Application of p-regularity theory to nonlinear boundary value problems

R E S E A R C H

Open Access

Application of

p

-regularity theory to

nonlinear boundary value problems

Wiesław Grzegorczyk

_{, Beata Medak}

_{and Alexey A Tret’yakov}

*_{Correspondence:}

[email protected]

1_{Siedlce University of Natural}

Sciences and Humanities, 3-go Maja 54, Siedlce, 08-110, Poland

†_{Equal contributors}

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

Abstract

The paper studies the question of solution existence to a nonlinear equation in the degenerate case. This question is studied for three particular boundary value problems for ordinary and partial second-order differential equations. The so-called

p-regularity theory is applied to these purposes as an effective apparatus to investigate many nonlinear mathematical, physical and numerical problems. All results obtained in the paper are based on the constructions of this theory whose basic concepts were described by Tret’yakov. We recall the main definitions and theorems ofp-regularity theory and illustrate the results by examples including singular boundary value problems. In the first and second ones, the description of solutions by a tangent cone at an initial point are given. In the third example, we formulate a sufficient condition forp-regularity (p= 2), which can be tested using the notion of resultant.

MSC: 47J05; 34B15; 34B16

Keywords: p-regularity;p-factor operator;p-kernel; singularity; differential equations; nonlinear boundary value problems

Introduction

In the paper, we study the question of solution existence to a nonlinear equation

F(x) = , ()

whereX,Yare Banach spaces andF∈Cp_{(X,Y) (p≥). Letx}

∗be a solution to this

equa-tion,i.e.,F(x∗) = . The above problem is called regular at the pointx∗ ifImF(x∗) =Y.

Otherwise, problem () is called irregular, degenerate or singular atx∗.

The construction ofp-regularity [–] gives new possibilities for solving or describing degenerate problems (see, for instance, [–]). We are going to use it to certain ques-tions that appear in many numerical applicaques-tions. Namely, we consider the equation of rod bending and the nonlinear Laplace equation that describe many mathematical physics problems like string oscillation, membrane oscillation and so on.

The following degenerate nonlinear boundary value problem:

F(x,ε) =x+ ( +ε)x+x= , ()

whereF:C

[,π]×R→C[,π] andC[,π] ={x∈C[,π] :x() =x(π) = }, is the first of them.

The second one is a modification of equation (),i.e.,

F(x,ε) =x+x+εxp–+xp= , ()

whereF:C_[,π]×R→C[,π],p≥.

The third one is the partial differential equation of the form

F(u,ε) =u– (ε– )g(u) = , ()

where F:C_[]×R→C[],= [,π]×[,π],is the Laplacian andC_[] ={u∈

C_{[] :u|}

∂= }; moreover,g:→Ris a certain function such thatg() = ,g() = ,

g()= .

Note that the numbers – and – are the eigenvalues of the operators (·)andin equations () and (), respectively. Moreover, in the first example, we could takeg(x) in-stead ofx+xsuch thatg() = ,g() = ,g()= . It is important that the -regularity condition is fulfilled for the mappingF(x,ε) at the point (, ). We chose equation () in order to expose our results not only forp= . In equation () we have takenλ= – as an arbitrary representative element of eigenvalues of the Laplacian of the form –(k_+n_), k,n∈N. Of course it is possible to take for exampleλ= –.

We begin with some notation. Suppose X andY are Banach spaces and denote the space of all continuous linear operators from X to Y by L(X,Y). Let p be a natural number, and let B:X×X× · · · ×X (p-copies ofX)→Y be a continuous symmetric p-multilinear mapping. Thepform associated toBis the mapB[·]p:X→Y defined by B[x]p_{=B(x,x, . . . ,x) forx∈X. Alternatively, we may simply viewB[·]}p_{as a homogeneous}

polynomialQ:X→Yof degreep, and henceQ(αx) =αpQ(x). Throughout this paper, we assume that the mappingF:X→Y isp-times continuously Fréchet differentiable onX and itspth order derivative atx∈Xwill be denoted asF(p)_{(x) (a symmetric multilinear} map ofpcopies ofXtoY) and the associatedp-form, also called apth-order mapping, is

F(p)(x)[h]p=F(p)(x)[h,h, . . . ,h].

We also use the notation

KerpF(p)(x) =h∈X:F(p)(x)[h]p= 

and refer to it as thep-kernel of thepth-order mapping.

The setM(x∗) ={x∈X:F(x) =F(x∗) = }is calledthe solution setfor the mappingF.

We callh a tangent vectorto the setM⊆Xatx∗∈Mif there existε>  and a function

r: [,ε]→Xwith the property that fort∈[,ε] we havex∗+th+r(t)∈Mand r(t) =o(t)

ast→ (for the sake of simplicity, the recordt→ will be omitted in the following text of paper). The set of all tangent vectors atx∗is calledthe tangent conetoMatx∗and is

denoted byTM(x∗). A mapF:X→Yis regular atx∗∈XifImF(x∗) =Y. In the regular

case, the tangent cone to the solution set coincides with the kernel of the first derivative of the mapF. Recall the following theorem.

Theorem  (Classical Lyusternic theorem) Let X and Y be Banach spaces and a map

F:X→Y be regular at x∗∈X.Then

The notion of regularity is generalized to the notion of the so-calledp-regularity which will be described in the next section.

Elements ofp-regularity theory

Assume thatx∗∈U⊆X,Uis a neighborhood of an elementx∗. Let a mappingF:U→Y

bep-times Fréchet differentiable inU andImF(x∗)=Y (the regularity condition does

not hold). In order to define the notion ofp-regularity, let us define the so-calledp-factor operator (see []). Assume that the spaceYis decomposed into a direct sum

Y=Y⊕ · · · ⊕Yp, ()

whereY=cl(ImF(x∗)) (the closure of the image of the first derivative ofFevaluated at x∗) and the next spaces are defined as follows. LetZbe a closed complementary subspace toY, that is,Y=Y⊕Z(we assume that such a closed complement exists), and letPZ :

Y→Zbe a projection operator ontoZalongY. LetY=cl(span ImPZF(x∗)[·]

_)⊆Z (the closure of the linear span of the image of the quadratic map PZF(x∗)[·]

_{). More}

generally, define

Yi=cl

span ImPZiF (i)_(x

∗)[·]i⊆Zi, i= , . . . ,p– ,

whereZiis a closed complementary subspace toY⊕ · · · ⊕Yi–,i= , . . . ,p, with respect

toY, andPZi:Y→Ziis a projection operator ontoZialongY⊕ · · · ⊕Yi–,i= , . . . ,p. Finally,Yp=Zp. The orderpis chosen as the minimal number (if it exists) for which the

above decomposition () holds.

Now, let us define the following mappings:

fi(x) :U→Yi, fi(x) =PYiF(x), i= , . . . ,p,

wherePYi:Y→Yiis a projection operator fromYalongY⊕ · · · ⊕Yi–⊕Yi+⊕ · · · ⊕Yp. Below we recall some important definitions for further considerations.

We introduce the following operator:

p(x∗,·) =f(x∗) +f(x∗)[·] +· · ·+fp(p)(x∗)[·]p–.

This means that

p(x∗,h) =f(x∗) +f(x∗)[h] +· · ·+fp(p)(x∗)[h]p–.

Definition  Leth∈X. The linear operator p(x∗,h)∈L(X,Y) is called thep-factor

op-erator.

Sometimes it is convenient to use the following representation of thep-factor operator:

p(x∗,h) =f(x∗),f(x∗)[h], . . . ,fp(p)(x∗)[h]p–

=PYF

_(x ∗),PYF

_(x

∗)[h], . . . ,PYpF (p)_(x

∗)[h]p–

We say thatFiscompletely degenerateatx∗up to orderpifF(i)(x∗) = ,i= , . . . ,p– .

In the completely degenerate case, thep-factor operator is equal toF(p)_(x

∗)[h]p–.

Definition  Thep-kernel of the operator p(x∗,·) is a set

Hp(x∗) =Kerp p(x∗,·) =h∈X: p(x∗,h)[h] = 

=h∈X:f_(x∗)[h] +f_(x∗)[h]+· · ·+f_p(p)(x∗)[h]p= .

Note that

Kerp p(x∗,·) =

_p

i=

Kerif_i(i)(x∗)

.

In the completely degenerate case,Hp(x∗) is equal to

KerpF(p)(x∗) =h∈X:F(p)(x∗)[h]p= .

Definition  A mappingFis calledp-regular atx∗alongh,p> , ifIm p(x∗,h) =Y(i.e.,

the operator p(x∗,h) is surjection).

Definition  A mappingFis calledp-regular atx∗,p> , if it isp-regular along everyh

belonging to the setHp(x∗)\{}.

Note that ifHp(x∗) ={}, thenFis automaticallyp-regular.

The following theorem gives a description of the tangent cone in the degenerate case.

Theorem (Generalized Lyusternik theorem []) Let X and Y be Banach spaces, F∈

Cp_{(X,Y)be p-regular at x}

∗∈U⊂X.Then

TM(x∗) =Hp(x∗).

Applications

The following lemma will be important in the study of surjectivity ofp-factor operators in the mentioned examples.

Lemma  Suppose that Y =Y ⊕Y, where Y, Y are closed subspaces in Y, A,B∈

L(X,Y),ImA=Y.Let also Pbe a projection onto Yalong Y.Then(A+PB)X=Y⇔ (PB)KerA=Y.

This lemma is a straightforward consequence of the following simple lemma.

Lemma  Suppose that Y =Y⊕Y, where Y,Y are closed subspaces in Y, A,A ∈

L(X,Y),AX⊂Y,AX⊂Y.Then(A+A)X=Y iff AKerA=Yand AKerA=Y.

Example  Now we apply the above theory to differential equation ()

F(x,ε) =x+ ( +ε)x+x= ,

whereF:C

[,π]×R→C[,π] andC[,π] ={x∈C[,π] :x() =x(π) = }. Observe thatx∗= (, ) is a trivial solution of this equation.

The first derivative of the mappingFis

F(x,ε) = F_x(x,ε),F_ε(x,ε)=

d

dt + ( +ε)( + x),x+x 

.

In our case,F_x(, ) = (·)+ (·) andFε(, ) = .

Note thatKerF_x(, ) ={x∈C[,π] :d_dtx+x= }. The general solution of the equation

x+x=  isx(t) =ccost+csint. Taking into account the boundary condition, we obtain c= ,x(t) =csintandKerFx(, ) =span{sint}.

The image ofF_x(, ) is defined as follows:

ImF_x(, ) =y∈C[,π] :∃x∈C[,π]F_x(, )x=y,x() =x(π) =  =y∈C[,π] :∃x∈C[,π] such thatx+x=y,x() =x(π) = . The general solution of the above equation has the form

x(t) =ccost+csint–cost

t



y(τ)sinτdτ+sint

t



y(τ)cosτdτ.

In view of the boundary conditionsx() =x(π) = , we obtain

ImF_x(, ) =

y∈C[,π] :

π



y(τ)sinτdτ= 

.

One can easily show that the boundary value problem x+x=sint, x() =x(π) =  does not have a solution. Moreover, since _πsin(t)dt= , it follows thaty(t) =sint∈/ ImF_x(, ). This implies that the operator F_x(, ) is not surjective, i.e., ImF_x(, )=

C[,π] =Y. ThenY=Y⊕Z, whereY=ImF_x(, ) andZ=KerF_x(, ). The projectorPZ:Y→ZalongYcan be described as

PZy=



πsint

π



y(τ)sinτdτ, y∈Y.

Since

F(x,ε) =

( +ε)  + x  + x 

, F(, ) =

   

,

F(, )[hz,hε] = [hz+hε,hz], F(, )[hz,hε]= hz+ hzhε,

then

Y=span

ImPZF(, )[·]



=span

y(t)∈Y:y(t) = 

πsint

π



F(, )[·]sinτdτ

and

Y=C[,π] =Y⊕Y=ImF_x(, )⊕KerF_x(, ).

Consequently, [ImF_x(, )]⊥=KerF_x(, ).

Usingp-regularity theory and the generalized Lyusternik theorem, we obtain the fol-lowing assertion. If the mapping F isp-regular (p= ) at the pointx∗= (, ) with re-spect to the elementh= [hz,hε], wherehz=zsint,z∈R,hε=ε, then there exist solutions

x=x(t,ε) =εzsint+r(ε), r(ε) =o(ε) of equation () forε∈(–σ,σ), whereσ>  is suffi-ciently small. Below we will describe a -factor operator and show its surjectivity.

LetP=PYbe a projection ontoY=ImFx(, ) andP=PYbe a projection ontoY=

KerF_x(, ) alongY. The -factor operator has the form



(, ), [hz,hε]

=f_(, ) +f_(, )[hz,hε]

=PF_x(, ) +PF(, )[hz,hε]

=F_x(, ) +PF(, )[hz,hε]

becausePF_x=F_x.

Using the second derivative ofF, we obtain, for [hu,hλ]∈C[,π]×R,



(, ), [hz,hε]

[hu,hλ] =

d_h

u

dt +hu+P[hzhu+hεhu+hzhλ]

=d _h

u

dt +hu+ 

πhzhu+hεhu+hzhλ,sintsint

=d _h

u

dt +hu+ 

πsint

π



(hzhu+hεhu+hzhλ)sint dt.

Now we describe the -kernel of the operator ((, ),·)

H(, ) =h= [hz,hε]∈C[,π]×R: 

(, ),h[h] = .

Assume thath= [hz,hε]∈KerF_x(, ),i.e.,F_x(, )[hz,hε] = . Thenhz=zsint,z∈R,hε∈ Rand



(, ), [hz,hε]

[hz,hε] =PF(, )[hz,hε]=



πsint

π



h_z+ hzhε

sint dt= 

⇔

π



h_z+ hzhε

sint dt= 

⇔ z

π



sint dt+zhε

π



sint dt= 

⇔ 

z ₊π

zhε=  ⇔ z= ∨z= –  πhε

⇔ hz= ∨hz= –



πhεsint.

Next we verify that the mapping ((, ), [hz,hε]) is surjective ontoC[,π] for elements

belonging to the setH(, ), that is,

∀y∈C[,π]∃[hu,hλ]∈C[,π]×Rsuch that 

(, ), [hz,hε]

[hu,hλ] =y.

Consider the first case [hz,hε] = [,ε],ε= .

Assume that

d_h

u(t)

dt +hu(t) + 

πsint

π



εhu(t)sint dt=y=y+y, ()

where

y=d _h

u(t)

dt +hu(t) =Py∈Y=ImF

x(, ),

y= 

πsint

π



εhu(t)sint dt=Py∈Y=KerFx(, ).

Puttingy=asintand using Lemma , it suffices to takehu=bsint∈KerFx(, ). Then

from () we obtainy=  and _πεsint_πbsint dt=asintfrom whereb=a_ε,hu=a_εsint.

The solutions of equation () exist, and hence ((, ), [hz,hε]) is surjective.

Verify the second case [hz,hε] = [–_π εsint,ε],ε= . Consider the equation

d_h

u(t)

dt +hu(t) + 

πsint

π



ε

– πsint

hu(t) +εhu(t)

+ε

– πsint

hλ

sint dt=y=y+y. ()

Puttingy=asint, using Lemma  and takinghu=bsint∈KerFx(, ), we havey=  and

–π

 hλεsint+ 

πεsint

π



– πsin

_t+sint

bsint dt=asint.

Henceb= –_πhλ–a_ε, andhu= (–_πhλ–a_ε)sint. The solutions of equation () exist, hence

((, ), [hz,hε]) is surjective in this case too. Therefore the mappingFis -regular at the

pointx∗= (, ) with respect to the elementh= [hz,hε].

Using the generalized Lyusternik theorem, we can describe the solutions belonging to the tangent coneTM(, ). For [hz,hε] = [,ε] we havex(t,ε) =r(ε) and for [hz,hε] =

[–

π εsint,ε] we havex(t,ε) = – 

π εsint+r(ε), where ri(ε) =o(ε) fori= ,  andε∈ (–σ,σ),σ>  is sufficiently small. Thus we obtain the following theorem.

Theorem  Equation()has two solutions xi(t,ε),i= , ,such that

x(t,ε) =r(ε),

x(t,ε) = –

π εsint+r(ε),

Example  Equation (),i.e.,

F(x,ε) =x+x+εxp–+xp= ,

where F:C_[,π]×R→C[,π],p≥, can be investigated analogously. Similarly, we obtain the following result.

Theorem  Equation()has a nonzero solution x(t,ε)such that

x(t,ε) = –ε

pcpsint+r(ε),

where

cp=

π

 sin

p_{t dt}

π

 sin

p+_{t dt}, r(ε)=o(ε)

andε∈(–σ,σ),σ> is sufficiently small.

Example  Consider now equation ()

F(u,ε) =u– (ε– )g(u) = ,

whereF:C

[]×R→C[],= [,π]×[,π],is the Laplacian, with the assump-tiong() = ,g() = ,g()= . The pointx∗= (, ) is the trivial solution of the above

equation.

We putu=u(y) =u(y,y).

The first derivative of the mappingFis the following:

F(u,ε) = F_u(u,ε),F_ε(u,ε)= – (ε– )g(u), –g(u).

In our caseF_u(, ) =+ .

Note thatKerF_u(, ) ={u∈C_[] :u+ u= }. This space is spanned by the L orthonormal functions (see [], p.)

u=

πsinysiny, u=



π sinysiny

(that is,ui,uj=

uiujdydyand

u



idydy= ,

uiujdydy=  fori=j).

More-over,

C[] =ImF_u(, )⊕KerF_u(, )

and

ImF_u(, )⊥=KerF_u(, ).

Fromp-regularity theory and the generalized Lyusternik theorem, we obtain the following assertion. If the mappingFisp-regular at the pointx∗= (, ) with respect to the element

ε(zu+zu) +r(ε), r(ε) =o(ε) of equation () forε∈(–σ,σ), whereσ>  is sufficiently small.

LetP=PYbe the projection ofYontoY=ImFu(, ), and letP=PYbe the

projec-tion ofYontoY=KerF_u(, ) alongY. We define the -factor operator:



(, ),h= 

(, ), [hz,hε]

=f_(, ) +f_(, )[hz,hε]

=PF_u(, ) +PF(, )[hz,hε]

=F_u(, ) +PF(, )[hz,hε]

becausePF_u=F_u.

Using the second derivative ofF, that is,

F(u,ε) =

–(ε– )g(u) –g(u) –g(u) 

, F(, ) =

g() – – 

,

F(, )[hz,hε] =

g() – – 

[hz,hε] = g()hz–hε, –hz

,

we obtain, for [hu,hλ]∈C[]×R,

F(, )[hz,hε][hu,hλ] = g()hz–hε, –hz

[hu,hλ] = g()hzhu–hεhu–hzhλ,

and hence

F(, )[hz,hε]= g()hz– hzhε.

From the above calculations it follows that the value of the -factor operator ((, ), [hz,hε]) on an elementx= [hu,hλ]∈C_[]×Ris equal to



(, ),h[x] = 

(, ), [hz,hε]

[hu,hλ]

=hu+ hu+P g()hzhu–hεhu–hzhλ

=hu+ hu+



i=

g()hzhu–hεhu–hzhλ,ui

ui.

Now we describe the -kernel of the operator ((, ),·), which is the set H(, ) =h= [hz,hε]∈C_[]×R: 

(, ),h[h] = .

Assuming that [hz,hε] = [zu+zu,ε]∈KerFu(, ), we have

F_u(, )[hz,hε] = 

and



(, ), [hz,hε]

[hz,hε]

=PF(, )[hz,hε]=



i=

F(, )[hz,hε],ui

=  i= ui

g()h_z– hzhε

uidydy

=  i= ui

g()(zu+zu)– (zu+zu)εuidydy

=  i= ui

g()

ui(zu+zu)dydy– ε

ui(zu+zu)dydy

=  i= ui

g()

ui(zu+zu)dydy– εzi

.

Next, introducing the quadratic forms

Qi(hz,ε) =Qi(z,z,ε) = g()

ui(zu+zu)dydy– εzi, i= , ,

we obtain the following system of equations for elements belonging to the setH(, ):

Q(hz,ε) =Q(z,z,ε) = g()

u(zu+zu)

_dy

dy– εz= , Q(hz,ε) =Q(z,z,ε) = g()

u(zu+zu)

_dy

dy– εz= 

()

or, equivalently,

g()(u_z_+ zzu_u+z_uu_)dydy– εz= , g()(uu_z_+ zzuu_+z_u_)dydy– εz= .

Simple calculations give us

a=

u_dydy=

u_dydy=  ·



π,

b=

u_udydy=

uu_dydy= – ·



π,

and we obtain the following form of system ():

g()(az

+ bzz+bz) – εz= , g()(bz

+ bzz+az) – εz= ,

()

or puttingC_ijk= g()uiujukdydy, we get



j,k=

Cjk_i zjzk– εzi= , i= , .

Note that ifM(z,z) is the × symmetric matrix whose (i,j) entry is

aij=



m=

uiujumdydy

that is,

M(z,z) =

az+bz bz+bz bz+bz bz+az

,

then system () is equivalent to the following equation:

g()M(z) – εI

z z

= , whereI=

   

.

Thus we get the following condition: elements [zu +zu,ε]∈H(, ) (that is, ele-ments (z,z,ε) are the solutions of system ()) if and only if either (z,z) is an eigenvec-tor of g()M(z) corresponding to the eigenvalue εor (z,z) = (, ). We will prove that for [u+ u,ε]∈H(, ), the mappingF is always -regular and for any other [zu+zu,ε]∈H(, ),Fis -regular ifεis not an eigenvalue of g()M(z).

Note thatQ(z,z,ε) = (Q(z,z,ε),Q(z,z,ε)) is a quadratic mapping associated to the -factor operator ((, ),h). If [zu+zu,ε]∈H(, ), then the first differentiation of Qgives us the matrix form of the operator ((, ),h). Consider the calculations:

Q(z,z,ε) =

g()(za+ zb) – ε g()(zb+ zb) –z g()(zb+ zb) g()(zb+ za) – ε –z

,

Q(z,z,ε) =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎣

g()a g()b – g()b g()b 

–   ⎤ ⎥ ⎦ ⎡ ⎢ ⎣

g()b g()b  g()b g()a –

 –  ⎤ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,

and ((, ),h) =Q(z,z,ε)

_z  z ε .

It follows that



(, ),h=

g()(za+ zb) – ε g()(zb+ zb) –z g()(zb+ zb) g()(zb+ za) – ε –z

, ()

that is, ((, ),h) =Q.

For an arbitrary elementh= (z,z,ε)∈H(, ), we examine the surjectivity of the map-ping ((, ),h), that is, the following condition:

∀η∈C[]∃ξ∈C_[]×Rsuch that 

(, ),h[ξ] =η.

Forξ= (ξ,ξ,λ),η= (η,η), we have



(, ),h

⎡ ⎢ ⎣ ξ ξ λ ⎤ ⎥ ⎦ =

[g()(za+ zb) – ε]ξ+ [g()(zb+ zb)]ξ– zλ g()(zb+ zb)ξ+ [g()(zb+ za) – ε]ξ– zλ

or

g()

az+ bz bz+ bz bz+ bz bz+ az

– ε

   

ξ

ξ

– λ

z z

=

η

η

and

 g()M(z) –εI

ξ

ξ

= λ

z z

+

η

η

. ()

Observe that for [u+ u,ε],ε= ,Fis always -regular (since the matrix () has the maximal rank). Now putε=  in (). Then we getz=z= . This means that in this case H(, ) ={}and thenFis -regular (in this trivial case, the equationF(u,ε) =  has the unique solutionu= ). Consider the other [zu+zu,ε]∈H(, ). Then equation () has the solution if det[g()M(z) –εI]= . This means thatεis not an eigenvalue of g()M(z).

From the above consideration, we obtain the following sufficient condition for -regular-ity: the matrix g()M(z) has no eigenvaluesεand ε. This condition can be tested using the notion of resultant. We evidently have g()= . Letλand λbe the eigenvalues, and letλ= . Then we obtain

det M(z,z) –λI

=det

az+bz bz+bz bz+bz bz+az

–λ

   

=λ+Aλ+A= , ()

whereA= –(a+b)(z+z) andA= (ab–b)z_+ (a–b)zz+ (ab–b)z_. Likewise,

det M(z,z) – λI

= λ+ Aλ+A= . ()

Subtracting equation () from () gives λ+A=  (becauseλ= ). The condition that an eigenvalue multiplied by two is not an eigenvalue means that the polynomialsf =λ+ Aλ+Aandg= λ+Ado not have a common root. This is equivalent to the nonvanishing of the resultant:

R(f,g) =

 A A  A    A

=Az

+ Bzz+Az.

This resultant is a quadratic form withA= (ab–b_{) – (a+b)}_,B= [(a

_–b_{) – (a+}

b)_{]. Thus a sufficient condition of -regularity is that this form is positively or negatively} defined. Therefore one obtainsA–B> . In our case, one findsA= [a– ab+ b]≈ .·(_π),B=_[b– a+ ba]≈.·(_π)andA>B, so the test with resultants

is effective.

Theorem  Equation()has a nonzero solution u(ε)such that

u(ε) =ε(zu+zu) +r(ε), r(ε)=o(ε)

forε∈(–σ,σ),whereσ> is sufficiently small. Conclusion

The paper was inspired by Buchner, Marsden and Schecter’s article []. The authors con-sider the bifurcation problemF(x,λ) =Lx+ (λ–λ)x+R(x) = , whereLis the elliptic self-adjoint operator on a suitable Banach spaceYof functions, with another suitable Banach space of functionX- the domain ofL⊂Y,R:X→Yis a smooth map withR() =  and R() = ,λ- is an eigenvalue ofLof multiplicityn,x∈Xandλ∈R. They use Lyapunov-Schmidt procedure to examine the above equation and show that (, ) is a bifurcation point and that there exist solutions different from (,λ). Our examples are special cases of the above problem, and we usep-regularity theory to prove the existence of solutions and give an approximative description of the solution set. The structure of the solution set is reduced to a study of the system of homogeneous algebraic equations.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1_{Siedlce University of Natural Sciences and Humanities, 3-go Maja 54, Siedlce, 08-110, Poland.}2_{System Research Institute,}

Polish Academy of Sciences, Newelska 6, Warsaw, 01-447, Poland.3_{Dorodnicyn Computer Center, Russian Academy of}

Sciences, Vavilova 40, Moscow, 119991, Russia.

Acknowledgements

This work was supported by the Russian Foundation for Basic Research Grant No. 11-01-00786-a, by the Leading Research Schools Grant No. NSH-5264.2012.1 and by the Russian Academy of Sciences Presidium Program P-18.

Received: 27 May 2013 Accepted: 28 October 2013 Published:22 Nov 2013

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10.1186/1687-2770-2013-251

Cite this article as:Grzegorczyk et al.:Application ofp-regularity theory to nonlinear boundary value problems.