Existence of solutions for a class of operator equations

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R E S E A R C H

Open Access

Existence of solutions for a class of

operator equations

Xiaojing Feng

*

*_{Correspondence:}

[email protected] School of Mathematical Sciences, Shanxi University, Wucheng Road, Taiyuan, 030006, People’s Republic of China

Abstract

In this paper we deal with the existence and multiplicity of nontrivial solutions to a class of operator equation. By using inﬁnite dimensional Morse theory, we establish some conditions which guarantee that the equation has many nontrivial solutions.

MSC: 47H10; 54E50

Keywords: inﬁnite dimensional Morse theory; nontrivial solutions; critical group

1 Introduction

LetE=C[, ] be the usual real Banach space with the normu=maxt∈[,]|u(t)|for all u∈C[, ], andH=L_{[, ] be the usual real Hilbert space with the inner product (}_·_,_·_{) and}

the norm · . Obviously,Eis embedded continuously intoH, denoted byE→H. This paper is concerned with the existence of nontrivial solutions for the following op-erator equation of the form

u=Kfu, (.)

wheref :R→R is continuous and that f_x, the ﬁrst-order derivative off inx, is also continuous onR_{, f :}_E_→_E_{is deﬁned as f}_u₍_t_{) =}_f₍_u₍_t_)),_∀_u_∈_E_; _K_:_H_→_E_→_H _{is a}

compact symmetric positive linear operator with  /∈σp(K), whereσp(K) denotes all the

eigenvalues ofK. In recent years, there have been many papers to study the existence of nontrivial solutions on higher order boundary value problems, see [–]. In [], by using spectral theory and the fixed point theorem, Li established some conditions forf to guar-antee that the problem has a unique solution. In a later paper [], by applying the strongly monotone operator principle and the critical point theory, some new existence theorems on unique, at least one nontrivial and infinitely many solutions were established. Moti-vated by the above papers, in this paper, we try to discuss equation (.) by using Morse theory. Specifically, we consider the existence and multiplicity of the solutions for (.) and obtain at least two nontrivial solutions, three nontrivial solutions and five nontrivial solutions, respectively. And then we apply the abstract results to a fourth-order boundary value problem.

In this paper, we consider the existence of solutions to equation (.) by applying Morse theory. Our methods are diﬀerent from those in the literature mentioned above. As is well

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known, this kind of theory is based on deformation lemmas. In general the functional needs to satisfy a compactness condition. In this article, we use the Palais-Smale (PS) con-dition: LetDbe a real Banach space,J∈C(D,R). If any sequence{vk}∞ ⊂Dfor which {J(vk)}is bounded anddJ(vk)→θ inDask→ ∞possesses a convergent subsequence,

then we sayJsatisﬁes the Palais-Smale (PS) condition.

The paper is organized as follows. In Section , we present some preliminary knowl-edge about Morse theory. In Section , we apply Morse theory to give the proofs of The-orems .-. and provide some examples to illustrate the results.

2 Preliminary

Assume that{λk}∞ is the sequence of all eigenvalues ofK, where each eigenvalue is

re-peated according to its multiplicity, and {ek}∞ ⊂E is the corresponding orthonormal

eigenvector sequence in H. In the following, we outline some preliminary knowledge about Morse theory, which will be used in the proofs of our main results. Please refer to [] for more details.

LetHbe a real Hilbert space with the norm·and the inner product (·,·),J∈C(H,R). Suppose that (X,Y) is a pair of topological spaces andHq(X,Y) is theqth singular relative

homology group with coeﬃcients in an Abelian groupG. Alsoβq=rankHq(X,Y) is called

theq-dimension Betti number. Letpbe an isolated critical point ofJwithJ(p) =c,c∈R_,

andUbe a neighborhood ofpin whichJhas no critical points exceptp. The group

Cq(J,p) =Hq

Jc∩U,

Jc\{p}

∩U, q= , , , . . .

is called theqth critical group ofJatp, whereJc={u∈H:J(u)≤c}. We call the dimension

of a negative space corresponding to the spectral decomposition ofd_J₍_p_{) the Morse index}

ofp, denoted byind(J,p) (it can be∞). Andpis called a nondegenerate critical point if

d_J₍_p_{) has a bounded inverse.}

LetAbe a bounded self-adjoint operator deﬁned onH. According to its spectral de-composition,H=H+⊕H⊕H–, whereH±,Hare invariant subspaces corresponding to the positive/negative, and zero spectrum ofArespectively. We shall study the number of critical points of the functional

J(u) = 

(Au,u) +g(u),

or equivalently, the number of solutions of the operator equation

Au+dg(u) = .

The following assumptions are given:

(i) A±=A|H±has a bounded inverse onH±.

(ii) γ =dim(H⊕H–) <∞.

(iii) g∈C₍_H_,_R₎_{has a bounded and compact diﬀerential}_dg₍_x₎_{. In addition, if dim} H= , we assume

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Lemma .[] Under assumptions(i), (ii)and(iii),we have that () Jsatisﬁes the(PS)condition,and

() Hq(H,Ja) =δqrGfor–alarge enough,asJa∩K=∅.

Lemma .[] Under assumptions(i), (ii)and(iii),if J has critical points{pi}ni=with

γ ∈/

n

i=

m–(pi),m–(pi) +m(pi)

,

where m–(p) =index(J,p)and m(p) =dim kerdJ(p),then J has a critical pdiﬀerent from p, . . . ,pnwith Cr(J,p)= .

Lemma .[] Under assumptions(i), (ii)and(iii),if f has a nondegenerate critical point pwith Morse index m–(p)=γ,then f has a critical point p=p.Moreover,if

m(p)≤m–(p) –γ,

then f has one more critical point p=p,p.

Remark . In Lemmas . and ., ifdimH= , the boundedness ofdgcan be replaced by the following condition:

dg(u)=ou asu → ∞.

Lemma .[] Let J∈C₍_H_,_R₎_{be a function satisfying the}_(PS)_condition_._{Assume that} dJ=I–T,where T is a compact mapping,and that pis an isolated critical point of J.

Then we have

ind(dJ,p) = ∞

q=

(–)q_rank_C q(J,p).

Lemma .[] Assume that J∈C(H,R)is bounded from below,satisﬁes the(PS) condi-tion.Suppose that dJ=I–T is a compact vector ﬁeld,and pis an isolated critical point but not the global minimum withindex(dJ,p) =±.Then J has at least three critical points.

3 Proofs of main results

In this section, we will prove the main results.

Lemma . Suppose that{vk}∞ ⊂His bounded and that dJ(vk) = (I–KfK)vk→θ inH

as k→ ∞.Then J satisﬁes the(PS)condition.

Proof SinceK:H→Eis completely continuous, f :E→Eis bounded and continuous, andvk–KfKvk→θask→ ∞, we have that{vk}∞ has a convergent subsequence. ThusJ

satisﬁes the (PS) condition.

Theorem . Assume that f satisﬁes the condition

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(H) there existμ∈(, /)andR> such that <F(x)≤μxf(x)for all|x| ≥R,where F(x) =_xf(y)dy.

Then equation(.)possesses at least two nontrivial solutions.

Proof By condition (H), there existε∈(, ) andδ>  such that

F(x)≤  λ

( –ε)|x|, x∈[–δ,δ]. (.)

Letρ≤δ/M, whereM=C( ∞

k=λk)/. Then it follows from [] thatKv≤Mv ≤δ

for allv∈Bρ. Hence by (.) we have

J(v) = 

v

_– 



FKv(t)dt

≥ 

v

_– 

λ_( –ε) 



Kv(t)dt

= 

v

_– 

λ_( –ε)(Kv,v)

≥ 

v

_– 

λ 

( –ε)λ_v= εv

_, _v_∈_B

ρ,

that is,

J(v)≥ 

εv, v∈Bρ. (.)

It follows from (.) thatθis a local minimum. We now ﬁnd two nontrivial solutions. Let us deﬁne

f+(x) =

⎧ ⎨ ⎩

f(x), x≥,

, x< ,

and

J+(v) = 

v

_– 



F+Kv(t)dt,

whereF+(x) = x

f+(y)dy.

By condition (H), there existC,C>  such that

F+(x)≥C|x|/μ–C, x∈R.

Thus,

J+(τv) =

 τ

_v_– 

 F+

τKv(t)dt

≤ 

τ

_v_–_C_τ/μ_Kv/μ

/μ+C, v∈H.

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Now we shall prove that J+ satisﬁes the (PS) condition on H. Let {vk}∞ ⊂H with |J+(vk)| ≤βfor allk∈N\ {}and someβ> , anddJ+(vk) = (I–Kf+K)vk→θask→ ∞.

By Lemma ., we only claim that{vk}∞ is bounded.

In fact, notice that

dJ+(vk),vk

= (vk–Kf+Kvk,vk) =vk–





f+Kvk(t)

Kvk(t)dt.

According to condition (H), there existsC>  such that

F+(x)≤μxf+(x) +C, x∈R,

thus, we have

β≥J+(vk) =

 vk

_– 



F+Kvk(t)

dt

≥ 

vk

_–_μ 



f+Kvk(t)

Kvk(t)dt–C

= (/ –μ)vk+μ

dJ+(vk),vk

–C

≥(/ –μ)vk–μdJ+(vk)vk–C, k∈N\ {}.

SincedJ+(vk)→θ ask→ ∞, there existsN∈N\ {}such that β≥(/ –μ)vk–vk–C, k>N.

Thus{vk}∞ ⊂His bounded. Again,J+∈C(H,R) satisﬁes the (PS) condition. We also

have

J+(se)→–∞, s→+∞.

On the other hand, by the same way to (.), we have

J+|∂Bρ> .

The mountain pass lemma is applied to obtain a critical pointv+∈HofJ+, with critical valuec+> , which satisﬁes

Kv+=Kf+Kv+.

By the positive property ofK, we haveKv+≥, sov+is again a critical point ofJ.

Analogously, we deﬁne

f–(x) = ⎧ ⎨ ⎩

f(x), x≤,

, x> ,

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Theorem . Assume that:

(H) f() = and≤f() < /λ;

(H) f(u) > and strictly increasing in u foru> ; (H) f(∞) =lim|u|→∞f(u)exists and lies in(/λ, /λ). Then(.)has at least three distinct solutions.

Proof Deﬁne a functionalJ:H→Ras

J(v) = 

v

_– 



FKv(t)dt, v∈H.

First, it is obvious thatθis a solution, which is also a strict local minimum of the functional

JonH.

By (H), for allε> , there existsδ> ,  <|x|<δ, such that|f(x)|< (/λ–ε)|x|, we have Kv≤Cv. Thus

J(u) = 

v

_– 



FKv(t)dt

≥ 

v

_–



/λ_–ε 



Kv(t)dt

≥ 

v

_–



/λ_–ελ_v

=ελv> ,  <u<δ/C.

Modifyf to be a new function

f(x) =

⎧ ⎨ ⎩

f(x), x≥,

, x< ,

and consider a new functional

J(v) = 

v

_– 



FKv(t)dt,

whereF(x) =_xf(s)ds. It is easily seen thatθis also a strict local minimum ofJ, which is aC_{functional with the (PS) condition.}

Indeed,

J(v) = 

v

_– 



FKv(t)dt

=

v

_– π



FKv(t)dt, v∈H.

It is well known thatθ is also a strict local minimum ofJ. We will demonstrate thatJ

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J(vn)→ asn→ ∞. We derive

o()v–_n=J(vn),v–n

=  v

– n  – π 

_f_Kv_n(t)Kv–_n(t)dt.

Hence{v–_n}is bounded. In the following, we will show that{v–_n}is also bounded by contra-diction. Settingun= v

+

n

v+

n.J(v

+

n) =J(vn) –J(v–n), andJ(v–n) is bounded. By (H), there exists C>  such that|f(x)|< /λ|x|+C, we have

  =

J(v+

n)

v+

n

+



 F(Kv+

n)

v+

n

dt

≤o() +  λ





Kun(t)



dt+ C

v+

n





Kun(t)dt

=o() + λ λ



 un(t)



dt+ C

v+

n





Kun(t)dt.

Thusun= ,n= , , . . . , and





unedt= (un,e) =

J(vn)

v+ n ,e – v– n v+ n ,e +   f(Kvn)

v+

n

edt

=o() +



 f(Kvn)

u+

n

Kedt

≥o() +/λ_+ελ_ 

 v+

n(t)

v+

n

edt–C   e v+ n dt

=o() + (/λ+ε)λ 



unedt.

Hence, we haveε_uedt≤ is a contradiction.

SinceJis unbounded from below, along the rayus=se(t),s> . Indeed,

J(us) =

 us

_– 



F(us)dt

=  us

_– 



F(us)dt

≤ 

us

_–

 λ_ +ε





Kus(t)

 dt+C





Kus(t)dt

= –ελ_s+Cλ 

 edt.

The mountain pass lemma is applied to obtain a critical pointu=θofJwhich solves the

equation

u(t) =Kfu(t), t∈[, ].

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Now we shall prove thatI–K_f₍_u₍_t_{)) has a bounded inverse operator on}_H_,_i.e._,_u_{is a}

nondegenerate critical point ofJ. Sinceusatisﬁes (.), it is also a solution of the equation

u(t) =Kg(t)u(t), t∈[, ],

whereg(t) =_f(su(t))ds. Letμ>μ>· · · be eigenvalues of the problem

Kfu(t)w(t) =μw(t), t∈[, ].

We shall prove thatμ>  >μ. This implies the invertibility of the operatorKf(u(t)).

In fact, according to assumption (H), we haveg(t) <f(u(t)),t∈[, ], such that

/μ=min

(w,w)



Kf(u(t))wdt

<min_ (w,w)

Kg(t)wdt ≤.

Again, by assumptions (H) and (H), we havef(u(t)) < /λ,t∈[, ]. According to the

Rayleigh quotient characterization of the eigenvalues

/μ=sup

E

inf

w∈E⊥_

(w,w)



Kf(u(t))wdt ≥ λ

 –λ_εsupE

inf

w∈E⊥_

(w,w)

π

 Kwdt

> ,

whereEis any one-dimensional subspace inH.

The Morse identity yields an odd number of critical points. Therefore there are at least

three solutions of (.). The proof is completed.

Theorem . Assume that:

(H) f() = and/λ<f() < /λ;

(H) f(∞) =limx→±∞f(x)andf(∞)–∈/σp(K)withf(∞) > /λ;

(H) |f(x)|< and≤f(x) < /λin the interval[–c,c],wherec=maxt∈[,]ϕ(t),andϕ(t)

is the solution of the equation

ϕ(t) =K.

Then(.)possesses at least ﬁve nontrivial solutions.

Proof Deﬁne

f(x) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

f(c), x>c,

f(x), |x| ≤c,

f(–c), u< –c,

and let

J(v) = 

v

_– 



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whereF(x) =_xf(y)dy. The truncated problem

u(t) =Kfu(t) (.)

possesses at least three solutionsθ,u,ubecause there are two pairs of subsolution and supersolution [εe,ϕ] and [–ϕ, –εe], whereeis the ﬁrst eigenfunction ofKandε>  is a small enough constant.

In fact, one may assume thatu(x),u(x) is a pair of sub- and supersolution of (.) with

u(x) <u(x) without loss of generality. Deﬁne a new function

f(u) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

f(u(t)), u>u(t),

f(u), u(x)≤u≤u(t),

f(u(t)), u<u(t).

By deﬁnition,f(x)∈C(R) is bounded and satisﬁes f(u) =f(u) for u(x)≤u≤u(t). Let

F(x) =_xf(y)dy. ThenF∈C(R), and the functional

J(v) = 

v

_– 



FKu(t)dt

defined onH is bounded from below and satisfies the (PS) condition. Hence there is a minimumuwhich satisfiesdJ(u) =θ. Sinceuis a strict sub-solution,

K(u–u)(t)≥, but not identical to  int∈[, ].

It follows from the maximum principle thatu>u; similarly we haveu>u. By a weak version of the mountain pass lemma, there is a mountain pass pointu. Thus, we have

Ck(J,u) = ⎧ ⎨ ⎩

G, k= ,

, k= .

But from (H), it easy to see that

Ck(J,θ) =

⎧ ⎨ ⎩

G, k= ,

, k= .

Hence,u=θ. It follows from [], Lemma ., p., that

Ck(J,ui) =

⎧ ⎨ ⎩

G, k= ,

, k= ,

wherei= , . Noticing thatJis bounded from below, we conclude that there is at least another critical pointuby using the Morse inequalities.

Obviously, all these critical pointsui,i= , , , , are solutions of problem (.).

On account of the ﬁrst condition in (H), in combination with the maximum principle,

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(.); moreover, all these solutionsu, because of their ranges, are included in [–c,c], and we conclude

ind(J,u) +dim kerdJ(u)≤ =dim



j=

K–λjI

,

provided by the second condition in (H).

Because of condition (H), we learned from Lemma . withγ> . Therefore there exists

another critical pointu, which yields the ﬁfth nontrivial solution for problem (.). The

proof is completed.

We now present some examples. Consider the following problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

u()₍_t_{) =}_f₍_u₍_t_)), _t_∈_{[, ],}

u() =u() = ,

u() =u() = .

(.)

It is well known that for eachv∈E, a solution inC_{[, ] of the boundary value problem}

–u(t) =v(t) for allt∈[, ] with u() =u() =  is equivalent to a solution inE of the following integral equation:

u(t) =





G(t,s)v(s)ds, t∈[, ],

where G: [, ]×[, ]→[, +∞) is the Green’s function of the linear boundary value problem –u(t) =  for allt∈[, ] withu() =u() = ,i.e.,

G(t,s) =

⎧ ⎨ ⎩

t( –s), ≤t≤s≤,

s( –t), ≤s≤t≤.

Now we deﬁne the operatorT:E→Eas follows:

Tu(t) =





G(t,s)u(s)ds, t∈[, ],u∈E.

It is easy to see thatT :E→E is linear completely continuous. Then problem (.) is equivalent to the operator equation

u=Tfu,

where fu(t) =f(u(t)),u∈E.

Example . Let

fu(t)=u(t), t∈[, ].

(11)

Example . Let

f(u) =

⎧ ⎨ ⎩

π[uarctanu– ln(u

_{+ )] + (}_π_{– )}_u_, _u_≥_,

–π_[_u_arctan_u_–

ln(u+ )] + (π– )u, u< .

We note that all the conditions of Theorem . are satisﬁed. It follows that (.) has at least three nontrivial solutions inE.

Example . Let

fu(t)=u(t) + πu(t) – .arctanu(t), t∈[, ].

Since all the conditions of Theorem . are satisﬁed, (.) has at least ﬁve distinct solutions inE.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author contributed equally in writing this article. He read and approved the ﬁnal manuscript.

Acknowledgements

The author greatly thanks Professor Fuyi Li for his great help and many valuable discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11271299 and 11301313), Natural Science Foundation of Shanxi Province (2012011004-2, 2013021001-4, 2014021009-1) and the Scientiﬁc and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2015101).

Received: 8 July 2015 Accepted: 14 October 2015

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