# Existence and Uniqueness of Solutions for a Third-Order Three-Point Boundary Value Problem via Measure of Noncompactness

Authors: Chen Yang
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we consider a nonlinear third-order three-point boundary value problem and give the existence and uniqueness of solutions by constructing Green’s function and using its properties. The methods used here are based on Darbo’s fixed point theorem combined with the technique of measure of noncompactness. Finally, as applications, two examples are given to illustrate our main results.


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## Body

## 1. Introduction

Multipoint boundary value problem (BVP) for the third- and higher-order differential equations plays an important part in various fields, such as fluid mechanics, physics, engineering, and many other branches of applied mathematics (see [1–14]). In the past decades, third-order three-point boundary value problems have been widely investigated. In particular, Anderson in [1] considered a right focal problem(1)x3t=0,t1≤t≤t3,xt1=x′t2=0,γxt3+δx″t3=0,where γ≥0, δ>0, t1<t2<t3 are real numbers and t2−t1>t3−t2, and the Krasnoselskii, Leggett, and Williams fixed point theorems were used to prove the existence of at least three solutions of (1).In [2], the authors solved the existence of the solutions of the BVP(2)x3t=ft,xt,x′t,x″t+et,0≤t≤1,x0=∑i=1m−2αixξi,x′0=0,x1=βxη,where f:0,1×R3⟶R is a continuous function, e∈L10,1, αi∈R1≤i≤m−2, β≥0, 0<ξ1<ξ2<⋯<ξm−2<1, η∈0,1. The coincidence degree theory of Mawhin is the fundamental tool to deduce the existence results.In 2019, the authors in [3] studied the existence and uniqueness of solutions of the BVP(3)x3t+ft,xt=0,a≤t≤b,xa=x′a=0,xb=kxη,where η∈a,b, k∈R, f:Ca,b⟶R, ft,0≠0, and its primary tools are contracting mapping theorem and variation of parameters formula; namely, it first deals with(4)u3t+ht=0,a≤t≤b,ua=u′a=0,ub=0.And then, the solution of BVP (3) can be expressed as(5)xt=ut+λ0+λ1t+λ2t2uη,where λ0,λ1,λ2 are constants determined by its boundary value conditions.Motivated greatly by above-mentioned works, in this paper, we consider the existence and uniqueness of solutions to the following BVP(6)x3t+ft,xt=0,a≤t≤b,xb=x″a=0,xa=kxη,where η∈b−b−a/k,b, k>1, f:a,b×R⟶R is continuous and ft,0≠0. The methods used are a measure of noncompactness and Darbo’s fixed point theorem, which prove the existence of solutions, and at the same time, its uniqueness also holds by Banach contraction principle, which is different from [3]. A new direction of research with the presentation of the notion of a measure of noncompactness was opened up in 1930 by Kuratowski, which can be applied to prove the existence results related to various integral, differential equations, integro-differential equations as well as their systems. As an important application of this measure, Darbo’s fixed point theorem generalizes the Schauder-fixed point theorem and Banach contraction principle, especially in proving the existence of solutions for classes of nonlinear equations (see[15–30]). Comparing with other papers, our advantage is that the solution we get is in a small sphere rather than in the whole space.The rest of the paper is organized as follows. In Section2, we give the noncompactness, Darbo’s theorem, and related notations. In Section 3, the corresponding Green’s function and its properties are given. In Section 4, we prove our main results on the existence and uniqueness for the solutions and give some examples to verify our results’ effectiveness and applicability.


## 2. Preliminaries

We start this section by introducing some necessary definitions, notations, and basic results required for further development.LetX be subset of a metric space E,d, we note X¯ and coX are closure and convex of X, respectively; ME and NE, respectively, denote the family of all nonempty and bounded subsets, and nonempty and relatively compact subsets of the metric space E,d. Then,(1)
finite setN⊂X is a finite ϵ− net of X if and only if for any ϵ>0, any x∈X; there exists y∈N, such that(7)dx,y<ε.(2)
X is totally bounded if and only if it has a finite ϵ− net.(3)
X is relatively compact if and only if any sequence of X has a convergent subsequence.(4)
X is compact if and only if any sequence of X has a convergent subsequence, and its limit is in M. M is compact if M is relatively compact and closed.Definition 1.
(see [31]). The Kuratowski measure of noncompactness α⋅ defined on bounded set S of Banach space E is(8)αS=infε>0:S=∪i=1mSi,Si⊂E,diamSi<εi=1,2⋯m,m∈N,where diamSi denotes the diameter of the set Si, that is,(9)diamSi=supdx,y:x,y∈Si.Definition 2.
(see [32]). The Hausdorff measure of noncompactness χ⋅ defined on bounded set S of Banach space E is(10)χS=infε>0:S⊂∪i=1mBxi,ri,xi∈S,ri<εi=1,2⋯m,m∈N,where(11)Bxi,ri=y∈X,dy,xi<ri.
The definition of the Hausdorff measure of noncompactness for the setS can equivalently be stated as follows:(12)χS=infε>0:Shas a finiteε−net inE.Definition 3.
(see [32]). The Istraˇtescu measure of noncompactness β⋅ defined on bounded set S of Banach space E is(13)βS=infε>0:Shas no infiniteε−discrete subsets.
The definition of theIstraˇtescu measure of noncompactness for the set S can also equivalently be stated as follows:(14)βS=supε>0:Scontains an infiniteε−discrete set.
Almost all known measures of noncompactness possess the property that they are equal to zero on the family of all relatively compact sets in a given space.Lemma 1.
(see [32]). Let E be a Banach space, S and T be bounded subsets of E, a∈R. Then,(1)
αS=0 if and only if S is a relatively compact(2)
the family kerμ=Y∈ME:μY=0 is nonempty and ker μ⊂NE(3)
S⊂T⇒αS≤αT(4)
αS∪T=maxαS,αT(5)
αS+T≤αS+αT, where S+T=x:x=y+z,y∈S,z∈T(6)
αaS=aαS, where aS=x:x=ay,y∈S(7)
co¯S=αS¯=αS(8)
αS+x=αS, for any x∈E, where S+x=z=s+x:s∈S(9)
αS−αT≤2dhS,TwheredhS,T denotes the distance of Hausdorff between S and T; i.e.,(15)dhS,T=maxsupx∈Sdx,T,supx∈Tdx,S.(10)
Xn⊂ME, Xn¯=Xn, Xn+1⊂Xn for n=1,2,…, if limn⟶∞αXn=0; then, X∞=∩i=1∞Xi≠∅
In this article, we work on infinite spaceCa,b, which denotes the space of all continuous functions defined on the interval a,b. For x∈Ca,b, we define the usual norm as follows:(16)x=maxxt:t∈a,b.
It is well known thatCa,b,⋅ is complete.
The measure of noncompactness inCa,b can be formulated as follows ([33]).
LetX be a nonempty and bounded subset of the space Ca,b and for x∈X, ε>0, T>0, let(17)ωTx,ε=supxt−xs:t,s∈0,T,t−s<ε,ωTX,ε=supωTx,ε:x∈X,ω0TX=limε⟶0ωTX,ε,ω0X=limT⟶∞ω0TX,Xt=xt:x∈X,t∈R,and(18)αX=ω0X+limsupt⟶∞diamXt,where(19)diamXt=supxt−yt:x,y∈X.Lemma 2 (Agarwal and O’Regan [34]).
LetZ be a closed, convex subset of a Banach space E. Then, every compact, continuous map T:Z⟶Z has at least one fixed point.
Using the measure of noncompactness, Darbo generalized Lemma2, namely, Darbo’s fixed point theorem.Lemma 3 (Darbo [35]).
LetZ be a nonempty, bounded, closed, and convex subset of a Banach space E and T:Z⟶Z be a continuous mapping. Assume that there is a constant ζ∈0,1 such that(20)αTM≤ζαM,M⊂Z.
Then,T has a fixed point.Lemma 4 (Ahmad et al. [36]).
LetZ be a closed subset of Banach space E and T:Z⟶Z be a strictly contractive operator; i.e., there exists a constant γ∈0,1 such that(21)Tu−Tv≤γu−v,∀u,v∈Z.
Then,T has a unique fixed point.


## 3. The Construction of Green’s Function

First of all, let us construct the Green’s function for the BVP(22)x3t+ht=0,a≤t≤b,xb=x″a=0,xa=kxη,where η∈b−b−a/k,b, k>1, h∈Ca,b.Lemma 5.
The above BVP (22) has the solution(23)xt=k2m∫aηt−bη−s2hsds+∫abGt,shsds,where(24)m=1−kb−a−kη,and(25)Gt,s=12m1−kt−a−kηb−s2−mt−s2,a≤s≤t≤b,1−kt−a−kηb−s2,a≤t≤s≤b.Proof.
We easily know thatxt=c1+c2t+c3t2−1/2∫att−s2hsds, where c1, c2, c3 are constants. As a result,(26)x″t=2c3−∫athsds,by using the boundary condition x″a=0, which yields that c3=0, and thus,(27)xt=c1+c2t−1/2∫att−s2hsds.Substituting the values of a,b and η into the above equation, one has(28)1−kc1+a−kηc2=−k2∫aηη−s2hsds,c1+bc2=12∫abb−s2hsds.Applying the Gaussian formula, we have(29)c1=−12mkb∫aηη−s2hsds+a−kη∫abb−s2hsds,and(30)c2=12mk∫aηη−s2hsds+1−k∫abb−s2hsds.
Therefore,(31)xt=−12mkb∫aηη−s2hsds+a−kη∫abb−s2hsds+t2mk∫aηη−s2hsds+1−k∫abb−s2hsds−12∫att−s2hsds=k2m∫aηt−bη−s2hsds+12m∫ab1−kt−a−kηb−s2hsds−12m∫atmt−s2hsds=k2m∫aηt−bη−s2hsds+∫abGt,shsds.
This completes the proof.Lemma 6.
The Green’s functionGt,s from Lemma 5 satisfies the following properties:(1)
Gt,s is continuous(2)
0≤Gt,s≤1/2mkη−ab−s2 for t∈a,bProof.
The continuity ofGt,s is obvious, and next, we consider (2). First, we note that(32)Gtt,s=12m1−kb−s2−2mt−s,a≤s≤t≤b,1−kb−s2,a≤t≤s≤b.(i)
Whena≤t≤s≤b,(33)Gtt,s=12m1−kb−s2<0.(ii)
Whena≤s≤t≤b,(34)Gtt,s=12m1−kb−s2−2mt−s<0.
Therefore,Gt,s is decreasing with respect to t for fixed s∈a,b, so(35)mina≤t≤bGt,s=Gb,s=0,and(36)maxa≤t≤bGt,s=Ga,s=12mkη−ab−s2.
This completes the proof.
Now, we shall present our main results concerning the existence and uniqueness of solutions for problem (6). Let us introduce the following conditions:
C1 There exists a nonnegative constant L such that(37)ft,x−ft,y≤Lx−y,x,y∈R.
C2ζ:=kL/6mb−aη−aη−a2+b−a2<1.
C3 There exists a positive number d0 satisfying the following inequality:(38)ζLLd0+γ≤d0,where γ=maxa≤t≤bft,0.


## 4. Main Results

We now give the existence and uniqueness of solutions for BVP (6).Theorem 1.
Under the conditionsC1−C3, BVP (6) has at least a solution.Proof.
First of all, define an operatorH on Ca,b by(39)Hxt=k2m∫aηt−bη−s2fs,xsds+∫abGt,sfs,xsds,t∈a,b.
Then,xt is a solution of BVP (6) if and only if it is a fixed point of H. Next, we are going to divide the progress into three steps.Step 1.
We need to show the operatorH is continuous.
According to Lemma6 and condition C1, for any ε>0, there exists δ=ε/ζ such that, for any x,y∈Bd0 and x−y<δ, we have(40)Hx−Hy<ε.
In fact,(41)Hxt−Hyt=k2m∫aηt−bη−s2fs,xsds+∫abGt,sfs,xsds−k2m∫aηt−bη−s2fs,ysds−∫abGt,sfs,ysds≤k2mb−a∫aηη−s2fs,xs−fs,ysds+k2mη−a∫abb−s2fs,xs−fs,ysds≤kL6mb−aη−aη−a2+b−a2x−y=ζx−y.
So,(42)Hx−Hy=supa≤t≤bHxt−Hyt≤ζx−y<ϵ.Step 2.
We proveH:Bd0⟶Bd0, where Bd0 denotes a closed sphere of Ca,b which center is 0 and radius is d0.
For anyx∈Ca,b, t∈a,b, from C1,C2 and Lemma 6 (2), we have(43)Hxt≤k2mb−a∫aηη−s2fs,xs−fs,0+fs,0ds+k2mη−a∫abb−s2fs,xs−fs,0+fs,0ds≤kb−a2m∫aηη−s2Lxs+γds+kη−a2m∫abb−s2Lxs+γds≤kb−aη−a6mη−a2+b−a2Lx+γ=ζLLx+γ.
This implies(44)Hx≤ζLLx+γ.
Combining conditionC3 with (44), we know that there exists d0>0 such that H maps Bd0 into itself.Step 3.
Now, we proveαHBd0≤ζαBd0.
First, for allx,y∈Bd0, t∈a,b, based on the proof process of inequality (40), we can obtain(45)Hxt−Hyt≤ζx−y=ζsups∈a,bxs−ys,and it follows that(46)limsupt⟶∞diamHXt≤ζlimsupt⟶∞diamXt.
Besides, for allt1−t2<ε, t1,t2∈a,b, by the continunity of Gt,s, we know that(47)Gt1,s−Gt2,s<ε,for alls∈a,b,combining (46) with conditions C1 and C2, for any x,y∈Bd0, we have(48)Hxt1−Hyt2=k2m∫aηt1−bη−s2fs,xsds+∫abGt1,sfs,xsds−k2m∫aηt2−bη−s2fs,ysds−∫abGt2,sfs,ysds≤k2mb−a∫aηη−s2fs,xs−fs,ysds+∫abGt1,s−Gt2,sfs,xsds+∫abGt2,sfs,xs−fs,ysds≤kL6mb−aη−a3sups∈a,bxs−ys+ε∫abfs,xs−fs,0+fs,0ds+12mη−a∫abb−s2fs,xs−fs,ysds≤kL6mb−aη−a3sups∈a,bxs−ys+εb−aLx+γ+kL6mb−a3η−asups∈a,bxs−ys=ζsups∈a,bxs−ys+εb−aLx+γ,which implies that(49)ωTHx,ε≤ζωTx,ε.
Consequently,(50)ω0THx≤ζω0Tx.
LetT⟶∞, we obtain(51)ω0Hx≤ζω0x.
In view of (45) and (50), we easily obtain(52)αHBd0≤ζαBd0.
Thus, by Lemma3, we conclude that H has at least a solution in Bd0⊂Ca,b; that is, BVP (6) has at least a solution in Bd0⊂Ca,b. This completes the proof.Corollary 1.
In fact, Theorem1 not only proves the existence of solutions of BVP (6) but also illustrates its uniqueness.Proof.
According to the proof of inequality (40) and condition C2, we know, namely, operator H:Ca,b⟶Ca,b is a contraction mapping, so H has a unique solution in Ca,b by construction mapping theorem in Lemma 4; that is, BVP (6) has a unique solution in the whole space Ca,b. This completes the proof.
To be honest, if we only need to know the BVP (6) has an unique solution in its domain, the Lipschitz conditions C1 and C2 are enough. But, it is better to find the more accurate range of the solution in practical matters and concrete situations, so we could choose Theorem 1 to identify the scope of the solution.
In the following, we give two concrete examples to illustrate our main results.Example 1.
Consider the following problem:(53)x″′t+1+t+sinxt=0,t∈1,2,x″1=0,x2=0,x1=32x53.
Here, we have(54)ft,x=1+t+sinx,a=1,b=2,k=32,η=53.
By easy calculation, we get(55)m=1−kb−a−kη=12,∂f∂xt,x=cosx≤1=L,and(56)γ=max1≤t≤21+t=3,ζ=1327<1,choose d0=2.8, with the help of Theorem 1, the problem (51) has at least a solution xt in Bd0.Example 2.
Consider the following problem(57)x″′t+etcosxt+sinxt=0,t∈0,1,x″0=0,x1=0,x0=2x32.
Here, we have(58)ft,x=etcost+sinx,a=0,b=1,k=2,η=34.
By easy calculation, we get(59)m=1−kb−a−kη=12,∂f∂xt,x=cosx≤1=L,and(60)γ=max0≤t≤1etcost=ecos1≈2.718,ζ=2532<1,choose d0=9.8, with the help of Theorem 1, the problem (55) has a solution xt in Bd0.


## 5. Conclusion

In this paper, we study the nonlinear third-order three-point boundary value problem (6). First, we construct Green’s function for the third-order three-point boundary value problem (22) and discuss its properties. Second, based on Green’s function and its properties, we define a solution operator H. Then, we prove that H has at least a fixed point by using Darbo’s fixed point theorem combined with the technique of measure of noncompactness. The fixed point is unique by Banach fixed point theorem. Therefore, the existence and uniqueness of solutions for (6) have been established. Finally, as applications, two examples are given to illustrate our main results. It should be pointed out that the method used here can be applied to impulsive differential equation boundary value problems, such as [37].



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