# Iteration for a Third-Order Three-Point BVP with Sign-Changing Green's Function

Authors: Ya-Hong Zhao; Xing-Long Li
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We are concerned with the following third-order three-point boundary value problem:u‴(t)=f(t,u(t)),t∈[0,1],u′(0)=u(1)=0,u″(η)+αu(0)=0, where α∈[0,2) and η∈[2/3,1). Although corresponding Green's function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions on f by applying iterative method. An example is also included to illustrate the main results obtained.


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## Body

## 1. Introduction

Third-order differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity driven flows, and so on [1].Recently, the existence of single or multiple positive solutions to some third-order three-point boundary value problems (BVPs) has received much attention from many authors; see [2–7] and the references therein. However, all the above-mentioned papers are achieved when corresponding Green’s functions are nonnegative, which is a very important condition. A natural question is that whether we can obtain the existence of positive solutions to some third-order three-point BVPs when corresponding Green’s functions are sign-changing.In 2008, Palamides and Smyrlis [8] studied the existence of at least one positive solution to the singular third-order three-point BVP with an indefinitely signed Green’s function
(1)u′′′(t)=a(t)f(t,u(t)),t∈(0,1),u(0)=u(1)=u′′(η)=0,
where η∈(17/24,1). Their technique was a combination of the Guo-Krasnoselskii fixed point theorem [9, 10] and properties of the corresponding vector field.Very recently, for the third-order three-point BVP with sign-changing Green’s function(2)u′′′(t)=f(t,u(t)),t∈[0,1],u′(0)=u(1)=u′′(η)=0,
Sun and Zhao proved the existence of single or multiple positive solutions when η∈(1/2,1) by using the Guo-Krasnoselskii and Leggett-Williams fixed point theorems in [11, 12] and obtained the existence of a positive solution when η∈[2-2,1) by using iterative technique in [13].In 2013, Li et al. [14] established the existence of at least one positive solution to the following third-order three-point BVP with sign-changing Green’s function:
(3)u′′′(t)=f(t,u(t)),t∈[0,1],u′(0)=u(1)=0,u′′(η)+αu(0)=0,
where α∈[0,2) and η∈[(121+24α-5)/3(4+α),1). The main tool used was the Guo-Krasnoselskii fixed point theorem [9, 10].It is worth mentioning that there are other types of works on sign-changing Green’s functions which prove the existence of sign-changing solutions, positive in some cases; see Infante and Webb’s papers [15–17].Motivated greatly by the above-mentioned works, in this paper, we will study the BVP (3) by applying iterative method. Throughout this paper, we always assume that α∈[0,2) and η∈[2/3,1). Although corresponding Green’s function is sign-changing, we still obtain the existence of monotone positive solution for the BVP (3) under some suitable conditions on f. Moreover, our iterative scheme starts off with zero function, which implies that the iterative scheme is feasible.


## 2. Main Results

For anyy∈C[0,1], we consider the BVP
(4)u′′′(t)=y(t),t∈[0,1],u′(0)=u(1)=0,u′′(η)+αu(0)=0.It follows from [14] that the expression of Green’s function G(t,s) of the BVP (4) is as follows:
(5)G(t,s)=g1(t,s)+g2(t,s)+g3(η,t,s),
where
(6)g1(t,s)=-(2-αt2)(1-s)22(2-α),(t,s)∈[0,1]×[0,1],g2(t,s)={0,0≤t≤s≤1,(t-s)22,0≤s≤t≤1,g3(η,t,s)={0,s≥η,1-t22-α,s<η,
and the G(t,s) has the following properties:
(7)G(t,s)≥0for0≤s<η,G(t,s)≤0forη≤s≤1.
Moreover, for s≥η,
(8)max⁡{G(t,s):t∈[0,1]}=G(1,s)=0,min⁡{G(t,s):t∈[0,1]}=G(0,s)=-(1-s)22-α≥-(1-η)22-α,
and, for s<η,
(9)max⁡{G(t,s):t∈[0,1]}=G(0,s)=2s-s22-α≤2η-η22-α,min⁡{G(t,s):t∈[0,1]}=G(1,s)=0.So, if we letM=max⁡{|G(t,s)|:t,s∈[0,1]}, then
(10)M=max⁡{(1-η)22-α,2η-η22-α}<12-α.Let Banach spaceE=C[0,1] be equipped with the norm ∥u∥=max⁡t∈[0,1]⁡|u(t)|⁡ and
(11)K={y∈E:y(t)hhhisnonnegativeanddecreasingon⁡[0,1]}.
Then K is a cone in E. Note that this induces an order relation “≲” in E by defining u≲v if and only if v-u∈K.In the remainder of this paper, we always assume thatf∈C([0,1]×[0,+∞),[0,+∞)) and satisfies the following two conditions:(H1)
for eachu∈[0,+∞), the mapping t↦f(t,u) is decreasing;(H2)
for eacht∈[0,1], the mapping u↦f(t,u) is increasing.Now, we define an operatorT as follows:
(12)(Tu)(t)=∫01‍G(t,s)f(s,u(s))ds,u∈K,t∈[0,1].
Obviously, if u is a fixed point of T in K, then u is a nonnegative and decreasing solution of the BVP (3).Lemma 1.
T
:
K
→
K is completely continuous.Proof.
Letu∈K. Then, for t∈[0,η], we have
(13)(Tu)(t)=∫0t‍[g1(t,s)+(t-s)22+1-t22-α]f(s,u(s))ds+∫tη‍[g1(t,s)+1-t22-α]f(s,u(s))ds+∫η1‍g1(t,s)f(s,u(s))ds,
which together with (H1) and (H2) implies that
(14)(Tu)′(t)=αt2-α∫0η‍(s2-2s)f(s,u(s))ds-∫0t‍sf(s,u(s))ds-t∫tη‍f(s,u(s))ds+αt2-α∫η1‍(1-s)2f(s,u(s))ds≤f(η,u(η))[αt2-α∫0η‍(s2-2s)ds-∫0t‍sds-t∫tη‍ds+αt2-α∫η1‍(1-s)2ds]=tf(η,u(η))[α(1-3η)3(2-α)-η+t2]≤tf(η,u(η))[α(1-3η)3(2-α)-η2]≤0.
For t∈[η,1], we have
(15)(Tu)(t)=∫0η‍[g1(t,s)+(t-s)22+1-t22-α]f(s,u(s))ds+∫ηt‍[g1(t,s)+(t-s)22]f(s,u(s))ds+∫t1‍g1(t,s)f(s,u(s))ds,
which together with (H1) and (H2) shows that
(16)(Tu)′(t)=αt2-α∫0η‍(s2-2s)f(s,u(s))ds+∫ηt‍(t-s)f(s,u(s))ds-∫0η‍sf(s,u(s))ds+αt2-α∫η1‍(1-s)2f(s,u(s))ds≤f(η,u(η))[αt2-α∫0η‍(s2-2s)ds+∫ηt‍(t-s)dshhhhhhhhhhhh-∫0η‍sds+αt2-α∫η1‍(1-s)2ds]=tf(η,u(η))[α(1-3η)3(2-α)+t-2η2]≤tf(η,u(η))[α(1-3η)3(2-α)+1-2η2]≤0.
So,(Tu)(t) is decreasing on [0,1]. At the same time, since (Tu)(1)=0, we know that (Tu)(t) is nonnegative on [0,1]. This indicates that Tu∈K. Furthermore, although G(t,s) is not continuous, it follows from known textbook results, for example, see [18], that T:K→K is completely continuous.Theorem 2.
Assume thatf(t,0)≢0 for t∈[0,1] and there exist two positive constants a and b such that the following conditions are satisfied:(H3)
f(0,a)≤(2-α)a;(H4)
b(u2-u1)≤f(t,u2)-f(t,u1)≤2b(u2-u1), 0≤t≤1, 0≤u1≤u2≤a.
If we construct an iterative sequencevn+1=Tvn, n=0,1,2,…, where v0(t)≡0 for t∈[0,1], then {vn}n=1∞ converges to v* in E and v* is a decreasing and positive solution of the BVP (3).Proof.
LetKa={u∈K:∥u∥≤a}. Then we may assert that T:Ka→Ka.
In fact, ifu∈Ka, then it follows from (H3) that
(17)0≤(Tu)(t)=∫01‍G(t,s)f(s,u(s))ds≤∫01‍|G(t,s)|f(0,a)ds≤(2-α)aM<a,t∈[0,1],
which shows that ∥Tu∥≤a. So, T:Ka→Ka.
Now, we prove that{vn}n=1∞ converges to v* in E and v* is a decreasing and positive solution of the BVP (3).
Indeed, in view ofv0∈Ka and T:Ka→Ka, we have vn∈Ka, n=1,2,…. Since the set {vn}n=0∞ is bounded and T is completely continuous, we know that the set {vn}n=1∞ is relatively compact. In what follows, we prove that {vn}n=0∞ is monotone by induction. First, it is obvious that v1-v0=v1∈K, which shows that v0≲v1. Next, we assume that vk-1≲vk. Then vk-vk-1 is decreasing and 0≤vk-1(t)≤vk(t)≤a, t∈[0,1]. So, it follows from (H4) that
(18)vk+1′(t)-vk′(t)=αt2-α∫0η‍(s2-2s)[f(s,vk(s))-f(s,vk-1(s))]ds-∫0t‍s[f(s,vk(s))-f(s,vk-1(s))]ds-t∫tη‍[f(s,vk(s))-f(s,vk-1(s))]ds+αt2-α∫η1‍(1-s)2[f(s,vk(s))-f(s,vk-1(s))]ds≤bαt2-α∫0η‍(s2-2s)[vk(s)-vk-1(s)]ds-b∫0t‍s[vk(s)-vk-1(s)]ds-bt∫tη‍[vk(s)-vk-1(s)]ds+2bαt2-α∫η1‍(1-s)2[vk(s)-vk-1(s)]ds≤b[vk(η)-vk-1(η)]×{αt2-α∫0η‍(s2-2s)dshhhhh-∫0t‍sds-t∫tη‍ds+2αt2-α∫η1‍(1-s)2ds}=b[vk(η)-vk-1(η)]t×[α(-η3+3η2-6η+2)3(2-α)-η+t2]≤b[vk(η)-vk-1(η)]t[α(-η3+3η2-6η+2)3(2-α)-η2]≤b[vk(η)-vk-1(η)]t[α(-3η+2)3(2-α)-η2]≤0,t∈[0,η],vk+1′(t)-vk′(t)=αt2-α∫0η‍(s2-2s)[f(s,vk(s))-f(s,vk-1(s))]ds+∫ηt‍(t-s)[f(s,vk(s))-f(s,vk-1(s))]ds-∫0η‍s[f(s,vk(s))-f(s,vk-1(s))]ds+αt2-α∫η1‍(1-s)2[f(s,vk(s))-f(s,vk-1(s))]ds≤bαt2-α∫0η‍(s2-2s)[vk(s)-vk-1(s)]ds+2b∫ηt‍(t-s)[vk(s)-vk-1(s)]ds-b∫0η‍s[vk(s)-vk-1(s)]ds+2bαt2-α∫η1‍(1-s)2[vk(s)-vk-1(s)]ds≤b[vk(η)-vk-1(η)]×[αt2-α∫0η‍(s2-2s)ds+2∫ηt‍(t-s)dshhhhhh-∫0η‍sds+2αt2-α∫η1‍(1-s)2ds]=b[vk(η)-vk-1(η)]×[αt(-η3+3η2-6η+2)3(2-α)+t2-2ηt+η22]≤b[vk(η)-vk-1(η)]t×[α(-η3+3η2-6η+2)3(2-α)+t-3η2]≤b[vk(η)-vk-1(η)]t[α(-3η+2)3(2-α)+2-3η2]≤0,t∈[η,1].
So,
(19)vk+1′(t)-vk′(t)≤0,t∈[0,1].
This together with vk+1(1)-vk(1)=∫01‍G(1,s)[f(s,vk(s))-f(s,vk-1(s))]ds=0 implies that
(20)vk+1(t)-vk(t)≥0,t∈[0,1].
In view of (19) and (20), we know that vk+1-vk∈K, which indicates that vk≲vk+1. Thus, we have shown that vn≲vn+1, n=0,1,2…. Since {vn}n=1∞ is relatively compact and monotone, there exists v*∈Ka such that lim⁡n→∞vn=v*, which together with the continuity of T and the fact that vn+1=Tvn implies that v*=Tv*. This indicates that v* is a decreasing and nonnegative solution of the BVP (3). Moreover, since f(t,0)≢0 for t∈[0,1], we know that zero function is not a solution of the BVP (3), which shows that v* is a positive solution of the BVP (3).


## 3. An Example

Consider the BVP(21)u′′′(t)=116u2(t)+14u(t)+(1-t),t∈[0,1],u′(0)=u(1)=0,u′′(η)+u(0)=0.If we letα=1, η=7/9, and f(t,u)=(1/16)u2+(1/4)u+(1-t), (t,u)∈[0,1]×[0,+∞), then all the hypotheses of Theorem 2 are fulfilled with a=2 and b=1/4. Therefore, it follows from Theorem 2 that the BVP (21) has a decreasing and positive solution. Moreover, the iterative scheme is v0(t)≡0 for t∈[0,1] and
(22)vn+1(t)={∫0t‍[-(2-t2)(1-s)22+(t-s)22+1-t2]×[116(vn(s))2+14vn(s)+(1-s)]ds+∫t7/9‍[-(2-t2)(1-s)22+1-t2]×[116(vn(s))2+14vn(s)+(1-s)]ds+∫7/91‍[-(2-t2)(1-s)22]×[116(vn(s))2+14vn(s)+(1-s)]ds,hhhhhhhhhhhht∈[0,79],n=0,1,2,…,∫07/9‍[-(2-t2)(1-s)22+(t-s)22+1-t2]×[116(vn(s))2+14vn(s)+(1-s)]ds+∫7/9t‍[-(2-t2)(1-s)22+(t-s)22]×[116(vn(s))2+14vn(s)+(1-s)]ds+∫t1‍[-(2-t2)(1-s)22]×[116(vn(s))2+14vn(s)+(1-s)]ds,hhhhhhhhhhhht∈[79,1],n=0,1,2,….
The first, second, third, and fourth terms of this scheme are as follows:
(3)v0(t)≡0,v1(t)=-124t4+16t3-227648t2+73324,v2(t)=155296t11-35165888t10+839746496t9-74892239488t8+13789340310784t7+1415521120932352t6-48187110077696t5+16264910077696t4+198720110077696t3-4814651119744t2+9514093779136,v3(t)=1293534171136t25-110871635968t24+93417925422620672t23-249492641807540224t22+184270335664401793024t21-20695715106993205379072t20+12614066172888816545234944t19-131313793962938848411648t18-463466345483155996093442686976t17+202016229531925877696823296t16-4124079293903467988280328060928t15-2138312338076994211894522952548352t14+18909403546379749755840167936t13-86168085379277526486815369068544t12-1448853654211119499511680335872t11+436989616007405175495605123022848t10-4235986460912329249267520503808t9-2156592823827731263243407684534272t8+1865260689801799749755840167936t7+413727883674101263243407684534272t6-13158305753867203119913336832t5+43268361481197710968475320188928t4+92539330261825457019805007872t3-18302851636706094113178245070848t2+11486059167752914503599627370496.


## 4. Conclusion

In [14], only the existence of at least one positive solution to the BVP (3) was obtained when α∈[0,2) and η∈[(121+24α-5)/3(4+α),1). In this paper, when α∈[0,2) and η∈[2/3,1), we have successfully constructed an iterative sequence, whose limit is just a desired monotone positive solution of the BVP (3). Moreover, our iterative scheme starts off with zero function, which implies that the iterative scheme is feasible.



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