Higher order functional boundary value problems without monotone assumptions

R E S E A R C H

Open Access

Higher order functional boundary value

problems without monotone assumptions

João F Fialho

_{and Feliz Minhós}

*_{Correspondence:} [email protected]

1_{School of Mathematics, Physics} and Technology, College of the Bahamas, Nassau, Bahamas 3_{Research Centre on Mathematics} and Applications, University of Évora (CIMA-UE), Rua Romão Ramalho, 59, Évora, 7000-671, Portugal Full list of author information is available at the end of the article

Abstract

In this paper, givenf: [a,b]×(C([a,b]))n–2_×R2_→RaL1_{-Carathéodory function, it is}

considered the functional higher order equation

u(n)(x) =f

(x

)

together with the nonlinear functional boundary conditions, for i= 0,. . .,n– 2

Li

(u

)

(u

)

Here,Li,i= 0,. . .,n– 1, are continuous functions. It will be proved an existence and

location result in presence of not necessarily ordered lower and upper solutions, without assuming any monotone properties on the boundary conditions and on the nonlinearityf.

1 Introduction

In this paper, it is considered the functional higher order boundary value problem, for n≥ composed by the equation

u(n)(x) =fx,u, . . . ,u(n–),u(n–)(x),u(n–)(x) ()

for a.a.x∈I:= [a,b], wheref :I×(C(I))(n–)_×R_{→Ris aL}_{-Carathéodory function,}

and the function boundary conditions

Liu,u, . . . ,u(n–),u(i)(a)= , i= , . . . ,n– , Ln–

u,u, . . . ,u(n–),u(n–)(b)= ,

()

whereLi,i= , . . . ,n– , are continuous functions without assuming monotone conditions or another type of variation.

The functional differential equation () can be seen as a generalization of several types of full differential and integro-differential equations and allow to consider delays, max-ima or minmax-ima arguments, or another kind of global variation on the unknown function or its derivatives until order (n– ). On the other hand, the functional dependence in () makes possible its application to a huge variety of boundary conditions, such as Lidstone,

separated, multipoint, nonlocal and impulsive conditions, among others. As example, we mention the problems contained in [–]. A detailed list about the potentialities of func-tional problems and some applications can be found in [].

Recently, functional boundary value problems have been studied by several authors fol-lowing several approaches, as it can be seen, for example, in [–]. In this work, the lower and upper solutions method is applied together with topological degree theory, ac-cording some arguments suggested in [–].

The novelty of this paper consists in the following items:

• There is no monotone assumptions on the boundary functionsLi,i= , . . . ,n– , by using adequate auxiliary functions and global arguments. This fact with the functional dependence on the unknown function and its derivatives till order(n– )will allow that problem ()-() can include the periodic and antiperiodic cases, which were not covered by the existent literature on functional boundary value problems. In this sense, the results in this area, as for instance [–], are improved, even forn= , where equation () loses its functional part.

• No extra condition on the nonlinear part of () is considered, besides a Nagumo-type growth assumption. In fact, as far as we know, it is the first time where lower and upper solutions technique is used without such hypothesis on functionf, by the use of stronger definitions for lower and upper solutions.

• No order between lower and upper solutions is assumed. Putting the ‘well ordered’ case on adequate auxiliary functions, it allows that lower and upper solutions could be well ordered, by reversed order or without a defined order.

The last section contains an example where the potentialities of the functional depen-dence on the equation and on the boundary conditions are explored.

2 Definitions and auxiliary functions

In this section, it will be introduced the notations and definitions needed forward together with some auxiliary functions useful to construct some ordered functions on the basis of the not necessarily ordered lower and upper solutions of the referred problem.

A Nagumo-type growth condition, assumed on the nonlinear part, will be an important tool to set ana prioribound for the (n– )th derivative of the corresponding solutions.

In the following, Wm,_{(I) denotes the usual Sobolev Spaces inI, that is, the subset of}

Cm–_{(I) functions, whose (m– )th derivative is absolutely continuous inIand themth}

derivative belongs toL_{(I) and the usual norms}

up= ⎧ ⎨ ⎩

(_|u(x)|pdx)/p, ≤p<∞, sup{|u(x)|:x∈I}, p=∞,

for spacesLp_{, ≤p≤ ∞.}

The function f : I × (C(I))(n–) _×R _{→ R is a L}_{-Carathéodory function, that is,}

f(x,·, . . . ,·,·,·) is a continuous function for a.e. x∈I; f(·,y, . . . ,yn–,yn–) is measurable

for (y, . . . ,yn–,yn–)∈(C(I))(n–)×R; and for everyM>  there is a real-valued function ψM∈L(I) such that

f(x,y,y,y,y)≤ψM(x), for a.e.x∈[, ]

The main tool to obtain the location part is the upper and lower solutions method. How-ever, in this case, they must be defined as a pair, which means that it is not possible to define them independently from each other. Moreover, it is pointed out that lower and upper functions, and the correspondent first derivatives, are not necessarily ordered.

To introduce ‘some order’, some auxiliary functions must be defined. For anyα,β∈Wn–,_{(I) define functionsαi,β}

i:I→R,i= , . . . ,n– , as it follows:

αn–(x) =min α(n–)(a),β(n–)(a)

+

x

a

α(n–)(s)ds,

βn–(x) =max α(n–)(a),β(n–)(a)

+

x

a

β(n–)(s)ds,

αi(x) =min α(i)(a),β(i)(a)+ x

a

αi+(s)ds,

βi(x) =max α(i)(a),β(i)(a)+ x

a

βi+(s)ds,

()

fori= , . . . ,n– .

The Nagumo-type condition is given by next definition.

Definition  Consideri,γi∈C(I),i= , . . . ,n– , such thati(x)≤γi(x),∀x∈I, and the set

E= (x,y, . . . ,yn–)∈I×Rn:γi(x)≤yi≤i(x),i= , . . . ,n– 

.

A functionf :I×Rn_{→Ris said to verify a Nagumo-type condition inEif there exists}

ϕE∈C([, +∞), (, +∞)) such that

f(x,y, . . . ,yn–)≤ϕE

|yn–|, ()

for every (x,y, . . . ,yn–)∈E, and

+∞

r t

ϕE(t)dt>maxx∈I n–(x) –minx∈I γn–(x), () wherer≥ is given by

r:=max

n–(b) –γn–(a)

b–a ,

n–(a) –γn–(b)

b–a

.

The next result gives ana prioriestimate for the (n– )th derivative of all possible solu-tions of ().

Lemma  There exists K >  such that for every L_{-Carathéodory function f : I ×}

(C(I))n–×R→Rsatisfying()and()and every solution u of()such that

for i= , . . . ,n– ,we have

u(n–)_∞<R. ()

Moreover,the constant R depends only on the functionsϕandγi,i(i= , . . . ,n– )and not on the boundary conditions.

Proof The proof is similar to [, Lemma .].

The upper and lower solution definition is then given by the following.

Definition  The functionsα,β ∈Wn,_{(I) are a pair of lower and upper solutions for}

problem ()-() ifα(n–)_(x)≤β(n–)_{(x), on I, for all (v}

, . . . ,vn–)∈A:= [α,β]× · · · ×

[αn–,βn–], and for every (w,w)∈B:= [α(n–),β(n–)]×[–K,K], for someK> , the

fol-lowing inequalities hold for a.e.x∈[a,b],

α(n)(x)≥fx,v, . . . ,vn–,α(n–)(x),α(n–)(x)

,

β(n)(x)≤fx,v, . . . ,vn–,β(n–)(x),β(n–)(x)

,

()

and forj= , . . . ,n– ,

Ljv, . . . ,vn–,w,w,αj(a)

≥,

Ln–

v, . . . ,vn–,w,w,α(n–)(a)

≥,

Ln–

v, . . . ,vn–,w,w,α(n–)(b)

≥,

Ljv, . . . ,vn–,w,w,βj(a)

≤,

Ln–

v, . . . ,vn–,w,w,β(n–)(a)

≤,

Ln–

v, . . . ,vn–,w,w,β(n–)(b)

≤.

()

3 Existence and location result

In this section, it is provided an existence and location theorem for the problem ()-(). More precisely, sufficient conditions are given for, not only the existence of a solutionu, but also to have information about the location ofu, and all its derivatives up to the (n– ) order.

The arguments of the proof require the following lemma, given on [].

Lemma  For v,w∈C(I)such that v(x)≤w(x),for every x∈I,define

q(x,u) =max v,min{u,w}.

Then,for each u∈C_{(I)the next two properties hold:}

(a) _dxdq(x,u(x))exists for a.e.x∈I.

(b) Ifu,um∈C_{(I)andum→uinC}_(I)then

d dxq

x,um(x)→ d dxq

Now, we are in a position to prove the main result of this paper.

Theorem  Assume that there exists a pair of lower and upper solutions(α,β)of problem ()-().

If f :I×(C(I))n–×R→Ris a L-Carathéodory function,satisfying a Nagumo-type condition in

E∗= ⎧ ⎨ ⎩

(x,y, . . . ,yn–)∈I×Rn–:αi(x)≤yi≤βi(x), i= , . . . ,n– ,

α(n–)_(x)≤yn

–≤β(n–)(x),

then problem()-()has at least one solution u such that

αi(x)≤u(i)(x)≤βi(x), i= , . . . ,n– ,

α(n–)(x)≤u(n–)(x)≤β(n–)(x), ∀x∈I

for every x∈I,and|u(n–)_{(x)| ≤K,∀x∈I,where}

K=max R,α(n–)(x),β(n–)(x) ()

and R> is given by().

Proof Define the continuous functions, fori= , . . . ,n– ,

δi(x,yi) =max αi(x),min yi,βi(x),

δn–(x,yn–) =max α(n–)(x),min yn–,β(n–)(x)

,

()

and the truncation, not necessarily continuous,

ξ(z) =max –K,min{z,K},

withKgiven by ().

Consider the modified problem composed by the equation

u(n)(x) =f

x,δ(·,u), . . . ,δn–(·,u(n–)), δn–(x,u(n–)(x)),ξ(_dxd(δn–(x,u(n–)(x))))

()

and the boundary conditions, fori= , . . . ,n– ,

u(i)(a) =δi

a,u(i)(a) +Li

δ(·,u), . . . ,δn–(·,u(n–)), ξ(_dxd(δn–(·,u(n–)))),u(i)(a)

,

u(n–)(b) =δn–

b,u(n–)(b) +Ln–

δ(·,u), . . . ,δn–(·,u(n–)), ξ(_dxd(δn–(·,u(n–)))),u(n–)(b)

.

()

Step . Every solutionuof problem ()-(), satisfies

α(n–)(x)≤u(n–)(x)≤β(n–)(x),

αi(x)≤u(i)(x)≤βi(x), i= , . . . ,n– ,

and|u(n–)(x)|<K, for everyx∈I, withK>  given in ().

Letube a solution of the modified problem ()-(). Assume, by contradiction, that there existsx∈Isuch thatα(n–)(x) >u(n–)_{(x) and letx}

∈Ibe such that

min x∈I(u–α)

(n–)_{(x) := (u–α)}(n–)_(x ) < .

As, by (),u(n–)_(a)≥α(n–)_{(a) andu}(n–)_(b)≥α(n–)_{(b), thenx}

∈(a,b). So, there is

(x,x)⊂(a,b) such that

u(n–)(x) <α(n–)(x), ∀x∈(x,x),

(u–α)(n–)(x) = (u–α)(n–)(x) = .

()

Therefore,

δn–

x,u(n–)(x)=α(n–)(x), ∀x∈(x,x),

and

d dxδn–

x,u(n–)(x)=α(n–)(x), a.e.x∈(x,x).

Now, since for allu∈Cn–_{(I) it is satisfied that (δ}

(·,u), . . . ,δn–(·,u))∈A, we deduce

that

u(n)(x) =f

x,δ(·,u), . . . ,δn–(·,u(n–)),δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x))))

=fx,δ(·,u), . . . ,δn–

·,u(n–),α(n–)(x),α(n–)(x)

≤α(n)(x) for a.e.x∈(x,x).

As (u–α)(n–)_(x

) =  and (u–α)(n–)is nonincreasing in (x,x), this contradicts the

definitions ofxandx.

The inequalityu(n–)_(x)≤β(n–)_{(x), inI, can be proved in same way and so,}

α(n–)(x)≤u(n–)(x)≤β(n–)(x), ∀x∈I. ()

By () and (), the following inequalities hold for everyx∈I:

u(n–)(x) =u(n–)(a) + x

a

u(n–)(s)ds≥αn–(a) +

x

a

α(n–)(s)ds

≥min α(n–)(a),β(n–)(a)+ x

a

α(n–)(s)ds=αn–(x).

Analogously, it can be obtainedu(n–)_(x)≤β

The remaining inequalities are obtained by the same integration process. Applying previous bounds in Lemma , and remarking that

K

r s

ϕ(s)ds≥ R

r s

ϕ(s)ds,

forKgiven by (), it is obtained, by Lemma , thea prioribound|u(n–)_{(x)|<K, forx∈I.}

For details, see [, Lemma ].

Step . Problem ()-() has at least one solution.

Forλ∈[, ] let us consider the homotopic problem given by

u(n)(x) =λf

x,δ(·,u), . . . ,δn–(·,u(n–)), δn–(x,u(n–)(x)),ξ(_dxd(δn–(x,u(n–)(x))))

()

and the boundary conditions, fori= , . . . ,n– ,

u(i)(a) =λδi

a,u(i)(a) +Li

δ(·,u), . . . ,δn–(·,u(n–)), ξ(_dxd(δn–(·,u(n–)))),u(i)(a)

:=λLAi,

u(n–)(b) =λδn–

b,u(n–)(b) +Ln–

δ(·,u), . . . ,δn–(·,u(n–)), ξ(d

dx(δn–(·,u

(n–)_))),u(n–)_(b)

:=λLB.

()

Let us consider the norms inCn–(I) and inL(I)×Rn, respectively,

vCn–=max v_∞, . . . ,v(n–)_∞

and

(h,h, . . . ,hn)=max hL,max |h|, . . . ,|hn|

.

Define the operatorsL:Wn,_(I)⊂Cn–_(I)→L_(I)×Rn_{byLu= (u}(n)_{,u(a), . . . ,u}(n–)_(a),

u(n–)(b)) and, forλ∈[, ],i= , . . . ,n– ,Nλ:Cn–(I)→L(I)×Rnby

Nλu=

⎛ ⎜ ⎝λf

x,δ(·,u), . . . ,δn–(·,u(n–)),δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x))))

,

λLA, . . . ,λLAn–,λLB

⎞ ⎟ ⎠.

SinceL, . . . ,Ln–are continuous andfis aL-Carathéodory function, then, from Lemma , Nλis continuous. Moreover, asL–is compact, it can be defined the completely

continu-ous operatorTλ:Cn–(I)→Cn–(I) byTλu=L–Nλ(u).

It is obvious that the fixed points of operatorTλcoincide with the solutions of problem

()-().

AsNλuis bounded inL(I)×Rnand uniformly bounded inCn–(I), we have that any

solution of the problem ()-(), verifies the followinga prioribound

uCn–≤L–

Cn–Nλ(u)≤ ¯K,

In the set={u∈Cn–_{(I) :u}

Cn–<K¯+ }, the degreed(I–Tλ,, ) is well defined for

everyλ∈[, ] and, by the invariance under homotopy,d(I–T,, ) =d(I–T,, ).

As the equationx=T(x) is equivalent to the problem

⎧ ⎨ ⎩

u(n)_{(x) = ,}

u(i)_{(a) =u}(n–)_{(b) = , i= , . . . ,n– ,}

which has only the trivial solution, thend(I–T,, ) =±. So, by degree theory, the

equationx=T(x) has at least one solution, that is, the problem ()-() has at least a

solution in.

Step . Every solutionuof problem ()-() is a solution of ()-().

Letube a solution of the modified problem ()-(). By previous steps, functionu ful-fills equation (). So, it will be enough to prove the following inequalities, fori= , . . . ,n– :

αi(a)≤u(i)(a) +Li

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(i)(a)

≤βi(a),

α(n–)(a)≤u(n–)(a) +Ln–

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(n–)(a)

≤β(n–)(a)

and

α(n–)(b)≤u(n–)(b) +Ln–

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(n–)(b)

≤β(n–)(b).

Assume that

u(a) +L

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(a)

>β(a). ()

Then, by (),u(a) =β(a). By previous steps, it is obtained the following contradiction

with ():

u(a) +L

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(a)

=β(a) +L

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),β(a)

≤β(a).

Applying similar arguments, it can be proved that

α(a)≤u(a) +L

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(a)

and analogously, forj= , . . . ,n– ,

αj(a)≤u(j)(a) +Lj

δ(·,u), . . . ,δn–(x·,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(j)(a)

≤βj(a).

Also, using the same arguments and the same techniques, it can be proved that

α(n–)(a)≤u(n–)(a) +Ln–

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(n–)(a)

≤β(n–)(a),

α(n–)(b)≤u(n–)(b) +Ln–

δ(·,u), . . . ,δn–(x,u(n–)(x)), ξ(_dxd(δn–(x,u(n–)(x)))),u(n–)(b)

≤β(n–)(b).

4 Example

This section contains a problem composed by an integro-differential equation with some functional boundary conditions, whose solvability is proved in presence of nonordered lower and upper solutions. We remark that such fact was not possible with the results in the current literature. This example does not model any particular problem arising in real phenomena. Our purpose consists on emphasizing the powerful of the developed theory in this paper by showing what kind of problems we can deal with.

Consider, forx∈[, ], the fourth-order equation

u(iv)_{(x) =}

x



u(s)ds+max x∈[,] u

_(x)+u(x)

–u(x) +  _()

coupled with the boundary value conditions

– min x∈[,]u

_{(x) – u() = ,}

u(s) –u()+  = ,

max

x∈[,]u(x) – u

_{() = ,}



u() = .

()

One can verify that functions

α(x) = –x 



– x+ x–  and β(x) = x 



+ x+ 

are, respectively, lower and upper solutions for the problem ()-(). Moreover, we de-duce that

α(x) = –

x 



– x–

, α(x) = – x 



– x– x– ,

β(x) =x+ x+ , β(x) =

x 



and

f(x,y,y,y,y) =

x



y(s)ds+ max

x∈[,] y(x)

+y(x)



–y(x) + 

 _,

L(z,z,z,z,z) = – min

x∈[,]z– z,

L(z,z,z,z,z) =z– (z)+ ,

L(z,z,z,z,z) = max

x∈[,]z– z,

L(z,z,z,z,z) = –√z.

As the continuous functionf verifies () and () forϕE∗(y) =,_ + (y+ )

  _in

E∗=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(x,y,y,y,y)∈[, ]×R:

–_x– x_–

x– ≤y≤

x

 

+ x_{+ x+} 

–x_– x–_≤y≤x+ x+ 

–x– ≤y≤x+ 

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ ,

then, by Theorem , there is a nontrivial solutionufor problem ()-() such that

–x 



– x–

x– ≤u(x)≤ x 



+ x+ x+  ,

–x 



– x– ≤u

_(x)≤x_{+ x+ ,}

–x– ≤u(x)≤x+ , for allx∈[, ].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally, read and approved the final version of the manuscript.

Author details

1_{School of Mathematics, Physics and Technology, College of the Bahamas, Nassau, Bahamas.}2_{Department of} Mathematics, School of Sciences and Technology, University of Évora, Évora, Portugal. 3_{Research Centre on Mathematics} and Applications, University of Évora (CIMA-UE), Rua Romão Ramalho, 59, Évora, 7000-671, Portugal.

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.

Received: 15 December 2012 Accepted: 22 March 2013 Published: 10 April 2013 References

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doi:10.1186/1687-2770-2013-81