Iterative solution to singular nth-order nonlocal boundary value problems

R E S E A R C H

Open Access

Iterative solution to singular

n

th-order

nonlocal boundary value problems

Xinan Hao

_{, Lishan Liu}

_{and Yonghong Wu}

*_{Correspondence:}

[email protected]

1_{School of Mathematical Sciences,}

Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China

Full list of author information is available at the end of the article

Abstract

By using the cone theory and the Banach contraction mapping principle, we study the existence and uniqueness of an iterative solution to the singularnth-order nonlocal boundary value problems.

Keywords: iterative solution; singular higher-order differential equation; nonlocal boundary conditions; cone theory

1 Introduction

The boundary value problems (BVPs for short) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of con-trol theory. The nonlocal BVPs have been studied extensively. The methods used therein mainly depend on the fixed-point theorems, degree theory, upper and lower techniques, and monotone iteration. Many existence, uniqueness, and multiplicity results have been obtained. For instance, see [–] and the references therein.

The purpose of this paper is to investigate the existence and uniqueness of iterative so-lution to the followingnth-order nonlocal BVP:

x(n)(t) +f(t,x(t),x(t), . . . ,x(n–)(t)) = , t∈(, ), x() =x() =· · ·=x(n–)_{() = , x}(n–)_{() =}

x(n–)(s)dA(s),

(.)

wheref ∈C((, )×Rn–,R),:=_t dA(t)= . _x(n–)(s)dA(s) denotes the Riemann-Stieltjes integral, whereAis of bounded variation.

In BVP (.),_x(n–)_{(s)dA(s) denotes the Riemann-Stieltjes integral with a signed}

mea-sure. This includes as special cases the two-point, three-point, multi-point problems and integral problems. Let us remark that the idea of using a Riemann-Stieltjes integral in the boundary conditions is quite old, see for example the review by Whyburn in []. The BVP (.) used to model various nonlinear phenomena in physics, chemistry and biology. Over the past decades, great efforts have been devoted to nonlinearnth-order nonlocal BVP (.) and its particular and related cases, and many results of the existence of solutions have been obtained by several authors; see [, , , , –] and references therein. For example, whenn= ,A(t)≡, the BVP (.) becomes the second-order two-point BVP

x(t) +f(t,x(t)) = , t∈(, ),

x() = , x() = . (.)

BVP (.) is the well-known second-order Dirichlet BVP, which has been extensively stud-ied and has important applications in physical sciences. Whenn= ,_x(s)dA(s) =αx(η), the BVP (.) reduces to the second-order three-point BVP

x(t) +f(t,x(t)) = , t∈(, ), x() = , x() =αx(η).

Whenn= , the BVP (.) reduces to the fourth-order nonlocal BVP

x(t) +f(t,x(t),x(t),x(t)) = , t∈(, ),

x() =x() =x() = , x() =_x(s)dA(s). (.)

In material mechanics, the BVP (.) describes the deflection or deformation of an elastic beam whose the ends are controlled.

Motivated by the works mentioned above, in this paper, we consider thenth-order non-local BVP (.). The existence and uniqueness of iterative solution is established by apply-ing the cone theory and the Banach contraction mappapply-ing principle. In comparison with previous works, this paper has several new features. Firstly, the nonlinearityf is allowed to depend on higher derivatives of unknown functionx(t) up ton–  order, and we allowf to be singular att= , . The second new feature is that the nonlinearityf is not monotone or convex, the conclusions and the proof used in this paper are different from the known papers. Thirdly, the scope of is not limited to ≤< , therefore, we do not need to suppose that the Green functionG(t,s) is nonnegative.

2 The preliminary lemmas

Lemma .([]) For any y∈L[, ],the BVP

–x(t) =y(t), t∈(, ), x() = , x() =_x(s)dA(s),

has a unique solution x(t) =_G(t,s)y(s)ds,where

G(t,s) =G(t,s) +

t

 –GA(s), t,s∈[, ],

G(t,s) =

t( –s), ≤t≤s≤,

s( –t), ≤s≤t≤, GA(s) = 



G(t,s)dA(t).

DenoteI= [, ],J= (, ), and for anyx∈C(I),t∈I, define

(Ix)(t) =

t



x(s)ds, (Iix)(t) = t



(t–s)i–

(i– )! x(s)ds, i= , . . . ,n– ,

and

(Fx)(t) = 



G(t,s)fs, (In–x)(s), . . . , (Ix)(s),x(s)

ds.

Lemma .

(i) Ifx∈Cn–_{(I)is a solution of BVP(.),theny(t) =x}(n–)_{(t)∈C(I)is a fixed point of}

the operatorF.

(ii) Ifx∈C(I)is a fixed point of the operatorF,then

y(t) = (In–x)(t) =

t

 (t–s)n–

(n–)! x(s)ds∈Cn–(I)is a solution of BVP(.).

Let

h(t,s) = 

s( –s)G(t,s),

e(t) =





h(t,s)ds,

em+(t) =max





h(t,s)em(s)ds, 



h(t,s) s



em(τ)dτds , m= , , . . . ,

G= lim m→∞

sup

t∈J em(t)

–m .

It is easy to see thatG> .

Lemma .([]) P is a generating cone in the Banach space(E, · )if and only if there exists a constantτ> such that every element x∈E can be represented in the form x=y–z, where y,z∈P and y ≤τ x , z ≤τ x .

3 Main results

Consider the Banach spaceC(I) of the usual real-valued continuous functionsu(t) defined onIwith the norm u =sup_t∈I|u(t)|for allu∈C(I). LetP={u∈C(I)|u(t)≥,∀t∈I}. Obviously,P is a normal solid cone ofC(I), by Lemma .. in [], we see thatP is a generating cone inC(I).

Theorem . Suppose that f(t,x,x, . . . ,xn–) =g(t,x,x,x,x, . . . ,xn–,xn–),and there

exist positive constants B,C,B,C, . . . ,Bn–,Cn–with B+C+B+C+B+C+· · ·+

Bn–+Cn–

(n–)! <G,such that for any t∈J,a,b,a,b,a,b,a,b, . . . ,a,n–,b,n–,a,n–,

b,n–∈R with a≤b,a≥b,a≤b,a≥b, . . . ,a,n–≤b,n–,a,n–≥b,n–,

–Bn–(b,n––a,n–) –Cn–(a,n––b,n–) –· · ·

–B(b–a) –C(a–b) –B(b–a) –C(a–b)

≤t( –t)g(t,a,n–,a,n–, . . . ,a,a,a,a)

–g(t,b,n–,b,n–, . . . ,b,b,b,b)

≤Bn–(b,n––a,n–) +Cn–(a,n––b,n–) +· · ·

+B(b–a) +C(a–b) +B(b–a) +C(a–b), (.)

and there exist x,y∈Cn–(I)such that

t( –t)gt,x(t),y(t),x(t),y(t), . . . ,x (n–)  (t),y

(n–)

 (t)

Then BVP(.)has a unique solution In–x∗ in Cn–(I).Moreover,for any x∈C(I),the

iterative sequence

x(t) =





G(t,s)fs, (In–x)(s), . . . , (Ix)(s),x(s)

ds,

xm(t) = 



G(t,s)fs, (In–xm–)(s), . . . , (Ixm–)(s),xm–(s)

ds (m= , , . . .)

converges to x∗in C(I).

Proof Byt( –t)g(t,x(t),y(t),x(t),y(t), . . . ,x (n–)  (t),y

(n–)

 (t))∈L[, ], it is easy to see

that for anyt∈J,





G(t,s)gs,x(s),y(s),x(s),y(s), . . . ,x (n–)  (s),y

(n–)

 (s)

ds

is well defined. Setp(t) =x(_n–)(t),q(t) =y(_n–)(t), then





G(t,s)gs, (In–p)(s), (In–q)(s), . . . , (Ip)(s), (Iq)(s),p(s),q(s)

ds< +∞.

For anyx,y∈C(I), letu(t) =|p(t)|+|x(t)|,v(t) = –|q(t)|–|y(t)|, thenu≥p,v≤q. By (.), we have

–Bn–(In–u–In–p)(t) –Cn–(In–q–In–v)(t) –· · ·

–B(Iu–Ip)(t) –C(Iq–Iv)(t) –B(u–p)(t) –C(q–v)(t)

≤t( –t)gt, (In–u)(t), (In–v)(t), . . . , (Iu)(t), (Iv)(t),u(t),v(t)

–t( –t)gt, (In–p)(t), (In–q)(t), . . . , (Ip)(t), (Iq)(t),p(t),q(t)

≤Bn–(In–u–In–p)(t) +Cn–(In–q–In–v)(t) +· · ·

+B(Iu–Ip)(t) +C(Iq–Iv)(t) +B(u–p)(t) +C(q–v)(t),

then

G(t,s)gs, (In–u)(s), (In–v)(s), . . . , (Iu)(s), (Iv)(s),u(s),v(s)

–G(t,s)gs, (In–p)(s), (In–q)(s), . . . , (Ip)(s), (Iq)(s),p(s),q(s)

≤h(t,s)Bn–(In–u–In–p)(s)+Cn–(In–q–In–v)(s)+· · ·

+B(Iu–Ip)(s)+C(Iq–Iv)(s)+B(u–p)(s)+C(q–v)(s)

≤h(t,s)(Bn–+· · ·+B+B) u–p + (Cn–+· · ·+C+C) q–v

.

Following the former inequality, we can easily get





G(t,s)gs, (In–u)(s), (In–v)(s), . . . , (Iu)(s), (Iv)(s),u(s),v(s)

–gs, (In–p)(s), (In–q)(s), . . . , (Ip)(s), (Iq)(s),p(s),q(s)

is convergent, and then





G(t,s)gs, (In–u)(s), (In–v)(s), . . . , (Iu)(s), (Iv)(s),u(s),v(s)

ds

= 



G(t,s)gs, (In–p)(s), (In–q)(s), . . . , (Ip)(s), (Iq)(s),p(s),q(s)

ds

+ 



G(t,s)gs, (In–u)(s), (In–v)(s), . . . , (Iu)(s), (Iv)(s),u(s),v(s)

–gs, (In–p)(s), (In–q)(s), . . . , (Ip)(s), (Iq)(s),p(s),q(s)

ds

is convergent. Similarly, byu≥x,v≤y, we get





G(t,s)gs, (In–x)(s), (In–y)(s), . . . , (Ix)(s), (Iy)(s),x(s),y(s)

ds< +∞.

Define the operatorA:C(I)×C(I)→C(I) by

A(x,y)(t) = 



G(t,s)gs, (In–x)(s), (In–y)(s), . . . , (Ix)(s), (Iy)(s),x(s),y(s)

ds.

ThenIn–xis the solution of BVP (.) if and only ifx=A(x,x). Let

(Kx)(t) =B





h(t,s)x(s)ds,

(Kix)(t) =Bi 



h(t,s)(Iix)(s)ds, i= , , . . . ,n– ,

(My)(t) =C





h(t,s)y(s)ds,

(Miy)(t) =Ci 



h(t,s)(Iiy)(s)ds, i= , , . . . ,n– ,

(Kx)(t) = (Kx+Kx+· · ·+Kn–x)(t),

(My)(t) = (My+My+· · ·+Mn–y)(t).

By (.), for anyx,x,y,y∈C(I),x≥x,y≤y, we have

–K(x–x) –M(y–y)≤A(x,y) –A(x,y)≤K(x–x) +M(y–y) (.)

and

(K+M)x(t)

= 



h(t,s)Bx+Cx+B(Ix) +C(Ix) +· · ·+Bn–(In–x) +Cn–(In–x)

(s)ds

≤

B+C+B+C+B+C+· · ·+

Bn–+Cn–

(n– )!

x e(t),

= 



h(t,s)B(K+M)x+C(K+M)x+BI(K+M)x+CI(K+M)x

+· · ·+Bn–In–(K+M)x+Cn–In–(K+M)x

(s)ds

≤

B+C+B+C+B+C+· · ·+

Bn–+Cn–

(n– )! 

x e(t).

By the method of mathematical induction, for any positive integermandt∈J,

(K+M)mx(t)≤

B+C+B+C+B+C+· · ·+

Bn–+Cn–

(n– )! m

x em(t).

Then

(K+M)m≤

B+C+B+C+B+C+· · ·+

Bn–+Cn–

(n– )! m

sup t∈J

em(t)

and

r(K+M)≤B+C+B+C+B+C+· · ·+

Bn–+Cn– (n–)!

G < .

Hence, we can choose aβ>  such that

lim

m→∞(K+M)

mm _{=r(K+M) <β< .}

So, there exists a positive integermsuch that

(K+M)m<βm< , m≥m. (.)

SincePis a generating cone inC(I), from Lemma ., there existsτ>  such that every elementx∈C(I) can be represented in the form

x=y–z, wherey,z∈Pand y ≤τ x , z ≤τ x , (.)

which implies

–(y+z)≤x≤y+z. (.)

Let

x =inf

u |u∈P, –u≤x≤u, (.)

by (.) we know that x is well defined for anyx∈C(I). It is easy to verify that · is

a norm inC(I). By (.)-(.), we get

x ≤ y+z ≤τ x , ∀x∈C(I). (.)

Sinceuis arbitrary, we have

x ≤(N+ ) x , ∀x∈C(I). (.)

It follows from (.) and (.) that the norms · and · are equivalent.

Now, for anyx,y∈C(I) andu∈Pwhich satisfies –u≤x–y≤u, we set

u=



(x+y–u), u= 

(x–y+u), u= 

(–x+y+u),

thenx≥u,y≥u, andx–u=u,y–u=u,u+u=u. It follows from (.) that

–Ku≤A(x,x) –A(u,x)≤Ku, (.)

–Ku–Mu≤A(y,u) –A(u,x)≤Ku+Mu, (.)

–Mu≤A(y,u) –A(y,y)≤Mu. (.)

By (.)-(.), we have

–(K+M)u≤A(x,x) –A(y,y)≤(K+M)u.

LetA(x) =A(x,x), then we obtain

–(K+M)u≤A(x) –A(y)≤(K+M)u.

As K andMare both positive linear bounded operators, soK+Mis a positive linear bounded operator, and therefore (K+M)u∈P. Hence, by mathematical induction, it is easy to see that for the natural numbermin (.), we have

–(K+M)m_u≤_Am_{(x) –A}m_(y)≤_(K+M)m_{u, (K+M)}m_u∈_P.

Since (K+M)m_u∈_{P, we see that}

_Am_{(x) –A}m_(y)

≤(K+M)

m _{u ,}

which implies by virtue of the arbitrariness ofuthat

Am_x–Am_y

≤(K+M)

m _x–y

≤βm x–y .

By  <β< , we have  <βm_{< . Thus the Banach contraction mapping principle}

im-plies thatAm_{has a unique fixed pointx}∗_{inC(I), and soAhas a unique fixed pointx}∗_in

C(I). By the definition ofA,Ahas a unique fixed pointx∗inC(I), then by Lemma ., In–x∗ is the unique solution of BVP (.). And, for any x∈C(I), let x =A(x,x),

xm=A(xm–,xm–) (m= , , . . .), we have xm–x∗ → (m→ ∞). By the equivalence of

Remark . For the casen= ,

A(t) =

, t∈[,η), α, t∈[η, ],

iff(t,x,y) =f(t,x), Theorem . is reduced to Theorem . in [], if  <α< 

ηandf(t,x,y) =

h(t)f(t,x), the existence results of nontrivial solutions are given by means of the topological degree theory in []. So our results extend the corresponding results of [, ] to some degree.

Example . To illustrate the applicability of our results, we consider the BVP (.) with n=  and

f(t,x,y) =  t( –t)

sin x

n

+cos x n

+ln( +y

₎

n

+sin y n

,

A(t) =

, t∈[,_), , t∈[_, ],

wheren,n,n,nare positive integral numbers. Then=



t dA(t) = ×  =

 , and

BVP (.) becomes the singular third-order three-point BVP

x(t) +_t(–_t)[sinx_n(t)

 +cos

x(t)

n +

ln(+(x(t)))

n +sin

x(t)

n ] = , t∈(, ),

x() =x() = , x() = x(_). (.)

Let

G(t,s) = t

( –s) – 

 –s

χ_[,

](s)

– (t–s)χ[,t](s),

whereχis the characteristic function,i.e.

χ[a,b](t) =

, t∈[a,b], , t∈/[a,b]

and

(Fx)(t) = 



G(t,s)fs, (Ix)(s),x(s)

ds, t∈(, ).

By Lemma ., if x ∈ C(I) is a fixed point of the operator F, then y(t) = (Ix)(t) =

t

x(s)ds∈C

_{(I) is a solution of BVP (.). Letf(t,x,y) =g(t,x,x,y,y), then for anyt∈J,}

a,b,a,b,a,b,a,b∈Rwitha≤b,a≥b,a≤b,a≥b, we have

– n

(b–a) –

 n

(a–b) –

 n

(b–a) –

 n

(a–b)

≤t( –t)g(t,a,a,a,a) –g(t,b,b,b,b)

=sina n

+cosa n

+ln( +a

 )

n

–

sinb n

+cosb n

+ln( +b

 )

n

+sinb n

≤ 

n

(b–a) +

 n

(a–b) +

 n

(b–a) +

 n

(a–b).

By Theorem ., BVP (.) has a unique solutionIx∗∈C(I) provided _n_ +_n_ +_n_ + 

n<G. Moreover, for anyx∈C(I), the iterative sequence

xm(t) = 



G(t,s)fs, (Ixm–)(s),xm–(s)

ds (m= , , , . . .)

converges tox∗(m→ ∞).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors wrote, read, and approved the final manuscript.

Author details

1_{School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, People’s Republic of China.} 2_{Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia.}

Acknowledgements

The authors would like to thank the referees for their pertinent comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China (11371221, 11201260), the Natural Science Foundation of Shandong Province of China (ZR2015AM022, ZR2013AQ014), the Specialized Research Fund for the Doctoral Program of Higher Education (20123705120004, 20123705110001) and Foundation of Qufu Normal University (BSQD20100103).

Received: 16 March 2015 Accepted: 16 July 2015 References

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