Two-Point Boundary Value Problems for a Class of Second-Order Ordinary Differential Equations

Abstract

We study the general semilinear second-order ODE 𝑢^′′+𝑔(𝑡,𝑢,𝑢^′)=0 under different two-point boundary conditions. Using the method of upper and lower solutions, we obtain an existence result. Moreover, under a growth condition on 𝑔, we prove that the set of solutions of 𝑢^′′+𝑔(𝑡,𝑢,𝑢^′)=0 is homeomorphic to the two-dimensional real space.

1. Introduction

The Dirichlet problem for the semilinear second-order ODE𝑢^′′+𝑔𝑡,𝑢,𝑢^′=0 (1.1) has been studied by many authors from the pioneering work of Picard [1], who proved the existence of a solution by an application of the well-known method of successive approximations under a Lipschitz condition on 𝑔 and a smallness condition on 𝑇. Sharper results were obtained by Hamel [2] in the special case of a forced pendulum equation (see also [3, 4]). The existence of periodic solutions for this case has been first considered by Duffing [5] in 1918. Variational methods have been also applied when 𝑔=𝑔(𝑡,𝑢) by Lichtenstein [6], who considered the functional𝐼(𝑢)=^𝑇₀𝑢′²2−𝐺(𝑡,𝑢)𝑑𝑡 (1.2) with 𝐺(𝑡,𝑢)=∫^𝑢₀𝑔(𝑡,𝑠)𝑑𝑠. When 𝑔 depends on 𝑢′,  the problem is nonvariational, and different techniques are required, for example, the shooting method introduced in 1905 by Severini [7] and the more general topological approach, which makes use of Leray-Schauder Degree theory. For an overview of the problem and further results, we refer the reader to [8]. A different kind of nonlinear boundary value PDE (quasilinear elliptic equations) was studied extensively in [9, 10].

This problem is recently studied in [11]. Also, this problem is generalized in [12–14]. Several much more general forms of the problem have been studied in [15–17] via lower and upper solution method. We will study the existence of solutions of (1.1) under Dirichlet, periodic, and nonlinear boundary conditions of the type𝑢^′(0)=𝑓₁(𝑢(0)),𝑢^′(𝑇)=𝑓₂(𝑢(𝑇)), (1.3) where 𝑓₁ and 𝑓₂ are given continuous functions. Note that if 𝑓_𝑖(𝑥)=𝑎_𝑖𝑥+𝑏_𝑖, then (1.3) corresponds to a particular case of Sturm-Liouville conditions and Neumann conditions when 𝑎₁=𝑎₂=0.

In the second section, we impose a growth condition on 𝑔 in order to obtain unique solvability of the Dirichlet problem. Furthermore, we prove that the trace mappingTr∶𝑢∈𝐻²(0,𝑇)∶𝑢^′′+𝑔𝑡,𝑢,𝑢^′=0⟶ℝ² (1.4) given by Tr(𝑢)=(𝑢(0),𝑢(𝑇)) is a homeomorphism, and we apply this result to obtain solutions for other boundary conditions in some specific cases.

In the third section, we construct solutions of the aforementioned problems by an iterative method based on the existence of an ordered couple (𝛼,𝛽) of a lower and an upper solution. This method has been successfully applied to different boundary value problems when 𝑔 does not depend on 𝑢^′. For general 𝑔, existence results have been obtained assuming that |𝑔(𝑡,𝑢,𝑢^′)|≤𝐵(𝑡,|𝑢^′|), where 𝐵∶[0,𝑇]×[0,+∞)→[0,+∞) is a Bernstein function for some 𝑁≥‖𝛼^′‖_∞,‖𝛽^′‖_∞, namely,(i)𝐵 is nondecreasing in 𝑢′,(ii)for 𝛼≤𝑢≤𝛽, if |𝑢^′′(𝑡)|≤𝐵(𝑡,|𝑢′(𝑡)|) for any 𝑡, then ‖𝑢^′‖_∞≤𝑁 (see, e.g., [18]).

We will assume instead a Lipschitz condition with respect to 𝑢′ and construct in each case a nonincreasing (resp., nondecreasing) sequence of upper (lower) solutions that converges to a solution of the problem.

2. A Growth Condition for 𝑔

For simplicity, let us assume that 𝑔 is continuous. We may write it as𝑔𝑡,𝑢,𝑢^′=𝑟(𝑡)𝑢^′+ℎ𝑡,𝑢,𝑢^′, (2.1) with 𝑟 being continuous. We will assume that ℎ satisfies a global Lipschitz condition on 𝑢^′, namely,ℎ(𝑡,𝑢,𝑥)−ℎ𝑡,𝑢,𝑦𝜋𝑥−𝑦≤𝑘<𝑇for𝑥≠𝑦. (2.2)

Remark 2.1. Without loss of generality, we may assume that 𝑟∈𝑊^1,∞(0,𝑇). Indeed, if not, we may multiply (1.1) for any positive 𝑝 such that 𝑟∶=𝑟𝑝−𝑝^′∈𝑊^1,∞(0,𝑇) in order to get the modified equation: 𝑝𝑢^′′+𝑟𝑢^′+𝑝ℎ𝑡,𝑢,𝑢^′=0. (2.3) Note that in this case the value 𝜋/𝑇 in (2.2) must be replaced by 𝜆₁/‖𝑝‖_∞, where 𝜆₁ is the first eigenvalue of (−𝑝𝑢^′)^′ for the Dirichlet conditions.
Furthermore, assume that ℎ satisfies the following one-sided growth condition on 𝑢: ℎ(𝑡,𝑢,𝑥)−ℎ(𝑡,𝑣,𝑥)𝑢−𝑣≤𝑐, (2.4) for 𝑢≠𝑣, with 𝜋𝑐+𝑘𝑇<𝜋𝑇²+12inf_{0≤𝑡≤𝑇}𝑟^′(𝑡). (2.5) Under these assumptions, the set 𝒮 of solutions of (1.1) is homeomorphic to ℝ². More precisely, one has the following.

Theorem 2.2. Assume that (2.2)–(2.4) hold and let 𝑎,𝑏∈ℝ. Then there exists a unique solution 𝑢_𝑎,𝑏 of (1.1) satisfying the nonhomogeneous Dirichlet condition: 𝑢_𝑎,𝑏(0)=𝑎,𝑢_𝑎,𝑏(𝑇)=𝑏. (2.6) Furthermore, the trace mapping Tr∶(𝒮,‖⋅‖_𝐻2)→ℝ² given by Tr(𝑢)=(𝑢(0),𝑢(𝑇)) is a homeomorphism.

Proof. For any 𝑢∈𝐻¹(0,𝑇) let 𝑢 be the unique solution of the linear problem: 𝑢^′′=−𝑟𝑢^′+ℎ𝑡,𝑢,𝑢^′,𝑢(0)=𝑎,𝑢(𝑇)=𝑏. (2.7) It is immediate that the operator 𝐴∶𝐻¹(0,𝑇)→𝐻¹(0,𝑇) given by 𝐴(𝑢)=𝑢 is compact. Moreover, if 𝑆_𝜎𝑢∶=𝑢^′′+𝜎[𝑟𝑢^′+ℎ(𝑡,𝑢,𝑢^′)] with 𝜎∈[0,1], a simple computation shows that the following a priori bound holds for any 𝑢,𝑣∈𝐻²(0,𝑇) with 𝑢−𝑣∈𝐻¹₀(0,𝑇): 𝑢′−𝑣′_𝐿2≤𝜇𝑆_𝜎𝑢−𝑆_𝜎𝑣_𝐿2, (2.8) where 1𝜇=1𝜋/𝑇−𝑘,if𝑐≤2inf𝑟^′,𝜋/𝑇𝜋/𝑇−𝑘+(1/2)inf𝑟^′−𝑐𝑇/𝜋,otherwise. (2.9) Hence, if 𝑢=𝜎𝐴𝑢 (i.e., 𝑆_𝜎(𝑢)=0) for some 𝜎∈[0,1], setting 𝑙_𝑎,𝑏(𝑡)=((𝑏−𝑎)/𝑇)𝑡+𝑎, we obtain 𝑢^′−𝜎𝑙^′_{𝑎,𝑏𝐿2}≤𝜇𝑆_𝜎𝜎𝑙_{𝑎,𝑏𝐿2}≤𝑀 (2.10) for some fixed constant 𝑀. Thus, existence follows from Leray-Schauder Theorem. Uniqueness is an immediate consequence of (2.8) for 𝜎=1.
Hence, Tr is bijective, and its continuity is clear. On the other hand, if (𝑎,𝑏)→(𝑎₀,𝑏₀), applying (2.8) to 𝑢=𝑢_𝑎,𝑏−𝑙_𝑎,𝑏 and 𝑣=𝑢_𝑎0,𝑏0−𝑙_𝑎0,𝑏0, it is easy to see that 𝑢_𝑎,𝑏→𝑢_𝑎0,𝑏0 for the 𝐻¹-norm. As 𝑢_𝑎,𝑏 and 𝑢_𝑎0,𝑏0 satisfy (1.1), we conclude that also 𝑢^′′_𝑎,𝑏→𝑢_𝑎^′′_0,𝑏0 for the 𝐿²-norm and the proof is complete.

Remark 2.3. The proof of Theorem 2.2 still holds under more general assumptions for 𝑔. In fact, if 𝑔 satisfies Caratheodory-type conditions, we may assume only that ℎ(𝑡,𝑢,𝑥)−ℎ𝑡,𝑣,𝑦𝑢−𝑣≤𝑐+𝑘𝑥−𝑦𝑢−𝑣 (2.11) (for 𝑢≠𝑣 and 𝑐,𝑘 as before), which is not equivalent to (2.2)–(2.4) when ℎ is noncontinuous. Thus, the result may be considered a slight extension of well-known results (see, e.g., [19], Corollary V.2).

As a simple consequence we have the following.

Corollary 2.4. Assume that (2.2) and (2.4) hold. Further, assume that there exists a constant 𝑀>0 such that ℎ()(𝑢)|𝑢|ℎ()𝑡,𝑢,0sgn≥0for≥𝑀,⋅,𝑢,0𝑢_𝐿²≤𝛿𝑇^1/2𝜇for|𝑢|≥𝑀, (2.12) where 𝜇 is the constant defined by (2.9) and 𝛿<1. Then (1.1) admits at least one 𝑇-periodic solution, which is unique if 𝑐<0 in (2.4).

Proof. With the notations of the previous theorem, let us consider the mapping 𝜑∶ℝ→ℝ given by 𝜑(𝑎)=𝑢^′_𝑎,𝑎(𝑇)−𝑢^′_𝑎,𝑎(0). (2.13) From Theorem 2.2, 𝜑 is continuous, and it is clear that 𝑢 is a periodic solution of the problem if and only if 𝑢=𝑢_𝑎,𝑎 for some 𝑎 with 𝜑(𝑎)=0.
From (2.8), if |𝑎|≥𝑀, we take 𝑣=𝑎 and 𝜎=1. Observe that 𝑆₁𝑢_𝑎𝑎=0. Therefore it follows that 𝑢^′_{𝑎,𝑎𝐿2}≤𝜇‖ℎ(⋅,𝑎,0)‖_𝐿2≤𝛿𝑇^1/2|𝑎|, (2.14) where in the last inequality we used (2.6). Hence |𝑢_𝑎,𝑎−𝑎|≤∫^𝑇₀|𝑢_𝑎,𝑎′|≤𝛿|𝑎| or, equivalently: (1−𝛿)|𝑎|≤𝑢_𝑎,𝑎≤(1+𝛿)|𝑎|. (2.15) Let 𝑝 be the unique solution (in distributional sense) of the following problem: 𝑝^′−(𝑟+𝜉)𝑝^′=0,𝑝(0)=𝑝(𝑇)=1, (2.16) where 𝜉∈𝐿^∞(0,𝑇) is given by 𝜉(𝑡)=ℎ𝑡,𝑢_𝑎,𝑎,𝑢^′_𝑎,𝑎−ℎ𝑡,𝑢_𝑎,𝑎,0𝑢^′_𝑎,𝑎, (2.17) with 𝜉(𝑡)=0 if 𝑢^′_𝑎,𝑎(𝑡)=0. A simple computation shows that 𝑝 is positive, and multiplying the equation by 𝑝, we obtain 𝜑(𝑎)=^𝑇₀𝑝𝑢^′_𝑎,𝑎^′=−^𝑇₀𝑝ℎ𝑡,𝑢_𝑎,𝑎,0. (2.18) Hence by (2.12) 𝜑(𝑎)≤0≤𝜑(−𝑎) for (1−𝛿)𝑎≥𝑀 and existence follows from the continuity of 𝜑. On the other hand, if 𝑢 and 𝑣 are periodic solutions of the problem, then (𝑢−𝑣)^′′+𝑟+𝜓(𝑢−𝑣)^′+ℎ𝑡,𝑢,𝑣^′−ℎ𝑡,𝑣,𝑣^′=0, (2.19) where 𝜓(𝑡)=ℎ𝑡,𝑢,𝑢^′−ℎ𝑡,𝑢,𝑣^′𝑢^′−𝑣^′∈𝐿^∞(0,𝑇). (2.20) Now take 𝑝>0 as the unique solution of the problem [𝑝^′−(𝑟+𝜓)𝑝]^′=0, 𝑝(0)=𝑝(𝑇)=1. Multiplying the previous equality by 𝑝(𝑢−𝑣) and applying the boundary conditions for 𝑝,𝑢, and 𝑣, we observe ^𝑇₀𝑝(𝑢−𝑣)(𝑢−𝑣)^′′=−^𝑇₀𝑝^′(𝑢−𝑣)+𝑝𝑢^′−𝑣^′(𝑢−𝑣)^′=−^𝑇₀𝑝^′(𝑢−𝑣)(𝑢−𝑣)^′+^𝑇₀𝑝𝑢^′−𝑣^′2,_𝑇0𝑝𝑟+𝜓(𝑢−𝑣)(𝑢−𝑣)^′=−^𝑇₀𝑝𝑟+𝜓^′(𝑢−𝑣)²2=−^𝑇₀𝑝^′′(𝑢−𝑣)²2=^𝑇₀(𝑢−𝑣)(𝑢−𝑣)^′𝑝^′. (2.21) Thus we obtain 0=^𝑇₀𝑝𝑢^′−𝑣^′2−^𝑇₀𝑝ℎ𝑡,𝑢,𝑣^′−ℎ𝑡,𝑣,𝑣^′)≥(𝑢−𝑣_𝑇0𝑝𝑢^′−𝑣^′2−𝑐^𝑇₀𝑝(𝑢−𝑣)². (2.22) If 𝑐<0, we conclude that 𝑢=𝑣.

Remark 2.5. In the previous proof, note that the sign condition on ℎ is only used for (1−𝛿)|𝑎|≤|𝑢|≤(1+𝛿)|𝑎|. Thus, (2.12) may be replaced by the weaker condition ℎ(𝑡,⋅,0)|_𝐼1≥0≥ℎ(𝑡,⋅,0)|_𝐼2 (2.23) where 𝐼_𝑗=[𝑎_𝑗−𝛿_𝑗|𝑎_𝑗|,𝑎_𝑗+𝛿_𝑗|𝑎_𝑗|] for some 𝑎_𝑗∈ℝ, 𝛿_𝑗<1 with ℎ(⋅,𝑢,0)𝑢_𝐿²≤𝛿_𝑗𝑇^1/2𝜇for𝑢∈𝐼_𝑗. (2.24)

Remark 2.6. As a particular case of Corollary 2.4, we deduce the existence of 𝑇-periodic solutions under the following Landesman-Lazer type conditions (see, e.g., [20]): liminf_|𝑢|→∞ℎ(𝑡,𝑢,0)sgn(𝑢)≥0,lim_|𝑢|→∞ℎ(⋅,𝑢,0)𝑢_𝐿²=0. (2.25) As in the standard Duffing equation 𝑢^′′+ℎ(𝑢)=𝜃(𝑡), the asymptotic condition (2.6) can be dropped if the sign in (2.12) is reversed. More precisely, we have the following.

Corollary 2.7. Assume that (2.2) and (2.4) hold. Further, assume that there exists a constant 𝑀>0 such that ℎ(𝑡,𝑢,0)sgn(𝑢)<0for|𝑢|≥𝑀. (2.26) Then (1.1) is solvable under periodic or Sturm-Liouville conditions: 𝑢^′(0)=𝑎₁𝑢(0)+𝑏₁,𝑢^′(𝑇)=𝑎₂𝑢(𝑇)+𝑏₂,𝑎₁≥0≥𝑎₂. (2.27) Furthermore, if 𝑐<0 in (2.4), then the respective solutions are unique.

Proof. For the periodic problem, define 𝜑 as in the previous corollary. For 𝑎≥𝑀, if 𝑢_𝑎,𝑎(𝑡₀)>𝑎 for some 𝑡₀, we may assume that 𝑡₀ is maximum, and hence 𝑢^′′𝑡₀=−𝑔𝑡₀,𝑢𝑡₀,0=−ℎ𝑡₀,𝑢𝑡₀,0>0, (2.28) a contradiction. Thus, 𝑢≤𝑎, which implies that 𝜑(𝑎)≥0. In the same way, we deduce that 𝜑(𝑎)≤0 for 𝑎≤−𝑀. Uniqueness follows as in Corollary 2.4.
For (2.27) conditions, let us first note that if 𝜆>0, the linear problem 𝑣^′′+𝑟𝑣^′−𝜆𝑣=0,𝑣^′(0)=𝑎₁𝑣(0)+𝑏₁,𝑣^′(𝑇)=𝑎₂𝑣(𝑇)+𝑏₂ (2.29) is uniquely solvable, and setting 𝑤=𝑢−𝑣 problem (1.1)–(2.27) is equivalent to 𝑤^′′+𝑔𝑡,𝑤,𝑤^′=0,𝑤^′(0)=𝑎₁𝑤(0),𝑤^′(𝑇)=𝑎₂𝑤(0), (2.30) where 𝑔(𝑡,𝑤,𝑤′)∶=𝑔(𝑡,𝑤+𝑣,𝑤′+𝑣′)−𝜆𝑣 satisfies the hypothesis. Hence, it suffices to consider only the homogeneous case 𝑏₁=𝑏₂=0. In the same way as before, define 𝜑∶ℝ²→ℝ² by 𝜑(𝑎,𝑏)=𝑢^′_𝑎,𝑏(0),𝑢^′_𝑎,𝑏(𝑇). (2.31) For 𝑎≥𝑀≥|𝑏|, we obtain that 𝑢^′_𝑎,𝑏(0)≤0≤𝑢^′_−𝑎,𝑏(0), and for 𝑏≥𝑀≥|𝑎|, it holds that 𝑢^′_𝑎,𝑏(𝑇)≥0≥𝑢^′_𝑎,−𝑏(𝑇). By the generalized intermediate value theorem, we deduce that 𝜑 has a zero in [−𝑀,𝑀]×[−𝑀,𝑀]. Uniqueness can be proved as in the periodic case.

3. Iterative Sequences of Upper and Lower Solutions

In this section, we construct solutions of (1.1) under the mentioned two-point boundary conditions by an iterative method. As before, consider 𝑔(𝑡,𝑢,𝑢^′)=𝑟(𝑡)𝑢^′+ℎ(𝑡,𝑢,𝑢^′) where 𝑟∈𝑊^1,∞(0,𝑇) and ℎ is globally Lipschitz on 𝑢^′ with constant 𝑘<𝜋/𝑇.

We will need the following auxiliary lemmas.

Lemma 3.1. Assume that (2.2) holds and let 𝜆>0 be large enough. Then for any 𝑧,𝜃∈𝐶([0,𝑇]), 𝑢^′′+𝑟𝑢^′+ℎ𝑡,𝑧,𝑢^′−𝜆𝑢=𝜃(𝑡) (3.1) is uniquely solvable under Dirichlet, periodic, or (2.27) conditions. Furthermore, the application 𝑇∶𝐶([0,𝑇])²→𝐶([0,𝑇]) given by 𝑇(𝑧,𝜃)=𝑢 is compact.

Proof. Taking 𝜆>𝑘𝜋/𝑇−(𝜋/𝑇)²−(1/2)inf𝑟^′, existence and uniqueness follow as a particular case of Theorem 2.2 and Corollary 2.7 for 𝑔𝑡,𝑢,𝑢^′=𝑟𝑢^′+ℎ𝑡,𝑧,𝑢^′−𝜆𝑢−𝜃(𝑡). (3.2) Let (𝑧,𝜃)→(𝑧₀,𝜃₀), and set 𝑢=𝑇(𝑧,𝜃), 𝑢₀=𝑇(𝑧₀,𝜃₀). Then 𝑢−𝑢₀^′′+𝑟+𝜓𝑢−𝑢₀^′−𝜆𝑢−𝑢₀=ℎ𝑡,𝑧,𝑢^′₀−ℎ𝑡,𝑧₀,𝑢^′₀+𝜃−𝜃₀, (3.3) where 𝜓(𝑡)=ℎ(𝑡,𝑧,𝑢^′)−ℎ(𝑡,𝑧,𝑢^′₀)/𝑢^′−𝑢^′₀. Hence, it suffices to prove that the following a priori bound holds for any 𝑤 satisfying periodic or homogeneous Dirichlet or (2.27) conditions: ‖𝑤‖_𝐻1≤𝑐𝑤^′′+𝑟+𝜓𝑤^′−𝜆𝑤_𝐿2, (3.4) where the constant 𝑐 depends only on 𝑘. For Dirichlet and (2.27) conditions, apply Cauchy-Schwartz inequality to the integral −∫^𝑇₀𝑝𝐿𝑤⋅𝑤 where 𝐿𝑤=𝑤^′′+(𝑟+𝜓)𝑤^′−𝜆𝑤 and 𝑝=𝑒^{∫𝑇0(𝑟+𝜓)}, and observe that 0<𝑚≤𝑝≤𝑀 for some 𝑚 and 𝑀 depending only on 𝑘 (note that under homogeneous (2.27) conditions, it holds that −𝑝𝑤𝑤^′|^𝑇₀=𝑝(0)𝑎₁𝑤(0)²−𝑝(𝑇)𝑎₂𝑤(𝑇)²≥0). For periodic conditions, take 𝑝 such that 𝑝^′=(𝑟+𝜓)𝑝−𝑟 with 𝑟 constant and 𝑝(0)=𝑝(𝑇)=1 and the proof follows.

Lemma 3.2. Let 𝜙∈𝐿^∞(0,𝑇) and assume that 𝑤^′′+𝜙𝑤^′−𝜆𝑤≥0 a.e. for 𝜆>0. Then 𝑤≤0, provided that 𝑤 satisfies one of the boundary conditions:(i)𝑤(0),𝑤(𝑇)≤0,(ii)𝑤(𝑇)−𝑤(0)=0≥𝑤^′(𝑇)−𝑤^′(0),(iii)𝑤^′(0)−𝑎₁𝑤(0)≥0≥𝑤^′(𝑇)−𝑎₂𝑤(𝑇),𝑎₁≥0≥𝑎₂.

Proof. For 𝑤(0),𝑤(𝑇)≤0, the result is the well-known maximum principle for Dirichlet conditions. If (ii) holds and 𝑤(0)=𝑤(𝑇)>0, as 𝑤 cannot achieve a positive maximum on (0,𝑇), we have that 𝑤^′(0)=𝑤^′(𝑇)=0 and 𝑤,𝑤^′≥0 over a maximal interval (𝑡₀,𝑇]. Taking 𝑝=𝑒^∫𝑡0𝜙, we deduce that 𝑝𝑤^′ is nondecreasing on [𝑡₀,𝑇], a contradiction. If (iii) holds and, for example, 𝑤(0)>0, restricting 𝑤 up to its first zero if necessary, it suffices to consider only the case 𝑤≥0. As before, we get a contradiction from the fact that 𝑝𝑤^′ is nondecreasing. The proof is similar if we assume that 𝑤(𝑇)>0.

In order to prove the main result of this section, we recall that (𝛼,𝛽) is an ordered couple of a lower and an upper solution for (1.1) if 𝛼≤𝛽 and𝛼^′′+𝑔⋅,𝛼,𝛼^′≥0≥𝛽^′′+𝑔⋅,𝛽,𝛽^′, (3.5) under the following boundary conditions.

For the Dirichlet problem,𝛼(0)≤𝑎≤𝛽(0),𝛼(𝑇)≤𝑏≤𝛽(𝑇). (3.6)For the periodic problem,𝛼(𝑇)−𝛼(0)=0=𝛽(𝑇)−𝛽(0),𝛼^′(𝑇)−𝛼^′(0)≤0≤𝛽^′(𝑇)−𝛽^′(0). (3.7)For the problem (1.3),𝛼^′(0)−𝑓₁(𝛼(0))≥0≥𝛽^′(0)−𝑓₁𝛽(0),𝛼^′(𝑇)−𝑓₂(𝛼(𝑇))≤0≤𝛽^′(𝑇)−𝑓₂𝛽(𝑇). (3.8)

We make the following extra assumption.

There exists a constant 𝑅>0 such thatℎ𝑡,𝑢,𝛼^′−ℎ𝑡,𝑣,𝛼^′ℎ𝑢−𝑣≤𝑅,𝑡,𝑢,𝛽^′−ℎ𝑡,𝑣,𝛽^′𝑢−𝑣≥−𝑅, (3.9) for any 𝑢,𝑣 such that 𝛼(𝑡)≤𝑢(𝑡),𝑣(𝑡)≤𝛽(𝑡), and for (1.3): there exists a constant 𝑅>0 such that𝑓₁(𝑥)−𝑓₁𝑦(0)(0),𝑓𝑥−𝑦≤𝑅for𝛼≤𝑥,𝑦≤𝛽₂(𝑥)−𝑓₂𝑦(𝑇)(𝑇).𝑥−𝑦≥−𝑅for𝛼≤𝑥,𝑦≤𝛽 (3.10)

Then we have the following.

Theorem 3.3. Assume that there exists an ordered couple (𝛼,𝛽) of a lower and an upper solution for Dirichlet, periodic, or (1.3) conditions. Further, assume that (2.2) and (3.9) hold (and also (3.10), for the (1.3) case). Then the respective boundary value problem admits at least one solution 𝑢 with 𝛼≤𝑢≤𝛽.

Remark 3.4. Observe that a Lipschitz condition is a particular case of a Nagumo condition. This result can also be obtained as a Corollary of Theorem  3.2 in [21].

Proof. For 𝜆≥𝑅 large enough and 𝑢∈𝐶([0,𝑇]) define 𝑇𝑢=𝑢 to be the unique solution of the following problem: 𝑢^′′+𝑟𝑢^′+ℎ𝑡,𝑢,𝑢^′−𝜆𝑢=−𝜆𝑢 (3.11) satisfying, respectively, Dirichlet, periodic, or the Sturm-Liouville condition: 𝑢^′(0)−𝑅𝑢(0)=𝑓₁(𝑢(0))−𝑅𝑢(0),𝑢^′(𝑇)+𝑅𝑢(𝑇)=𝑓₂(𝑢(𝑇))+𝑅𝑢(𝑇). (3.12) Compactness of 𝑇 follows easily from Lemma 3.1. Moreover, if 𝑢≤𝛽, then 𝑢^′′+𝑟𝑢^′+ℎ𝑡,𝑢,𝑢^′+𝑅𝑢−𝜆𝑢=(𝑅−𝜆)()𝛽≥()𝑢≥𝑅−𝜆𝑅−𝜆𝛽+𝛽^′′+𝑟𝛽′+ℎ𝑡,𝛽,𝛽^′. (3.13) Hence, setting 𝜓(𝑡)=ℎ𝑡,𝑢,𝑢^′−ℎ𝑡,𝑢,𝛽^′𝑢^′−𝛽′, (3.14) we deduce that 𝑢−𝛽^′′+𝑟+𝜓𝑢−𝛽^′−𝜆𝑢−𝛽≥ℎ𝑡,𝛽,𝛽^′+𝑅𝛽−ℎ𝑡,𝑢,𝛽^′+𝑅𝑢≥0. (3.15) For Dirichlet and periodic cases, it follows that 𝑢≤𝛽. For (1.3), note that 𝑢^′(0)−𝑅𝑢(0)=𝑓₁(𝑢(0))−𝑅𝑢(0)≥𝑓₁𝛽(0)(0),𝑢−𝑅𝛽_′(𝑇)+𝑅𝑢(𝑇)=𝑓₂(𝑢(𝑇))+𝑅𝑢(𝑇)≤𝑓₂𝛽(𝑇)+𝑅𝛽(𝑇). (3.16) Hence, 𝑢−𝛽^′(0)−𝑅𝑢−𝛽(0)≥0≥𝑢−𝛽^′(𝑇)−𝑅𝑢−𝛽(𝑇), (3.17) and from Lemma 3.2, we also obtain that 𝑢≤𝛽. In the same way, if 𝑢≥𝛼, we obtain that 𝑢≥𝛼 and the proof follows from Schauder Fixed Point Theorem.

Remark 3.5. Existence conditions in Corollary 2.7 are easily improved by applying Theorem 3.3. Indeed, under condition (2.26), it is immediate that (−𝑀,𝑀) is an ordered couple of a lower and an upper solution.

Remark 3.6. In the context of the previous theorem, from Lemma 3.1, we deduce the existence of a constant 𝐾 such that if 𝑢=𝑇𝑢 for 𝛼≤𝑢≤𝛽, then 𝑢^′_𝐿∞≤𝐾. (3.18)

Example 3.7. It is easy to see that the problem 𝑢^′′𝑢−𝑢=0,_′(0)=𝑢(0),𝑢^′(𝑇)=𝑢(𝑇)+1 (3.19) has no solution, although 𝛼=−1 is a lower solution. From the previous theorem, we deduce that no upper solution 𝛽≥−1 exists. This can be proved directly from the following conditions: 𝛽^′′−𝛽≤0,𝛽^′(0)≤𝛽(0),𝛽^′(𝑇)≥𝛽(𝑇)+1. (3.20) Indeed, as no negative minimum exists, if 𝛽(0),𝛽(𝑇)≥0, we may take 𝑡₀ maximum such that 𝛽^′>0 over (𝑡₀,𝑇]. Hence, (𝛽^′)²(𝑇)−𝛽²(𝑇)≤(𝛽^′)²(𝑡₀)−𝛽²(𝑡₀)≤0, a contradiction. On the other hand, if 𝛽(0)<0, then 𝛽^′<0 on [0,𝑇) and 𝛽^′(𝑇)=0, a contradiction since 𝛽^′′≤𝛽. The case 𝛽(0)≥0>𝛽(𝑇) can be easily reduced to the previous one.

In order to construct solutions by iteration, we need a stronger assumption on ℎ.

There exists a constant 𝑅 such that|ℎ(𝑡,𝑢,𝑥)−ℎ(𝑡,𝑣,𝑥)|≤𝑅|𝑢−𝑣| (3.21) for 𝑢,𝑣 such that 𝛼(𝑡)≤𝑢(𝑡),𝑣(𝑡)≤𝛽(𝑡), and 𝑥∈ℝ.

Corollary 3.8. Assume that there exists an ordered couple (𝛼,𝛽) of a lower and an upper solution for Dirichlet, periodic, or (1.3) conditions. Further, assume that (2.2) and (3.21) hold (and also (3.10), for the (1.3) case). Set 𝜆≥𝑅 large enough, and define the sequences {𝑢_𝑛} and {𝑢_𝑛} given by 𝑢₀=𝛼,𝑢₀=𝛽, (3.22) and 𝑢_𝑛+1, 𝑢_𝑛+1 are the (unique) solutions of the following problems: 𝑢^′′_𝑛+1+𝑟𝑢^′_𝑛+1+ℎ𝑡,𝑢_𝑛,𝑢^′_𝑛+1−𝜆𝑢_𝑛+1=−𝜆𝑢_𝑛,𝑢^′′_𝑛+1+𝑟𝑢^′_𝑛+1+ℎ𝑡,𝑢_𝑛,𝑢^′_𝑛+1−𝜆𝑢_𝑛+1=−𝜆𝑢_𝑛, (3.23) under the respective boundary conditions. Then (𝑢_𝑛𝑢_𝑛) is an ordered couple of a lower and an upper solution. Furthermore, {𝑢_𝑛} (resp., {𝑢_𝑛}) is nonincreasing (nondecreasing) and converges to a solution of the problem.

Remark 3.9. Observe that this is also a classical result that can be found in the works of Adje [22] or Cabada [23], for example.

Proof. From the previous theorem, we know that 𝛼≤𝑢₁≤𝛽. Moreover, 𝑢₁^′′+𝑟𝑢^′₁+ℎ𝑡,𝑢₁,𝑢^′₁=(𝜆−𝑅)𝑢₁−𝛽+ℎ𝑡,𝑢₁,𝑢^′₁+𝑅𝑢₁−ℎ𝑡,𝛽,𝑢^′₁+𝑅𝛽≤0. (3.24) Hence, 𝑢₁ is an upper solution of the problem. Inductively, it follows that 𝑢_𝑛 is an upper solution for every 𝑛, with 𝛼≤𝑢_𝑛+1≤𝑢_𝑛. Hence 𝑢_𝑛 converges pointwise to a function 𝑢.
From 𝑢^′′_𝑛+1+𝑟𝑢^′_𝑛+1+ℎ𝑡,𝑢_𝑛,𝑢^′_𝑛+1⟶0 (3.25) pointwise. Moreover, by Lemma 3.1, we know that {𝑢_𝑛} is bounded in 𝐻¹(0,𝑇); hence in 𝐻²(0,𝑇), it follows easily that 𝑢^′′+𝑟𝑢^′+ℎ𝑡,𝑢,𝑢^′=0. (3.26) Thus, 𝑢 is a solution of the problem. The proof for 𝑢_𝑛 is analogous. Moreover, if we assume that 𝑢_𝑛≤𝑢_𝑛, it is immediate that 𝑢_𝑛+1≤𝑢_𝑛+1.