3.1(1) that, in Theorem 3.6(3,5), if u1,u2 are distinct positive (resp. negative) solutions to problem (1.1) different from u∗ (resp. u∗), then conditions (3.4) hold. Remark 3.8. Let ω > 0 and p(t) := −a, h(t) := b, f(t) := 1 for t∈[0, ω], where a, b > 0. Then, conditions (3.1) and (3.9) hold, p6∈ V−(ω)∪ V0(ω) (see Remark 2.4), and all the conclusions of Theorem 3.6 are in compliance with those in Proposition 1.2. We showed in [11, Example 2.8] that assuming p6∈ V−(ω)∪ V0(ω), hypothesis (3.1) in Theorems 3.1 and 3.6 (i.e. the positivity of ha. e. on [0, ω]) is essential for the existence of a positive solution to problem (1.1) with µ= 0 and cannot be weakened to the non-negativity of h. However, under a stronger assumption on the coefficient p, namely, p∈ V+(ω), hypothesis (3.1) of Theorems 3.1 and 3.6 can be relaxed to h(t)≥0 for a.e. t∈[0, ω], h(t)6≡ 0.(3.11) Theorem 3.9. Let λ > 1,p∈ V+(ω),hsatisfy (3.11), and (p, f)∈ U(ω),Zω 0 f(s) ds > 0.(3.12) Then, there exist −∞ ≤ µ∗<0and 0< µ∗<+∞such that the following conclusions hold: (1) For any µ > µ∗, problem (1.1) has no positive solution. (2) For µ=µ∗, problem (1.1) has a unique positive solution u∗and, moreover, every solution uto problem (1.1) satisfies (3.2). (3) For µ∈]0, µ∗[, problem (1.1) has exactly two positive solutions u1,u2and these solutions satisfy u1(t)> u2(t)>0for t∈[0, ω].(3.13) Moreover, every solution uto problem (1.1) different from u1is such that u(t)< u1(t)for t∈[0, ω].(3.14) (4) For µ= 0, problem (1.1) has exactly three solutions: a positive solution u0, the trivial solution, a negative solution −u0. 8 J. ˇ SREMR EJDE-2023/65 (5) For µ∈]µ∗,0[ , problem (1.1) has either one or two positive solutions. Moreover, (1.1) has a positive solution u∗such that every solution to problem (1.1) satisfies (3.2). (6) If µ∗>−∞, then, for any µ<µ∗, problem (1.1) has no positive solution. Remark 3.10. Assume that hypotheses of Theorem 3.9 hold and µ∗>−∞. If, moreover, h(t)>0 for a. e. t∈[0, ω], then it follows from Theorem 3.1(3b) that problem (1.1) with µ=µ∗has a unique non-negative solution u∗and, moreover, every solution to (1.1) with µ=µ∗satisfies (3.2). Open questions. The following two questions remain open in Theorem 3.9: (1) Does the inequality µ∗>−∞ hold without any additional assumption? (2) What happens in the case of µ=µ∗, if µ∗>−∞ and h(t) = 0 on a set of positive measure? Remark 3.11. It is proved in [10, Theorem 16.4] that, if p∈Int V+(ω), then the inclusion (p, f)∈ U(ω) holds for every function f∈L([0, ω]) satisfying f(t)6≡ 0 and Zω 0 [f(s)]+ds≥Γ(p)Zω 0 [f(s)]−ds, where Γ is given by (2.3). On the other hand, if p∈ V+(ω) and fsatisfies (3.9), then (p, f)∈ U(ω) as well (see [10, Remark 9.2]). Remark 3.12. In [1], to show a possible use of the main results, the authors consider the parameter-dependent periodic problem for the forced Mathieu-Duffing equation z00 =−(e+bcos(t))z+νz3+c(t); z(0) = z(2π), z0(0) = z0(2π),(3.15) where e≥0 and b∈Rare such that e+|b|>0 and k[e+bcos(·)]+kLα≤max K(2α∗,2π) : α≥1, Kis the so-called best Sobolev constant, csatisfies −(e+bcos(·)), c∈ U(2π), and ν∈Ris a parameter. It is proved in [1, Corollary 45] that there exits ν0>0such that problem (3.15) has at least two positive solutions provided that 0< ν < ν0. Putting u(t) := √ν z(t), problem (3.15) is equivalent, in some sense, with problem (1.1) in which p(t) := −(e+bcos(t)), h(t) := 1, f(t) := c(t), λ:= 3, and µ:= √ν. Since −(e+bcos(·)) ∈ V+(ω) in the case considered, Theorem 3.9 complements the conclusion of [1, Corollary 45] as follows: There exists ν0>0such that problem (3.15) has exactly two positive solutions provided that 0< ν < ν0, a unique positive solution provided that ν=ν0, and no positive solution provided that ν > ν0. Theorem 3.9 guarantees the existence of certain “critical” values µ∗,µ∗of the parameter µsuch that crossing these values, a bifurcation of positive solutions to problem (1.1) occurs. From an application point of view, the estimates of these numbers are also needed. Proposition 3.13. Let λ > 1,p∈Int V+(ω),hsatisfy (3.11), and Zω 0 [f(s)]+ds > Γ(p)Zω 0 [f(s)]−ds > 0,(3.16) EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 9 where Γis given by (2.3). Then, the numbers µ∗,µ∗appearing in the conclusion of Theorem 3.9 satisfy µ∗≤ − (λ−1) [∆(p)]−λ λ−1 λλRω 0h(s) ds1 λ−1Rω 0[f(s)]−ds ,(3.17) µ∗≥(λ−1) [∆(p)]−λ λ−1 λλRω 0h(s) ds1 λ−1Rω 0[f(s)]+ds ,(3.18) where ∆is defined in Remark 2.5, and µ∗<(λ−1)[Γ(p)Rω 0[p(s)]−ds−Rω 0[p(s)]+ds]λ λ−1 λ[λRω 0h(s) ds]1 λ−1Rω 0[f(s)]+ds−Γ(p)Rω 0[f(s)]−ds.(3.19) If the forcing term fis non-negative, then, similarly as in Theorem 3.6, the conclusions of Theorem 3.9 can be extended as follows. Theorem 3.14. Let λ > 1,p∈ V+(ω), and conditions (3.9) and (3.11) be fulfilled. Then, there exists 0< µ0<+∞such that the following conclusions hold: (1) For any µ > µ0, problem (1.1) has a unique solution which is negative. (2) For µ=µ0, problem (1.1) has exactly two solutions: one positive and one negative. (3) For µ∈]0, µ0[, problem (1.1) has exactly three solutions u1,u2,u3and these solutions satisfy u1(t)> u2(t)>0, u3(t)<0for t∈[0, ω]. (4) For µ= 0, problem (1.1) has exactly three solutions: a positive solution u0, the trivial solution, a negative solution −u0. (5) For µ∈]−µ0,0[ , problem (1.1) has exactly three solutions u1,u2,u3and these solutions satisfy u1(t)< u2(t)<0, u3(t)>0for t∈[0, ω]. (6) For µ=−µ0, problem (1.1) has exactly two solutions: one positive and one negative. (7) For any µ < −µ0, problem (1.1) has a unique solution which is positive. Remark 3.15. Theorem 3.14 extends the conclusions of Theorem 1.1 for the case of c= 0 and confirms a conjecture formulated in [2, Remark 3, p. 2502] because, at least in case of c= 0, the conclusions of Theorem 1.1 (except for the asymptotic stability) are still true for dwhich changes its sign (and belongs to a certain class of functions). We finally provide the upper and lower estimates of the number µ0appearing in Theorem 3.14, which follow immediately from Proposition 3.13. Proposition 3.16. Let λ > 1,p∈Int V+(ω), and conditions (3.9) and (3.11) hold. Then, the number µ0appearing in the conclusion of Theorem 3.14 satisfies µ0≥(λ−1) [∆(p)]−λ λ−1 λλRω 0h(s) ds1 λ−1Rω 0f(s) ds , 16 J. ˇ SREMR EJDE-2023/65 α2(0) = α2(ω), α0 2(0) = α0 2(ω),(4.33) α00 2(t) = p(t)α2(t) + µ∗ µλ−1h(t)αλ 2(t) + µf(t) ≥p(t)α2(t) + h(t)αλ 2(t) + µf(t) for a.e. t∈[0, ω], (4.34) meas t∈[0, ω] : α00 2(t)> p(t)α2(t) + h(t)αλ 2(t) + µf(t)>0,(4.35) because 0 < µ < µ∗and hsatisfies (3.11). Therefore, Lemma 4.3(2) (with α(t) := α2(t)), problem (1.1) has a solution α1 such that α1(t)≥α2(t) for t∈[0, ω].(4.36) Consequently, the functions α1,α2satisfy conditions (4.5) and (4.6). We finally show that (4.4) is fulfilled as well. Suppose on the contrary that (4.4) does not hold. Extend the functions p,h,f,α1,α2periodically to the whole real axis denoting them by the same symbols. Then, in view of (4.32) and (4.36), there exists a∈[0, ω[ such that α1(a) = α2(a), α0 1(a) = α0 2(a).(4.37) Put w(t) := α1(t)−α2(t) for t∈[a, a +ω], ϕ(t) := gα1(t), α2(t)for t∈[a, a +ω], where g(x, y) := (xλ−yλ x−yfor x, y ∈R, x 6=y, λ|x|λ−1sgn xfor x, y ∈R, x =y. It is not difficult to verify that g:R2→Ris a continuous function and, thus, the function ϕis continuous and non-negative on [a, a +ω]. By (4.36) and (4.37), w satisfies (4.31) with b=a+ω. Since α1is a solution to problem (1.1) and α2 satisfies (4.34), we have w00(t)≤p(t)w(t) + h(t)αλ 1(t)−αλ 2(t) ≤|p(t)|+h(t)ϕ(t)w(t) for a.e. t∈[a, a +ω]. Therefore, Lemma 4.11 (with `(t) := |p(t)|+h(t)ϕ(t) and b:= a+ω) yields w(t)≡0, i., e., α1(t)≡α2(t). However, this contradicts condition (4.35), because α1is a solution to problem (1.1).  Lemma 4.13. Let λ > 1,µ∗>0,p, h, f ∈L([0, ω]),hsatisfy (3.11), and there exist functions α1, α2∈AC1([0, ω]) such that (4.5) with µ=µ∗and (4.6) hold and 0≤α2(t)< α1(t)for t∈[0, ω].(4.38) Then, there exist µ>µ∗and a positive function α∈AC1([0, ω]) satisfying (4.1) with µ=µ∗and (4.2). The proof of the above lemma is similar to the proof of [14, Lemma 4.9] and thus, it is omitted. Lemma 4.14. Let λ > 1,µ∗∈R,p6∈ V−(ω)∪V0(ω), and hsatisfy (3.1). Then, for any c > 0, there exists a function β∈AC 1([0, ω]) such that β00(t)≤p(t)β(t) + h(t)βλ(t) + µ∗f(t)for a.e. t∈[0, ω],(4.39) EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 17 β(0) = β(ω), β0(0) = β0(ω),(4.40) β(t)≥cfor t∈[0, ω].(4.41) Proof. Put q0(t, x) := h(t)|x|λ−1for a.e. t∈[0, ω] and all x∈R. Since limx→+∞xλ−1REh(s) ds= +∞for every E⊆[0, ω], meas E > 0, it follows from [15, Lemma 4.15] that there exists R > 0 such that p+q0(·, R)∈ V−(ω). Therefore, the conclusion of the lemma follows from [15, Proposition 4.21] (with q(t, x) := q0(t, x) and x0:= 0).  Lemma 4.15 ([10, Proposition 10.8, Remark 0.7]).If p∈ V−(ω)∪ V0(ω), then either Rω 0p(s) ds > 0or p(t)≡0. 5. Proofs of main results Proof of Theorem 3.1. Put A:= µ∈R: problem (1.1) has a positive solution.(5.1) In view of Lemmas 4.1(1) and 4.2, there exists ε > 0 such that ]−ε, ε[∩A 6=∅. Let µ∗:= inf A, µ∗:= sup A.(5.2) Then, −∞ ≤ µ∗<0 and 0 < µ∗≤+∞. Conclusion (1):Let µ0∈ A\{0}be arbitrary and µ∈Rbe such that 0 <|µ| ≤ |µ0| and sgn µ= sgn µ0. Let, moreover, u0be a positive solution to problem (1.1) with µ=µ0. Put α(t) := µ µ0 u0(t) for t∈[0, ω].(5.3) Clearly, α(t)>0 for t∈[0, ω]. It follows from (1.1) with µ=µ0that αsatisfies (4.2) and α00(t) = p(t)α(t) + µ0 µλ−1h(t)αλ(t) + µf(t) ≥p(t)α(t) + h(t)αλ(t) + µf(t) for a.e. t∈[0, ω], (5.4) because |µ0|≥|µ|>0 and (3.1) holds. Therefore, Lemma 4.1(1) yields µ∈ A. Consequently, ]µ∗, µ∗[⊆ A and, thus, conclusion (1) of the theorem follows from Lemma 4.1(1). Conclusion (2):Assume that µ∗<+∞. Then, it follows immediately from (5.1) and (5.2) that conclusion (2a) of the theorem holds. Let {µn}∞ n=1 be a sequence of positive numbers such that µn∈ A for n∈N,lim n→+∞µn=µ∗(5.5) and, for any n∈N, let unbe a solution to problem (1.1) with µ=µn. Lemma 4.10 yields (4.13). By the standard arguments using in the proof of a well-possedness of the periodic problems for second-order ODEs, one can show that there exists a subsequence {unk}∞ k=1 of {un}∞ n=1 such that lim k→+∞u(i) nk(t) = (u∗)(i)(t) uniformly on [0, ω], i = 0,1,(5.6) 18 J. ˇ SREMR EJDE-2023/65 where u∗∈AC1([0, ω]) is a solution to problem (1.1) with µ=µ∗. All the functions unkare positive and, thus, it is clear that u∗(t)≥0 for t∈[0, ω]. We now prove that u∗is a unique non-negative solution to problem (1.1) with µ=µ∗. Suppose on the contrary that u∗is a non-negative solution to (1.1) with µ=µ∗such that u∗(ξ)6=u∗(ξ) for some ξ∈[0, ω].(5.7) Put α(t) := max u∗(t), u∗(t)for t∈[0, ω]. It is not difficult to verify that α∈AC`([0, ω]), condition (4.2) with µ=µ∗holds, and α(a) = α(ω), α0(a)≥α0(ω).(5.8) Let us show that α(t)>0 for t∈[0, ω].(5.9) If this condition does not hold, then, in view of the non-negativity of u∗,u∗, there exists t0∈[0, ω] such that u∗(t0)=0, u∗(t0)=0.(5.10) Extend the functions p,h,f,u∗,u∗periodically to the whole real axis denoting them by the same symbols. Then, using (5.10) and the non-negativity of u∗,u∗, we obtain u0 ∗(t0)=0,(u∗)0(t0)=0.(5.11) Since the function x7→ |x|λsgn xis Lipschitz on every compact interval, for any c1, c2∈R, the Cauchy problem u00 =p(t)u+h(t)|u|λsgn u+µ∗f(t); u(t0) = c1, u0(t0) = c2(5.12) is uniquely solvable. Therefore, (5.10) and (5.11) yield u∗(t)≡u∗(t), which contradicts (5.7). Hence, (5.9) holds. Now, in view of (4.2) with µ=µ∗, (5.8), and (5.9), it follows from Lemma 4.1(1) that problem (1.1) with µ=µ∗has a positive solution ˜u∗such that 0≤u∗(t)<˜u∗(t) for t∈[0, ω] or 0 ≤u∗(t)<˜u∗(t) for t∈[0, ω]. Therefore, Lemma 4.13 guarantees that there exist ˜µ > µ∗and a positive function ˜α∈AC1([0, ω]) satisfying ˜α00(t)≥p(t)˜α(t) + h(t)˜αλ(t) + ˜µf(t) for a.e. t∈[0, ω],(5.13) ˜α(0) = ˜α(ω),˜α0(0) = ˜α0(ω).(5.14) Consequently, it follows from Lemma 4.1 (with α(t) := ˜α(t) and µ:= ˜µ) that problem (1.1) with µ= ˜µhas at least one positive solution, which contradicts the above-proved conclusion (2a). The contradiction obtained proves that u∗is a unique non-negative solution to problem (1.1) with µ=µ∗. It remains to show that every solution uto problem (1.1) with µ=µ∗satisfies (3.2). Indeed, suppose on the contrary that uis a solution to problem (1.1) with µ=µ∗such that (3.2) does not hold. We have mentioned above that, for any t0∈[0, ω] and c1, c2∈R, the Cauchy problem (5.12) is uniquely solvable and, thus, the solution usatisfies max u(t)−u∗(t) : t∈[0, ω]>0.(5.15) EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 19 Put α(t) := max u(t), u∗(t)for t∈[0, ω].(5.16) It is not difficult to verify that α∈AC`([0, ω]) and conditions (4.2) with µ=µ∗and (5.8) hold. Moreover, it follows from Lemma 4.14 that there exists β∈AC 1([0, ω]) satisfying (4.39), (4.40), and β(t)≥α(t) for t∈[0, ω].(5.17) Therefore, by (4.2) with µ=µ∗, (4.39), (4.40), (5.8), and (5.17), we conclude that αand βform a well-ordered pair of lower and upper functions and, thus, problem (1.1) with µ=µ∗has a solution ˆusuch that α(t)≤ˆu(t)≤β(t) for t∈[0, ω]. However, this condition, together with (5.15) and (5.16), implies that ˆuis a nonnegative solution to problem (1.1) with µ=µ∗different from u∗, which contradicts the above-proved fact concerning the uniqueness of the non-negative solution u∗. Conclusion (3):It can be proved in much the same way as conclusion (2) considering −µand −finstead of µand f. Conclusion (4):It follows immediately from Lemma 4.9.  Proof of Corollary 3.2. It is clear that uis a solution to problem (1.1) if and only if −uis a solution to problem (4.10). Therefore, the conclusion of the corollary follows from Theorem 3.1(1).  Proof of Proposition 3.4. Let µ∗,µ∗be the numbers appearing in the conclusion of Theorem 3.1. Assume that [f(t)]+6≡ 0 and suppose on the contrary that (3.6) does not hold, i.e., µ∗<1 Rω 0[f(s)]+dssup r ∆p+rλ−1h:r > 0, p +rλ−1h∈ V+(ω), where ∆ is defined by Remark 2.5. Then, µ∗∈]0,+∞[ and there exists ε > 1 such that Zω 0 [εµ∗f(s)]+ds < sup r ∆p+rλ−1h:r > 0, p +rλ−1h∈ V+(ω). Therefore, from Lemmas 4.2 and 4.1(1) that problem (1.1) with µ=εµ∗has at least one positive solution, which contradicts conclusion (2a) of Theorem 3.1. Assuming [f(t)]−6≡ 0, estimate (3.7) can be proved analogously to (3.6).  Proof of Theorem 3.6. It follows from Theorem 3.1 and Lemmas 4.6 and 4.9 that there exists µ0∈]0,∞[ such that conclusions (1), (2), and (3) of the theorem hold. Since uis a solution to problem (1.1) if and only if −uis a solution to problem (4.10), conclusions (5), (6), and (7) of the theorem hold as well. Finally, conclusion (4) of the theorem follows from Lemma 4.6 and the above-mentioned equivalence.  Proof of Theorem 3.9. Let the set Abe given by formula (5.1). In view of Lemmas 4.3(2,3) and 4.4(1), there exists ε > 0 such that ] −ε, ε[∩A 6=∅. Define the numbers µ∗and µ∗by (5.2). Then, −∞ ≤ µ∗<0, µ∗>0, and Lemma 4.8 implies that µ∗<+∞. Conclusion (1):It follows from (5.1), (5.2), and the condition µ∗∈]0,+∞[ . 20 J. ˇ SREMR EJDE-2023/65 Conclusion (2):We first show that µ∗∈ A.(5.18) Indeed, let {µn}∞ n=1 be a non-decreasing sequence of positive numbers such that µn∈ A for n∈N,lim n→+∞µn=µ∗. Moreover, for any n∈N, let unbe a positive solution to problem (1.1) with µ=µn. It follows from Lemma 4.8 that condition (4.13) holds. By the standard arguments using in the proof of a well-possedness of the periodic problems for second-order ODEs, one can show that there exists a subsequence {unk}∞ k=1 of {un}∞ n=1 such that (5.6) is satisfied, where u∗∈AC1([0, ω]) is a solution to problem (1.1) with µ=µ∗. Since the functions unk,k∈N, are positive, it is clear that u∗(t)≥0 for t∈[0, ω].(5.19) In view of the hypothesis (p, f)∈ U(ω) and the positivity of µ∗, problem (4.23) has a unique solution v, which is positive. By (1.1) with µ=µ∗, (3.11), (4.23), and (5.19), we obtain z00(t)≥p(t)z(t) for a.e. t∈[0, ω], z(0) = z(ω), z0(0) = z0(ω), where z(t) := u∗(t)−µ∗v(t) for t∈[0, ω]. Therefore, the hypothesis p∈ V+(ω) yields z(t)≥0 for t∈[0, ω]. Hence, we have u∗(t)≥µ∗v(t)>0 for t∈[0, ω] and, thus condition (5.18) holds. Since u∗is a positive solution to problem (1.1) with µ=µ∗, in view of Lemma 4.3(2), to prove conclusion (2) of the theorem, it is sufficient to show that problem (1.1) with µ=µ∗does not have more than one positive solution. Suppose on the contrary that problem (1.1) with µ=µ∗has a positive solution different from u∗. Then, it follows from Lemma 4.3(2) (with α(t) := u∗(t) and µ:= µ∗) that problem (1.1) with µ=µ∗possesses solutions ˜u∗, ˜u∗such that ˜u∗(t)>˜u∗(t)>0 for t∈[0, ω]. Therefore, Lemma 4.13 (with α1(t) := ˜u∗(t) and α2(t) := ˜u∗(t)) guarantees that there exist ˜µ > µ∗and a positive function ˜α∈AC1([0, ω]) satisfying (5.13) and (5.14). Consequently, by Lemma 4.1(1) (with α(t) := ˜α(t) and µ:= ˜µ), we conclude that problem (1.1) with µ= ˜µhas at least one positive solution, which contradicts the above-proved conclusion (1). Conclusions (3):Having a positive solution u∗to problem (1.1) with µ=µ∗, it is clear that all the hypotheses of Lemma 4.12 (with α(t) := u∗(t)) are fulfilled. Consequently, for any µ∈]0, µ∗[ , (p, µf)∈ U(ω) and there exist functions α1, α2∈ AC1([0, ω]) satisfying conditions (4.4), (4.5), and (4.6) and, therefore, conclusion (3) of the theorem follows from Lemma 4.3(3). Conclusion (4):It follows immediately from [11, Corollary 2.31(2)]. Conclusion (5):Let µ0∈ A∩]−∞,0[ and µ∈[µ0,0[ be arbitrary and let u0be a positive solution to problem (1.1) wigth µ=µ0. Define the function αby (5.3). Clearly, α(t)>0 for t∈[0, ω]. It follows from (1.1) with µ=µ0that αsatisfies (4.2) and (5.4), because µ0≤µ < 0 and (3.11) holds. Therefore, Lemma 4.3(2) yields µ∈ A. Consequently, ]µ∗,0[ ⊆ A and, thus, conclusion (5) of the theorem follows from Lemma 4.3(1,2). EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 21 Conclusion (6):Assume that µ∗>−∞. Then, it follows immediately from (5.1) and (5.2) that, for any µ < µ∗, problem (1.1) has no positive solution.  Proof of Proposition 3.13. By Remark 3.11, it follows from (3.16) that condition (3.12) holds. Let µ∗,µ∗be the numbers appearing in the conclusion of Theorem 3.9. We first show that µ∗satisfies (3.17), where ∆ is defined in Remark 2.5. Suppose on the contrary that (3.17) does not hold, i.e., µ∗>−(λ−1) [∆(p)]−λ λ−1 λλRω 0h(s) ds1 λ−1Rω 0[f(s)]−ds . Then, µ∗∈]−∞,0[ and there exists ε > 1 such that 0<Zω 0 [εµ∗f(s)]+ds=−εµ∗Zω 0 [f(s)]−ds≤(λ−1) [∆(p)]−λ λ−1 λλRω 0h(s) ds1 λ−1 . Therefore, it follows from Lemmas 4.4(1) and 4.3(2) that problem (1.1) with µ=εµ∗has a positive solution, which contradicts conclusion (6) of Theorem 3.9. Now we show that µ∗satisfies (3.18), where ∆ is defined in Remark 2.5. Suppose on the contrary that (3.18) does not hold, i.e., µ∗<(λ−1) [∆(p)]−λ λ−1 λλRω 0h(s) ds1 λ−1Rω 0[f(s)]+ds .(5.20) By the conditions (p, f)∈ U(ω) and µ∗>0, we obtain (p, µ∗f)∈ U(ω). Therefore, in view of (5.20), it follows from Lemmas 4.4(2) and 4.3(3) that problem (1.1) with µ=µ∗has exactly two positive solutions, which contradicts conclusion (2) of Theorem 3.9. We finally show that µ∗satisfies (3.19), where Γ is given by (2.3). Suppose on the contrary that (3.19) does not hold, i.e., µ∗≥(λ−1)[Γ(p)Rω 0[p(s)]−ds−Rω 0[p(s)]+ds]λ λ−1 λλRω 0h(s) ds1 λ−1[Rω 0[f(s)]+ds−Γ(p)Rω 0[f(s)]−ds] . Then, it follows from Lemma 4.5 that problem (1.1) with µ=µ∗has no positive solution, which contradicts conclusion (2) of Theorem 3.9.  Proof of Theorem 3.14. We first note that, by Remark 3.11, condition (3.12) holds. Therefore, it follows from Theorem 3.9(1,2,3) and Lemmas 4.6 and 4.7 that there exists µ0∈]0,+∞[ such that conclusions (1), (2), and (3) of the theorem hold. Since uis a solution to problem (1.1) if and only if −uis a solution to problem (4.10), conclusions (5), (6), and (7) of the theorem hold as well. Finally, the validity of conclusion (4) of the theorem follows immediately from Theorem 3.9(4).  6. Conclusions The existence and exact multiplicity of solutions to problem (1.1) was studied depending on the choice of the parameter µ. We extended the conclusions stated in [2, Theorem 1.1] for the case of undamped Duffing equation (1.2) with c:= 0 and weakened hypotheses (1.3) and (1.4). Our results confirm a conjecture formulated in [2, Remark 3, p. 2502] because, at least in the case of c= 0, the conclusions of Theorem 1.1 (except for the asymptotic stability) are still true for dwhich changes its sign (and belongs to a certain class of functions). We also provided both lower 22 J. ˇ SREMR EJDE-2023/65 and upper estimates of the “critical” values µ∗,µ∗(resp. µ0) of the parameter µ appearing in the conclusions of Theorems 3.1 and 3.9 (resp. Theorems 3.6 and 3.14). The approach used in [2] employs identifying the fold point on bifurcation curves and the continuation method combined with the Sturm’s comparison theorem, topological degree, and the maximum principle. We used a slightly different approach; we proved our results by using the method of lower and upper functions only, which was combined with the the maximum and anti-maximum principles. The results obtained substantially generalize the results available in the literature because they are not only specific sufficient conditions. Our general results hold for all the equations of the type studied whose coefficient in the linear part belongs to a certain sufficiently wide class of functions. Such a class is described in terms of the behavior of the corresponding linear periodic problem and does not exclude the so-called resonant cases. Finally, it is worth mentioning that if the results concerning the maximum and anti-maximum principles are known for the periodic linear problem u00 =p(t)u+g(t)u0;u(0) = u(ω), u0(0) = u0(ω) with p, g ∈L([0, ω]), the parameter-dependent problem u00 =p(t)u+g(t)u0+h(t)|u|λsgn u+µf(t); u(0) = u(ω), u0(0) = u0(ω) might be also studied in a similar way as (1.1). The first steps are already done for the Duffing equation with a constant damping coefficient g(see, e. g., [2, 8]). Acknowledgments. This research was supported by the internal grant FSI-S-206187 of FME BUT. References [1] A. Cabada. J. ´ A. Cid, L. L´opez-Somoza; Maximum principles for the Hill’s equation, Academic Press, London, 2018. [2] H. Chen, Y. Li; Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity,21 (2008), No. 11, 2485–2503. [3] C. Fabry, J. Mawhin, M. N. Nkashama; A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. Soc.,18 (1986), No. 2, 173–180. [4] S. Gaete, R. F. Man´asevich; Existence of a pair of periodic solutions of an O.D.E. generalizing a problem in nonlinear elasticity, via variational methods, J. Math. Anal. Appl.,134 (1988), No. 2, 257–271. [5] P. Habets, C. De Coster; Two-point boundary value problems: lower and upper solutions, Mathematics in Science and Engineering, 205, Elsevier B.V., Amsterdam, 2006. [6] X. Han, Y. He, H. Wei; Existence of positive periodic solutions for a nonlinear system of second-order ordinary differential equations, Electron. J. Differential Equations,2022 (2022), No. 83, 1–11. [7] I. Kiguradze; Initial and boundary value problems for systems of ordinary differential equations. Vol. I. Linear theory, Metsniereba, Tbilisi, 1997, in Russian. [8] S. Liang; Exact multiplicity and stability of periodic solutions for Duffing equation with bifurcation method, Qual. Theory Dyn. Syst.,18 (2019), No. 2, 477–493. [9] A. Lomtatidze; On periodic boundary value problem for second order ordinary differential equations, Commun. Contemp. Math.,22 (2020), No. 6, 1950049. [10] A. Lomtatidze; Theorems on differential inequalities and periodic boundary value problem for second-order ordinary differential equations, Mem. Differential Equations Math. Phys., 67 (2016), 1–129. [11] A. Lomtatidze, J. ˇ Sremr; On periodic solutions to second-order Duffing type equations, Nonlinear Anal. Real World Appl.,40 (2018), 215–242. EJDE-2023/65 PARAMETER-DEPENDENT PERIODIC PROBLEMS 23 [12] F. I. Njoku, P. Omari; Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions, Appl. Math. Comput.,135 (2003), Nos. 2–3, 471–490. [13] R. Ortega; Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B (7),3(1989), No. 3, 533–546. [14] J. ˇ Sremr; Bifurcation of positive periodic solutions to non-autonomous undamped duffing equations, Math. Appl. 10 (2021), No. 1, 79–92. [15] J. ˇ Sremr; Existence and exact multiplicity of positive periodic solutions to forced nonautonmous Duffing type differential equations, Electron. J. Qual. Theory Differ. Equ.,2021 (2021), No. 62, 1–33. [16] P. J. Torres; Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. Differential Equations,190 (2003), 643–662. [17] T. Wa˙zewski; Syst`emes des ´equations et des in´egalit´es diff´erentielles ordinaires aux deuxi`emes membres monotones et leurs applications, Ann. Soc. Polon. Math.,23 (1950), 112–166, in French. Jiˇ r´ ıˇ Sremr Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technick´ a 2, 616 69 Brno, Czech Republic Email address:[email protected]