Boundary value problems for nonlinear second-order functional differential equations with piecewise constant arguments

Received: 19 July 2022 Revised: 2 November 2022 Accepted: 3 November 2022 DOI: 10.1002/mma.8878 SPECIAL ISSUE PAPER Boundary value problems for nonlinear second-order functional differential equations with piecewise constant arguments Sebastián Buedo-Fernández1Daniel Cao Labora2,1 Rosana Rodríguez-López2,1 1Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, Santiago de Compostela, Spain 2CITMAga, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Correspondence Rosana Rodríguez-López, CITMAga and Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. Email: [email protected] Communicated by: C. Pinto Funding information Agencia Estatal de Investigación, Grant/Award Number: PID2020-113275GB-I00 and MTM2016-75140-P; Ministerio de Educación, Cultura y Deporte, Grant/Award Number: FPU16/04168 and FPU16/04416; Xunta de Galicia, Grant/Award Number: ED431C 2019/02 In this paper, we consider a class of nonlinear second-order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. KEYWORDS boundary value problems, monotone iterative technique, piecewise constant functional dependence, second-order functional differential equations, upper and lower solutions MSC CLASSIFICATION 34K10, 34K07, 34K05, 34K12 1INTRODUCTION In the study of nonlinear differential equations, the existence of solutions is sometimes achieved through the development of the method of upper and lower solutions, and the approximation of the extremal solutions in the functional interval defined by those functions is performed by the monotone iterative technique. A fundamental reference on monotone method is Ladde et al1(see also previous works2–6). To mention some other related works, the application of the monotone method to functional differential equations can be found in Nieto et al,7and second-order periodic boundary value problems were considered in Cabada and Nieto.8 Dedicated to the memory of Professor J.A. Tenreiro Machado. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2022 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd. Math Meth Appl Sci. 2022;1–35. wileyonlinelibrary.com/journal/mma 1 2BUEDO-FERNÁNDEZ ET AL. The existence of solution to second-order functional differential equations with a functional dependence given by a piecewise constant argument has attracted the attention of many authors. We cite, for instance, previous works.9–13 For the mentioned works, the delay is given by the integer part function but the nonlinearity in the equation is independent of x′. Chen and Sun14 considered a class of linear boundary value problems for nonlinear second-order impulsive functional differential equations with continuous delay function, and they applied the upper and lower solutions method and the monotone iterative technique to obtain the existence of solution. However, the nonlinearity in the equation is also independent of the derivative of x. On the other hand, the class of functional differential equations considered in Corduneanu15 presents a linear dependence on x′. Some other recent works include a nonlinearity depending on x′but avoid the introduction of delay in the equation. In Guo and Guo,16 the authors studied the existence and multiplicity of periodic solutions for a class of second-order delay differential equations with no explicit dependence on x′. More recently, some results based on Avery–Peterson fixed point theorem were provided in Shen et al17 for a thermostat model including the first-order derivative in the nonlinearity. Other references relevant to the topic are, for instance, the article by Henríquez and Hernández,18 which was devoted to the analysis of the approximate controllability of control systems given by second-order semilinear functional differential equations with infinite delay; the work by Sakthivel et al,19 which is focused on the study of the exact controllability of certain second-order nonlinear impulsive control differential systems; the study of Shoukaku,20 about the oscillatory behavior of certain hyperbolic equations with continuous distributed deviating arguments; or Liu and Huang,21 where the coincidence degree theory was applied to obtain results on the existence and uniqueness of T-periodic solutions for a class of second-order neutral functional differential equations. The same approach, coincidence degree theory, was applied in22 to analyze the existence of periodic solutions for higher-order differential equations with deviating arguments. In this last case, the dependence on the different derivatives x(i)is linear. In particular, in Zhang,13 the authors considered the following periodic boundary value problem for second-order functional differential equations: {−x′′(t)=𝑓(t,x(t),x([t])),t∈J=[0,T], x(0)=x(T),x′(0)=x′(T), while previous studies23,24 were focused on the analysis of similar problems for the case of first-order differential equations. On the other hand, Nieto and Rodríguez-López25 presented some results on the study of the existence of solution to second-order functional differential equations with piecewise constant arguments of the type ⎧ ⎪ ⎨ ⎪ ⎩ x′′(t)+ax′(t)+bx(t)+cx′([t]) + dx([t]) = 𝜎(t),t∈J=[0,T], x(0)=x(T), x′(0)=x′(T)+𝜆, (1) by proving the existence of solutions to the following linear impulsive periodic boundary value problems: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x′′(t)+ax′(t)+bx(t)+cx′([t]) + dx([t]) = 0,t∈R, x(0)=x(T), x′(0−)=x′(T+), x′(s+)=x′(s−)+1, (2) where s∈J,a,b,c,d,𝜆∈R,T>0and𝜎is a piecewise continuous function. More specifically, it was illustrated how the solution to (1) can be obtained by means of a Green's function which is given by the solution of (2) (see also Yang et al26). Further, in Buedo-Fernández et al,27 conditions were provided in order to prove the existence of solutions to (1) with a constant sign, deducing comparison results which will be useful to prove in this paper the existence of solutions to nonlinear second-order functional differential equations with piecewise constant arguments, for which the nonlinearity depends on the x′term and the delay is also introduced in the derivative. Our main motivations for the study of this problem are, on one hand, its applicability to the modeling of a thermostat including the dependence on the first-derivative of the state variable, similarly to the one considered in Shen et al,17 but controlled through the introduction of functional terms in xand x′, and on the other hand but also related, the need of 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.3 determining the behavior of the solutions for such models controlled by the temperature and the speed of change of the temperature at some fixed instants. Some other models for thermostats have been studied in previous studies28,29 (see also the study in Webb30). For some models with this type of applications from the perspective of fractional calculus, we refer to previous works,31,32 and also Rezapour et al33 as example of variable order fractional thermostat models. The paper is organized as follows. In Section 2, we present the problem of study and recall the comparison results for second-order functional differential equations extracted from Buedo-Fernández et al27 that will be useful to our purposes, and then, in Section 3, we provide results on the existence of solution for nonlinear second-order functional problems with boundary value conditions by using the upper and lower solutions' method. In Section 4, we include the development of the monotone iterative technique, and finally, in Sections 5 and 6, we present, respectively, some examples and conclusions. 2PRELIMINARIES We consider the problem {x′′(t)=g(t,x(t),x′(t),x([t]),x′([t])),t∈J=[0,T], x(0)=x(T), x′(0)=x′(T)+𝜆, (3) where 𝜆∈R,andg∶J×R4→Ris continuous on (J∖{1,2,…,[T]}) × R4and such that the following limits are finite: lim t→n−g(t,x,𝑦,u,v),g(n,x,𝑦,u,v) = lim t→n+g(t,x,𝑦,u,v),n∈{1,2,…,[T]}. This kind of problems is useful in the study of phenomena which are self-regulated at fixed equidistant instants, for instance, at the positive integer numbers. Note that the consideration of this problem allows to consider functional dependence on the derivative x′, hence the context is more general than that in other previous works. We denote by C(J)the space of continuous functions defined on J C(J)={u∶J→R∶ucontinuous}, furnished with the supremum norm. Definition 1 (Definition 2 of Nieto and Rodríguez-López25).Consider the spaces Λ∶={𝑦∶J→R∶𝑦continuous on J∖{1,2,…,[T]},and there exist 𝑦(n−)∈R,𝑦(n+)=𝑦(n),∀n∈{1,2,…,[T]}} and E∶= {x∶J→R∶x,x′are continuous and x′′ ∈Λ}. Definition 2. A function xis a solution to (3) if x∈Eand satisfies the conditions in (3). Definition 3 (Nieto and Rodríguez-López25).Similarly, a function xis a solution to (1) if x∈Eand satisfies the conditions in (1), where x′′(n)=x′′(n+),∀n∈{0,1,2,…,[T]}. In the sequel, Idenotes the identity mapping, H(z)∶=(h1(z)h2(z) h′ 1(z)h′ 2(z)),for z∈[0,1],and C∶= H(1)=(C1C2 C′ 1C′ 2), 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4BUEDO-FERNÁNDEZ ET AL. where h1(s)is given by 1−d as+d a2(1−e−as),if b=0,a≠0, 1−d 2s2,if b=0,a=0, (1+d b)(1+a 2s)e−a 2s−d b,if b≠0,a2=4b, (1+d b)𝛽e𝛼s−𝛼e𝛽s 𝛽−𝛼−d b,if b≠0,a2>4b, (1+d b)e−a 2s⎧ ⎪ ⎨ ⎪ ⎩ cos √b−a2 4s+a 2√b−a2 4 sin √b−a2 4s⎫ ⎪ ⎬ ⎪ ⎭ −d b,if b≠0,a2<4b, and h2(s)is given by 1 a(1−e−as −cs +c a(1−e−as)),if b=0,a≠0, s−c 2s2,if b=0,a=0, e−a 2s[c b(1+a 2s)+s]−c b,if b≠0,a2=4b, (𝛽c b−1)e𝛼s+(1−𝛼c b)e𝛽s 𝛽−𝛼−c b,if b≠0,a2>4b, e−a 2s⎧ ⎪ ⎨ ⎪ ⎩ c bcos √b−a2 4s+ 1+ac 2b √b−a2 4 sin √b−a2 4s⎫ ⎪ ⎬ ⎪ ⎭ −c b,if b≠0,a2<4b. Here, for b≠0,a2>4b,wedenote 𝛼=− a 2+√(a 2)2 −b,𝛽=− a 2−√(a 2)2 −b.(4) Note that, according to the previous notation, H(T−[T]) = (h1(T−[T]) h2(T−[T]) h′ 1(T−[T]) h′ 2(T−[T]) ). Theorem 1. (Theorem 3.2 of Nieto and Rodríguez-López25). If hypothesis det (I−H(T−[T])C[T])≠0(5) holds, then problem (1) has a unique solution, for all 𝜎∈Λ(see Definition 1) and 𝜆∈R, which can be obtained by the expression x(t)= T ∫ 0 K(t,s)𝜎(s)ds +𝜆K(t,0),t∈J,(6) where, for all s ∈J,K(·,s)is the unique solution to (2). See Nieto and Rodríguez-López25 for the expression of the Green's function Kin Theorem 1. We also fix the notation Nfor the set of positive integer numbers and Z+=N∪{0}for the set of nonnegative integer numbers. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.5 Moreover, the following comparison results are useful in the proof of the main results. Consider the set ∶= {(x,𝑦)∈R2∶x≥0,𝑦≥−Mx}, where M∶= inf z∈(0,1) h1(z) h2(z). Define also g(z)∶=⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1−e−az a,if b=0,a≠0, z,if b=0,a=0, ze−a 2z,if b≠0,a2=4b, e𝛽z−e𝛼z 𝛽−𝛼,if b≠0,a2>4b, e−a 2zsin √b−a2 4z √b−a2 4 ,if b≠0,a2<4b. (7) Theorem 2 (Theorem of Buedo-Fernández et al27).Suppose that the hypothesis (5) holds. Assume that 𝜎∈Λis nonnegative on J, 𝜆≥0, and that the following conditions hold: (I) h1(1)>0,andh 2>0on (0,1). (II) The vector V0given by V0∶= [I−H(T−[T]) C[T]]−1(0 1)(8) satisfies that CkV0∈for every k =0,1,…,[T]. (III) The function g given by (7) is nonnegative on (0,1). (IV) For each 0<s<Twiths∈(n,n+1)for some n ∈Z+, the vector V0,sgiven, for T <n+1,by V0,s∶= [I−H(T−[T]) C[T]]−1(g(T−s) g′(T−s)), and for n +1≤T, by V0,s∶= [I−H(T−[T]) C[T]]−1H(T−[T]) C[T]−n−1(g(n+1−s) g′(n+1−s)), satisfies that CkV0,s∈, for every k =0,1,…,[T]. (V) For each 0<s<Twiths∈(n,n+1)for some n ∈Z+,ifT≥n+1, we also assume that the vector V1,s∶= [Cn+1V0,s+(g(n+1−s) g′(n+1−s))] (with V0,sgiven in IV), satisfies that CkV1,s∈, for every k =0,1,…,[T]−n−1. Then the unique solution to problem (1) is nonnegative on J. Theorem 3 (Theorem 10 of Buedo-Fernández et al27).Suppose that the hypothesis (5) holds. Assume also that the conditions (I)–(V) in Theorem 2 are satisfied. If 𝜎∈Λis nonpositive on J and 𝜆≤0, then the unique solution to problem (1) is nonpositive on J. Remark 1 (Remark 6 of Buedo-Fernández et al27).Condition (I) in Theorems 2–3 is satisfied under the following circumstances: Case b=0,a≠0:1−d a+d a2(1−e−a)>0, and one of the following conditions holds: ∗c=0; or ∗c≠0, and a+c≤0; or ∗c≠0,a+c>0, and (a+c)e−a−c a≥0; or ∗c≠0,a+c>0,(a+c)e−a−c a<0, and 1 a(1−e−a−c+c a(1−e−a))≥0. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6BUEDO-FERNÁNDEZ ET AL. Case a=b=0:d<2, and one of the following conditions holds: ∗c≤1; or ∗c>1, and c≤2. Case b≠0,a2=4b:(1+d b)(1+a 2)e−a 2−d b>0, and one of the following conditions holds: ∗a 2+c≤1; or ∗a 2+c>1, and e−a 2[c b(1+a 2)+1]−c b≥0. Case b≠0,a2>4b:(1+d b)𝛽e𝛼−𝛼e𝛽 𝛽−𝛼−d b>0, and one of the following conditions holds: ∗a+2c≤√a2−4b(1+2 e√a2−4b−1);or ∗a+2c>√a2−4b(1+2 e√a2−4b−1),and(𝛽c b−1)e𝛼+(1−𝛼c b)e𝛽 𝛽−𝛼−c b≥0. Case b≠0,a2<4b:(1+d b)e−a 2{cos  R+a 2 Rsin  R}−d b>0, and one of the following conditions holds: ∗ R≤𝜋 2and a 2+c≤0; or ∗ R<𝜋 2,and0<a 2+c≤ Rcot ( R);or ∗𝜋 2< R<𝜋,anda 2+c≤ Rcot ( R)<0; or ∗e−a 2{c bcos  R+1+ac 2b  Rsin  R}−c b≥0, and one of the following restrictions holds: ⋆ R=𝜋 2and a 2+c>0; or ⋆ R<𝜋 2and a 2+c> Rcot ( R);or ⋆ R∈( 𝜋 2,𝜋],anda 2+c>0; or ⋆ R∈(𝜋, 3𝜋 2),and0<a 2+c≤ Rcot ( R);or ⋆3𝜋 2≥ R≥𝜋,anda 2+c<0; or ⋆𝜋 2< R<𝜋,and0>a 2+c> Rcot ( R); ∗e−a 2s1{c bcos  Rs1+1+ac 2b  Rsin  Rs1}−c b>0 and one of the following restrictions holds: ⋆ R∈(𝜋, 3𝜋 2),anda 2+c> Rcot ( R)>0; or ⋆ R=3𝜋 2,anda 2+c>0. ∗a 2+c=0, and  R≤𝜋 2. where  R∶= √b−a2 4>0. Remark 2 (Remark 7 of Buedo-Fernández et al.27).Condition (III) in Theorems 2–3 is satisfied if one of the following conditions holds: •b=0. •b≠0,a2≥4b. •b≠0,a2<4b,and  R≤𝜋. We present a new comparison result which will also be useful in the proof of the main results. Lemma 1. Suppose that x ∈C([0,T]) satisfies that x′∈Λand x′(t)+Lx(t)+Fx([t]) ≤𝜎(t), x(0)≤x(T),(9) 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.7 where L ≠0,F∈R, and assume that e−Ls −F L(1−e−Ls)≥0,∀s∈(0,1],(10) and (e−L−F L(1−e−L))[T](e−L(T−[T]) −F L(1−e−L(T−[T])))<1.(11) Then, for t ∈[m,m+1)∩[0,T],withm=0,…,[T],weget x(t)≤1 1−A⎛⎜⎜⎝ [T] ∑ 𝑗=1(e−L−F L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds (e−L(T−[T]) −F L(1−e−L(T−[T]))) + T ∫ [T] 𝜎(s)eL(s−T)ds⎞⎟⎟⎠(e−L−F L(1−e−L))m(e−L(t−m)−F L(1−e−L(t−m))) + m ∑ 𝑗=1(e−L−F L(1−e−L))m−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds (e−L(t−m)−F L(1−e−L(t−m))) + t ∫ m 𝜎(s)eL(s−t)ds, where A ∶= (e−L−F L(1−e−L))[T](e−L(T−[T]) −F L(1−e−L(T−[T]))). Proof. For t∈[𝑗,𝑗 +1),𝑗=0,…,[T],wehave x′(t)+Lx(t)+Fx(𝑗)≤𝜎(t), which implies that (x(t)eL(t−𝑗))′≤(𝜎(t)−Fx(𝑗))eL(t−𝑗). Then, by integrating between 𝑗and t,wehave,fort∈[𝑗,𝑗 +1),𝑗=0,…,[T], x(t)eL(t−𝑗)−x(𝑗)≤t ∫ 𝑗 (𝜎(s)−Fx(𝑗))eL(s−𝑗)ds; thus, we obtain x(t)eL(t−𝑗)≤x(𝑗)−Fx(𝑗) t ∫ 𝑗 eL(s−𝑗)ds + t ∫ 𝑗 𝜎(s)eL(s−𝑗)ds and x(t)≤x(𝑗)(1−F L(eL(t−𝑗)−1))e−L(t−𝑗)+ t ∫ 𝑗 𝜎(s)eL(s−t)ds. In particular, for 𝑗=0,…,[T], by the continuity of x, x(𝑗+1)≤x(𝑗)(1−F L(eL−1))e−L+ 𝑗+1 ∫ 𝑗 𝜎(s)eL(s−𝑗−1)ds. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8BUEDO-FERNÁNDEZ ET AL. By using the estimate on the constants, taking 𝑗=0, we get x(1)≤x(0)(1−F L(eL−1))e−L+ 1 ∫ 0 𝜎(s)eL(s−1)ds, and for 𝑗=1, x(2)≤x(1)(1−F L(eL−1))e−L+ 2 ∫ 1 𝜎(s)eL(s−2)ds ≤x(0)(1−F L(eL−1))2 e−2L+(1−F L(eL−1))e−L 1 ∫ 0 𝜎(s)eL(s−1)ds + 2 ∫ 1 𝜎(s)eL(s−2)ds. By induction, it is easy to check that, for k=1,…,[T], x(k)≤x(0)(1−F L(eL−1))k e−kL + k ∑ 𝑗=1(1−F L(eL−1))k−𝑗 e−(k−𝑗)L 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds =x(0)(e−L−F L(1−e−L))k + k ∑ 𝑗=1(e−L−F L(1−e−L))k−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds. Hence, x([T]) ≤x(0)(e−L−F L(1−e−L))[T] + [T] ∑ 𝑗=1(e−L−F L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds and for t∈[[T],T],weobtain x(t)≤x([T])e−L(t−[T]) (1−F L(eL(t−[T]) −1))+ t ∫ [T] 𝜎(s)eL(s−t)ds =x([T]) (e−L(t−[T]) −F L(1−e−L(t−[T])))+ t ∫ [T] 𝜎(s)eL(s−t)ds. This proves that x(T)≤x([T]) (e−L(T−[T]) −F L(1−e−L(T−[T])))+ T ∫ [T] 𝜎(s)eL(s−T)ds. Therefore, x(0)≤x(T) ≤x(0)(e−L−F L(1−e−L))[T](e−L(T−[T]) −F L(1−e−L(T−[T]))) + [T] ∑ 𝑗=1(e−L−F L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds (e−L(T−[T]) −F L(1−e−L(T−[T]))) + T ∫ [T] 𝜎(s)eL(s−T)ds. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.9 By recalling the expression of A, the previous inequality implies that (1−A)x(0)≤[T] ∑ 𝑗=1(e−L−F L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds (e−L(T−[T]) −F L(1−e−L(T−[T]))) + T ∫ [T] 𝜎(s)eL(s−T)ds. Since condition (11) is written as A<1, then, for m=0,…,[T]and t∈[m,m+1)∩[0,T],weobtain x(t)≤x(m)(e−L(t−m)−F L(1−e−L(t−m)))+ t ∫ m 𝜎(s)eL(s−t)ds ≤x(0)(e−L−F L(1−e−L))m(e−L(t−m)−F L(1−e−L(t−m))) + m ∑ 𝑗=1(e−L−F L(1−e−L))m−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds (e−L(t−m)−F L(1−e−L(t−m))) + t ∫ m 𝜎(s)eL(s−t)ds ≤1 1−A⎛⎜⎜⎝ [T] ∑ 𝑗=1(e−L−F L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds (e−L(T−[T]) −F L(1−e−L(T−[T]))) + T ∫ [T] 𝜎(s)eL(s−T)ds⎞⎟⎟⎠(e−L−F L(1−e−L))m(e−L(t−m)−F L(1−e−L(t−m))) + m ∑ 𝑗=1(e−L−F L(1−e−L))m−𝑗 𝑗 ∫ 𝑗−1 𝜎(s)eL(s−𝑗)ds (e−L(t−m)−F L(1−e−L(t−m))) + t ∫ m 𝜎(s)eL(s−t)ds. □ Remark 3. The proof of Lemma 1 has been written by assuming that T∉Z,sothat[T]<T. However, there is no contradiction with the case [T]=T, where the inequalities are deduced on each interval [m,m+1),form= 0,…,[T]−1. It is obvious that, if T∈N, condition (11) is reduced to A∶= (e−L−F L(1−e−L))[T] <1. Remark 4. It is easily checked that condition (10) is satisfied for F+L≤0. Besides, (10) is valid for F+L>0, by assuming that e−L−F L(1−e−L)≥0, that is, (10) holds if F+L>0,F L(eL−1)≤1. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 16 BUEDO-FERNÁNDEZ ET AL. By applying Lemma 1, we get, for t∈[m,m+1), 𝑦(t)≤(d−Rc) 1−A ×⎛⎜⎜⎝ [T] ∑ 𝑗=1(e−L−c L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 (𝛽([s]) − x([s]))eL(s−𝑗)ds (e−L(T−[T]) −c L(1−e−L(T−[T]))) + T ∫ [T] (𝛽([s]) − x([s]))eL(s−T)ds⎞⎟⎟⎠(e−L−c L(1−e−L))m(e−L(t−m)−c L(1−e−L(t−m))) + m ∑ 𝑗=1(e−L−c L(1−e−L))m−𝑗 𝑗 ∫ 𝑗−1 (d−Rc)(𝛽([s]) − x([s]))eL(s−𝑗)ds (e−L(t−m)−c L(1−e−L(t−m))) + t ∫ m (d−Rc)(𝛽([s]) − x([s]))eL(s−t)ds. Since all the coefficients in the last expression are greater than or equal to zero and 0≤𝛽(s)−x(s)≤𝛽(s)−𝛼(s),for every s∈[0,T], we obtain that 𝑦(t)=x′(t)−𝛽′(t)+R(x(t)−𝛽(t)) ≤Km,for t∈[m,m+1); therefore, x′(t)≤𝛽′(t)−R(x(t)−𝛽(t)) + Km≤𝛽′(t)−R(𝛼(t)−𝛽(t)) + Km=k2(t), for every t∈[m,m+1)and every m=0,1,…,S∗,whereS∗is given in Lemma 2. Finally, we prove that problem (15) is solvable. Note that, due to the properties of functions 𝛼, 𝛽, k1,k2,g, and the definition of the operators pand q, it is deduced that function 𝜎x(t)∈Λ, for every x∈E. Hence, problem (15) can be written equivalently (see Theorem 1) as x(t)= T ∫ 0 K(t,s)𝜎x(s)ds +𝜆K(t,0),t∈J, where for each s∈J,K(·,s)is the unique solution to an auxiliary problem of the type (2). We define the mapping ∶C1(J)→C1(J)given by [x](t)∶= T ∫ 0 K(t,s)𝜎x(s)ds +𝜆K(t,0),t∈J, in such a way that the set of solutions of the modified problem (15) is the set of fixed points of . Let M>0besuchthat|𝛼(t)|≤M,|𝛽(t)|≤M,|𝛼′(t)|≤M,|𝛽′(t)|≤M, for every t∈J(it is possible since 𝛼, 𝛽 ∈E). 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.17 On the other hand, for t∈[m,m+1),m=0,1,…,S∗, |k1(t)|=|𝛼′(t)+R(𝛼(t)−𝛽(t)) − Km|≤M+2RM +|Km| ≤M+2RM +(d−Rc) 1−A2M ×⎛⎜⎜⎝ [T] ∑ 𝑗=1(e−L−c L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 eL(s−𝑗)ds (e−L(T−[T]) −c L(1−e−L(T−[T]))) + T ∫ [T] eL(s−T)ds⎞⎟⎟⎠(e−L−c L(1−e−L))m(e−L(t−m)−c L(1−e−L(t−m))) +2M(d−Rc) m ∑ 𝑗=1(e−L−c L(1−e−L))m−𝑗 𝑗 ∫ 𝑗−1 eL(s−𝑗)ds (e−L(t−m)−c L(1−e−L(t−m))) +2M(d−Rc) t ∫ m eL(s−t)ds =M1. Similarly, |k2(t)|≤M1,t∈[m,m+1),m=0,1,…,S∗. This proves that ||k1||=supt∈J|k1(t)|≤M1and ||k2||≤M1. Consider the compact set in R5 D∶= {(t,x,𝑦,z,w)∈R5∶t∈[0,T],𝛼(t)≤x≤𝛽(t),k1(t)≤𝑦≤k2(t),𝛼([t]) ≤z≤𝛽([t]),k1([t]) ≤w≤k2([t])}. By the hypotheses on g, it is possible to choose N>0suchthat |g(t,x,𝑦,z,w)|≤N,for every (t,x,𝑦,z,w)∈D. Let 𝜇∈(0,1)and xbe such that x=𝜇x. ||x||C1(J)=𝜇||x||C1(J)=𝜇(||x||+||(x)′||). From the expression of x,wededucethat (x)′(t)= T ∫ 0 𝜕 𝜕tK(t,s)𝜎x(s)ds +𝜆𝜕 𝜕tK(t,0),t∈J. Note that, by definition, p(t,x(t)) is between 𝛼(t)and 𝛽(t)and q(t,x′(t)) is between k1(t)and k2(t). Hence, (t,p(t,x(t)),q(t,x′(t)),p([t],x([t])),q([t],x′([t]))) ∈ D, for every t,and |𝜎x(t)|=|g(t,p(t,x(t)),q(t,x′(t)),p([t],x([t])),q([t],x′([t]))) +aq(t,x′(t)) + bp(t,x(t)) + cq([t],x′([t])) + dp([t],x([t]))| ≤N+|a|M1+|b|M+|c|M1+|d|M=N+(|a|+|c|)M1+(|b|+|d|)M. This proves that ||x||≤(N+(|a|+|c|)M1+(|b|+|d|)M)sup t∈[0,T] T ∫ 0 K(t,s)ds +|𝜆|sup t∈[0,T] K(t,0), 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 18 BUEDO-FERNÁNDEZ ET AL. and ||(x)′||≤(N+(|a|+|c|)M1+(|b|+|d|)M)sup t∈[0,T] T ∫ 0|||| 𝜕 𝜕tK(t,s)||||ds +|𝜆|sup t∈[0,T]|||| 𝜕 𝜕tK(t,0)||||. Furthermore, Kand 𝜕 𝜕tKare bounded, since their expressions depend on the functions h1,h2and function gin Buedo-Fernández et al27 and their respective derivatives, which are obviously continuous and hence bounded on [0,1]. We remark that the nonnegative character of functions K(·,0)and K(·,s)for almost every s∈(0,T)is guaranteed by the conditions in Theorem 2 (see Buedo-Fernández et al27). Then, by Schauder's Fixed Point Theorem, there exists at least one fixed point xof which is a solution to (15). This solution xsatisfies that 𝛼≤x≤𝛽and k1≤x′≤k2on J, and in consequence, it is a solution to (3) and the proof is complete. □ 4MONOTONE METHOD In this section, we develop the monotone iterative technique for problem (3). Theorem 6. Suppose that g ∶J×R4→Ris continuous on (J∖{1,2,…,[T]}) × R4and such that the following limits are finite lim t→n−g(t,x,𝑦,u,v),g(n,x,𝑦,u,v) = lim t→n+g(t,x,𝑦,u,v). Assume that hypothesis (H1)holds. Suppose that (H6)There exist constants a,b,c,d∈R,L ≠0,andR >0with R+L=a,LR =b,Rc ≤d, c+L>0,and e−L−c L(1−e−L)≥0, in such a way that, for those values of a,b,c,d∈R, the following inequality holds: g(t,x,𝑦,u,v)−g(t, x,𝑦,  u, v)≥−a(𝑦−𝑦)−b(x− x)−c(v− v)−d(u− u), for t ∈J, and 𝛼(t)≤ x≤x≤𝛽(t),k1(t)≤𝑦, 𝑦 ≤k2(t),𝛼([t]) ≤ u≤u≤𝛽([t]),k1([t]) ≤ v,v≤k2([t]), where k1(t),k2(t)and Km,m=0,1,…,S∗, are given in the statement of Lemma 2, S∗=[T]if T ∉Zand S∗=[T]−1 if T ∈Z. Assume also that the constants a,b,c,d∈Rin (H6)are such that the hypothesis (5) and conditions (I)–(V) in Theorem 2 hold (see condition (H3)). Suppose that 𝛼′(t)−𝛽′(t)≤R(𝛽(t)−𝛼(t)),∀t,(16) or, more generally, 𝛼′(t)−𝛽′(t)≤R(𝛽(t)−𝛼(t)) + Km,∀t.(17) Then there exist monotone sequences {𝛼n},{𝛽n}in E such that 𝛼0=𝛼, 𝛽0=𝛽,and{𝛼n},{𝛽n}are uniformly convergent to 𝜌, 𝛾, which are the extremal solutions to (3) in the set {𝜂∈C1(J)|𝛼≤𝜂≤𝛽and k1≤𝜂′≤k2on J}. Furthermore, there exist subsequences {𝛼′ nk}→𝜌′,{𝛽′ nk}→𝛾′,ask→+∞. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.19 Proof. For each 𝜂∈C1(J), we consider the modified problem {x′′(t)+ax′(t)+bx(t)+cx′([t]) + dx([t])=𝜎𝜂(t),t∈J=[0,T], x(0)=x(T), x′(0)=x′(T)+𝜆, (18) where 𝜎𝜂(t)∶=g(t,𝜂(t),𝜂′(t),𝜂([t]),𝜂′([t])) + a𝜂′(t)+b𝜂(t)+c𝜂′([t]) + d𝜂([t]),fort∈J. Note that 𝜎𝜂is continuous on J∖{1,2,…,[T]}, and there exist 𝜎𝜂(n−)∈R,𝜎 𝜂(n+)=𝜎𝜂(n), for every n∈ {1,2,…,[T]}, hence 𝜎𝜂∈Λ. We define the operator as (𝜂)(t)∶= T ∫ 0 K(t,s)𝜎𝜂(s)ds +𝜆K(t,0),t∈J, for 𝜂∈C1(J). We choose 𝛼0=𝛼and 𝛼1the unique solution to (18), which exists since 𝜎𝛼∈Λand condition (5) holds. Hence, 𝛼1(t)= T ∫ 0 K(t,s)𝜎𝛼(s)ds +𝜆K(t,0),t∈J. The sequences 𝛼n,𝛽 nare defined as 𝛼n∶= 𝛼n−1,𝛽 n∶= 𝛽n−1,∀n≥1. We check that {𝛼n}is monotone nondecreasing and {𝛽n}is monotone nonincreasing. We proceed in different steps: (i) 𝛼′(t),𝛽 ′(t)∈[k1(t),k2(t)],∀t∈J. (ii) 𝛼′′ n,𝛽′′ nare bounded. (iii) If 𝜂∈C1(J)is such that 𝛼≤𝜂≤𝛽,andk1≤𝜂′≤k2on J,then𝜂belongs to [𝛼,𝛽]and (𝜂)′is between k1 and k2. (iv) is nondecreasing on the set {𝜂∈C1(J)|𝛼≤𝜂≤𝛽and k1≤𝜂′≤k2on J}. (v) {𝛼n}is uniformly convergent towards 𝜌and {𝛽n}is uniformly convergent towards 𝛾. For the derivatives, we only have convergence of a certain subsequence. (vi) 𝜌, 𝛾 are extremal solutions to (3) in {𝜂∈C1(J)|𝛼≤𝜂≤𝛽and k1≤𝜂′≤k2on J}. First, we prove (i). Note that, for every m, k1(t)=𝛼′(t)+R(𝛼(t)−𝛽(t)) − Km≤𝛼′(t),t∈[m,m+1), k2(t)=𝛽′(t)−R(𝛼(t)−𝛽(t)) + Km≥𝛽′(t),t∈[m,m+1). If 𝛼′(t)≤𝛽′(t),thenk1≤𝛼′(t)≤𝛽′(t)≤k2, but if this condition does not hold, we deduce 𝛼′(t)+R(𝛼(t)−𝛽(t)) − Km≤𝛽′(t), and 𝛼′(t)≤𝛽′(t)−R(𝛼(t)−𝛽(t)) + Km, 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 20 BUEDO-FERNÁNDEZ ET AL. for t∈[m,m+1), and every m, from hypothesis (16), or more generally, (17). We check (ii). Indeed, 𝛼′ n(t)= T ∫ 0 𝜕 𝜕tK(t,s)𝜎𝛼n−1(s)ds +𝜆𝜕 𝜕tK(t,0),t∈J,(19) and 𝛼′′ n(t)= T ∫ 0 𝜕2 𝜕t2K(t,s)𝜎𝛼n−1(s)ds +𝜆𝜕2 𝜕t2K(t,0),t∈J,(20) so that 𝛼′′ n∈Λ, and similarly, 𝛽′′ n(t)= T ∫ 0 𝜕2 𝜕t2K(t,s)𝜎𝛽n−1(s)ds +𝜆𝜕2 𝜕t2K(t,0),t∈J, and hence, 𝛼′′ nand 𝛽′′ nare bounded on each Jk=[k,k+1], and thus, bounded. We prove (iii). Consider 𝜂∈C1(J)such that 𝛼≤𝜂≤𝛽,andk1≤𝜂′≤k2on J,thenweprovethat𝜂belongs to [𝛼,𝛽]and (𝜂)′is between k1and k2. Note that 𝜂is the unique solution to (18). Consider m∶= 𝛼−𝜂, then, by using that 𝛼≤𝜂≤𝛽, k1≤𝜂′≤k2,and(H6),weget,fort∈[0,T], m′′(t)+am′(t)+bm(t)+cm′([t]) + dm([t]) =𝛼′′(t)+a𝛼′(t)+b𝛼(t)+c𝛼′([t]) + d𝛼([t]) −g(t,𝜂(t),𝜂′(t),𝜂([t]),𝜂′([t])) − a𝜂′(t)−b𝜂(t)−c𝜂′([t]) − d𝜂([t]) ≤g(t,𝛼(t),𝛼′(t),𝛼([t]),𝛼′([t])) + a𝛼′(t)+b𝛼(t)+c𝛼′([t]) + d𝛼([t]) −g(t,𝜂(t),𝜂′(t),𝜂([t]),𝜂′([t])) − a𝜂′(t)−b𝜂(t)−c𝜂′([t]) − d𝜂([t]) ≤0. Moreover, m(0)=𝛼(0)−(𝜂)(0)=𝛼(T)−(𝜂)(T)=m(T), m′(0)=𝛼′(0)−(𝜂)′(0)≤𝛼′(T)+𝜆−(𝜂)′(T)−𝜆=m′(T). By the comparison result Theorem 3, m≤0on[0,T], hence 𝛼≤𝜂on J. On the other hand, by taking  m∶= 𝜂−𝛽,wehave  m′′(t)+a m′(t)+b m(t)+c m′([t]) + d m([t]) ≤0,t∈[0,T], and  m(0)=  m(T), m′(0)≤ m′(T), hence we deduce again, by the comparison result Theorem 3, that  m≤0onJ,and 𝜂≤𝛽on J. Now, we prove that (𝜂)′∈[k1,k2]by using Lemma 1. We consider the function 𝑦(t)∶=𝛼′(t)−(𝜂)′(t)+R(𝛼(t)−(𝜂)(t)),t∈J, and check that 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.21 𝑦′(t)+L𝑦(t)=𝛼′′(t)−(𝜂)′′(t) +R(𝛼′(t)−(𝜂)′(t)) + L(𝛼′(t)−(𝜂)′(t)) + LR(𝛼(t)−(𝜂)(t)) ≤g(t,𝛼(t),𝛼′(t),𝛼([t]),𝛼′([t])) + R𝛼′(t)+L𝛼′(t)+LR𝛼(t) −(𝜂)′′(t)−R(𝜂)′(t)−L(𝜂)′(t)−LR(𝜂)(t) =g(t,𝛼(t),𝛼′(t),𝛼([t]),𝛼′([t])) + a𝛼′(t)+b𝛼(t) −(𝜂)′′(t)−a(𝜂)′(t)−b(𝜂)(t) =g(t,𝛼(t),𝛼′(t),𝛼([t]),𝛼′([t])) + a𝛼′(t)+b𝛼(t) −g(t,𝜂(t),𝜂′(t),𝜂([t]),𝜂′([t])) − a𝜂′(t)−b𝜂(t) −c𝜂′([t]) − d𝜂([t]) + c(𝜂)′([t]) + d(𝜂)([t]) ≤−c𝛼′([t]) − d𝛼([t]) + c(𝜂)′([t]) + d(𝜂)([t]) =c((𝜂)′([t]) − 𝛼′([t])) + d((𝜂)([t]) − 𝛼([t])),t∈J, and 𝑦(0)=𝛼′(0)−(𝜂)′(0)+R(𝛼(0)−(𝜂)(0)) ≤𝛼′(T)+𝜆−(𝜂)′(T)−𝜆+R(𝛼(T)−(𝜂)(T)) = 𝑦(T). This proves that 𝑦′(t)+L𝑦(t)+c𝑦([t]) ≤(d−cR)((𝜂)([t]) − 𝛼([t])) ≤(d−cR)(𝛽([t]) − 𝛼([t])), since (𝜂)≤𝛽and d≥cR. By Lemma 1, we get, for t∈[m,m+1),that 𝑦(t)≤(d−Rc) 1−A ×⎛⎜⎜⎝ [T] ∑ 𝑗=1(e−L−c L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 (𝛽([s]) − 𝛼([s]))eL(s−𝑗)ds (e−L(T−[T]) −c L(1−e−L(T−[T]))) + T ∫ [T] (𝛽([s]) − 𝛼([s]))eL(s−T)ds⎞⎟⎟⎠(e−L−c L(1−e−L))m(e−L(t−m)−c L(1−e−L(t−m))) + m ∑ 𝑗=1(e−L−c L(1−e−L))m−𝑗 𝑗 ∫ 𝑗−1 (d−Rc)(𝛽([s]) − 𝛼([s]))eL(s−𝑗)ds (e−L(t−m)−c L(1−e−L(t−m))) + t ∫ m (d−Rc)(𝛽([s]) − 𝛼([s]))eL(s−t)ds. Hence, 𝑦(t)=𝛼′(t)−(𝜂)′(t)+R(𝛼(t)−(𝜂)(t)) ≤Km,for t∈[m,m+1). Therefore, 𝛼′(t)−(𝜂)′(t)+R(𝛼(t)−𝛽(t)) ≤Km, and in consequence, k1(t)=𝛼′(t)+R(𝛼(t)−𝛽(t)) − Km≤(𝜂)′(t), for every t∈[m,m+1)and every m. On the other hand, to prove that (𝜂)′≤k2,take𝑦(t)∶=(𝜂)′(t)−𝛽′(t)+R((𝜂)(t)−𝛽(t)),t∈J, and check the hypotheses of Lemma 1. Indeed, 𝑦(0)=(𝜂)′(0)−𝛽′(0)+R((𝜂)(0)−𝛽(0)) ≤(𝜂)′(T)+𝜆−𝛽′(T)−𝜆+R((𝜂)(T)−𝛽(T)) = 𝑦(T), 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 22 BUEDO-FERNÁNDEZ ET AL. and 𝑦′(t)+L𝑦(t)=(𝜂)′′(t)−𝛽′′(t) +R((𝜂)′(t)−𝛽′(t)) + L((𝜂)′(t)−𝛽′(t)) + LR((𝜂)(t)−𝛽(t)) =(𝜂)′′(t)−𝛽′′(t)+a(𝜂)′(t)−a𝛽′(t)+b(𝜂)(t)−b𝛽(t) ≤g(t,𝜂(t),𝜂′(t),𝜂([t]),𝜂′([t])) + a𝜂′(t)+b𝜂(t)+c𝜂′([t]) + d𝜂([t]) −c(𝜂)′([t]) − d(𝜂)([t]) − g(t,𝛽(t),𝛽′(t),𝛽([t]),𝛽′([t])) − a𝛽′(t)−b𝛽(t) ≤c𝛽′([t]) + d𝛽([t]) − c(𝜂)′([t]) − d(𝜂)([t]) =c(𝛽′([t]) − (𝜂)′([t])) + d(𝛽([t]) − (𝜂)([t])),t∈J, which implies that 𝑦′(t)+L𝑦(t)+c𝑦([t]) ≤(d−cR)(𝛽([t]) − (𝜂)([t])) ≤(d−cR)(𝛽([t]) − 𝛼([t])),for t∈[0,T]. Since the function 𝜎is the same as in the previous case, then, by Lemma 1, we get 𝑦(t)≤Km,for t∈[m,m+1), and (𝜂)′(t)−𝛽′(t)+R(𝛼(t)−𝛽(t)) ≤(𝜂)′(t)−𝛽′(t)+R((𝜂)(t)−𝛽(t)) ≤Km,for t∈[m,m+1), thus (𝜂)′(t)≤𝛽′(t)+R(𝛽(t)−𝛼(t)) + Km=k2(t), for every t∈[m,m+1)and every m=0,1,…,[T], hence (𝜂)′≤k2on J. To check (iv), consider that 𝜂, 𝜉 ∈C1(J)are such that 𝛼≤𝜂≤𝜉≤𝛽,andk1≤𝜂′,𝜉′≤k2on J,then𝜂≤𝜉, which is deduced similarly to Theorem 4, since, by using (H6),weget,fort∈J, 𝜎𝜂(t)−𝜎𝜉(t)=g(t,𝜂(t),𝜂′(t),𝜂([t]),𝜂′([t])) + a𝜂′(t)+b𝜂(t)+c𝜂′([t]) + d𝜂([t]) −g(t,𝜉(t),𝜉′(t),𝜉([t]),𝜉′([t])) − a𝜉′(t)−b𝜉(t)−c𝜉′([t]) − d𝜉([t]) ≤0. Indeed, w∶= 𝜂−𝜉satisfies that w(0)=w(T),w′(0)=(𝜂)′(0)−(𝜉)′(0)=(𝜂)′(T)+𝜆−(𝜉)′(T)−𝜆=w′(T), and w′′(t)+aw′(t)+bw(t)+cw′([t]) + dw([t]) ≤0,t∈J. By the comparison result Theorem 3, we deduce that w≤0andthus,𝜂≤𝜉on J. Now, to prove (v), note that {𝛼n}is nondecreasing, and {𝛽n}is nonincreasing. Indeed, it is obvious (by iii)) that 𝛼≤𝛼=𝛼1≤𝛽, k1≤(𝛼)′=𝛼′ 1≤k2and 𝛼≤𝛽=𝛽1≤𝛽and k1≤(𝛽)′=𝛽′ 1≤k2on J.Byiv),wehave 𝛼1≤𝛽1. By applying iv) recursively, we derive the monotonicity of the sequences {𝛼n}and {𝛽n}. On the other hand, by the previous considerations, the integral expressions 𝛼n(t)= T ∫ 0 K(t,s)𝜎𝛼n−1(s)ds +𝜆K(t,0),t∈J, 𝛽n(t)= T ∫ 0 K(t,s)𝜎𝛽n−1(s)ds +𝜆K(t,0),t∈J, imply that {𝛼n},{𝛽n}⊂C(J)are uniformly bounded. We prove that these sets are equicontinuous. Consider the expressions of the corresponding derivatives included in (19) 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.23 𝛼′ n(t)= T ∫ 0 𝜕 𝜕tK(t,s)𝜎𝛼n−1(s)ds +𝜆𝜕 𝜕tK(t,0),t∈J, 𝛽′ n(t)= T ∫ 0 𝜕 𝜕tK(t,s)𝜎𝛽n−1(s)ds +𝜆𝜕 𝜕tK(t,0),t∈J. By using the compact set Ddefined in the proof of Theorem 5, we obtain that the set {𝜎𝜂(s)∶s∈J,𝜂∈C1(J)with 𝛼≤𝜂≤𝛽, and k1≤𝜂′≤k2} is bounded, where 𝜎𝜂(s)∶=g(s,𝜂(s),𝜂′(s),𝜂([s]),𝜂′([s])) + a𝜂′(s)+b𝜂(s)+c𝜂′([s]) + d𝜂([s]),s∈J; hence, {𝛼n}and {𝛽n}are equicontinuous sets on J. Hence, there exist 𝜌, 𝛾 ∈C(J)and subsequences {𝛼nk}→𝜌,and {𝛽nk}→𝛾, uniformly as k→+∞ (moreover, we can affirm that there exist 𝜌, 𝛾 ∈C1(J)and subsequences {𝛼nk}→𝜌 and {𝛽nk}→𝛾in ||·||1,ask→+∞,thatis,{𝛼nk}→𝜌,{𝛽nk}→𝛾, {𝛼′ nk}→𝜌′,and{𝛽′ nk}→𝛾′uniformly on J). Since {𝛼n}and {𝛽n}are monotone, then {𝛼n}→𝜌,and{𝛽n}→𝛾,asn→+∞. Note that, obviously, 𝛼≤𝜌≤𝛾≤𝛽on J. Furthermore, since k1≤(𝛼nk)′=(𝛼nk−1)′≤k2,andk1≤(𝛽nk)′= (𝛽nk−1)′≤k2, for every k,thenk1≤𝜌′,𝛾 ′≤k2on J. To prove (vi), we use that 𝛼n(t)= T ∫ 0 K(t,s)𝜎𝛼n−1(s)ds +𝜆K(t,0),t∈J. Since 𝛼′ nkis uniformly convergent towards 𝜌′, then, by taking the sequence 𝛼nk+1(t)= T ∫ 0 K(t,s)𝜎𝛼nk(s)ds +𝜆K(t,0),t∈J, and by using that 𝜎𝛼nk(t)=g(t,𝛼 nk(t),𝛼′ nk(t),𝛼 nk([t]),𝛼′ nk([t])) + a𝛼′ nk(t)+b𝛼nk(t)+c𝛼′ nk([t]) + d𝛼nk([t]),fort∈J,is uniformly convergent on Jtowards 𝜎𝜌(t)=g(t,𝜌(t),𝜌 ′(t),𝜌([t]),𝜌 ′([t])) + a𝜌′(t)+b𝜌(t)+c𝜌′([t]) + d𝜌([t]),for t∈J, we have 𝜌(t)= T ∫ 0 K(t,s)𝜎𝜌(s)ds +𝜆K(t,0),t∈J, hence, 𝜌′′(t)+a𝜌′(t)+b𝜌(t)+c𝜌′([t]) + d𝜌([t]) = 𝜎𝜌(t), which implies that 𝜌′′(t)=g(t,𝜌(t),𝜌 ′(t),𝜌([t]),𝜌 ′([t])),for t∈J, and besides, 𝜌(0)=𝜌(T),and𝜌′(0)=𝜌′(T)+𝜆. Therefore, 𝜌is a solution to (3). A similar reasoning is valid for the sequence {𝛽n}and 𝛾. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 24 BUEDO-FERNÁNDEZ ET AL. On the other hand, if x∈Eis a solution such that 𝛼≤x≤𝛽and k1≤x′≤k2on J,then𝛼1=𝛼≤x=x≤ 𝛽=𝛽1and k1≤(𝛼)′,(x)′,(𝛽)′≤k2on J.Byinduction,wehave𝛼n≤x≤𝛽non J, for every n,then𝜌≤x≤𝛾 on J, and the solutions 𝜌and 𝛾are extremal in {𝜂∈C1(J)|𝛼≤𝜂≤𝛽and k1≤𝜂′≤k2on J}. □ Remark 8. Once the condition 𝛼≤𝛽is satisfied on J, condition (16) is trivially valid at the points t0∈Jwith 𝛼(t0)= 𝛽(t0)and 𝛼′(t0)≤𝛽′(t0), which is consistent with the condition 𝛼≤𝛽(if 𝛼(t0)=𝛽(t0)and 𝛼′(t0)>𝛽 ′(t0), then there would exist a neighborhood where 𝛼>𝛽). On an interval where 𝛼<𝛽, condition (16) is equivalent to 𝛼′(t)−𝛽′(t) 𝛽(t)−𝛼(t)≤R, and integrating, we get −ln(𝛽(t)−𝛼(t)) ≤Rt +C,or𝛽(t)≥Ke−Rt +𝛼(t). Hence, if 𝛼(t0)<𝛽(t0), the validity of (16) for t≥t0implies that 𝛽(t)≥Ke−Rt +𝛼(t)for t≥t0(in particular, this implies that 𝛼(t)<𝛽(t)for all t≥t0). Condition (16) is equivalent to the differential inequality 𝛾′(t)≤−R𝛾(t),where𝛾=𝛼−𝛽≤0. If 𝛾(t0)<0, it implies that 𝛾(t)<0,∀t≥t0.If𝛾(t0)=0, we have 𝛾′(t0)≤0. Remark 9. The constants a,b,c,d∈Rmust be chosen in such a way that there exist L≠0, and R>0withR+L=a, and LR =b≠0. From the relations specified, once calculated R,Lis obtained as L=b R. In consequence, Rcan be chosen by solving the equation R+b R=a, which leads to the quadratic equation R2−aR +b=0. We easily derive two possibilities: R=a±√a2−4b 2. If a2−4b<0, there exists no R; hence, we must assume that a2≥4b. This expression is, in consequence, a restriction on the choice of a,b. If a2=4b,wededucethatb>0 (since b≠0), and hence, L>0, and a>0. In this case, we have only one possibility for R=a 2(which is consistent with a>0), and L=b R=2b a. If a2>4b, we have two possibilities for R, and we seek for a positive value of R. Note that, for at least one of those values to be positive, we have to check a>−√a2−4b, which is trivial for a>0. Hence, one choice is to assume that a>0andtakeR=a+√a2−4b 2>0. This option is also possible if a≤0andb<0. In the other choice, R=a−√a2−4b 2is positive if and only if a>√a2−4b, which is obviously satisfied for a>0,b>0. In summary, we have two options: •a2=4b>0,R=a 2with a>0, and L=2b a>0(here,a,b>0). •a2>4b, and two options for R, with the corresponding value of L: ∗R=a+√a2−4b 2>0,L=b R=2b a+√a2−4b(valid for a>0 independently of the sign of b≠0, and for a≤0and b<0). The sign of Lis the sign of b. ∗R=a−√a2−4b 2>0,L=b R=2b a−√a2−4b>0(validfora>0andb>0). Note that we need to impose additional restrictions on the constants L,R. These restrictions will give information about the way of choosing the rest of the constants c,d∈R. For instance, d−cR ≥0 is written as •a2=4b:d−ca 2≥0, that is, ca ≤2d. •a2>4b: ∗If R=a+√a2−4b 2>0, such condition is written as c(a+√a2−4b)≤2d. ∗If R=a−√a2−4b 2>0, such condition is written as c(a−√a2−4b)≤2d. Note that, taking into account the type of one-sided Lipschitz condition assumed, we are interested in the case b>0 (hence, L>0 and, therefore, a>0). 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.25 In general, the sequences {𝛼′ n}and {𝛽′ n}are not monotonic. If 𝛼′≤𝛽′on Jand the sequences {𝛼′ n}and {𝛽′ n}are monotone, then we would deduce that {𝛼′ n}→𝜌′and {𝛽′ n}→𝛾′uniformly on J,asn→+∞. Despite of this, we can prove the following estimates for the derivatives {𝛼′ n}and {𝛽′ n}. Theorem 7. For the sequences given in Theorem 6, the following estimates are valid: 𝛼′ n(t)−R(𝛼n+1(t)−𝛼n(t)) − Γm(𝛽−𝛼,t) ≤𝛼′ n(t)−R(𝛼n+1(t)−𝛼n(t)) − Γm(S−𝛼n,t) ≤𝛼′ n(t)−R(𝛼n+1(t)−𝛼n(t)) − Γm(𝛼n+1−𝛼n,t)≤𝛼′ n+1(t), (21) where S may be either 𝜌, 𝛾,or𝛽,and 𝛽′ n+1(t)≤𝛽′ n(t)+R(𝛽n(t)−𝛽n+1(t)) + Γm(𝛽n−𝛽n+1,t) ≤𝛽′ n(t)+R(𝛽n(t)−𝛽n+1(t)) + Γm(𝛽n−S,t) ≤𝛽′ n(t)+R(𝛽n(t)−𝛽n+1(t)) + Γm(𝛽−𝛼,t), (22) where S may be either 𝛼, 𝜌,or𝛾,fort∈[m,m+1),m=0,1,…,[T],with Γm(u,t)∶=(d−Rc) 1−A ×⎛⎜⎜⎝ [T] ∑ 𝑗=1(e−L−c L(1−e−L))[T]−𝑗 𝑗 ∫ 𝑗−1 u([s])eL(s−𝑗)ds (e−L(T−[T]) −c L(1−e−L(T−[T]))) + T ∫ [T] u([s])eL(s−T)ds⎞⎟⎟⎠(e−L−c L(1−e−L))m(e−L(t−m)−c L(1−e−L(t−m))) + m ∑ 𝑗=1(e−L−c L(1−e−L))m−𝑗 𝑗 ∫ 𝑗−1 (d−Rc)u([s])eL(s−𝑗)ds (e−L(t−m)−c L(1−e−L(t−m))) + t ∫ m (d−Rc)u([s])eL(s−t)ds. Proof. Hereafter, we will also consider 𝛼−1∶= 𝛼and 𝛽−1∶= 𝛽. Following the reasoning of Theorem 6, assume that n is a nonnegative integer, and take z(t)∶=𝛼′ n(t)−𝛼′ n+1(t)+R(𝛼n(t)−𝛼n+1(t)),t∈J. We deduce that z′(t)+Lz(t) =𝛼′′ n(t)−𝛼′′ n+1(t)+R(𝛼′ n(t)−𝛼′ n+1(t)) + L(𝛼′ n(t)−𝛼′ n+1(t)) + LR(𝛼n(t)−𝛼n+1(t)) =𝛼′′ n(t)+a𝛼′ n(t)+b𝛼n(t)−𝛼′′ n+1(t)−a𝛼′ n+1(t)−b𝛼n+1(t) ≤g(t,𝛼 n−1(t),𝛼′ n−1(t),𝛼 n−1([t]),𝛼′ n−1([t])) + a𝛼′ n−1(t)+b𝛼n−1(t) +c𝛼′ n−1([t]) + d𝛼n−1([t]) − c𝛼′ n([t]) − d𝛼n([t]) − g(t,𝛼 n(t),𝛼′ n(t),𝛼 n([t]),𝛼′ n([t])) −a𝛼′ n(t)−b𝛼n(t)−c𝛼′ n([t]) − d𝛼n([t]) + c𝛼′ n+1([t]) + d𝛼n+1([t]) ≤−c𝛼′ n([t]) − d𝛼n([t]) + c𝛼′ n+1([t]) + d𝛼n+1([t]). The first of the previous two inequalities is in fact an equality in case n≥1, while, if n=0, it comes from the properties of 𝛼. We also recall that 𝛼−1=𝛼0=𝛼. Furthermore, the last inequality is an equality provided that n=0, whereas for 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 32 BUEDO-FERNÁNDEZ ET AL. 𝛼n(t)∶=(𝛼n−1)(t)= 1 2 ∫ 0 K(t,s)(es+1 16 cos(𝛼n−1(s)) + 1 16𝛼n−1(s))ds,t∈[0,1 2],n≥1, 𝛽n(t)∶=(𝛽n−1)(t)= 1 2 ∫ 0 K(t,s)(es+1 16 cos(𝛽n−1(s)) + 1 16𝛽n−1(s))ds,t∈[0,1 2],n≥1. This way, we have 𝛼0≡0,𝛽 0≡7, and, for instance, for t∈[0,1 2], 𝛼1(t)∶=(𝛼0)(t)= 1 2 ∫ 0 K(t,s)(es+1 16 cos(𝛼0(s)) + 1 16𝛼0(s))ds = 1 2 ∫ 0 K(t,s)(es+1 16)ds ≈(9(1+1 4t)e−1 4t−8) 1 2 ∫ 0 e−1 4(1 2−s)(3.6088 (1 2−s)+3.0880 (7 8+1 4s))(es+1 16)ds +(e−1 4t[8(1+1 4t)+t]−8) 1 2 ∫ 0 e−1 4(1 2−s)(−1.9974 (1 2−s)+0.5208 (7 8+1 4s))(es+1 16)ds + t ∫ 0 (t−s)e−1 4(t−s)(es+1 16)ds. Similarly, 𝛽1(t)∶=(𝛽0)(t)= 1 2 ∫ 0 K(t,s)(es+1 16 cos(𝛽0(s)) + 1 16𝛽0(s))ds = 1 2 ∫ 0 K(t,s)(es+1 16 cos(7)+ 7 16)ds,t∈[0,1 2], and so on. For this example, by Theorem 7, we can deduce the following estimates: 𝛼′ n(t)−1 4(𝛼n+1(t)−𝛼n(t)) − Γ0(7,t) ≤𝛼′ n(t)−1 4(𝛼n+1(t)−𝛼n(t)) − Γ0(S−𝛼n,t)≤𝛼′ n(t)−1 4(𝛼n+1(t)−𝛼n(t)) − Γ0(𝛼n+1−𝛼n,t)≤𝛼′ n+1(t), where Smay be either 𝜌, 𝛾,or𝛽≡7, and 𝛽′ n+1(t)≤𝛽′ n(t)+1 4(𝛽n(t)−𝛽n+1(t)) + Γ0(𝛽n−𝛽n+1,t) ≤𝛽′ n(t)+1 4(𝛽n(t)−𝛽n+1(t)) + Γ0(𝛽n−S,t)≤𝛽′ n(t)+1 4(𝛽n(t)−𝛽n+1(t)) + Γ0(7,t), 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.33 where Smay be either 𝛼≡0,𝜌,or𝛾,fort∈[0,1 2), and every n,with Γ0(u,t)∶= 1 8(1−e−1 8)⎛⎜⎜⎜⎝ 1 2 ∫ 0 u([s])e 1 4(s−1 2)ds⎞⎟⎟⎟⎠(3e−1 4t−2)+ t ∫ 0 3 8u([s])e1 4(s−t)ds =u(0) 8(1−e−1 8)⎛⎜⎜⎜⎝ 1 2 ∫ 0 e 1 4(s−1 2)ds⎞⎟⎟⎟⎠(3e−1 4t−2)+u(0) t ∫ 0 3 8e1 4(s−t)ds =u(0) 2(3e−1 4t−2)+3 2u(0)(1−e−1 4t)=u(0) 2. Therefore, the previous inequalities mean 𝛼′ n(t)−1 4(𝛼n+1(t)−𝛼n(t)) − 7 2 ≤𝛼′ n(t)−1 4(𝛼n+1(t)−𝛼n(t)) − S(0)−𝛼n(0) 2≤𝛼′ n(t)−1 4(𝛼n+1(t)−𝛼n(t)) − 𝛼n+1(0)−𝛼n(0) 2≤𝛼′ n+1(t), where Smay be either 𝜌, 𝛾,or𝛽≡7, and 𝛽′ n+1(t)≤𝛽′ n(t)+1 4(𝛽n(t)−𝛽n+1(t)) + 𝛽n(0)−𝛽n+1(0) 2 ≤𝛽′ n(t)+1 4(𝛽n(t)−𝛽n+1(t)) + 𝛽n(0)−S(0) 2≤𝛽′ n(t)+1 4(𝛽n(t)−𝛽n+1(t)) + 7 2, where Smay be either 𝛼≡0,𝜌,or𝛾,fort∈[0,1 2), and every n. 6CONCLUSIONS In this paper, we presented a study about boundary value problems for nonlinear second-order functional differential equations with piecewise constant arguments of type (3), and obtained the following results: •We started the research work by recalling several preliminary notions about some suitable spaces of piecewise regular functions, together with some results concerning the expressions of the solutions to the linearized versions of the aforementioned problem (3). In these results, extracted from previous works,25,27 the deviating argument was based on the integer part function. •Then, after mentioning some useful results from Buedo-Fernández et al,27 we obtained some other new comparison results for boundary value problems associated to linear delay differential equations. These results were useful to determine the adequate relation (in terms of order) between two functions and also between their corresponding derivatives. •Moreover, the notions of upper and lower solutions (𝛽and 𝛼) for the problem of interest were presented, and it was proved that, under certain assumptions, the solutions to two comparable linear problems are also comparable. •Later, in Theorem 5, sufficient conditions were provided in order to prove that the boundary value problem (3) has at least one solution in the functional interval determined by 𝛼and 𝛽, and whose derivative lies on the functional interval [k1,k2](see Lemma 2, which provided the development of the upper and lower method). The proof of this result was based on the definition of truncation operators by using the functions 𝛼, 𝛽 for the function x,andk1,k2for the derivative x′. Afterwards, the definition of an integral map, whose kernel is given by solutions to linear problems, was a key point to complete the proof by applying comparison-type results, as well as Schauder's Fixed Point Theorem. •Besides, in Theorem 6, we developed the monotone iterative technique for the nonlinear second-order functional differential problem of interest, by considering certain constraints on the specific upper and lower functions considered. In particular, we proved the existence of monotone sequences {𝛽n},{𝛼n}starting at the upper and lower solutions and 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 34 BUEDO-FERNÁNDEZ ET AL. converging uniformly, respectively, to the maximal and minimal solutions to (3) in the region determined by 𝛼≤x≤𝛽 and k1≤x′≤k2. •Finally, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we cannot deduce their uniform convergence towards the corresponding derivatives of the extremal solutions. However, in Theorem 7, we obtained some estimates for these derivatives. The main limitations of the study are the existence of different auxiliary expressions to be used depending on the values of the parameters selected, and the consequent number of regions where it is guaranteed a suitable sign for the solutions to some linear associated problems. Besides, we mention the complexity of the expressions for the solutions of the auxiliary problems considered during the development of the monotone technique. Despite these obstacles, as some of the benefits, we mention that the approach followed allows to present a wide and complete range of cases for the one-sided Lipschitz condition of the nonlinearity to be satisfied in order to develop the procedure, and the use of suitable maximum principles that allow to give a priori bounds for the derivative, so that we can control not only the values of the solutions but also their rate of change. These results are applicable, for instance, to the study of a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. ACKNOWLEDGEMENTS We are grateful to the Editors and the anonymous Reviewers for their interesting comments. The research was partially supported by grant numbers PID2020-113275GB-I00 (AEI/FEDER, UE), MTM2016-75140-P (AEI/FEDER, UE) and ED431C 2019/02 (GRC Xunta de Galicia). The research of S. Buedo-Fernández and D. Cao Labora was also partially supported by PhD scholarships from the former Ministerio de Educación, Cultura y Deporte of Spain (FPU16/04416 and FPU16/04168, respectively). AUTHOR CONTRIBUTIONS All the authors contributed equally to the paper. CONFLICTS OF INTEREST The authors declare no potential conflicts of interest. ORCID Sebastián Buedo-Fernández https://orcid.org/0000-0002-5485-5667 Rosana Rodríguez-López https://orcid.org/0000-0001-5852-9845 REFERENCES 1. Ladde GS, Lakshmikantham V, Vatsala AS. Monotone Iterative Techniques for Nonlinear Differential Equations. Boston: Pitman; 1985. 2. Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. SIAM Rev. 1976;18:620-709. 3. Liz E. Monotone iterative techniques in ordered banach spaces. Nonlin Anal Theory Methods Appl. 1997;30(8):5179-5190. 4. Nieto JJ. An abstract monotone iterative technique. Nonlinear Anal. 1997;28(12):1923-1933. 5. Pao CV. Nonlinear Parabolic and Elliptic Equations. New York: Plenum Press; 1992. 6. Zeidler E. Nonlinear Functional Analysis and its Applications. Vol I: Fixed-point theorems. New York: Springer; 1993. 7. Nieto JJ, Jiang Y, Jurang Y. Monotone iterative method for functional differential equations. Nonlin Anal Theory Methods Appl. 1998;32(6):741-747. 8. Cabada A, Nieto JJ. A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems. J Math Anal Appl. 1990;151(1):181-189. 9. Seifert G. Second order scalar functional differential equations with piecewise constant arguments. J Differ Equ Appl. 2002;8:427-445. 10. Seifert G. Second-order neutral delay-differential equations with piecewise constant time dependence. JMathAnalAppl. 2003;281:1-9. 11. Yuan R. Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument. Nonlin Anal: Theory Methods Appl. 2000;41:871-890. 12. Yuan R. On the second-order differential equation with piecewise constant argument and almost periodic coefficients. Nonlinear Anal: Theory Methods Appl. 2003;52:1411-1440. 13. Zhang FQ. Boundary value problems for second order differential equations with piecewise constant arguments. Ann Differ Equ. 1993;9:369-374. 14. Chen L, Sun J. Boundary value problem of second order impulsive functional differential equations. J Math Anal Appl. 2006;323:708-720. 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License BUEDO-FERNÁNDEZ ET AL.35 15. Corduneanu C. Some classes of second order functional differential equations. Nonlinear Anal: Theory, Methods Appl. 2009;71(12):e865-e871. 16. Guo C, Guo Z. Existence of multiple periodic solutions for a class of second-order delay differential equations. Nonlinear Anal: Real World Appl. 2009;10(5):3285-3297. 17. Shen C, Zhou H, Yang L. Existence of positive solutions of a nonlinear differential equation for a thermostat model. Math Methods Appl Sci. 2018;41(16):6145-6154. 18. Henríquez HR, Hernández EM. Approximate controllability of second-order distributed implicit functional systems. Nonlinear Anal: Theory, Methods Appl. 2009;70:1023-1039. 19. Sakthivel R, Mahmudov NI, Kim JH. On controllability of second order nonlinear impulsive differential systems. Nonlinear Anal: Theory, Methods Appl. 2009;71:45-52. 20. Shoukaku Y. Forced oscillation of certain neutral hyperbolic equations with continuous distributed deviating arguments. Math Comput Modell. 2009;49:1211-1220. 21. Liu B, Huang L. Existence and uniqueness of periodic solutions for a kind of second order neutral functional differential equations. Nonlinear Anal: Real World Appl. 2007;8:222-229. 22. Pan L. Periodic solutions for higher order differential equations with deviating argument. J Math Anal Appl. 2008;343:904-918. 23. Cabada A, Ferreiro JB, Nieto JJ. Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments. J Math Anal Appl. 2004;291:690-697. 24. Nieto JJ. A comparison result for a linear differential equation with piecewise constant delay. Glas Mat Ser III. 2004;39(59):73-76. 25. Nieto JJ, Rodríguez-López R. Green's function for second-order periodic boundary value problems with piecewise constant arguments. J Math Anal Appl. 2005;304:33-57. 26. Yang P, Liu Y, Ge W. Green's function for second order differential equations with piecewise constant arguments. Nonlinear Anal: Theory, Methods Appl. 2006;64(8):1812-1830. 27. Buedo-Fernández S, Cao Labora D, Rodríguez-López R, Tersian S. Analysis of the sign of the solution for certain second-order periodic boundary value problems with piecewise constant arguments. Mathematics. 2020;8:1953. 28. Glashoff K, Sprekels J. The regulation of temperature by thermostats and set-valued integral equations. J Integral Equations. 1982;4:95-112. 29. Webb JRL. Existence of positive solutions for a thermostat model. Nonlinear Anal: Real World Appl. 2012;13(2):923-938. 30. Webb JRL. Non-local second-order boundary value problems with derivative-dependent nonlinearity: Nonlocal boundary value problems. Philos Trans R Soc A: Math, Phys Eng Sci. 2021;379(2191):20190383. 31. Baleanu D, Etemad S, Rezapour S. A hybrid caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound Value Prob. 2020;2020:64. 32. Shen C, Zhou H, Yang L. Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter. Math Methods Appl Sci. 2016;39(15):4504-4511. 33. Rezapour S, Souid MS, Bouazza Z, Hussain A, Etemad S. On the fractional variable order thermostat model: Existence theory on cones via piece-wise constant functions. Adv Nonlinear Anal Appl. 2022;2022:8053620. Howtocitethisarticle: Buedo-Fernández S, Cao Labora D, Rodríguez-López R. Boundary value problems for nonlinear second-order functional differential equations with piecewise constant arguments. Math Meth Appl Sci. 2022;1-35. doi:10.1002/mma.8878 10991476, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/mma.8878 by Universidade de Santiago de Compostela, Wiley Online Library on [15/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License