Periodic Boundary Value Problems for First-Order Impulsive Functional Integrodifferential Equations with Integral-Jump Conditions

Abstract

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results.

1. Introduction

The theory of impulsive differential equations is now being recognized as not only being richer than the corresponding theory of differential equations without impulses, but also representing a more natural framework for mathematical modelling of many real world phenomena and applications; see [1–5] and the references therein. Monotone iterative technique coupled with the method of upper and lower solutions has provided an effective mechanism to prove constructive existence results for initial and boundary value problems for nonlinear differential equations; see [6]. However, many papers have studied applications of the monotone iterative technique to impulsive problems; see, for example, [7–15]. In those articles, the authors assumed that Δ𝑥(𝑡_𝑘)=𝐼_𝑘(𝑥(𝑡⁻_𝑘)), that is, a short-term rapid change of the state (jump condition) at impulse point 𝑡_𝑘, depends on the left side of the limit of 𝑥(𝑡_𝑘).

In [16–18] the authors discussed some classes of first-order impulsive problems with the impulsive integral conditions: Δ𝑥𝑡_𝑘=𝐼_𝑘^𝑡𝑘_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠−^{𝑡𝑘−1+𝜎𝑘−1}_𝑡𝑘−1𝑥(𝑠)𝑑𝑠, (1) where 0<𝜎_𝑘−1≤(𝑡_𝑘−𝑡_𝑘−1)/2 and 0≤𝜏_𝑘≤(𝑡_𝑘−𝑡_𝑘−1)/2, 𝑘=1,2,…,𝑚. Furthermore, Thaiprayoon et al. [19] have used such technique to investigate the existence criteria of extremal solutions of multipoint impulsive problems to include multipoint jump conditions Δ𝑥𝑡_𝑘=𝐼_𝑘^𝑐𝑘_𝑙=1𝜌^𝑘_𝑙𝑥𝜂^𝑘_𝑙,𝑐_𝑘∈{1,2,…}, (2) for 𝑡_𝑘−1<𝜂^𝑘₁<𝜂^𝑘₂<⋯<𝜂^𝑘_𝑐𝑘≤𝑡_𝑘, 𝑘=1,2,…,𝑚. Recently, Thiramanus and Tariboon [20] have given some results on impulsive differential inequalities with integral-jump conditions of the form: 𝑥^′(𝑡)≤𝑝(𝑡)𝑚(𝑡)+𝑞(𝑡),𝑡≠𝑡_𝑘,𝑥𝑡⁺_𝑘≤𝑑_𝑘𝑥𝑡_𝑘+𝑐_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝑏_𝑘,𝑘=1,2,…, (3) where 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1. We note that if 𝑑_𝑘=1, 𝜎_𝑘<𝜏_𝑘, 𝑘=1,2,…, then the above inequalities mean that the bound of jump condition at 𝑡_𝑘 is a functional of past states on the interval (𝑡_𝑘−𝜏_𝑘,𝑡_𝑘−𝜎_𝑘] before the impulse point 𝑡_𝑘.

In spirit of the results from [20], this paper considers the periodic boundary value problem for first-order impulsive functional integrodifferential equation (PBVP) with integral-jump conditions: 𝑥^′(𝑡)((𝑡)(𝜃(𝑡,(𝑡),(𝑡,[]=𝑓𝑡,𝑥,𝑥))𝐾𝑥)(𝑆𝑥)())𝑡∈𝐽=0,𝑇,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑥(0)(𝑇),𝑑𝑠,𝑘=1,2,…,𝑚,=𝑥 (4) where 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, 𝑓∈𝐶(𝐽×𝑅⁴,𝑅), 𝜃∈𝐶(𝐽,𝐽), 𝐼_𝑘∈𝐶(𝑅,𝑅), Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, (𝐾𝑥)(𝑡)=^𝑡₀𝑘()𝑥(𝑠)(𝑡)=𝑡,𝑠𝑑𝑠,𝑆𝑥)(^𝑇₀ℎ(𝑡,𝑠)𝑥(𝑠)𝑑𝑠, (5)𝑘(𝑡,𝑠)∈𝐶(𝐷,𝑅⁺), ℎ(𝑡,𝑠)∈𝐶(𝐽×𝐽,𝑅⁺), 𝐷={(𝑡,𝑠)∈𝑅², 0≤𝑠≤𝑡≤𝑇}, and 𝑅⁺=[0,+∞).

We first introduce a new concept of lower and upper solutions, then establish a new comparison principle, and discuss the existence and uniqueness of the solutions for first-order impulsive functional integrodifferential equations with integral-jump conditions. By using the method of upper and lower solutions and monotone iterative technique, we obtain the existence of extreme solution of PBVP (4). Finally, we give an example to illustrate the obtained results.

2. Preliminaries

Let 𝐽₀=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑚}, 𝑘₀=max{𝑘(𝑡,𝑠);(𝑡,𝑠)∈𝐷}, ℎ₀=max{ℎ(𝑡,𝑠);(𝑡,𝑠)∈𝐽×𝐽}, and 𝑃𝐶(𝐽,𝑅)={𝑥∶𝐽→𝑅;𝑥(𝑡) be continuous everywhere except for some 𝑡_𝑘, at which 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) exist and 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘), 𝑘=1,2,…,𝑚}, and let 𝑃𝐶¹(𝐽,𝑅)={𝑥∈𝑃𝐶(𝐽,𝑅);𝑥^′(𝑡) be continuous everywhere except for some 𝑡_𝑘 at which 𝑥^′(𝑡⁺_𝑘) and 𝑥^′(𝑡⁻_𝑘) exist and 𝑥^′(𝑡⁻_𝑘)=𝑥^′(𝑡_𝑘), 𝑘=1,2,…,𝑚}. Clearly, 𝑃𝐶(𝐽,𝑅) is a Banach space with the norm ‖𝑥‖_𝑃𝐶=sup{𝑥(𝑡)∶𝑡∈𝐽}. Let 𝐸=𝑃𝐶(𝐽,𝑅)∩𝑃𝐶¹(𝐽,𝑅). A function 𝑥∈𝐸 is called a solution of PBVP (4) if it satisfies (4).

Definition 1. We say that the functions 𝛼, 𝛽∈𝐸 are lower and upper solutions of PBVP (4) if there exist 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, 𝐿_𝑘≥0, and 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1 such that 𝛼^′(𝑡)≤𝑓(𝑡,𝛼(𝑡),𝛼(𝜃(𝑡)),(𝐾𝛼)(𝑡),(𝑆𝛼)(𝑡))−𝑎_𝛼(𝑡),𝑡∈𝐽₀,Δ𝛼𝑡_𝑘≤𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼(𝑠)𝑑𝑠−𝑏_𝛼𝑘,𝑘=1,2,…,𝑚, (6) where 𝑎_𝛼(𝑡)=(0)(𝑇),(𝑀()×((𝑡)×(𝑇)0,if𝛼≤𝛼𝑇−𝑡+𝑊𝑇−𝜃))+1⁻¹+𝑁^𝑡₀𝑘(𝑡,𝑠)(𝑇−𝑠)𝑑𝑠𝑇+𝐿^𝑇₀ℎ(𝑡,𝑠)(𝑇−𝑠)𝑑𝑠𝑇×[𝛼(0)(𝑇(0)(𝑇),𝑏−𝛼)],if𝛼>𝛼_𝛼𝑘=(0)(𝑇),𝐿0,if𝛼≤𝛼_𝑘𝜏_𝑘−𝜎_𝑘×22𝑇𝑇−𝑡_𝑘+𝜏_𝑘+𝜎_𝑘×[𝛼(0)(𝑇(0)(𝑇),𝛽−𝛼)],if𝛼>𝛼_′(𝑡)≥𝑓𝑡,𝛽(𝑡),𝛽(𝜃(𝑡)),𝐾𝛽(𝑡),𝑆𝛽(𝑡)+𝑎_𝛽(𝑡),𝑡∈𝐽₀,Δ𝛽𝑡_𝑘≥𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽(𝑠)𝑑𝑠+𝑏_𝛽𝑘,𝑘=1,2,…,𝑚, (7) where 𝑎_𝛽(𝑡)=(0)(𝑇),(𝑀()×((𝑡)×(𝑇)0,if𝛽≥𝛽𝑇−𝑡+𝑊𝑇−𝜃))+1⁻¹+𝑁^𝑡₀𝑘(𝑡,𝑠)(𝑇−𝑠)𝑑𝑠𝑇+𝐿^𝑇₀ℎ(𝑡,𝑠)(𝑇−𝑠)𝑑𝑠𝑇×𝛽(𝑇)−𝛽(0),if𝛽(0)<𝛽(𝑇),𝑏_𝛽𝑘=(0)(𝑇),𝐿0,if𝛽≥𝛽_𝑘𝜏_𝑘−𝜎_𝑘×22𝑇𝑇−𝑡_𝑘+𝜏_𝑘+𝜎_𝑘×𝛽(𝑇)−𝛽(0),if𝛽(0)<𝛽(𝑇). (8)
Denote 𝑙=max{𝑘∶𝑡≥𝑡_𝑘,k=1,2,…}. We prove the comparison principle by using the following lemma (see [20]).

Lemma 2. Let 𝑟∈{𝑡₀,𝑡₁,…,𝑡_𝑚}, 𝑐_𝑘≥0,and0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝛾_𝑘, 𝑘=1,…,𝑚, be constants and let 𝑞∈𝑃𝐶(𝐽,𝑅), 𝑥∈𝑃𝐶¹(𝐽,𝑅). If 𝑥^′(𝑡)≥𝑞(𝑡),𝑡∈(𝑟,𝑇),𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘≥𝑐_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝛾_𝑘,𝑡_𝑘∈(𝑟,𝑇), (9) then, for 𝑡∈(𝑟,𝑇], 𝑥(𝑡)≥𝑥𝑟⁺_{𝑟<𝑡𝑘<𝑡}1+𝑐_𝑘𝜏_𝑘−𝜎_𝑘+_{𝑟<𝑡𝑘<𝑡𝑡𝑘<t𝑗<𝑡}1+𝑐_𝑗𝜏_𝑗−𝜎_𝑗×1+𝑐_𝑘𝜏_𝑘−𝜎_𝑘^{𝑡𝑘−𝜏𝑘}_𝑡𝑘−1𝑞(𝑠)+𝑑𝑠_{𝑡𝑘−𝜎𝑘𝑡𝑘−𝜏𝑘}1+𝑐_𝑘𝑡_𝑘−𝜎_𝑘𝑞(𝑠)+−𝑠𝑑𝑠_{𝑡𝑘𝑡𝑘−𝜎𝑘}𝑞(𝑠)𝑑𝑠+𝛾_𝑘+^𝑡_𝑡𝑙𝑞(𝑠)𝑑𝑠. (10)

Now we are in the position to establish a new comparison principle, which plays an important role in monotone iterative technique.

Lemma 3. Assume that 𝑥∈𝐸 satisfies𝑥^′(𝑡)≥𝑀𝑥(𝑡)+𝑊𝑥(𝜃(𝑡))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝑎_𝑥(𝑡),𝑡∈𝐽₀,Δ𝑥𝑡_𝑘≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝑏_𝑥𝑘,𝑘=1,2,…,𝑚, (11) where constants 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, 𝐿_𝑘≥0, and 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝑘=1,2,…,𝑚. If 𝐿+^𝑚_𝑘=1𝐵_𝑘+^𝑇₀𝑟(𝑠)𝑑𝑠^𝑚_𝑘=1𝐴_𝑘≤1, (12) where 𝑎_𝑥(𝑡), 𝑡∈𝐽₀, and 𝑏_𝑥𝑘 are given by Definition 1 with 𝛽=𝑥 and 𝐿=max𝐿_𝑘𝜏_𝑘−𝜎_𝑘,𝑟(𝑡);𝑘=1,2,…,𝑚=𝑀+𝑊+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑑𝑠+𝐿^𝑇₀ℎ()𝐴𝑡,𝑠𝑑𝑠,_𝑘=1+𝐿_𝑘𝜏_𝑘−𝜎_𝑘,𝐵_𝑘=𝐴_𝑘^{𝑡𝑘−𝜏𝑘}_𝑡k−1𝑟(𝑠)+𝑑𝑠_{𝑡𝑘−𝜎𝑘𝑡𝑘−𝜏𝑘}1+𝐿_𝑘𝑡_𝑘−𝜎_𝑘𝑟(𝑠)+−𝑠𝑑𝑠_{𝑡𝑘𝑡𝑘−𝜎𝑘}𝑟(𝑠)𝑑𝑠, (13) then 𝑥(𝑡)≤0, 𝑡∈𝐽.

Proof. Case 1. One has 𝑥(0)≥𝑥(T). Suppose that there exists 𝑡^∗∈𝐽 such that 𝑥(𝑡^∗)>0 and distinguish two cases.
Case (a). 𝑥(𝑡)≥0 for all 𝑡∈𝐽, 𝑥≢0; then 𝑥^′(𝑡)≥𝑀𝑥(𝑡)+𝑊𝑥(𝜃(𝑡))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑥(𝑠)𝑑𝑠≥0,𝑡∈𝐽₀,𝑥𝑡⁺_𝑘≥𝑥𝑡_𝑘+𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠≥𝑥𝑡_𝑘,𝑘=1,2,…,𝑚, (14) so that 𝑥 is nondecreasing in 𝐽, and then 𝑥(𝑇)≥𝑥(0). Since 𝑥(0)≥𝑥(𝑇), then 𝑥 is a constant function 𝑥(𝑡)=𝐶>0, which implies that 𝑥^′(𝑡)=0≥𝑀𝐶+𝑊𝐶+𝑁𝐶^𝑡₀𝑘(𝑡,𝑠)𝑑𝑠+𝐿𝐶^𝑇₀ℎ(𝑡,𝑠)𝑑𝑠≥𝑀𝐶>0, (15) getting a contradiction.
Case (b). 𝑥(𝑡)<0 for some 𝑡∈𝐽. Let inf_𝑡∈𝐽𝑥(𝑡)=−𝜆<0; then there exists 𝑡∈(𝑡_𝑖,𝑡_𝑖+1], for some 𝑖 such that 𝑥(𝑡)=−𝜆 or 𝑥(𝑡⁺_𝑖)=−𝜆. Without loss of generality, we only consider 𝑥(𝑡)=−𝜆, and for the case 𝑥(𝑡⁺_𝑖)=−𝜆 the proof is similar.
From (11), it is easy to see that 𝑥^′(𝑡)≥−𝜆𝑀+𝑊+𝑁^𝑡₀𝑘()∫𝑡,𝑠𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑑𝑠,𝑡∈𝐽₀. (16) We consider the inequalities 𝑥^′(𝑡)≥−𝜆𝑟(𝑡),𝑡∈𝐽₀,Δ𝑥𝑡_𝑘≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠,𝑘=1,2,…,𝑚. (17) From Lemma 2, we have 𝑥(𝑡)≥𝑥(0)_{0<𝑡𝑘<𝑡}𝐴_𝑘−𝜆_{0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑙𝑟(𝑠)𝑑𝑠,𝑡∈𝐽. (18)
Let 𝑡=𝑡 in (18); then 𝑥𝑡≥𝑥(0)_{0<𝑡𝑘<𝑡}𝐴_𝑘−𝜆_{0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑖𝑟(𝑠)𝑑𝑠, (19)
so that 𝑥(0)_{0<𝑡𝑘<𝑡}𝐴_𝑘≤−𝜆+𝜆_{0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑖𝑟(𝑠)𝑑𝑠. (20)
If 𝑥(0)>0, then (20) with 𝐴_𝑘≥1, 𝐵_𝑘≥0 for all 𝑘 implies 1<_{0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑖𝑟(𝑠)≤𝑑𝑠_{𝑚𝑘=1𝑘<𝑗<𝑚}𝐴_𝑗𝐵_𝑘+^𝑇₀𝑟(𝑠)𝑑𝑠. (21)
This contradicts the condition (12).
Suppose that 𝑥(0)≤0. If 𝑡^∗<𝑡 for 𝑡^∗∈(𝑡_𝜈,𝑡_𝜈+1], then Lemma 2 provides that 𝑥(𝑡)≥𝑥𝑡⁺_{𝜈+1𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘−𝜆_{𝑡𝜈+1<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑙𝑟(𝑠)𝑑𝑠,𝑡≥𝑡⁺_𝜈+1. (22) Since 𝑥𝑡⁺_𝜈+1≥𝑥𝑡_𝜈+1+𝐿_𝜈+1^{𝑡𝜈+1−𝜎𝜈+1}_{𝑡𝜈+1−𝜏𝜈+1}𝑥(𝑠)𝑡𝑑𝑠≥𝑥_𝜈+1−𝜆𝐿_𝜈+1𝜏_𝜈+1−𝜎_𝜈+1, (23) and integrating (11) from 𝑡^∗ into 𝑡_𝜈+1, we obtain 𝑥𝑡_𝜈+1≥𝑥𝑡^∗−𝜆^𝑡𝜈+1_𝑡∗𝑟(𝑠)𝑑𝑠. (24) Hence, 𝑥(𝑡)≥𝑥𝑡^∗_{𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘−𝜆𝐿_𝜈+1𝜏_𝜈+1−𝜎_𝜈+1×_{𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘−𝜆_{𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘^t𝜈+1_𝑡∗𝑟(𝑠)𝑑𝑠−𝜆_{𝑡𝜈+1<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑙𝑟(𝑠)𝑡𝑑𝑠≥𝑥^∗_{𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘−𝜆𝐿_𝜈+1𝜏_𝜈+1−𝜎_{𝜈+1𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘−𝜆_{𝑡𝜈<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑙𝑟(𝑠)𝑑𝑠,𝑡≥𝑡^∗. (25) We note that if 𝑡∈(𝑡_𝑣,𝑡_𝑣+1], then 𝑡_𝑙=𝑡^∗.
Let 𝑡=𝑡 in (25); then 𝑥𝑡≥𝑥𝑡^∗_{𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘−𝜆𝐿_𝜈+1𝜏_𝜈+1−𝜎_{𝜈+1𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘−𝜆_{𝑡𝜈<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑙𝑟(𝑠)𝑑𝑠. (26) From (26), we have 0<𝑥𝑡^∗_{𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘≤−𝜆+𝜆𝐿_𝜈+1𝜏_𝜈+1−𝜎_{𝜈+1𝑡𝜈+1<𝑡𝑘<𝑡}𝐴_𝑘+𝜆_{𝑡𝜈<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝐴_𝑗𝐵_𝑘+^𝑡_𝑡𝑙𝑟(𝑠)𝑑𝑠, (27) which gives 1<𝐿^𝑚_𝑘=1𝐴_𝑘+_{0<𝑡𝑘<𝑇𝑡𝑘<𝑡𝑗<𝑇}𝐴_𝑗𝐵_𝑘+^𝑇₀𝑟(𝑠)𝑑𝑠, (28) contradicting condition (12). For the case 𝑡<𝑡^∗, the proof is similar.
Case 2. One has 𝑥(0)<𝑥(𝑇). Set 𝑣(𝑡)=𝑥(𝑡)+((𝑇−𝑡)/𝑇)(𝑥(𝑇)−𝑥(0)). It follows that 𝑣(0)=𝑥(𝑇)=𝑣(𝑇), and for 𝑡∈𝐽₀, 𝑣^′(𝑡)≥𝑀𝑥(𝑡)+𝑊𝑥(𝜃(𝑡))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝑎_𝑥(𝑡)−1𝑇(𝑥(𝑇)(0(𝑡)(𝜃(𝑡−𝑥))=𝑀𝑣+𝑊𝑣))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑣(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑣(𝑠)𝑑𝑠+𝑎_𝑥(𝑡)−𝑀(𝑇−𝑡)+𝑊(𝑇−𝜃(𝑡))+1𝑇[𝑥(𝑇)(0−𝑁[𝑥(𝑇)(0−𝑥)]−𝑥)]𝑇^𝑡₀𝑘()−𝐿[𝑥(𝑇)(0𝑡,𝑠)(𝑇−𝑠𝑑𝑠−𝑥)]𝑇^𝑡₀ℎ()(𝑡)(𝜃(𝑡𝑡,𝑠)(𝑇−𝑠𝑑𝑠≥𝑀𝑣+𝑊𝑣))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑣(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑣(𝑠)𝑑𝑠, (29) and for 𝑡=𝑡_𝑘, 𝑘=1,2,…,𝑚, Δ𝑣𝑡_𝑘=Δ𝑥𝑡_𝑘≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)+𝐿𝑑𝑠_𝑘𝜏_𝑘−𝜎_𝑘2𝑇−𝑡_𝑘+𝜏_𝑘+𝜎_𝑘×(𝑥(𝑇)(02𝑇−𝑥))=𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑣(𝑠)𝑑𝑠. (30) In view of Case 1, we see that 𝑣(𝑡)≤0 on 𝐽, and therefore 𝑥(𝑡)≤0 on 𝐽. This completes the proof.

Corollary 4. Assume that 𝑥∈𝐸 satisfies 𝑥^′(𝑡)≥𝑀𝑥(𝑡)+𝑊𝑥(𝜃(𝑡))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑥(𝑠)𝑑𝑠,𝑡∈𝐽₀,Δ𝑥𝑡_𝑘≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑥(0)(𝑇),𝑑𝑠,𝑘=1,2,…,𝑚,≥𝑥 (31) where constants 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, 𝐿_𝑘≥0, and 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝑘=1,2,…,𝑚. If 𝐿+^𝑚_𝑘=1𝐵⁰_𝑘+𝑟₀𝑇^𝑚_𝑘=1𝐴_𝑘≤1, (32) where 𝑟₀=𝑀+𝑊+𝑘₀𝑁𝑇+ℎ₀𝐵𝐿𝑇,_0𝑘=𝑟₀𝐴_𝑘^{𝑡𝑘−𝜏𝑘}_𝑡𝑘−1𝑑𝑠+𝑟₀^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}1+𝐿_𝑘𝑡_𝑘−𝜎_𝑘−𝑠𝑑𝑠+𝑟₀^𝑡𝑘_{𝑡𝑘−𝜎𝑘}𝑑𝑠=𝑟₀𝑡_𝑘−𝑡_𝑘−1+𝐿_𝑘2𝜏_𝑘−𝜎_𝑘×2𝑡_𝑘−𝑡_𝑘−1−𝜏_𝑘+𝜎_𝑘, (33) and 𝐿, 𝐴_𝑘 are given by Lemma 3, then 𝑥(𝑡)≤0, for 𝑡∈𝐽.

Corollary 5. Assume that 𝑥∈𝐸 satisfies 𝑥^′(𝑡)≥𝑀𝑥(𝑡)+𝑊𝑥(𝜃(𝑡))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ()𝑥(𝑠)+𝑀()((𝑡𝑡,𝑠𝑑𝑠𝑇−𝑡+𝑊𝑇−𝜃))+1𝑇+𝑘₀𝑁2𝑇𝑡−𝑡²+ℎ2𝑇₀𝐿𝑇2[𝑥(𝑇)−𝑥(0)],𝑡∈𝐽₀,Δ𝑥𝑡_𝑘≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)+𝐿𝑑𝑠_𝑘𝜏_𝑘−𝜎_𝑘2𝑇−𝑡_𝑘+𝜏_𝑘+𝜎_𝑘×(𝑥(𝑇)(0𝑥(0)(𝑇),2𝑇−𝑥)),𝑘=1,2,…,𝑚,<𝑥 (34) where constants 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, 𝐿_𝑘≥0, and 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝑘=1,2,…,𝑚, and condition (32) holds. Then 𝑥(𝑡)≤0, for 𝑡∈𝐽.

Let us consider the following linear problem of PBVP (4): 𝑥^′(𝑡)−𝑀𝑥(𝑡)=𝑊𝑥(𝜃(𝑡))+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝐿^𝑇₀ℎ(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝑔(𝑡),𝑡∈𝐽₀,Δ𝑥𝑡_𝑘=𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑥(0)(𝑇),𝑑𝑠,𝑘=1,2,…,𝑚,=𝑥 (35) where 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, 𝐿_𝑘≥0, 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝜃∈(𝐽,𝐽), 𝐼_𝑘∈𝐶(𝐽,𝑅),  𝑘=1,2,…,𝑚, 𝑔∈𝑃𝐶(𝐽,𝑅), and 𝜂∈𝑃𝐶¹(𝐽,𝑅).

Lemma 6. A function 𝑥∈𝐸 is a solution of (35) if and only if 𝑥∈𝑃𝐶(𝐽,𝑅) is a solution of the following impulsive integral equation: 𝑥(𝑡)=−^𝑇₀𝐺(𝑡,𝑠)𝑅(𝑠)𝑑𝑠−^𝑚_𝑘=1𝐺𝑡,𝑡_𝑘×𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠,𝑡∈𝐽, (36) where 𝑅(𝑡)=𝑊𝑥(𝜃(𝑡))+𝑁∫^𝑡₀𝑘(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝐿∫^𝑇₀ℎ(𝑡,𝑠)𝑥(𝑠)𝑑𝑠+𝑔(𝑡) and 𝐺()=1𝑡,𝑠𝑒^𝑀𝑇𝑒−1𝑒^{𝑀(𝑡−𝑠)},0≤𝑠<𝑡≤𝑇,_{𝑀(𝑇+𝑡−𝑠)},0≤𝑡≤𝑠≤𝑇. (37)

Proof. If 𝑥(𝑡) is a solution of (35), by directly integrating, we obtain 𝑥(𝑡)=−^𝑇₀𝐺(𝑡,𝑠)𝑅(𝑠)𝑑𝑠−^𝑚_𝑘=1𝐺𝑡,𝑡_𝑘×𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠,𝑡∈𝐽. (38)
If 𝑥(𝑡) is a solution of the above-mentioned integral equation, then 𝑥^′(𝑡)=𝑀−^𝑡₀𝑒^{𝑀(𝑡−𝑠)}𝑒^𝑀𝑇𝑅(𝑠)−1𝑑𝑠−^𝑇_𝑡𝑒^{𝑀(𝑇+𝑡−𝑠)}𝑒^𝑀𝑇𝑅(𝑠)−1𝑑𝑠−_{0<𝑡𝑘<𝑡}𝑒^{𝑀(𝑡−𝑡𝑘)}𝑒^𝑀𝑇𝐿−1^{𝑘𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+I_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−_{𝑡≤𝑡𝑘<𝑡}𝑒^{𝑀(𝑇+𝑡−𝑡𝑘)}𝑒^𝑀𝑇𝐿−1^{𝑘𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)(𝑡)−𝑑𝑠+𝑅=𝑀^𝑇₀𝐺()𝑅(𝑠)−𝑡,𝑠𝑑𝑠_𝑚𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)(𝑡)(𝑡)(𝑡),𝑡𝑑𝑠+𝑅=𝑀𝑥+𝑅Δ𝑥_𝑘=𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝑒^𝑀𝑇𝑒^𝑀𝑇𝐿−1^{𝑘𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)−1𝑑𝑠𝑒^𝑀𝑇𝐿−1^{𝑘𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠),𝑥(0)𝑑𝑠=−^𝑇₀𝑒^{𝑀(𝑇−𝑠)}𝑒^𝑀𝑇𝑅(𝑠)−1𝑑𝑠+^𝑚_𝑘=1𝑒^{𝑀(𝑇−𝑡𝑘)}𝑒^𝑀𝑇×𝐿−1_{𝑘𝑡𝑘−𝜎𝑘𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠=𝑥(𝑇). (39)
The proof is complete.

Lemma 7. Let 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, 𝐿_𝑘≥0, 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝜃∈(𝐽,𝐽), 𝐼_𝑘∈𝐶(𝐽,𝑅), 𝑘=1,2,…,𝑚, 𝑔∈𝑃𝐶(𝐽,𝑅), and 𝜂∈𝑃𝐶¹(𝐽,𝑅) and assume that sup_𝑡∈𝐽^𝑇₀𝐺(𝑡,𝑠)𝑊+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑑𝑟+𝐿^𝑇₀ℎ()+1𝑠,𝑟𝑑𝑟𝑑𝑠1−𝑒^{𝑚−𝑀𝑇}_𝑘=1𝐿_𝑘𝜏_𝑘−𝜎_𝑘<1. (40)
Then problem (35) has a unique solution in 𝑃𝐶(𝐽,𝑅).

Proof. We define the mapping 𝐹∶𝑃𝐶(𝐽,𝑅)→𝑃𝐶(𝐽,𝑅) by (𝐹𝑥)(𝑡)=−^𝑇₀𝐺(𝑡,𝑠)𝑊𝑥(𝜃(𝑠))+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑥(𝑟)𝑑𝑟+𝐿^𝑇₀ℎ()𝑥(𝑟)(𝑠)−𝑠,𝑟𝑑𝑟+𝑔𝑑𝑠_𝑚𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝜂(𝑠)𝑑𝑠,𝑡∈𝐽, (41) for any 𝑥∈𝑃𝐶(𝐽,𝑅) and 𝐺 is given by Lemma 6. Then 𝐹𝑥−𝐹𝑦_𝑃𝐶=sup_𝑡∈𝐽−^𝑇₀𝐺(𝑡,𝑠)𝑊𝑥(𝜃(𝑠))−𝑦(𝜃(𝑠))+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑥(𝑟)−𝑦(𝑟)𝑑𝑟+𝐿^𝑇₀ℎ(𝑥(𝑟)(𝑟−𝑠,𝑟)−𝑦)𝑑𝑟𝑑𝑠_𝑚𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)−𝑦(𝑠)𝑑𝑠≤sup_𝑡∈𝐽^𝑇₀𝐺(𝑡,𝑠)𝑊𝑥(𝜃(𝑠))−𝑦(𝜃(𝑠))+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑥(𝑟)−𝑦(𝑟)𝑑𝑟+𝐿^𝑇₀ℎ(𝑥(𝑟)(𝑟+𝑠,𝑟)−𝑦)𝑑𝑟𝑑𝑠_𝑚k=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑟)(𝑟≤−𝑦)𝑑𝑠𝑥−𝑦_𝑃𝐶×sup_𝑡∈𝐽^𝑇₀𝐺()×𝑡,𝑠𝑊+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑑𝑟+𝐿^𝑇₀ℎ()+1𝑠,𝑟𝑑𝑟𝑑𝑠1−𝑒^{𝑚−𝑀𝑇}_𝑘=1𝐿_𝑘𝜏_𝑘−𝜎_𝑘. (42) The above result and condition (40) imply that 𝐹 is a contractive mapping, which completes the proof.

Corollary 8. Let 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, 𝐿_𝑘≥0, 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝜃∈(𝐽,𝐽), 𝐼_𝑘∈𝐶(𝐽,𝑅), 𝑘=1,2,…,𝑚, 𝑔∈𝑃𝐶(𝐽,𝑅), and 𝜂∈𝑃𝐶¹(𝐽,𝑅) and assume that 𝑊+𝑘₀𝑁+ℎ₀𝐿𝑇𝑀+11−𝑒^{𝑚−𝑀𝑇}_𝑘=1𝐿_𝑘𝜏_𝑘−𝜎_𝑘<1. (43) Then problem (35) has a unique solution in 𝑃𝐶¹(𝐽,𝑅).

3. Main Results

In this section, we establish existence criteria for solutions of the PBVP (4) by the method of lower and upper solutions and monotone iterative technique. For 𝛼, 𝛽∈𝐸, we write 𝛽≤𝛼 if 𝛽(𝑡)≤𝛼(𝑡) for all 𝑡∈𝐽. In such a case, we denote [𝛽,𝛼]={𝑥∈𝐸∶𝛽(𝑡)≤𝑥(𝑡)≤𝛼(𝑡), 𝑡∈𝐽}.

Theorem 9. Assume the existence of lower and upper solutions for PBVP (4) and also suppose that the following conditions hold.
(𝐻₁) The function 𝑓∈𝐶(𝐽×𝑅⁴,𝑅) satisfies 𝑓𝑡,𝑧₁,𝑧₂,𝑧₃,𝑧₄−𝑓𝑡,𝑦₁,𝑦₂,𝑦₃,𝑦₄≤𝑀𝑧₁−𝑦₁+𝑊𝑧₂−𝑦₂+𝑁𝑧₃−𝑦₃+𝐿𝑧₄−𝑦₄,𝑓𝑡,𝑦₁,𝑦₂,𝑦₃,𝑦₄−𝑓𝑡,𝑥₁,𝑥₂,𝑥₃,𝑥₄≤𝑀𝑦₁−𝑥₁+𝑊𝑦₂−𝑥₂+𝑁𝑦₃−𝑥₃+𝐿𝑦₄−𝑥₄, (44) where 𝛽(𝑡)≤𝑥₁≤𝑦₁≤𝑧₁≤𝛼(𝑡), 𝛽(𝜃(𝑡))≤𝑥₂≤𝑦₂≤𝑧₂≤𝛼(𝜃(𝑡)), (𝐾𝛽)(𝑡)≤𝑥₃≤𝑦₃≤𝑧₃≤(𝐾𝛼)(𝑡), and (𝑆𝛽)(𝑡)≤𝑥₄≤𝑦₄≤𝑧₄≤(𝑆𝛼)(𝑡), 𝑡∈𝐽, where 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, and 𝜃∈𝐶(𝐽,𝐽).
(𝐻₂) The functions 𝐼_𝑘∈𝐶(𝑅,𝑅) satisfy 𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼(𝑠)𝑑𝑠−𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑦(𝑠)𝑑𝑠≤𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼(𝑠)(𝑠)𝐼−𝑦𝑑𝑠,_{𝑘𝑡𝑘−𝜎𝑘𝑡𝑘−𝜏𝑘}𝑦(𝑠)𝑑𝑠−𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽(𝑠)𝑑𝑠≤𝐿_𝑘∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑦(𝑠)−𝛽(𝑠)𝑑𝑠, (45) where ∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑦(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼(𝑠)𝑑𝑠, 𝑘=1,2,…,𝑚, where 𝐿_𝑘≥0 and 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝑘=1,2,…,𝑚.
If inequalities (12) and (40) hold, then there exists a solution 𝑥 of PBVP (4) such that 𝛽(𝑡)≤𝑥(𝑡)≤𝛼(𝑡), for 𝑡∈𝐽.

Proof. We consider the following modified problem relative to PBVP (4): 𝑥^′(𝑡)(𝑡)(𝜃(𝑡(𝑡)(𝑡)−𝑀𝑥=𝑊𝑥))+𝑁𝐾𝑥)(+𝐿𝑆𝑥)(+𝑔_𝑞(𝑡),𝑡∈𝐽₀,Δ𝑥𝑡_𝑘=𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑞(𝑠,𝑥(𝑠))𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑞((𝑠𝑥(0)(𝑇),𝑠,𝑥))𝑑𝑠,𝑘=1,2,…,𝑚,=𝑥 (46) where 𝑔_𝑞(𝑡)=𝑓(𝑡,𝑞(𝑡),𝑞(𝜃(𝑡)),(𝐾𝑞)(𝑡),(𝑆𝑞)(𝑡))−𝑀𝑞(𝑡)−𝑊𝑞(𝜃(𝑡))−𝑁(𝐾𝑞)(𝑡)−𝐿(𝑆𝑞)(𝑡) and 𝑞((𝑡𝛽(𝑡){𝑥(𝑡)(𝑡=𝛽(𝑡)(𝑡),𝑥(𝑡)(𝑡)(𝑡),𝛼(𝑡)(𝑡).𝑡,𝑥))=max,min,𝛼)},if𝑥<𝛽,if𝛽≤𝑥≤𝛼,if𝑥>𝛼 (47) If 𝑥∈𝐸 is such that 𝛽≤𝑥≤𝛼 on 𝐽, then 𝑥 is a solution of PBVP (4) if and only if 𝑥 is a solution of (46). We will show that (46) is solvable and that every solution of (46) satisfies 𝛽≤𝑥≤𝛼 on 𝐽. Suppose that 𝑥∈𝐸 is a solution of (46). We will show that 𝛽≤𝑥. Let 𝑝=𝛽−𝑥. Then, we have 𝑝(0)−𝑝(𝑇)=𝛽(0)−𝛽(𝑇) since 𝑥(0)=𝑥(𝑇) and 𝑝^′(𝑡)=𝛽^′(𝑡)−𝑥^′(𝑡)(𝑡)(𝜃(𝑡,(𝑡),(𝑡)+≥𝑓𝑡,𝛽,𝛽))𝐾𝛽𝑆𝛽𝑎_𝛽−(𝑡)(𝜃(𝑡(𝑡)(𝑡)(𝑡)(𝜃(𝑡,(𝑡),(𝑡)(𝑡)(𝜃(𝑡𝑡)𝑡(𝑡)(𝜃(𝑡(𝑡)(𝑡)+𝑀𝑥+𝑊𝑥))+𝑁𝐾𝑥)(+𝐿𝑆𝑥)(+𝑓𝑡,𝑞,𝑞))𝐾𝑞𝑆𝑞−𝑀𝑞−𝑊𝑞))−𝑁𝐾𝑞(−𝐿𝑆𝑞()≥𝑀𝑝+𝑊𝑝))+𝑁𝐾𝑝+𝐿𝑆𝑝𝑎_𝑝(𝑡),𝑡∈𝐽₀.Δ𝑝𝑡_𝑘=Δ𝛽𝑡_𝑘−Δ𝑥𝑡_𝑘≥𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽(𝑠)𝑑𝑠+𝑏_𝛽𝑘−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑞(𝑠,𝑥(𝑠))𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑞(𝑠,𝑥(𝑠))𝑑𝑠≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑝(𝑠)𝑑𝑠+𝑏_𝑝𝑘,𝑘=1,2,…𝑚, (48) By Lemma 3, we get 𝑝(𝑡)≤0 on 𝐽; that is, 𝛽≤𝑥. Similar arguments show that 𝑥≤𝛼.
It remains to prove that (46) possesses at least one solution. By Lemma 6, PBVP (46) is equivalent to the following impulsive integral equation: 𝑥(𝑡)=−^𝑇₀𝐺()×(𝜃(𝑠(𝑠)(𝑠)𝑡,𝑠𝑊𝑥))+𝑁𝐾𝑥)(+𝐿𝑆𝑥)(+𝑔_𝑞(𝑠−)𝑑𝑠_𝑚𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝑒_𝑘,𝑡∈𝐽, (49) where 𝑒_𝑘=𝐼_𝑘(∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑞(𝑠,𝑥(𝑠))𝑑𝑠)−𝐿_𝑘∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑞(𝑠,𝑥(𝑠))𝑑𝑠. For any 𝑥∈𝐸, define a continuous compact operator 𝐹 by (𝐹𝑥)(𝑡)=−^𝑇₀𝐺()×(𝜃(𝑠(𝑠)(𝑠)𝑡,𝑠𝑊𝑥))+𝑁𝐾𝑥)(+𝐿𝑆𝑥)(+𝑔_𝑞(𝑠−)𝑑𝑠_𝑚𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠+𝑒_𝑘,𝑡∈𝐽. (50) Let 𝛿>0, such that |𝛼(𝑡)|<𝛿, |𝛽(𝑡)|<𝛿, 𝑡∈𝐽, and take the compact sets𝐵={(𝑡,𝑦₁,𝑦₂,𝑦₃,𝑦₄)∈𝑅⁵∶𝑡∈𝐽, 𝛽(𝑡)≤𝑦₁≤𝛼(𝑡), 𝛽(𝜃(𝑡))≤𝑦₂≤𝛼(𝜃(𝑡)), (𝐾𝛽)(𝑡)≤𝑦₃≤(𝐾𝛼)(𝑡), and (𝑆𝛽)(𝑡)≤𝑦₄≤(𝑆𝛼)(𝑡)}. Since 𝑓 is continuous, then we can choose a constant 𝜌>0, such that |𝑓(𝑡,𝑦₁,𝑦₂,𝑦₃,𝑦₄)|≤𝜌, (𝑡,𝑦₁,𝑦₂,𝑦₃,𝑦₄)∈𝐵. For 𝜆∈(0,1), we see that any solution of 𝑥=𝜆𝐹𝑥 (51) satisfies ‖𝑥‖_𝑃𝐶=𝜆‖𝐹𝑥‖_𝑃𝐶≤𝜆sup_𝑡∈𝐽^𝑇₀𝐺()×𝑊|𝑥(𝜃(𝑠𝑠𝑠+𝑔𝑡,𝑠))|+𝑁|(𝐾𝑥)()|+𝐿|(𝑆𝑥)()|_𝑞(𝑠)𝑑𝑠+𝜆^𝑚_𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}|𝑥(𝑠)|𝑑𝑠+𝑒_𝑘≤sup_𝑡∈𝐽^𝑇₀𝐺()×𝑡,𝑠𝑊+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑑𝑟+𝐿^𝑇₀ℎ(𝑠,𝑟)𝑑𝑟𝑑𝑠‖𝑥‖_𝑃𝐶+𝜌+𝑀𝛿𝑀+sup^𝑚_{𝑡∈𝐽𝑘=1}𝐺𝑡,𝑡_𝑘×𝐿_𝑘𝜏_𝑘−𝜎_𝑘‖𝑥‖_𝑃𝐶+𝑒_𝑘+sup_𝑡∈𝐽^𝑇₀𝐺()×𝑡,𝑠𝑊+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑑𝑟+𝐿^𝑇₀ℎ(𝑠,𝑟)𝑑𝑟𝑑𝑠𝛿. (52) From the continuity of 𝐼_𝑘, 𝑘=1,2,…,𝑚, and 𝛽(𝑡)≤𝑞(𝑡,𝑥(𝑡))≤𝛼(𝑡) on 𝐽, we can choose some 𝜔>0 such that |𝑒_𝑘|<𝜔, 𝑘=1,2,…,𝑚. Then we have ‖𝑥‖_𝑃𝐶≤11−𝜇𝜌+𝑀𝛿𝑀+sup_𝑡∈𝐽^𝑇₀𝐺()×𝑡,𝑠𝑊+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑑𝑟+𝐿^𝑇₀ℎ()𝛿+𝑠,𝑟𝑑𝑟𝑑𝑠𝜔𝑚1−𝑒^−𝑀𝑇, (53) where 𝜇=sup_𝑡∈𝐽^𝑇₀𝐺()×𝑡,𝑠𝑊+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑑𝑟+𝐿^𝑇₀ℎ()+1𝑠,𝑟𝑑𝑟𝑑𝑠1−𝑒^{𝑚−𝑀𝑇}_𝑘=1𝐿_𝑘𝜏_𝑘−𝜎_𝑘<1. (54) Hence, by Schaefer’s theorem [21], we get that 𝐹 has at least a fixed point 𝑥∈𝐸, which is a solution of (46). Such a solution lies between 𝛽 and 𝛼 and, consequently, is a solution of (4). Thus, the proof is complete.

Theorem 10. Assume that there exist lower and upper solutions for PBVP (4) and assume the following.
(𝐻₃) The function 𝑓∈𝐶(𝐽×𝑅⁴,𝑅) satisfies 𝑓𝑡,𝑧₁,𝑧₂,𝑧₃,𝑧₄−𝑓𝑡,𝑧₁,𝑧₂,𝑧₃,𝑧₄≤𝑀𝑧₁−𝑧₁+𝑊𝑧₂−𝑧₂+𝑁𝑧₃−𝑧₃+𝐿𝑧₄−𝑧₄, (55) where 𝛽(𝑡)≤𝑧₁≤𝑧₁≤𝛼(𝑡), 𝛽(𝜃(𝑡))≤𝑧₂≤𝑧₂≤𝛼(𝜃(𝑡)), (𝐾𝛽)(𝑡)≤𝑧₃≤𝑧₃≤(𝐾𝛼)(𝑡), and (𝑆𝛽)(𝑡)≤𝑧₄≤𝑧₄≤(𝑆𝛼)(𝑡), 𝑡∈𝐽, where 𝑀>0, 𝑊≥0, 𝑁≥0, 𝐿≥0, and 𝜃∈𝐶(𝐽,𝐽).
(𝐻₄) The functions 𝐼_𝑘∈𝐶(𝑅,𝑅) satisfy 𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠−𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑦(𝑠)𝑑𝑠≤𝐿_𝑘∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)−𝑦(𝑠)𝑑𝑠, (56) where ∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑦(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼(𝑠)𝑑𝑠, 𝑘=1,2,…,𝑚, where 𝐿_𝑘≥0 and 0≤𝜎_𝑘≤𝜏_𝑘≤𝑡_𝑘−𝑡_𝑘−1, 𝑘=1,2,…,𝑚.
Suppose that inequalities (12) and (40) hold. Then there exist monotone sequences {𝛼_𝑛}, {𝛽_𝑛} with 𝛼₀=𝛼, 𝛽₀=𝛽, which converge uniformly on 𝐽 to the extremal solutions of the periodic boundary value problem (4) in [𝛽,𝛼].

Proof. For any 𝜂∈[𝛽,𝛼], consider PBVP (35) with 𝑔(𝑡)(𝑡)(𝜃(𝑡,(𝑡),(𝑡)(𝑡)(𝜃(𝑡𝑡)𝑡).=𝑓𝑡,𝜂,𝜂))𝐾𝜂𝑆𝜂−𝑀𝜂−𝑊𝜂))−𝑁𝐾𝜂(−𝐿𝑆𝜂( (57) By Lemmas 6 and 7, PBVP (35) possesses a unique solution 𝑥∈𝐸. We define an operator 𝐴 by 𝑢=𝐴𝜂; then the operator 𝐴 has the following properties:(i)𝛽≤𝐴𝛽, 𝐴𝛼≤𝛼;(ii)𝐴𝜂₁≤𝐴𝜂₂ for any 𝜂₁, 𝜂₂∈[𝛽,𝛼] with 𝜂₁≤𝜂₂.First we prove (i). Let 𝑝=𝛽₀−𝛽₁, where 𝛽₁=𝐴𝛽₀. Then, we have 𝑝(0)−𝑝(𝑇)=𝛽₀(0)−𝛽₀(𝑇) since 𝛽₁(0)=𝛽₁(𝑇) and 𝑝^′(𝑡)=𝛽^′₀(𝑡)−𝛽^′₁(𝑡)(𝑡)(𝜃(𝑡(𝑡)(𝑡)+≥𝑀𝑝+𝑊𝑝))+𝑁𝐾𝑝+𝐿𝑆𝑝𝑎_𝑝(𝑡),𝑡∈𝐽₀,Δ𝑝𝑡_𝑘=Δ𝛽₀𝑡_𝑘−Δ𝛽₁𝑡_𝑘≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑝(𝑠)𝑑𝑠+𝑏_𝑝𝑘,𝑘=1,2,…𝑚. (58) By Lemma 3, we get 𝑝(𝑡)≤0 on 𝐽; that is, 𝛽≤𝐴𝛽. Analogously, we have 𝐴𝛼≤𝛼.
Now, we claim (ii). Set 𝑢₁=𝐴𝜂₁, 𝑢₂=𝐴𝜂₂, where 𝜂₁, 𝜂₂∈[𝛽,𝛼] with 𝜂₁≤𝜂₂. Let 𝑝=𝑢₁−𝑢₂; by (𝐻₃)-(𝐻₄), we have 𝑝^′(𝑡)=𝑢^′₁(𝑡)−𝑢^′₂(𝑡)(𝑡)(𝜃(𝑡(𝑡)(𝑡)≥𝑀𝑝+𝑊𝑝))+𝑁𝐾𝑝+𝐿𝑆𝑝,𝑡∈𝐽₀,Δ𝑝𝑡_𝑘=Δ𝑢₁𝑡_𝑘−Δ𝑢₂𝑡_𝑘=𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑝(𝑠)𝑝(0)(𝑇).𝑑𝑠,𝑘=1,2,…𝑚,=𝑝 (59) By Lemma 3, we have 𝑝(𝑡)≤0 on 𝐽 and so 𝑢₁≤𝑢₂. Thus we may define the sequences {𝛼_𝑛}, {𝛽_𝑛} by 𝛼_𝑛+1=𝐴𝛼_𝑛, 𝛽_𝑛+1=𝐴𝛽_𝑛, 𝛼₀=𝛼, and 𝛽₀=𝛽. From (i), (ii), we obtain 𝛽₀≤𝛽₁≤𝛽₂≤⋯≤𝛽_𝑛≤⋯≤𝛼_𝑛≤⋯≤𝛼₂≤𝛼₁≤𝛼₀on𝐽, (60) and each 𝛼_𝑛, 𝛽_𝑛∈𝐸(∀𝑛∈𝑁) satisfies 𝛼_𝑛(𝑡)=−^𝑇₀𝐺(𝑡,𝑠)𝑊𝛼_𝑛(𝜃(𝑠))+𝑁𝐾𝛼_𝑛(𝑠)+𝐿𝑆𝛼_𝑛(𝑠)+𝑔_𝑛−1(𝑠−)𝑑𝑠_𝑚𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{t𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼_𝑛(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼_𝑛−1(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼_𝑛−1(𝑠)𝛽𝑑𝑠,𝑡∈𝐽,_𝑛(𝑡)=−^𝑇₀𝐺(𝑡,𝑠)𝑊𝛽_𝑛(𝜃(𝑠))+𝑁𝐾𝛽_𝑛(𝑠)+𝐿𝑆𝛽_𝑛(𝑠)+𝑔_𝑛−1(𝑠−)𝑑𝑠_𝑚𝑘=1𝐺𝑡,𝑡_𝑘𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽_𝑛(𝑠)𝑑𝑠+𝐼_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽_𝑛−1(𝑠)𝑑𝑠−𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽_𝑛−1(𝑠)𝑑𝑠,𝑡∈𝐽, (61) where 𝑔_𝑛−1(𝑡)=𝑓𝑡,𝛼_𝑛−1(𝑡),𝛼_𝑛−1(𝜃(𝑡)),𝐾𝛼_𝑛−1(𝑡),𝑆𝛼_𝑛−1(𝑡)−𝑀𝛼_𝑛−1(𝑡)−𝑊𝛼_𝑛−1(𝜃(𝑡))−𝑁𝐾𝛼_𝑛−1(𝑡)−𝐿𝑆𝛼_𝑛−1(𝑡),𝑔_𝑛−1(𝑡)=𝑓𝑡,𝛽_𝑛−1(𝑡),𝛽_𝑛−1(𝜃(𝑡)),𝐾𝛽_𝑛−1(𝑡),𝑆𝛽_𝑛−1(𝑡)−𝑀𝛽_𝑛−1(𝑡)−𝑊𝛽_𝑛−1(𝜃(𝑡))−𝑁𝐾𝛽_𝑛−1(𝑡)−𝐿𝑆𝛽_𝑛−1(𝑡). (62) Hence, there exist 𝜉, 𝜓 such that lim_𝑛→∞𝛼_𝑛(𝑡)=𝜓(𝑡) and lim_𝑛→∞𝛽_𝑛(𝑡)=𝜉(𝑡) uniformly on 𝐽. Clearly, 𝜉, 𝜓 satisfy PBVP (4). We will prove that 𝜉, 𝜓 are extreme solutions of PBVP (4). Let 𝑥(𝑡) be any solution of PBVP (4), which satisfies 𝛽(𝑡)≤𝑥(𝑡)≤𝛼(𝑡), 𝑡∈𝐽. Also suppose there exists a positive integer 𝑛 such that for 𝑡∈𝐽, 𝛽_𝑛(𝑡)≤𝑥(𝑡)≤𝛼_𝑛(𝑡). Setting 𝑝(𝑡)=𝛽_𝑛+1(𝑡)−𝑥(𝑡), then for 𝑡∈𝐽, 𝑝^′(𝑡)=𝛽^′_𝑛+1(𝑡)−𝑥^′(𝑡)≥𝑀𝑝(𝑡)+𝑊𝑝(𝜃(𝑡))+𝑁𝐾𝑝(𝑡)+𝐿𝑆𝑝(𝑡),𝑡∈𝐽₀,Δ𝑝𝑡_𝑘=Δ𝛽_𝑛+1𝑡_𝑘−Δ𝑥𝑡_𝑘≥𝐿_𝑘^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑝(𝑠)(0)(𝑇).𝑑𝑠,𝑘=1,2,…𝑚,𝑝=𝑝 (63) According to Lemma 3, we get that 𝑝(𝑡)≤0 on 𝐽. Similarly, we obtain 𝑥(𝑡)≤𝛼_𝑛+1(𝑡) on 𝐽. Since 𝛽₀≤𝑥(𝑡)≤𝛼₀(𝑡) on 𝐽, by induction, we have 𝛽_𝑛(𝑡)≤𝑥(𝑡)≤𝛼_𝑛(𝑡) on 𝐽 for all 𝑛. Therefore, 𝜉(𝑡)≤𝑥(𝑡)≤𝜓(𝑡) on 𝐽 by taking limit as 𝑛→∞. The proof is complete.

4. An Example

In this section, in order to illustrate our results, we consider an example.

Example 11. Consider the following periodic boundary value problem: 𝑢^′(𝑡)=16𝑡²((𝑡+11+𝑢))116𝑡𝑢2𝑡+136^𝑡₀𝑡𝑠³𝑢(𝑠)𝑑𝑠³+124¹₀𝑡³𝑠𝑢(𝑠)𝑑𝑠²[]1,𝑡∈𝐽=0,1,𝑡≠2,1Δ𝑢2=12^1/4_1/6𝑢(𝑠)𝑑𝑠,𝑘=1,𝑢(0)=𝑢(1), (64) where 𝑘(𝑡,𝑠)=𝑡𝑠³, ℎ(𝑡,𝑠)=𝑡³𝑠, 𝑚=1, 𝑡₁=1/2, 𝜎₁=1/4, 𝜏₁=1/3, and 𝑇=1.
Obviously, 𝛼₀=0, 𝛽₀=−5 are lower and upper solutions for (64), respectively, and 𝛽₀≤𝛼₀.
Let 𝑓𝑡,𝑧₁,𝑧₂,𝑧₃,𝑧₄=16𝑡²1+𝑧₁+116𝑡𝑧₂+1𝑧36³³+1𝑧24²⁴. (65) We have 𝑓𝑡,𝑧₁,𝑧₂,𝑧₃,𝑧₄−𝑓𝑡,𝑧₁,𝑧₂,𝑧₃,𝑧₄≤16𝑧₁−𝑧₁+1𝑧16²−𝑧₂+13𝑧₃−𝑧₃+14𝑧₄−𝑧₄, (66) where 𝛽(𝑡)≤𝑧₁≤𝑧₁≤𝛼(𝑡), 𝛽((1/2)𝑡)≤𝑧₂≤𝑧₂≤𝛼((1/2)𝑡), (𝐾𝛽)(𝑡)≤𝑧₃≤𝑧₃≤(𝐾𝛼)(𝑡), and (𝑆𝛽)(𝑡)≤𝑧₄≤𝑧₄≤(𝑆𝛼)(𝑡), 𝑡∈𝐽. It is easy to see that 𝐼₁^1/4_1/6𝑥(𝑠)𝑑𝑠−𝐼₁^1/4_1/6𝑦(𝑠)=1𝑑𝑠2^1/4_1/6𝑥(𝑠)−𝑦(𝑠)𝑑𝑠, (67) where ∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛽(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑦(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝑥(𝑠)𝑑𝑠≤∫^{𝑡𝑘−𝜎𝑘}_{𝑡𝑘−𝜏𝑘}𝛼(𝑠)𝑑𝑠, 𝑘=1.
Taking 𝐿₁=1/2, 𝑀=1/6, 𝑊=1/16, 𝑁=1/3, and 𝐿=1/4, it follows that 𝐿=𝐿_𝑘𝜏_𝑘−𝜎_𝑘=1,𝑟(𝑡)24=𝑀+𝑊+𝑁^𝑡₀𝑘(𝑡,𝑠)𝑑𝑠+𝐿^𝑇₀ℎ()=1𝑡,𝑠𝑑𝑠6+1+1163^𝑡₀𝑡𝑠³1𝑑𝑠+4¹₀𝑡³𝐴𝑠𝑑𝑠,_𝑘=1+𝐿_𝑘𝜏_𝑘−𝜎_𝑘1=1+213−14=25,𝐵24_𝑘=𝐴_𝑘^{𝑡𝑘−𝜏𝑘}_𝑡𝑘−1𝑟(𝑠)+𝑑𝑠_{𝑡𝑘−𝜎𝑘𝑡𝑘−𝜏𝑘}1+𝐿_𝑘𝑡_𝑘−𝜎_𝑘𝑟(𝑠)+−𝑠𝑑𝑠_{𝑡𝑘𝑡𝑘−𝜎𝑘}𝑟(𝑠)𝑑𝑠,=𝐴_𝑘0^{(1/2)−(1/3)}𝑟(𝑠)+𝑑𝑠_{()−()(1/2)−(1/4)1/21/3}11+212−14𝑟(𝑠)+−𝑠𝑑𝑠_{()−()1/21/21/4}𝑟(𝑠)𝑑𝑠. (68) Thus, 𝐿+^𝑚_𝑘=1𝐵_𝑘+^𝑇₀𝑟(𝑠)𝑑𝑠^𝑚_𝑘=1𝐴_𝑘≈0.4528309326≤1,sup_𝑡∈𝐽^𝑇₀𝐺(𝑡,𝑠)𝑊+𝑁^𝑠₀𝑘(𝑠,𝑟)𝑑𝑟+𝐿^𝑇₀ℎ()+1𝑠,𝑟𝑑𝑟𝑑𝑠1−𝑒^{𝑚−𝑀𝑇}_𝑘=1𝐿_𝑘𝜏_𝑘−𝜎_𝑘=sup_𝑡∈𝐽¹₀𝐺(1𝑡,𝑠)+1163^𝑠₀𝑠𝑟³+1𝑑𝑟4¹₀𝑠³+1𝑟𝑑𝑟𝑑𝑠1−𝑒^−1/61213−14≈0.9256866285<1. (69) Therefore, (64) satisfies all conditions of Theorem 10. So PBVP (64) has minimal and maximal solutions in the segment [𝛽₀,𝛼₀].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand, and King Mongkut’s University of Technology North Bangkok, Thailand.