# Positive Solutions of Three-Order Delayed Periodic Boundary Value Problems

Authors: Na Wang
Journal: Discrete Dynamics in Nature and Society
 (2017)

---

## Abstract

Our main purpose is to consider the existence of positive solutions for three-order two-point boundary value problem in the following form:u′′′(t)+ρ3u(t)=f(t,u(t-τ)),0≤t≤2π,u(i)(0)=u(i)(2π),i=1,2,u(t)=σ,-τ≤t≤0, where σ,ρ, and τ are given constants satisfying τ∈(0,π/2). Some inequality conditions on ρ3u-f(t,u) guaranteeing the existence and nonexistence of positive solutions are presented. Our discussion is based on the fixed point theorem in cones.


---

## Body

## 1. Introduction and Preliminaries

We consider the existence of positive solutions for the following two-point BVP:(1)u′′′t+ρ3ut=ft,ut-τ,0≤t≤2πui0=ui2π,i=1,2,ut=σ,-τ≤t≤0,where ρ3u-f(t,u)∈C([0,2π],[0,+∞)),ρ3u-f(t,u)≥0. σ, ρ, and τ are given constants satisfying τ∈(0,π/2).BVPs play an important role in many branches of mathematics, physics, and engineering and have been a focus in tens of years. They have special importance in the theory and applications, and significant progress has been made on the existence, multiplicity, and nonexistence of positive solutions. Many methods and theorems based on the fixed point theory on cone, the upper and lower solution, Mawhin’s coincidence degree, and variational method and so on have been employed commonly in recent years to show the existence of positive or multiple positive solutions [1–9]. For example, Chu and Zhou [3] studied the boundary value problem:(2)u′′′t+ρ3ut=ft,ut,0≤t≤2πui0=ui2π,i=0,1,2.Here, ρ∈(0,1/3) is a positive constant and the nonlinearity f(t,u) may be singular at u=0. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones. In [5], Cabada investigated the solvability of two-point BVP:(3)Lnut=ft,ut,ui0-ui2π=μi∈R,where Ln is an nth-order linear operator and f is a Carathéodory function. By using the method of lower and upper solutions, the existence results are obtained.As far as the author knows, however, there are no results which contain the existence criteria of positive solutions to problem (1) with delay. Since the time delays in some equations with practical application background are often very small, they are easier to miss. As we know, even a small delay is also likely to have an important impact on the stability of the system. Therefore it is necessary for us to consider the influence of delays. Our work is based on such a background. Motivated by this fact, we will study mainly this problem here. By using fixed point theorem, we obtain some new results. The interest is that the result (Theorem 2) is related to the deviating argument τ. Meanwhile, we give an example to demonstrate our result.Our main results will hinge on an application of the Leggett-Williams fixed point theorem. For the convenience of the reader, we include here the necessary definitions from cone theory in Banach space.Throughout this paper, the sign[] stands for deviation operation; namely, [u]≡u(t-τ). Let 0<d<a be given and suppose that f satisfies(H1)
limu→+∞((ρ3u-ft,u)/u)<1/4π3;(H2)
ρ3u-f(t,[u])<d/4π3 for all 0≤u≤d;(H3)
ρ3u-f(t,[u])>96a/π for all a≤u≤8a;(H2′)
limu→0((ρ3u-ft,u)/u)<1/4π3.Green’s function for BVP (1) is (4)Gt,s=ss-t2π-t4π,0≤s≤t≤2π,tt-s2π-s4π,0≤t≤s≤2π.Suppose that u(t) is a solution of BVP (1); we have (5)ut=∫02πGt,sfs,us-τ-ρ3utds=∫0tst-s2π-t4πρ3ut-fs,us-τds+∫t2πts-t2π-s4πρ3ut-fs,us-τds≥∫0ts2π-s216π3ρ3ut-fs,us-τds+∫t2πs2π-s216π3ρ3ut-fs,us-τds=116π3∫02πs2π-s2ρ3ut-fs,us-τds.Sinceρ3u-f(t,[u])≥0, there exists an interval J=[δ1,δ2]⊂(0,2π) such that mint∈δ1,δ2(ρ3u-ft,u)>0. Thus u(t)>0.Next we state the Leggett-Williams fixed point theorem.Lemma 1 (see [6]).
LetΦ:Pc¯→Pc¯ be a completely continuous operator and let η be a nonnegative continuous concave functional on P such that η(u)≤u for all u∈Pc¯. Suppose that there exist 0<d<a<b≤c such that (a)
{u∈P(η,a,b)∣η(u)>a}≠∅ and η(Φu)>a for u∈P(η,a,b),(b)
Φu<d for u≤d,(c)
η(Φu)>a for u∈P(η,a,c) with Φu>b.Then Φ has at least three fixed points u1,u2, and u3 such that u1<d,a<η(u2) and u3>d with η(u3)<a.


## 2. Main Result

Theorem 2.
Suppose that conditions(H1), (H2), and (H3) hold. Then BVP (1) has at least three positive solutions y1, y2, and y3 such that y1<a, η(y2)>d, y3>a, and η(y3)<a.Proof.
LetZ denote the Banach space C[0,2π] with the norm ∗u0,2π=sup⁡ut:0≤t≤2πand define the cone P⊂Z by(6)P=u∈Z:ut≥0.Let η:P→[0,+∞) be the nonnegative continuous concave functional (7)ηu=minπ/4≤t≤π/2ut,∀u∈P,and let Φ:P→P be the operator∗∗Φut=∫02πGt,sfs,us-τ-ρ3usds.
It is easy to see thatΦ:P8a¯→P¯(b=8a) is completely continuous and u(t)≡(5a/2)(>a) is an element of P(η,a,8a). Since η(u)=η(5a/2)>a, we have {u∣u∈P(η,a,8a),η(u)>a}≠∅.
Chooseu∈P(η,a,8a) and τ∈(0,π/4)∪(π/4,π/2); then (8)ηΦu=minπ/4≤t≤π/2∫02πGt,sfs,us-τ-ρ3usds≥minπ/4≤t≤π/2∫02πs2π-s216π3ρ3us-fs,us-τds≥6aπ4∫π/4+τπ/2+τs2π-s2ds=6aπ42π2s2-43πs3+14s4π/4+τπ/2+τ>6aπ41772π4+2364π3τ+π4τ3-2332π2τ2>a.From (H1), there exist two positive constants λ and M with 0<λ<1/4π3 such that (9)ρ3u-ft,u≤λu,u>M.If we take L=max0≤u≤M(ρ3u-ft,u), then (10)0≤ρ3u-ft,u≤λu+L,0≤u≤+∞.Let r>max⁡{4π3L/(1-4π3λ),8a} and u∈Pr¯(b=8a,c=r); from (10) we get (11)Φu=max0≤t≤2π∫02πGt,sfs,us-τ-ρ3usds≤4π3λu+L≤4π3λr+L<r.So Φu∈Pr and Φ:Pr¯→Pr¯ is completely continuous. From (8), condition (a) of Leggett-Williams theorem is satisfied.
Chooseu∈Pd¯; then (12)Φu=max0≤t≤2π∫02πGt,sfs,us-τ-ρ3usds<d4π3max0≤t≤2π∫0tst-s2π-t4πds+∫t2πts-t2π-s4πds<d4π3∫0t8π34πds+∫t2π8π34πds=d.Therefore, condition (b) is satisfied.
Choosey∈P8a¯ and Φy≠0; then (13)ηΦu=minπ/4≤t≤π/2∫02πGt,sfs,us-τ-ρ3usds=minπ/4≤t≤π/2∫0tst-s2π-t4πρ3us-fs,us-τds+∫t2πts-t2π-s4πρ3us-fs,us-τds=minπ/4≤t≤π/2∫0t2π-t2π-s·st-s2π-t4πρ3us-fs,us-τds+∫t2πts·ss-t2π-s4πρ3us-fs,us-τds>2π-π/22π∫0tst-s2π-t4πρ3us-fs,us-τds+π/42π∫t2πss-t2π-s4πρ3us-fs,us-τds>18Φu.So, if Φu>8a, then η(Φu)>(1/8)Φu>a, which implies that condition (c) is also satisfied. An application of Leggett-Williams fixed point theorem yields the result.Theorem 3.
Suppose that conditions(H1), (H2′), and (H3) hold; then BVP (1) has at least three positive solutions y1,y2, and y3 such that y1<a, η(y2)>d, y3>a, and η(y3)<a.The proof of Theorem3 is similar to that of Theorem 2 and is hence omitted.Remark 4.
The extensions to Theorem2 can be obtained by Theorem 5, where 1/4π3 of (H1) and (H2) is replaced by δ and γ, respectively. We omit the detail.Theorem 5.
Suppose that conditions(H1∗)  limu→+∞((ρ3u-ft,u)/u)<δ,(H2∗)  limu→0((ρ3u-ft,u)/u)<γ,and(H3) hold.Then BVP (1) has at least three positive solutions; here δ,γ<1/4π3.Now we present a result on the nonexistence of positive solutions of BVP (1).Theorem 6.
BVP (1) has no positive solutions if (i)
limu→+∞inft∈0,2π((ρ3u-ft,u)/u)>1/4π3;(ii)
limu→+∞supt∈0,2π(ρ3u-ft,u/u)<1/4π3;(iii)
(ρ3u-ft,u)/u>1/4π3; or(iv)
(ρ3u-ft,u)/u<1/4π3,where u is defined by ∗.Proof.
(a) From condition (i), there exist two positive constantsλ3 and M3 with 0<λ3<1/4π3 such that (14)ρ3ut-ft,u>λ3u,u>M3.If we take L1∗=min0≤u≤M3(ρ3u(t)-f(t,[u])), then (15)ρ3ut-ft,u>λ3u+L1∗,0≤u≤+∞.Assume that BVP (1) has a positive solution u with u=r. Then u is a fixed point of the map Φ defined by ∗∗ and hence (16)r=Φu=max0≤t≤2π∫02πGt,sfs,us-τ-ρ3utds>4π3λ3u+L1∗=4π3λ3r+L1∗>r,which is a contradiction.
(b) From condition (ii), there exist two positive constantsλ4 and M4 with 0<λ4<1/4π3 such that (17)ρ3ut-ft,u<λ4u,u>M4.If we take L2∗=min0≤u≤M3(ρ3u(t)-f(t,[u])), then (18)ρ3ut-ft,u<λ4u+L2∗,0≤u≤+∞.Assume that BVP (1) has a positive solution u with u=r. Then u is a fixed point of the map Φ defined by ∗∗ and hence (19)r=Φu=max0≤t≤2π∫02πGt,sfs,us-τ-ρ3utds>4π3λ4u+L2∗=4π3λ4r+L2∗<r,which is a contradiction.
(c) Assume that BVP (1) has a positive solution u with u=r. Then u is a fixed point of the map Φ and hence (20)r=Φu=max0≤t≤2π∫02πGt,sfs,us-τ-ρ3utds>4π3·14π3u=r,which is a contradiction.
(d) The proof of case (iii) is similar to that of case (iv) and is hence omitted.Remark 7.
The extensions to Theorem6 can be obtained by Theorem 8, where 1/4π3 of (i)–(iv) is replaced by α,β,θ, and ν, respectively. Here α,β,θ,ν<1/4π3. We omit the detail.Theorem 8.
BVP (1) has no positive solutions if (i)
limu→+∞inft∈0,2π((ρ3u-ft,u)/u)>α;(ii)
limu→+∞supt∈0,2π((ρ3u-ft,u)/u)<β;(iii)
(ρ3u-ft,u)/u>θ; or(iv)
(ρ3u-ft,u)/u<ν,where u is defined by ∗.Example 9.
Let us consider the following equation:(21)u′′′t+ρ3ut=ft,ut-π8,0≤t≤2πui0=ui2π,i=1,2,u0=0,-π8≤t≤0,where (22)ρ3ut-ft,ut-π8=eut-π/832π3,0≤t≤2π,0≤u≤12;6ut-π8-3,0≤t≤2π,12≤u≤1;π2eut-π/8+u2t+17,0≤t≤2π,1≤u≤8;ut-π/85π30≤t≤2π,u>8.Let d=1/2, a=1, c=8, and r>8; then we have ρ3u(t)-f(t,u(t-π/8))∈C([0,2π],[0,+∞)), ρ3u-f(t,[u])≥0, and ρ3u(t)-f(t,u(t-π/8))<1/8π3 for all 0≤u≤1/2, ρ3u(t)-f(t,u(t-π/8))>96/π for all 1≤u≤8, and limu→+∞((ρ3u-ft,u)/u)<1/4π3. One can easily see that assumptions (H1),(H2), and (H3) hold. So by applying Theorem 2, BVP (21) has at least three positive solutions u1,u2, and u3 and satisfied u1<1/2, η(u2)>8, u3>1/2, and η(u3)<8.Example 10.
Let us consider the following equation:(23)u′′′t+ρ3ut=ft,ut-τ,0≤t≤2πui0=ui2π,i=1,2,u0=1,-π3≤t≤0.(1) Let ρ3u(t)-f(t,u(t-τ))=4u(t)+u(t-π/3); it is clear that (24)limu→+∞inft∈0,2πρ3u-ft,uu>14π3.Therefore, by applying Theorem 6(i), BVP (23) has no positive solutions.
(2) Letρ3u(t)-f(t,u(t-τ))=(1/8π3)(u(t)+u(t-π/3)); it is clear that (25)limu→+∞supt∈0,2πρ3u-ft,uu<14π3.Therefore, by applying Theorem 6(ii), BVP (23) has no positive solutions.The numerical results of Example10 by Mathematica 7.0 are in the following:(26)In18≔sol=NDSolvex′′′t==-4xt-xt-Pi/3,xt/;t≤0==1,x′0==x′2π,x′′0==x′′2π,x,t,0,2πFunction::slotn:Slotnumber5in#3,#4,-4#2-#5&cannotbefilledfrom#3,#4,-4#2-#5&0.,0.,0.,0..≫NDSolve::ndnum:Encounterednon-numericalvalueforaderivativeatt==0∴≫Out18=NDSolvex3t==-4xt-x-π3+t,xt/;t≤0==1,x′0==x′2π,x′′0==x′′2π,x,t,0,2πIn19≔sol=NDSolvex′′′t==-xt-xt-Pi/3/8π3,xt/;t≤0==1,x′0==x′2π,x′′0==x′′2π,x,t,0,2πFunction::slotn:Slotnumber5in#3,#4,-#2-#58π3&cannotbefilledfrom#3,#4,-#2-#58π3&0.,0.,0.,0..≫NDSolve::ndnum:Encounterednon-numericalvalueforaderivativeatt==0∴≫Out19=NDSolvex3t==-xt-x-π/3+t8π3,xt;t≤0==1,x′0==x′2π,x′′0==x′′2π,x,t,0,2π.Remark 11.
From the above example, we see that the result is related to the deviating argumentτ, which is different from the Theorem in papers [1–3, 5, 6, 10] and the references therein. The studies indicate that this kind of system with time delays can exhibit triple positive solutions, which shows that three-order two-point boundary value problems have the potential to reproduce the complex dynamics of real applied background in mechanics, engineering, physics, and so on.Remark 12.
The above problem can be extended to the corresponding vector case. For example,(27)εu′′′t+μρ3ut=ft,ut-τ,0≤t≤2π,ui0=ui2π,i=1,2,ut=σ∗,-τ≤t≤0,where u(t)∈C(R,Rn), f(t,u(t-τ))∈C(R×Rn,Rn), σ∗∈Rn, and ε and μ are two given parameters. However, if ε≪μρ3, such equation is a singular perturbation system; it is hard to obtain the positive solution of (27) by using the fixed point theorem.



---

*Source: 4143267-2017-10-04.xml*