Weak homoclinic solutions of anisotropic difference equation with variable exponents

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R E S E A R C H

Open Access

Weak homoclinic solutions of anisotropic

difference equation with variable exponents

Aboudramane Guiro

1

_{, Blaise Kone}

2

_{and Stanislas Ouaro}

3*

*_{Correspondence: [email protected];}

[email protected]

3_{Laboratoire d’Analyse}

Mathématique des Equations (LAME), UFR, Sciences Exactes et Appliquées, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso Full list of author information is available at the end of the article

Abstract

In this paper, we prove the existence of homoclinic solutions for a family of anisotropic diﬀerence equations. The proof of the main result is based on a minimization method and a discrete Hölder type inequality.

MSC: 47A75; 35B38; 35P30; 34L05; 34L30

Keywords: diﬀerence equations; homoclinic solutions; discrete Hölder type inequality

1 Introduction

In this paper, we study the following nonlinear discrete anisotropic problem: ⎧

⎨ ⎩

–(a(k– ,u(k– ))) +|u(k)|p(k)–_u₍_k_{) =}_f₍_k_), _k_∈_Z_, lim_|k|→∞u(k) = ,

(.)

whereu(k) =u(k+ ) –u(k) is the forward diﬀerence operator.

The problem (.) is a class of partial difference equations which usually describe the evo-lution of certain phenomena over the course of time. Elementary but relevant examples of partial difference equations are concerned with heat diffusion, heat control, temperature distribution, population growth, cellular neural networks,etc.(see [–]). Our interest for problems of the type (.) is motivated by major applications of differential and difference operators to various applied fields such as electrorheological (smart) fluids, space tech-nology, robotics, image processing,etc.On the other hand, they are strongly motivated by their applicability to mathematical physics and biology.

The goal of the present paper is to establish the existence of homoclinic solutions for the problem (.). In the theory of diﬀerential equations, a trajectoryx(t), which is asymptotic to a constant as|t| → ∞, is called a doubly asymptotic or homoclinic orbit. The notion of a homoclinic orbit was introduced by Poincaré [] for continuous Hamiltonian systems. Since we are seeking for solutionsuof the problem (.) satisfyinglim|k|→∞u(k) = , then

according to the notion of a homoclinic orbit by Poincaré [], we are interested in ﬁnding homoclinic solutions for the problem (.). We remember that boundary value problems involving diﬀerence operators with constant exponents have been intensively studied in the last decade (see [–] for details). The existence of homoclinic solutions where stud-ied in particular by the authors in []. The variable exponent cases were studstud-ied by some authors in [–] and the references therein. Other very recent applications of variable

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exponent equations are presented in [–]. The study of such kind of problems can be placed at the interface of certain mathematical ﬁelds such as nonlinear diﬀerential equa-tions and numerical analysis.

The variational approach to the study of homoclinic solutions in the context of variable exponent was ﬁrstly done by Mihailescu, Radulescu and Tersian in []. They studied the following problem:

⎧ ⎨ ⎩

_p₍_k_–)u(k– ) –V(k)|u(k)|q(k)–_u₍_k_{) +}_f₍_k_,_u₍_k_{)) = ,} _for_k_∈_Z_, lim_|k|→∞u(k) = ,

(.)

where_p₍_·₎stands for thep(·)-Laplace diﬀerence operator, that is,

_p₍_k_–)u(k– ) =u(k)p(k)–u(k) –u(k– )p(k–)–u(k– ),

for eachk∈Z.

In this paper, we use the minimization technique to get the existence of homoclinic solutions of (.).

The paper is organized as follows. In Section , we deﬁne the functional spaces and prove some of their useful properties, and ﬁnally, in Section , we prove the existence of homoclinic solutions of (.).

2 Auxiliary results

We will use the following notations from now on:

p+=sup

k∈Z

p(k) and p–=inf

k∈Zp(k).

For the dataf anda, we assume the following:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

a(k,·) :R→R ∀k∈Zand there exists a mapping

A:Z×R→Rwhich satisﬁes:

a(k,ξ) =_∂ξ∂A(k,ξ), ∀k∈Z and A(k, ) = , ∀k∈Z.

(.)

|ξ|p(k)≤a(k,ξ)ξ≤p(k)A(k,ξ) ∀k∈Zandξ∈R. (.)

There exists a positive constantCsuch that

a(k,ξ)≤C

j(k) +|ξ|p(k)–, (.)

for allk∈Zandξ∈R, wherej∈lp(·)_{(a space to be deﬁned later) with} 

p(k)+ 

p(k)= .

f ∈lp(·), (.)

with _p₍_k₎+_p₍_k₎ = , for allk∈Z.

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We now introduce the spaces:

lp(·)= u:Z→R;ρp(·)(u) :=

k∈Z

_u(k)p(k)<∞

and

W,p(·)₌ _u_:_Z_→_R_;_ρ

,p(·)(u) :=

k∈Z

u(k)p(k)+

k∈Z

u(k)p(k)<∞

=u:Z→R;u∈lp(·)andu(k)∈lp(·). Onlp(·)_{, we introduce the Luxemburg norm}

up(·):=inf λ> ;

k∈Z

u(_λk)p(k)≤

.

Then

u,p(·):=inf λ> ;

k∈Z

u(_λk)p(k)+

k∈Z

u_λ(k)p(k)≤

=up(·)+up(·) is a norm on the spaceW,p(·). Remark . Ifu∈lp(·)_{, then}_lim

|k|→+∞u(k) = . Indeed, ifu∈lp(·), thenk∈Z|u(k)|p(k)< ∞. Let

k∈Z

u(k)p(k)=

k∈S

u(k)p(k)+

k∈S

u(k)p(k),

where

S=

k∈Z;u(k)< 

and

S=

k∈Z;u(k)≥.

Sis necessarily a ﬁnite set and|u(k)|<∞for anyk∈Ssinceu∈lp(·). We also have that_k_∈_S

|u(k)|

p+_≤

k∈Z|u(k)|p(k), then

k∈S|u(k)|

p+ _<_∞_{. As}_S

is a ﬁnite set, then_k_∈_S

|u(k)|

p+

<∞, which implies that

k∈Z

u(k)p+<∞.

Thus,

lim

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Proposition . Under condition (.),ρp(·)satisﬁes: (a) ρp(·)(u+v)≤p+(ρp(·)(u) +ρp(·)(v));∀u,v∈lp(·). (b) Foru∈lp(·)_{, if}_λ_{> , we have}

ρp(·)(u)≤λρp(·)(u)≤λp

–

ρp(·)(u)≤ρp(·)(λu)≤λp

+ ρp(·)(u), and if <λ< , we have

λp+ρp(·)(u)≤ρp(·)(λu)≤λp

–

ρp(·)(u)≤λρp(·)(u)≤ρp(·)(u).

(c) For every ﬁxedu∈lp(·)\ {},ρp(·)(λu)is a continuous convex even function inλ, and it increases strictly whenλ∈[,∞).

Proof

(a) Letu,v∈lp(·)_{, we have}

ρp(·)(u+v) =

k∈Z

u(k) +v(k)p(k)

=

k∈Z

p(k) u(k) +

 v(k)

p(k)

≤

k∈Z

p(k)

 u(k)

p(k) +

v(k)

p(k)

≤p+–ρp(·)(u) +ρp(·)(v)

≤p+ρp(·)(u) +ρp(·)(v).

(b) Foru∈lp(·)_{, if}_λ_{> , we have}

ρp(·)(λu) =

k∈Z

λu(k)p(k)

=

k∈Z

λp(k)u(k)p(k)

≥λp–

k∈Z

u(k)p(k)=λp–ρp(·)(u).

We also haveρp(·)(λu) =k∈Zλp(k)|u(k)|p(k)≤λp

+

k∈Z|u(k)|p(k)=λp +

ρp(·)(u). If  <λ< , we have

ρp(·)(λu) =

k∈Z

λu(k)p(k)

=

k∈Z

λp(k)u(k)p(k)

≥λp+

k∈Z

u(k)p(k)=λp+ρp(·)(u).

We also haveρp(·)(λu) =k∈Zλp(k)|u(k)|p(k)≤λp

–

k∈Z|u(k)|p(k)=λp –

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(c) For every ﬁxedu∈lp(·)_{\ {}__}_and_α_∈_{(, ), we have}_∀_v_∈_lp(·)

ρp(·)

αu+ ( –α)v≤

k∈Z

αu(k)p(k)+( –α)v(k)p(k)

≤

k∈Z

αp(k)u(k)p(k)+ ( –α)p(k)v(k)p(k)

≤

k∈Z

αu(k)p(k)+ ( –α)v(k)p(k)

≤αρp(·)(u) + ( –α)ρp(·)(v).

This proves thatρp(·)(u) is convex. Letλ,λ≥ such thatλ<λ. We have

ρp(·)(λu) =

k∈Z

λu(k)

p(k)

=

k∈Z

λp_(k)u(k)p(k)

<

k∈Z

λp_(k)u(k)p(k)=ρp(·)(λu).

Thus, for every ﬁxedu∈lp(·)\ {},ρp(·)(λu) increases strictly whenλ∈[, +∞).

For the continuity ofρp(·)(λu), let (λn)n∈Nbe a real sequence such thatλn→λasn→

+∞. We denotegn,k=|λnu(k)|p(k). We have

k∈Z

|gn,k| ≤ |λn|β

k∈Z

u(k)< +∞,

where

β= ⎧ ⎨ ⎩

p+ if|λn| ≥,

p– _if_|_λ

n|< .

Then,

lim

n→+∞

k∈Z

|λnu|p(k)

=

k∈Z

lim

n→+∞|λnu|

p(k)₌

k∈Z |λu|p(k).

Therefore, we have the continuity ofρp(·).

Proposition . Let u∈lp(·)_{\ {}__}_{, then}_u

p(·)=a if and only ifρp(·)(ua) = .

Proof Let us denoteA={λ> ;ρp(·)(uλ)≤}.

Case :up(·)=a. Then, there exists a sequence (λn)n∈N⊂Asuch thatλn↓aasn→

+∞.

Therefore, asρp(·)(λun)≤ for alln∈Nandρp(·)(λu) is continuous with respect toλ, then

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Suppose now thatρp(·)(ua) < . Asa> , then there existsα>  such that  <α<a. Then

which is a contradiction. Thus,

up(·)=a.

Proposition . If u∈lp(·)and p+< +∞, then the following properties hold:

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where

β= ⎧ ⎨ ⎩

p– ifunp(·)≥,

p+ _if_u

np(·)< . Sounp(·)→ asn→+∞.

Case .unp(·)→ asn→+∞, then

ρp(·)(un) =ρp(·)

un

unp(·) ×

unp(·)

≤ un

β

p(·)ρp(·)

un

unp(·)

≤ un

β

p(·), where

β= ⎧ ⎨ ⎩

p+ _if_u

np(·)≥,

p– _if_u

np(·)< .

Soρp(·)(un)→ asn→+∞.

Proposition . Let u∈W,p(·)_{\ {}__}_{, then}_u

,p(·)=a⇔ρ,p(·)(ua) = .

Proof

Case .u,p(·)=a. Then

u,p(·)=a ⇔ up(·)+up(·)=a

⇔ up(·)=a and up(·)=a witha+a=a,

wherea,a> . Thus

ρ,p(·)

u a

=ρp(·)

u a

+ρp(·)

u a

≤a

aρp(·)

u a

+a

aρp(·)

u a

≤a

a + a

a = .

Therefore, by mimicking the proof of Proposition ., we deduce thatρ,p(·)(ua) = .

Case .ρ,p(·)(u_a) = . As in the ﬁrst case, we getu,p(·)=a. Proposition . If u∈W,p(·)_{and p}+_{< +}_∞_{, then the following properties hold:}

() u,p(·)<  (= ; > )⇔ρ,p(·)(u) <  (= ; > ); () u,p(·)> ⇒ up

–

,p(·)≤ρ,p(·)(u)≤ up,+p(·); () u,p(·)< ⇒ up

+

,p(·)≤ρ,p(·)(u)≤ up,–p(·); () un,p(·)→⇔ρ,p(·)(un)→asn→+∞.

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Theorem .(Discrete Hölder type inequality) Let u∈lp(·)_{and v}_∈_lq(·)_{be such that}  Young inequality, we deduce that

3 Existence of weak homoclinic solutions

In this section, we investigate the existence of weak homoclinic solutions of (.).

Deﬁnition . A weak homoclinic solution of (.) is a functionu∈W,p(·)_{such that}

Note that weak solutions are usual solutions of the problem (.). It can be seen by taking the test elementsvk= (. . . , , , , , , . . .) with  on (kth) place.

The main result of this work is the following.

Theorem . Assume that (.)-(.) hold. Then, there exists at least one weak homoclinic solution of (.).

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Proposition . The functional J is well deﬁned onW,p(·) _{and is of class C}₍_W,p(·)_,_R₎

with the derivative given by

J(u),v=

k∈Z

ak– ,u(k– )v(k– )

+

k∈Z

u(k)p(k)–u(k)v(k) –

k∈Z

f(k)v(k), (.)

for all u,v∈W,p(·)_.

Proof We denote by

I(u) =

k∈Z

Ak– ,u(k– ), L(u) =

k∈Z



p(k)u(k)

p(k)

and

(u) =

k∈Z

f(k)u(k).

We have by using Young inequality and assumptions (.) and (.) that

I(u)=

k∈Z

Ak– ,u(k– )

≤

k∈Z

Ak– ,u(k– )

≤

k∈Z

C

j(k– ) + 

p(k– )u(k– )

p(k–)–

u(k– )

≤

k∈Z

Cj(k– )u(k– )+

k∈Z C

p(k– )u(k– )

p(k–) <∞,

L(u)=

k∈Z



p(k)u(k)

p(k)

≤  p–

k∈Z

u(k)p(k)<∞

and

(u)=

k∈Z

f(k)u(k)≤

k∈Z

f(k)u(k)<∞.

Therefore,Jis well deﬁned. Clearly,I,Land are inC(W,p(·),R). Let us chooseu,v∈W,p(·)_{. We have}

I(u),v= lim

δ→+

I(u+δv) –I(u)

δ ,

L(u),v= lim

δ→+

L(u+δv) –L(u)

δ

and

(u),v= lim

δ→+

(u+δv) – (u)

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Let us denotegδ=A(k–,u(k–)+δv(k_δ–))–A(k–,u(k–)). We get, by using Young inequality,

k∈Z |gδ| ≤

δ

k∈Z

Ak– ,u(k– ) +δv(k– )+

δ

k∈Z

Ak– ,u(k– )< +∞.

Thus

lim

δ→+

I(u+δv) –I(u)

δ =δlim→+

k∈Z

A(k– ,u(k– ) +δv(k– )) –A(k– ,u(k– ))

δ

=

k∈Z

lim

δ→+

A(k– ,u(k– ) +δv(k– )) –A(k– ,u(k– ))

δ

=

k∈Z

ak– ,u(k– )v(k– ).

By the same method, we deduce that

lim

δ→+

L(u+δv) –L(u)

δ =δlim→+

k∈Z

|u(k) +δv(k)|p(k)_–_|_u₍_k₎_|p(k)

p(k)δ

=

k∈Z

lim

δ→+

|u(k) +δv(k)|p(k)_–_|_u₍_k₎_|p(k)

p(k)δ

=

k∈Z

u(k)p(k)–u(k)v(k)

and

lim

δ→+

(u+δv) – (u)

δ =δlim→+

k∈Z

f(k)(u(k) +δv(k)) –f(k)u(k)

δ

=

k∈Z

lim

δ→+

f(k)(u(k) +δv(k)) –f(k)u(k)

δ

=

k∈Z

f(k)v(k).

Lemma . The functional I is weakly lower semi-continuous.

Proof From (.),Iis convex with respect to the second variable. Thus, by Corollary III. in [], it is enough to show thatI is lower semi-continuous. For this, we ﬁxu∈W,p(·) and> . SinceIis convex, we deduce that, for anyv∈W,p(·)_,

I(v)≥I(u) +I(u),v–u

≥I(u) +

k∈Z

ak– ,u(k– )v(k– ) –u(k– )

≥I(u) –C



p– + 

p–

gp(·)(u–v)_p₍_·₎, whereg(k) =j(k) +u(k)p(k)–

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≥I(u) –Ku–v,p(·)

≥I(u) –,

for allv∈W,p(·)_with_u_–_v

,p(·)<δ=_K.

Hence, we conclude thatIis weakly lower semi-continuous.

Proposition . The functional J is bounded from below, coercive and weakly lower semi-continuous.

Proof By Lemma .,Jis weakly lower semi-continuous. We will only prove the coercive-ness of the energy functionalJand its boundedness from below.

J(u) =

k∈Z

Ak– ,u(k– )+

k∈Z



p(k)u(k)

p(k)

–

k∈Z

f(k)u(k)

≥

k∈Z



p(k– )u(k– )

p(k–)

+

k∈Z



p(k)u(k)

p(k)

–

k∈Z

f(k)u(k)

≥ 

p+

k∈Z

u(k– )p(k–)+

k∈Z



p(k)u(k)

p(k)

–cfp(·)up(·)

≥ 

p+ρ,p(·)(u) –cfp(·)u,p(·).

To prove the coerciveness ofJ, we may assume thatu,p(·)>  and we deduce from the above inequality that

J(u)≥ 

p+u

p–

,p(·)–cfp(·)u,p(·). Thus,

J(u)→+∞ as u,p(·)→+∞.

As J(u)→+∞whenu,p(·)→+∞, then foru,p(·) > , there existsc∈Rsuch that

J(u)≥c. Foru,p(·)≤, we have

J(u)≥ 

p+ρ,p(·)(u) –cfp(·)u,p(·)

≥–cfp(·)u,p(·)

≥–cfp(·)> –∞.

ThusJis bounded below.

We can now give the proof of Theorem ..

Proof of Theorem . By Proposition .,Jhas a minimizer which is a weak homoclinic

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

AG conceived and carried out the study. BK conceived and carried out the study. SO conceived and carried out the study.

Author details

1_{Laboratoire d’Analyse Mathématique des Equations (LAME), Institut des Sciences Exactes et Appliquées, Université}

Polytechnique de Bobo Dioulasso, 01 BP 1091 Bobo-Dioulasso 01, Bobo Dioulasso, Burkina Faso.2_{Laboratoire d’Analyse}

Mathématique des Equations (LAME), Institut Burkinabé des Arts et Métiers, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso.3_{Laboratoire d’Analyse Mathématique des Equations (LAME), UFR, Sciences}

Exactes et Appliquées, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso.

Acknowledgements

This work was done within the framework of the visit of the authors at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy. The authors thank the mathematics section of ICTP for their hospitality and for ﬁnancial support and all facilities.

Received: 22 May 2012 Accepted: 9 August 2012 Published: 5 September 2012 References

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doi:10.1186/1687-1847-2012-154