Special Issue in honor of John W. Neuberger Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 151–160. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu OPTIMAL CONDITIONS FOR THE MAXIMUM PRINCIPLE FOR SECOND-ORDER PERIODIC PROBLEMS GABRIELA HOLUBOVÁ Abstract. We provide alternate necessary and/or sufficient conditions on the sign-changing coefficient p(t) for the maximum principle for the second-order periodic problem u′′ = p(t)u + q(t) to hold, i.e., for nonnegative q to yield a nonpositive periodic solution u. 1. Introduction and problem formulation We consider the linear second-order periodic problem u′′ = p(t)u+ q(t), u(0) = u(ω), u′(0) = u′(ω) (1.1) and provide new answers to the fundamental question For which p a nonnegative q results in a nonpositive u? We can find several necessary and/or sufficient conditions on p in the extensive relevant literature. See for example [2, 6, 10] and references therein. Optimality and especially applicability and verifiability of these conditions are crucial for further studies of related nonlinear problems (see, e.g., [7]). Inspired by our previous results concerning the fourth-order problems [3, 4, 5], we state alternative series of (optimal and/or verifiable) conditions on p based on the principal weighted eigenvalue of the corresponding linear operator. For details, see our main results in Section 2 and their consequences in Section 3. In Section 4 we present two examples and comparison with known conditions. Since our text is mainly based on [6], we keep the same notation and terminology as much as possible. Throughout this article, we consider p, q ∈ Lω which is the space of ω-periodic functions that are Lebesgue integrable on (0, ω). By a solution to (1.1) we mean a differentiable function u with absolutely continuous derivative in [0, ω] that satisfies (1.1) almost everywhere on [0, ω]. We let Cω denote the space of ω-periodic continuous functions with norm given by ‖u‖ = maxt∈[0,ω] |u(t)|. Definition 1.1 (Lomtatidze [6]). We say that the function p ∈ Lω belongs to the set V−(ω) if for any differentiable function u with absolutely continuous derivative on [0, ω], and satisfying u′′(t) ≥ p(t)u(t) for a.e. t ∈ [0, ω], u(0) = u(ω), u′(0) = u′(ω), 2020 Mathematics Subject Classification. 34B05, 34B09, 34B27. Key words and phrases. Periodic boundary value problem; maximum principle; constant-sign solution; principle eigenvalue; Green function. ©2023 This work is licensed under a CC BY 4.0 license. Published March 27, 2023. 151 152 G. HOLUBOVÁ EJDE/SI/02 the inequality u(t) ≤ 0 holds for all t ∈ [0, ω]. In fact (see [6]), if p ∈ V−(ω), then for any q ∈ Lω, q 6≡ 0, q(t) ≥ 0 a.e. on [0, ω], the periodic problem (1.1) possesses a unique solution u satisfying u(t) < 0 on [0, ω]. Let us note that p ∈ V−(ω) is equivalent to the statement that the maximum principle is satisfied by (1.1), or that the linear periodic operator u 7→ −u′′+ p(t)u is (strictly) inverse positive. The following basic properties of V−(ω) can be found in [6]: (i) If p(t) ≥ 0 for a.e. t ∈ [0, ω] and p 6≡ 0, then p ∈ V−(ω). (ii) If p0(t) ∈ V−(ω) and p(t) ≥ p0(t) for a.e. t ∈ [0, ω], then p ∈ V−(ω) as well. (iii) The set V−(ω) is unbounded, open and convex. (iv) If p ∈ V−(ω), then ∫ ω 0 p(t) dt > 0. Because of conditions (i) and (iv), it makes sense to consider only sign-changing functions p for further study. 2. Main results Theorem 2.1. Let p ∈ Lω with p(t) = p1(t) − p2(t), pi(t) ≥ 0 for a.e. t ∈ [0, ω], pi 6≡ 0, i = 1, 2. Then p ∈ V−(ω) if and only if the principal (weighted) eigenvalue λ0 of the problem − y′′ + p1(t)y = λp2(t)y, y(0) = y(ω), y′(0) = y′(ω) (2.1) satisfies λ0 > 1. Theorem 2.1 can be reformulated as follows. Theorem 2.2. Let p ∈ Lω with p(t) = p1(t) − p2(t), pi(t) ≥ 0 for a.e. t ∈ [0, ω], pi 6≡ 0, i = 1, 2. Let Gp1(t, s) be the Green function related to the left hand side of (2.1), and let T : Cω → Cω be a linear operator defined by Ty(t) := ∫ ω 0 Gp1(t, s)p2(s)y(s) ds. (2.2) Then p ∈ V−(ω) if and only if the principal characteristic value λ0 of T (equal to 1/r(T ) with r(T ) being the spectral radius of T ) satisfies λ0 > 1. Before we prove Theorems 2.1 and 2.2, let us note that because of the non- negativity and nontriviality of p1, the Green function Gp1 exists, it is unique and continuous in both variables, hence the operator T is well defined. Moreover, it is compact. Since the eigenvalue problem (2.1) is equivalent to y = λTy, i.e., λ is an eigenvalue of (2.1) if and only if it is the characteristic value of T , we will prove both Theorems 2.1 and 2.2 together. Finally, let us recall that we speak about the principal eigenvalue (or principal characteristic value), if (at least one, in general) corresponding eigenfunction is of constant sign in [0, ω]. Proof of Theorems 2.1 and 2.2. From the assumptions on p1, p2, the Green func- tion Gp1 satisfies Gp1(t, s) > 0 a.e. on [0, ω]× [0, ω] and the spectral radius r(T ) of T is positive as well. Moreover, 1/r(T ) corresponds to the principal characteristic value λ0 of T . That is, (2.1) possesses the principal eigenvalue λ0 > 0 with the constant-sign eigenfunction y0 (cf., e.g., [8, Theorem 2.6 and Remark 2.1]). EJDE-2023/SI/02 MAXIMUM PRINCIPLE FOR PERIODIC PROBLEMS 153 First, we prove the necessity. Let p = p1 − p2 ∈ V−(ω), i.e., for any q ∈ Lω, q(t) ≥ 0, q 6≡ 0, the problem − u′′ + p1(t)u = p2(t)u− q(t), u(0) = u(ω), u′(0) = u′(ω) (2.3) possesses a strictly negative solution u. Multiplying the equation in (2.3) by the eigenfunction y0 > 0 and integrating over (0, ω), we easily obtain∫ ω 0 (−u′′ + p1u)y0 = ∫ ω 0 (−y′′0 + p1y0)u = λ0 ∫ ω 0 p2uy0 = ∫ ω 0 p2uy0 − ∫ ω 0 qy0. The last equality gives (1− λ0) ∫ ω 0 p2uy0 = ∫ ω 0 qy0 and the sign properties of p2, q, u and y0 imply λ0 > 1. To prove the sufficiency, we use successive iterations. Let us consider q ∈ Lω, q 6≡ 0, q(t) ≥ 0 arbitrary but fixed, and denote q∗(t) := ∫ ω 0 Gp1(t, s)q(s) ds. We have q∗ ∈ Cω, q∗(t) > 0 for t ∈ [0, ω] and (2.3) is equivalent to u = Tu − q∗. Now, we are ready to define a sequence {un}∞n=0 ⊂ Cω using the recurrence formula un+1 = Tun − q∗ with u0 ≡ 0. (2.4) From the assumptions on p1, p2, the operator T is strictly monotone increasing on Cω ordered by the cone C+ ω = {u ∈ Cω, u(t) ≥ 0 for every t ∈ [0, ω]}, and we obtain u0 ≡ 0 > u1 = −q∗ > u2 > · · · > un > . . . . Moreover, un = Tun−1 − q∗ = T (Tun−2 − q∗)− q∗ = − n−1∑ k=0 T kq∗ and hence ‖un‖ ≤ ‖q∗‖ n−1∑ k=0 ‖T k‖ ≤ ‖q∗‖ ∞∑ k=0 ‖T k‖. Using Gelfand’s formula r(T ) = limk→∞ ‖T k‖1/k and the assumption 1/λ0 = r(T ) < 1, we gain convergence of the series ∑ ‖T k‖ and hence uniform boundedness of {un} in Cω. Compactness of T and monotonicity of {un} yield the convergence un → u in Cω with u(t) < 0 for all t ∈ [0, ω] being the solution of (2.3). Thus p = p1 − p2 ∈ V−(ω). � The following lemma shows that the sign of (λ0 − 1) does not depend on the choice of p1, p2. Lemma 2.3. Let λ0 be the principal eigenvalue of (2.1) with some p1 and p2 satisfying p1(t) − p2(t) = p(t), pi(t) ≥ 0 for a.e. t ∈ [0, ω], pi 6≡ 0, i = 1, 2. Then sgn(λ0 − 1) is independent on the choice of p1, p2. Proof. Let p(t) = p1(t) − p2(t) = p̄1(t) − p̄2(t) with p1,2, p̄1,2 all nonnegative and nontrivial. Let λ0 be the principal eigenvalue of (2.1) with p1,2 and let y(t) be the corresponding positive eigenfunction. Similarly, let λ̄0 be the principal eigenvalue 154 G. HOLUBOVÁ EJDE/SI/02 of (2.1) with p̄1,2 and ȳ(t) be the corresponding positive eigenfunction. Multiplying (2.1) by ȳ, integrating over (0, ω) and using ∫ ω 0 y′′ȳ = ∫ ω 0 yȳ′′ yields∫ ω 0 p1(t)y(t)ȳ(t) dt− λ0 ∫ ω 0 p2(t)y(t)ȳ(t) dt = ∫ ω 0 p̄1(t)y(t)ȳ(t) dt− λ̄0 ∫ ω 0 p̄2(t)y(t)ȳ(t) dt. Since p1 − p̄1 = p2 − p̄2, we obtain (1− λ0) ∫ ω 0 p2(t)y(t)ȳ(t) dt = (1− λ̄0) ∫ ω 0 p̄2(t)y(t)ȳ(t) dt. Positivity of both y, ȳ and nonnegativity of p2, p̄2 means that sgn(1 − λ0) = sgn(1− λ̄0), which we wanted to prove. � Remark 2.4. The natural decomposition of sign-changing p is p1(t) = p+(t), p2(t) = p−(t), with p±(t) = max{±p(t), 0} being the positive and negative parts of p. However this choice is not convenient for computational purposes since min p+(t) = 0 (cf. Corollary 3.7 and the comment above). In further text, we will mainly use decomposition p1(t) = (p(t)− c)+ + c, p2(t) = (p(t)− c)− (2.5) with some fixed real constant 0 < c ≤ pM := ess supt∈[0,ω] p(t). Notice that for c = pM ∈ R, p1 is constant (p1(t) ≡ pM ). 3. Estimates of λ0 and consequences of main results To find the precise value of λ0 is not an easy task, however, for its estimates we can exploit, e.g., the following results. Lemma 3.1 (Webb and Lan [8]). Let λ0 be the principal eigenvalue of (2.1) and p1, p2, and Gp1 be as in Theorem 2.2. Then m ≤ λ0 ≤M , where m = ( sup 0≤t≤ω ∫ ω 0 Gp1(t, s)p2(s) ds )−1 , M = inf 0≤a 0 for all t ∈ [0, ω], we can write ‖y(t)‖ = max t∈[0,ω] y(t) = max t∈[0,ω] ∫ ω 0 Gp1(t, s) (−y′′(s) + p1(s)y(s)) ds ≤ max (t,s) Gp1(t, s) ∫ 1 0 (−y′′(s) + p1(s)y(s)) ds. Similarly, y(t) ≥ min (t,s) Gp1(t, s) ∫ 1 0 (−y′′(s) + p1(s)y(s)) ds ≥ min(t,s)Gp1(t, s) max(t,s)Gp1(t, s) ‖y‖. Hence, we can take σ0 = min(t,s)Gp1(t, s)/max(t,s)Gp1(t, s) > 0. Now, we are ready to formulate several corollaries of our main results that provide verifiable necessary and sufficient conditions for p ∈ V−(ω). The first one is a direct consequence of Theorem 2.2 and Lemma 3.1. Corollary 3.5. Let p ∈ Lω be decomposed to p(t) = p1(t) − p2(t) with pi(t) ≥ 0 for a.e. t ∈ [0, ω], pi 6≡ 0, i = 1, 2. If sup 0≤t≤ω ∫ ω 0 Gp1(t, s)p2(s) ds < 1, (3.1) then p ∈ V−(ω). If for some [a, b] ⊂ [0, ω] we have inf a≤t≤b ∫ b a Gp1(t, s)p2(s) ds > 1, (3.2) then p 6∈ V−(ω). Similarly, Lemma 3.2 provides the following conditions. Corollary 3.6. Let p ∈ Lω be decomposed to p(t) = p1(t) − p2(t) with pi(t) ≥ 0 for a.e. t ∈ [0, ω], pi 6≡ 0, i = 1, 2, and let θn, σn be defined as in Lemma 3.2. If there exists n ∈ N such that sup 0≤t≤ω θn(t) < 1, (3.3) then p ∈ V−(ω). Also, if there exists n ∈ N such that sup 0≤t≤ω σn(t) > 1, (3.4) then p 6∈ V−(ω). If p1 is bounded away from zero, i.e., if there exists c > 0 such that p1(t) ≥ c for a.e. t ∈ [0, ω], then from the comparison principle, Gp1(t, s) ≤ Gc(t, s) for all 156 G. HOLUBOVÁ EJDE/SI/02 (t, s) ∈ [0, ω]× [0, ω]. Moreover, Gc with constant positive c can be given explicitly, namely (see, e.g., [1]) Gc(t, s) =  cosh √ c(t−s−ω 2 ) 2 √ c sinh √ cω 2 , 0 ≤ s ≤ t ≤ ω, cosh √ c(t−s+ ω 2 ) 2 √ c sinh √ cω 2 , 0 ≤ t ≤ s ≤ ω. (3.5) Hence, we can state stricter but easier to apply sufficient condition. In particular, taking p1(t) = (p(t)− c)+ + c, p2(t) = (p(t)− c)− with c > 0, Corollary 3.5 directly implies the following assertion. Corollary 3.7. Let p ∈ Lω and Gc be given by (3.5). If there exists c > 0 such that sup 0≤t≤ω ∫ ω 0 Gc(t, s)(p(s)− c)− ds < 1, (3.6) then p ∈ V−(ω). Finally, using that maxGc(t, s) = Gc(t, t) = ( 2 √ c tanh √ cω2 )−1 and Gc(t, s) is non-constant, we can obtain the simplest sufficient condition that fits exactly Theorem 11.4 in [6]. Corollary 3.8. Let p ∈ Lω. If there exists c > 0 such that∫ ω 0 (p(s)− c)−(s) ds ≤ 2 √ c tanh √ c ω 2 , (3.7) then p ∈ V−(ω). Remark 3.9. From the opposite point of view, if p is bounded from above, taking p1(t) ≡ pM = ess supt∈[0,ω] p(t), p2(t) = pM − p(t) and GpM given by (3.5) with c = pM , the latter statement in Corollary 3.5 directly implies that if for some [a, b] ⊂ [0, ω] inf a≤t≤b ∫ b a GpM (t, s)(pM − p(s)) ds > 1, (3.8) then p 6∈ V−(ω). Since ∫ ω 0 GpM (t, s) ds = 1/pM , condition (3.8) with [a, b] = [0, ω] reads as follows sup 0≤t≤ω ∫ ω 0 GpM (t, s)p(s) ds < 0 . Similarly, the latter statement in Corollary 3.6 implies that if sup 0≤t≤ω ∫ ω 0 GpM (t, s)(pM − p(s))σ0(t) ds > 1, (3.9) then p 6∈ V−(ω). Since minGpM (t, s) = GpM (ω2 , 0) = ( 2 √ pM sinh √ pM ω 2 )−1 , it follows (3.9) with the choice σ0 = minGpM /maxGpM = cosh−1√pM ω 2 (cf. Remark 3.4) reads as follows inf 0≤t≤ω ∫ ω 0 GpM (t, s)p(s) ds < 1− cosh √ pM ω 2 . From the profile of GpM , neither of these conditions provide better information than the already known implication ∫ ω 0 p(t) dt ≤ 0 ⇒ p 6∈ V−(ω). EJDE-2023/SI/02 MAXIMUM PRINCIPLE FOR PERIODIC PROBLEMS 157 4. Examples and comparison of results In this section, we consider ω = 2π and two 2π-periodic test functions: p(t) = α2 + β cos t and p(t) = { α2 + β for t ∈ (0, π2 ) ∪ ( 3π 2 , 2π) α2 − β for t ∈ (π2 , 3π 2 ). and their various decompositions. The first function was used as an example in [6]. The latter (step) function has intentionally the same parameters and nodal do- mains. Its advantage is that we are able to find directly the corresponding principal eigenvalue. Using Wolfram Mathematica, we illustrate our results in Sections 2 and 3 for these functions in the αβ-plane, and compare them with sufficient conditions stated in [6, Theorems 11.1, 11.3, 11.4, 11.5]. Example 4.1. Let ω = 2π and p(t) = α2 + β cos t (4.1) with α, β ∈ R+. For this function, the best sufficient condition in [6] is provided by Theorem 11.1 therein. That’s why we take it as the reference set for our comparison. Let us also recall that [6, Theorem 11.4] coincides with our Corollary 3.8 and [6, Theorem 11.5] gives the sufficient condition α ≥ β for p ∈ V−(ω). We start with the decomposition p1(t) ≡ pM = α2 + β, p2(t) = pM − p(t) = β(1− cos t). For this choice, condition (3.1) of Corollary 3.5 coincides with condition (3.6) of Corollary 3.7 and with (3.3) for n = 1 of Corollary 3.6. These are visualized in αβ-plane in Figure 1 left (orange area) and we see that they do not provide better results than [6, Theorem 11.1] (blue area). Figure 1 right (orange area) illustrates (3.3) for n = 3, i.e. the third iteration of the lower estimate of λ0. Here we can see a considerable improvement. Next we use decomposition (2.5) with c = α2, i.e., c is the mean value of p(t) and p1(t) = α2 + β(cos t)+, p2(t) = β(cos t)−. In this case, we are not able to determine Gp1 and have to be content with the weaker condition (3.6) of Corollary 3.7 involving Gα2 . The obtained region in αβ- plane for which p(t) ∈ V−(2π) is visualized in Figure 2 in green color. Picture on the left illustrates comparison with result of [6, Theorem 11.1] (blue area), picture on the right illustrates comparison with our third iteration (3.3) for the previous choice c = pM (orange area). In general, we can observe that smaller c improves the sufficient condition for larger values of α, β, but spoils the result close to the origin. Moreover, for greater c, the higher iterations in (3.3) are easier to compute. Example 4.2. Let ω = 2π and p(t) = { α2 + β for t ∈ (0, π2 ) ∪ ( 3π 2 , 2π) α2 − β for t ∈ (π2 , 3π 2 ). (4.2) As we mentioned above, for this step function p, we are able to find directly the corresponding principal eigenvalue λ0. In particular, choosing, e.g., p1(t) ≡ pM = α2 + β, p2(t) = pM − p(t) = { 2β for t ∈ (π2 , 3π 2 ), 0 otherwise, (4.3) 158 G. HOLUBOVÁ EJDE/SI/02 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 β α 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 β α Figure 1. Values of α, β for which p(t) = α2 + β cos t ∈ V−(2π). Blue areas correspond to the condition given by [6, Theorem 11.1]. Orange areas correspond to the decomposition with p1(t) ≡ pM and the condition (3.3) of Corollary 3.6 for n = 1 (left) and n = 3 (right). 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 β α 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 β α Figure 2. Values of α, β for which p(t) = α2 + β cos t ∈ V−(2π). Green areas correspond to the decomposition (2.5) with c = α2 and the condition (3.6) of Corollary 3.7. Blue area on the left corresponds to the condition given by [6, Theorem 11.1]. Orange area on the right is the same as in Figure 1 (right). we easily find out that λ0 corresponds to the first positive root of the equation√ α2 + β tanh √ α2 + β π 2 = √ 2βx− α2 − β tan √ 2βx− α2 − β π 2 . (4.4) EJDE-2023/SI/02 MAXIMUM PRINCIPLE FOR PERIODIC PROBLEMS 159 The corresponding normalized constant-sign eigenfuction takes the form y(t) =  A cosh √ α2 + β t for t ∈ (0, π2 ), cos √ 2βλ0 − α2 − β(t− π) for t ∈ (π2 , 3π 2 ), A cosh √ α2 + β(2π − t) for t ∈ ( 3π 2 , 2π), with A = cos √ 2βλ0 − α2 − β π 2 / cosh √ α2 + β π 2 . The curve λ0 = 1 is plotted in red in Figure 3. Hence, according to Theorems 2.1 and/or 2.2, we have p ∈ V−(2π) above this curve, and p 6∈ V−(2π) below this curve. For comparison, we visualize also the approximate condition (3.3) of Corollary 3.6 for the same decomposition (4.3), see the orange areas in Figure 3 (n = 1 on the left, n = 3 on the right). Similarly as in Example 4.1, the blue regions in Figure 3 correspond to the best result of [6], which is in this case provided by Theorem 11.3 therein. Again, we can observe a considerable improvement for higher iterations. 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 β α 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 β α Figure 3. Values of α, β for which p given by (4.2) belongs to V−(2π). Blue areas correspond to the condition given by [6, The- orem 11.3]. Orange areas correspond to the decomposition with p1(t) ≡ pM and the condition (3.3) of Corollary 3.6 for n = 1 (left) and n = 3 (right). Red curve depicts the precise border λ0 = 1 of V−(2π). Acknowledgements. This work was supported by the Grant Agency of the Czech Republic, Grant No. 22-18261S. References [1] Alberto Cabada; Green’s functions in the theory of ordinary differential equations, Green’s Functions in the Theory of Ordinary Differential Equations, Springer, 2014, pp. 1–139. 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[7] Alexander Lomtatidze, Jǐŕı Šremr; On positive periodic solutions to second-order differential equations with a sub-linear non-linearity, Nonlinear Analysis: Real World Applications 57 (2021), 103200. [8] J. R. L. Webb, K. Q. Lan; Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 91–115. MR 2236412 [9] Bo Yang; Positive solutions of the (n− 1, 1) conjugate boundary value problem, Electron. J. Qual. Theory Differ. Equ. (2010), No. 53, 13. MR 2684108 [10] Meirong Zhang; Optimal conditions for maximum and antimaximum principles of the peri- odic solution problem, Boundary value problems 2010 (2010), 1–26. Gabriela Holubová Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitńı 8, 301 00 Plzeň, Czech Republic Email address: gabriela@kma.zcu.cz 1. Introduction and problem formulation 2. Main results 3. Estimates of 0 and consequences of main results 4. Examples and comparison of results Acknowledgements References