Skip to main content

Multiple solutions of boundary value problems with ϕ-Laplacian operators and under a Wintner-Nagumo growth condition

Abstract

In this paper, we establish the existence of multiple solutions to second-order differential equations with ϕ-Laplacian satisfying periodic, Dirichlet or Neumann boundary conditions. The right-hand side is a Carathéodory function satisfying a growth condition of Wintner-Nagumo type. The existence of upper and lower solutions is assumed. The proofs rely on the fixed point index theory.

MSC: 34B15, 34C25, 47H10, 37C25, 47H11.

1 Introduction

In this paper, we consider boundary value problems for second-order nonlinear differential equations with ϕ-Laplacian of the form:

( ϕ ( u ( t ) ) ) = f ( t , u ( t ) , u ( t ) ) a.e.  t [ 0 , T ] , u B ,
(1.1)

where denotes the Dirichlet, periodic, or Neumann boundary conditions:

u(0)=μ,u(T)=ν;
(1.2)
u(0)=u(T),ϕ ( u ( 0 ) ) =ϕ ( u ( T ) ) ;
(1.3)
ϕ ( u ( 0 ) ) =μ,ϕ ( u ( T ) ) =ν.
(1.4)

Here, f:[0,T]× R 2 R is a Carathéodory function, and ϕ:RR is a bijective increasing homeomorphism.

Such problems have been studied by many authors. The method of upper and lower solutions was widely used to obtain existence results; see, for instance, [116] and the references therein.

Many multiplicity results were obtained with the method of strict lower and upper solutions and for problems with the right member being a Carathéodory map not depending on the derivative, f(t,u). To our knowledge, De Coster [7] was the first to obtain multiplicity results for this problem with the Dirichlet boundary condition. Ben-Naoum and De Coster [1] considered the case where ϕ is the p-Laplacian with Sturm-Liouville boundary condition. Bereanu and Mawhin [2] established the existence of multiple periodic solutions in the case where f(t,u)=e(t)g(u)+c and ϕ can have a bounded domain or a bounded range. Zhang et al. [16] considered the periodic problem with the right member being a continuous map depending also on the derivative f(t,u, u ). A Nagumo growth condition was imposed on f. The case where f(t,u, u ) is a Carathéodory map and ϕ( u )= u was studied by El Khattabi [8] under a linear growth condition, and by Goudreau [11] under a Wintner-Nagumo growth condition.

Existence results were established for problem (1.1) under a growth condition of Nagumo type of the form:

|f(t,x,y)|θ ( | y | ) k(t),
(1.5)

with

min { ϕ ( d ) ϕ 1 ( s ) ( p 1 ) / p θ ( ϕ 1 ( s ) ) d s , ϕ ( d ) ϕ 1 ( s ) ( p 1 ) / p θ ( ϕ 1 ( s ) ) d s } >c,

for c, d suitable constants. In particular, the existence of a solution was obtained by O’Regan [13] when the right member has the form q(t)f(t,u, u ) with f continuous, and with either Dirichlet or mixed boundary conditions. His result was extended in [14] and in [15] for a Carathéodory function f. Cabada and Pouso [3] considered also a Carathéodory map f, and they established the existence of a solution to the problem with Neumann or periodic boundary conditions. More general boundary conditions or more general operators ϕ were considered in [46, 9, 12]. All those results rely on the Schauder fixed point theorem.

In this paper, we consider a Carathéodory map f satisfying a growth condition different from (1.5). Namely, we impose the growth condition of Wintner-Nagumo type

|f(t,x,y)|ψ ( | y | ) ( l ( t ) + c ( t ) | y | ( p 1 ) / p ) .

Using the method of upper and lower solutions and the fixed point index theory, we establish existence and multiplicity results for problem (1.1) with Dirichlet, Neumann or periodic boundary value conditions. Our proofs rely on the fixed point index theory. This theory is particularly convenient for the Neumann problem where we use the contraction property of the fixed point index to reduce the computation of the fixed point index in an affine space.

2 Preliminaries

In what follows, we denote I=[0,T]. The space of continuous functions C(I), and the space of continuously derivable functions C 1 (I) are equipped with the usual norms 0 and u 1 =max{ u 0 , u 0 }, respectively. We denote the usual norm in L p (I) by u L p , where 1p. We set

W(I)= { u C 1 ( I ) : ϕ ( u )  is absolutely continuous and  ( ϕ ( u ) ) L 1 ( I ) } .

Definition 2.1 A map f:I× R 2 R is a Carathéodory function if

  1. (i)

    f(t,,) is continuous for almost every tI;

  2. (ii)

    f(,x,y) is measurable for all (x,y) R 2 ;

  3. (iii)

    for all R>0, there exists h R L 1 (I) such that |f(t,x,y)| h R (t) for all (x,y) R 2 such that |x|R, |y|R, and for almost every tI.

Definition 2.2 We say that αW(I) is a lower solution of (1.1) if

( ϕ ( α ( t ) ) ) f ( t , α ( t ) , α ( t ) ) ,a.e. tI;
(2.1)

and, in addition,

  1. (i)

    if denotes Dirichlet boundary condition (1.2), it satisfies

    α(0)μandα(T)ν;
  2. (ii)

    if denotes periodic boundary condition (1.3), it satisfies

    α(0)=α(T)andϕ ( α ( 0 ) ) ϕ ( α ( T ) ) ;
  3. (iii)

    if denotes Neumann boundary condition (1.4), it satisfies

    ϕ ( α ( 0 ) ) μandϕ ( α ( T ) ) ν.

Similarly, we define an upper solution of (1.1) if the previous conditions are satisfied with the reversed inequalities.

Definition 2.3 We say that αW(I) is a strict lower solution of (1.1) if for any t 0 ]0,T[, there exist ε>0 and U t 0 , a neighborhood of t 0 in I, such that for almost every t U t 0 and all (x,y)[α(t),α(t)+ε]×[ α (t)ε, α (t)+ε],

( ϕ ( α ( t ) ) ) f(t,x,y);
(2.2)

and, in addition,

  1. (i)

    if denotes Dirichlet boundary condition (1.2), it satisfies

    α(0)<μandα(T)<ν;
  2. (ii)

    if denotes periodic boundary condition (1.3), it satisfies

    α(0)=α(T)andϕ ( α ( 0 ) ) >ϕ ( α ( T ) ) ;
  3. (iii)

    if denotes Neumann boundary condition (1.4), it satisfies

    ϕ ( α ( 0 ) ) >μandϕ ( α ( T ) ) <ν.

Similarly, we define a strict upper solution of (1.1) if the previous conditions are satisfied with the reversed inequalities.

We will use the following general assumptions.

(H ϕ ) The map ϕ:RR is a bijective increasing homeomorphism.

(H f ) The map f:I× R 2 R is Carathéodory.

( H B ) There exist α,βW(I), respectively lower and upper solutions of (1.1), such that α(t)β(t) for all tI.

(WN) There exist k>0, p]1,], c L p (I,[0,[), l L 1 (I,[0,[) and ψ:[0,[[k,[ such that

1 ψ ϕ 1 L l o c 1 (R), d s ψ ( | ϕ 1 ( s ) | ) =,

and

| f ( t , x , y ) | ψ ( | y | ) ( l ( t ) + c ( t ) | y | ( p 1 ) / p ) x [ α ( t ) , β ( t ) ] , y R ,  and a.e.  t I ,

with (p1)/p=1 if p=.

In what follows, ( H B ) will be replaced by (H D ), (H N ) or (H P ) if denotes (1.2), (1.3) or (1.4), the Dirichlet, periodic or Neumann boundary conditions, respectively.

We present some properties of operators that will be used later. Here is a particular case of Lemmas 2.3 and 2.6 in [10].

Lemma 2.4 Let f:I× R 2 R be a Carathéodory function and σ:I× R 2 R such that

  1. (i)

    for any u C 1 (I), the map tσ(t,u(t), u (t)) is measurable;

  2. (ii)

    for any sequence { u n } converging to u 0 in C 1 (I), there exists c>0 such that

    σ ( t , u n ( t ) , u n ( t ) ) ca.e. tI and n0;

and

σ ( t , u n ( t ) , u n ( t ) ) σ ( t , u 0 ( t ) , u 0 ( t ) ) a.e. tI.

Then the operator N g : C 1 (I)C(I) defined by

N g (t)= 0 t g ( s , u ( s ) , u ( s ) ) ds,

with

g ( t , u ( t ) , u ( t ) ) =f ( t , σ ( t , u ( t ) , u ( t ) ) ) ,

is continuous and completely continuous. Moreover, for every u C 1 (I), N g (u) is absolutely continuous and

d d t N g (u)(t)=g ( t , u ( t ) , u ( t ) ) a.e. tI.

Observe that the previous lemma holds with σ(t,x,y)=(x,y). In this case, N g = N f .

Now, we present some results related to the homeomorphism ϕ. The first one is a lemma due to Manásevich and Mawhin [17].

Lemma 2.5 Let ϕ:RR satisfy (H ϕ ), and let G:R×C(I)R be defined by

G(a,h)= 0 T ϕ 1 ( a + h ( t ) ) dt.

Then the following statements hold:

  1. (1)

    For any cR and any hC(I), the equation

    G(a,h)=c

has a unique solution a ¯ c (h).

  1. (2)

    For any cR, the function a ¯ c :C(I)R defined in (1) is continuous, and it sends bounded sets into bounded sets.

It is easy to show the following result.

Lemma 2.6 Assume that ϕ:RR satisfies (H ϕ ). Let Φ:R×C(I) C 1 (I) be defined by

Φ(r,h)(t)=r+ 0 t ϕ 1 ( h ( s ) ) ds.

Then Φ is continuous.

We will use the following maximum principle type result.

Lemma 2.7 Assume that (H ϕ ) is satisfied. Let v,wW(I) be such that

( ϕ ( v ( t ) ) ) ( ϕ ( w ( t ) ) ) a.e. on  { t I : v ( t ) < w ( t ) } .

Assume that one of the following conditions holds:

  1. (i)

    v(0)w(0), v(T)w(T);

  2. (ii)

    ϕ( v (0))ϕ( w (0)), ϕ( v (T))ϕ( w (T));

  3. (iii)

    v(0)=v(T), w(0)=w(T), ϕ( v (0))ϕ( v (T))ϕ( w (0))ϕ( w (T)).

Then v(t)w(t) for all tI, or there exists c>0 such that v(t)=w(t)c for all tI.

Proof Assume that

A= { t I : v ( t ) < w ( t ) } .

Condition (i) (resp. (ii) or (iii)) implies that one of the following statements holds:

there exists  [ t 0 , t 1 ] A  such that v ( t 0 ) w ( t 0 ) < v ( t 1 ) w ( t 1 ) and ϕ ( v ( t 0 ) ) ϕ ( w ( t 0 ) ) ;
(2.3)
there exists  [ t 0 , t 1 ] A  such that v ( t 0 ) w ( t 0 ) > v ( t 1 ) w ( t 1 ) and ϕ ( v ( t 1 ) ) ϕ ( w ( t 1 ) ) ;
(2.4)

or

there exists c>0 such thatv(t)=w(t)ctI.
(2.5)

Indeed, if (2.5) does not hold, let τ,ρI be such that

v(τ)w(τ)=min { v ( t ) w ( t ) : t I } <v(ρ)w(ρ),

and

v(t)w(t)<0for all t between τ and ρ.

If τ]0,T[, then ϕ( v (τ))=ϕ( w (τ)) and (2.3) or (2.4) hold.

If τ{0,T}, then (i) does not hold. If (ii) holds, then

ϕ ( v ( 0 ) ) ϕ ( w ( 0 ) ) andϕ ( v ( T ) ) ϕ ( w ( T ) ) .

Similarly, if (iii) holds, since

v(0)w(0)=v(T)w(T)=v(τ)w(τ),

one has v (0) w (0) and v (T) w (T). Using (iii) and the fact that ϕ is increasing, we get

ϕ ( v ( 0 ) ) ϕ ( w ( 0 ) ) 0ϕ ( v ( T ) ) ϕ ( w ( T ) ) ϕ ( v ( 0 ) ) ϕ ( w ( 0 ) ) .

Hence, (2.3) or (2.4) are satisfied.

Therefore, by assumption,

( ϕ ( v ( t ) ) ) ( ϕ ( w ( t ) ) ) a.e. t[ t 0 , t 1 ].
(2.6)

If (2.3) holds, for every t] t 0 , t 1 ],

ϕ ( v ( t ) ) ϕ ( v ( t 0 ) ) = t 0 t ( ϕ ( v ( s ) ) ) d s t 0 t ( ϕ ( w ( s ) ) ) d s = ϕ ( w ( t ) ) ϕ ( w ( t 0 ) ) ϕ ( w ( t ) ) ϕ ( v ( t 0 ) ) .

Since ϕ is increasing, we deduce that vw is nonincreasing in [ t 0 , t 1 ]. This is a contradiction. Similarly, we obtain a contradiction if (2.4) holds.

Therefore, v(t)w(t) for all tI, or (2.5) is satisfied. □

Lemma 2.8 Assume (H ϕ ) and (H f ). Let α,βW(I) be respectively strict lower and upper solutions of (1.1) such that α(t)<β(t) for all tI. If uW(I) is a solution of (1.1) such that α(t)u(t)β(t) for all tI, then α(t)<u(t)<β(t) for all tI.

Proof Let uW(I) be a solution of (1.1) such that α(t)u(t)β(t) for all tI. Assume that

A= { t I : α ( t ) = u ( t ) } .

First, we claim that 0,TA. This is obviously the case if denotes Dirichlet boundary condition (1.2). If denotes periodic boundary condition (1.3), then uα attains a minimum at 0 and T. So, by (H ϕ ) and Definition 2.3(ii),

ϕ ( u ( T ) ) ϕ ( α ( T ) ) 0ϕ ( u ( 0 ) ) ϕ ( α ( 0 ) ) <ϕ ( u ( T ) ) ϕ ( α ( T ) ) ,

a contradiction. If denotes Neumann boundary condition (1.4), then uα attains a minimum at 0 or at T. So, by (H ϕ ) and Definition 2.3(iii),

0ϕ ( u ( 0 ) ) ϕ ( α ( 0 ) ) <μμor0ϕ ( u ( T ) ) ϕ ( α ( T ) ) >νν,

a contradiction.

Let t 0 =maxA]0,T[. So, ϕ( u ( t 0 ))=ϕ( α ( t 0 )). By Definition 2.3, there exist ε>0 and U t 0 , a neighborhood of t 0 , such that ( ϕ ( α ( t ) ) ) f(t,x,y) a.e. t U t 0 and all x, y such that α(t)xα(t)+ε and |y α (t)|ε. Since u C 1 (I), there exists t 1 ] t 0 ,T[ such that u(t)]α(t),α(t)+ε] and | u (t) α (t)|ε for all t] t 0 , t 1 ]. Since t 0 =maxA, there exists t] t 0 , t 1 ] such that u (t)> α (t). Using the fact that ϕ is increasing, we deduce that

ϕ ( α ( t ) ) ϕ ( α ( t 0 ) ) < ϕ ( u ( t ) ) ϕ ( u ( t 0 ) ) = t 0 t f ( s , u ( s ) , u ( s ) ) d s t 0 t ( ϕ ( α ( s ) ) ) d s = ϕ ( α ( t ) ) ϕ ( α ( t 0 ) ) .

This is a contradiction. Therefore, α(t)<u(t) for all tI.

Similarly, we show that β(t)>u(t) for all tI. □

The following result establishes the existence of an a priori bound on the derivative of functions satisfying a suitable inequality.

Lemma 2.9 Assume that (H ϕ ) is satisfied. Let k>0 and ψ:[0,[[k,[ be such that

1 ψ ϕ 1 L l o c 1 (R)and 0 ± d s ψ ( | ϕ 1 ( s ) | ) =±.
(2.7)

Then, for every d 0 0, p]1,], c 0 L p (I,[0,[), l 0 L 1 (I,[0,[), and BC(I) bounded, there exists M 0 > d 0 such that for every

u { u W ( I ) : u B , min { | u ( t ) | : t I } d 0 }

satisfying

| ( ϕ ( u ( t ) ) ) |ψ ( | u ( t ) | ) ( l 0 ( t ) + c 0 ( t ) | u ( t ) | ( p 1 ) / p ) a.e. tI,

one has u 0 < M 0 .

Proof Let

m=max { u ( t 1 ) u ( t 2 ) : t 1 , t 2 I , u B } .

Assumptions (H ϕ ) and (2.7) imply that there exists M 0 > d 0 such that

ϕ( M 0 )>ϕ( d 0 )ϕ( d 0 )>ϕ( M 0 ),

and

max { ϕ ( d 0 ) ϕ ( M 0 ) d s ψ ( | ϕ 1 ( s ) | ) , ϕ ( M 0 ) ϕ ( d 0 ) d s ψ ( | ϕ 1 ( s ) | ) } > l 0 L 1 + m ( p 1 ) / p c 0 L p .
(2.8)

Assume that there exists

u { u W ( I ) : u B , min { | u ( t ) | : t I } d 0 , u and  | ( ϕ ( u ( t ) ) ) | ψ ( | u ( t ) | ) ( l 0 ( t ) + c 0 ( t ) | u ( t ) | ( p 1 ) / p )  a.e.  t I }

such that u 0 M 0 . If max{ u (t):tI} M 0 , there exist t 0 , t 1 I such that ϕ( u ( t 0 ))=ϕ( d 0 ), ϕ( u ( t 1 ))=ϕ( M 0 ), ϕ( u (t))]ϕ( d 0 ),ϕ( M 0 )[ and u (t)>0 for all t between t 0 and t 1 . Without loss of generality, we assume that t 0 < t 1 . Then, by assumption,

( ϕ ( u ( t ) ) ) ψ ( u ( t ) ) l 0 (t)+ c 0 (t) ( u ( t ) ) ( p 1 ) / p a.e. t[ t 0 , t 1 ].

Integrating from t 0 to t 1 and using the Hölder inequality and the change of variable formula in an integral give us

ϕ ( d 0 ) ϕ ( M 0 ) d s ψ ( | ϕ 1 ( s ) | ) = t 0 t 1 ( ϕ ( u ( t ) ) ) ψ ( u ( t ) ) d t t 0 t 1 l 0 ( t ) + c 0 ( t ) ( u ( t ) ) ( p 1 ) / p d t l 0 L 1 + c 0 L p ( u ( t 1 ) u ( t 0 ) ) ( p 1 ) / p l 0 L 1 + m ( p 1 ) / p c 0 L p .

This contradicts (2.8).

Similarly, if min{ u (t):tI} M 0 , there exist t 0 , t 1 I such that ϕ( u ( t 0 ))=ϕ( d 0 ), ϕ( u ( t 1 ))=ϕ( M 0 ), ϕ( u (t))]ϕ( M 0 ),ϕ( d 0 )[ and u (t)<0 for all t between t 0 and t 1 . Arguing as above leads to a contradiction. □

3 The Dirichlet problem

In this section, we consider problem (1.1) with the Dirichlet boundary condition. In order to establish the existence of a solution to (1.1), (1.2), we consider the following family of problems defined for λ[0,1]:

(3.1λ)

where f ˆ :[0,1]× R 2 R is defined by

f ˆ (t,x,y)={ f ( t , β ( t ) , β ( t ) ) h ˆ ( t ) ( x β ( t ) ) if  x > β ( t ) , f ( t , x , y ) if  α ( t ) x β ( t ) , f ( t , α ( t ) , α ( t ) ) + g ˆ ( t ) ( x α ( t ) ) if  x < α ( t ) ;
(3.2)

with g ˆ , h ˆ L 1 (I) chosen such that

g ˆ (t)>max { 0 , f ( t , α ( t ) , α ( t ) ) } and h ˆ (t)<min { 0 , f ( t , β ( t ) , β ( t ) ) } .
(3.3)

We show that the solutions to these problems are a priori bounded.

Proposition 3.1 Assume that (H ϕ ), (H f ), (H D ) and (WN) hold. Then there exists M>max{ α 0 , β 0 } such that any solution u of (3.1 λ ) satisfies u 1 <M.

Proof Fix M 1 >0 such that

M 1 min t I α(t)1and M 1 max t I β(t)+1.

We claim that any solution u of (3.1 λ ) is such that u 0 M 1 . Indeed, by (H D ),

M 1 <u(0)=μ< M 1 and M 1 <u(T)=ν< M 1 .

From the definition of f ˆ , one has, almost everywhere on {tI:u(t)< M 1 },

( ϕ ( u ( t ) ) ) = λ ( f ( t , α ( t ) , α ( t ) ) + g ˆ ( t ) ( u ( t ) α ( t ) ) ) λ ( f ( t , α ( t ) , α ( t ) ) g ˆ ( t ) ) 0 .

Similarly,

( ϕ ( u ( t ) ) ) 0a.e. on  { t I : u ( t ) > M 1 } .

It follows from Lemma 2.7 that M 1 u(t) M 1 for all tI.

We look for an a priori bound on the derivative of any solution u of (3.1 λ ). Let

l 1 (t)=max { | f ( t , α ( t ) , α ( t ) ) | + 2 M 1 | g ˆ ( t ) | , | f ( t , β ( t ) , β ( t ) ) | + 2 M 1 | h ˆ ( t ) | } .

Observe that, by (WN),

| ( ϕ ( u ( t ) ) ) | max { l 1 ( t ) , ψ ( | u ( t ) | ) ( l ( t ) + c ( t ) | u ( t ) | ( p 1 ) / p ) } ψ ( | u ( t ) | ) ( l 0 ( t ) + c ( t ) | u ( t ) | ( p 1 ) / p ) a.e.  t I ,

with

l 0 (t)=max { l ( t ) , l 1 ( t ) k } .
(3.4)

It follows from Lemma 2.9 applied with d 0 =(νμ)/T that there exists M 0 such that any solution u of (3.1 λ ) satisfies u 0 < M 0 .

Finally, set M=max{ M 0 ,1+ M 1 }. We have that u 1 <M for any solution u of (3.1 λ ). □

Proposition 3.2 Assume that (H ϕ ), (H f ), (H D ) and (WN) hold. Then, for every λ[0,1], problem (3.1 λ ) has at least one solution.

Proof Let us define N f ˆ : C 1 (I)C(I) and S D :[0,1]× C 1 (I)R×C(I) by

N f ˆ (u)(t)= 0 t f ˆ ( s , u ( s ) , u ( s ) ) ds,
(3.5)

and

S D (λ,u)= ( μ , a ¯ ν μ ( λ N f ˆ ( u ) ) + λ N f ˆ ( u ) ) ,
(3.6)

where a ¯ ν μ is obtained in Lemma 2.5. Now, we define D:[0,1]× C 1 (I) C 1 (I) by

D=Φ S D ,
(3.7)

where Φ is defined in Lemma 2.6. We deduce that S D is continuous and completely continuous from Lemma 2.5 and from Lemma 2.4 applied with

σ(t,x,y)={ ( β ( t ) , β ( t ) ) if  x > β ( t ) , ( x , y ) if  α ( t ) x β ( t ) , ( α ( t ) , α ( t ) ) if  x < α ( t ) .

This combined with Lemma 2.6 implies that D is continuous and completely continuous.

Now, we study the fixed points of D. Let u C 1 (I) and λ[0,1] be such that u=D(λ,u). One has, by Lemma 2.5,

u(0)=μ,u(T)=μ+G ( a ¯ ν μ ( λ N f ˆ ( u ) ) , λ N f ˆ ( u ) ) =μ+(νμ)=ν.

Also, ϕ( u )= a ¯ ν μ (λ N f ˆ (u))+λ N f ˆ (u). Hence, by Lemma 2.4, it is absolutely continuous and

( ϕ ( u ( t ) ) ) =λ f ˆ ( t , u ( t ) , u ( t ) ) a.e. tI.

So, fixed points of D are solutions of (3.1 λ ).

Let M>max{ α 0 , β 0 }max{μ,ν} be the constant obtained in Proposition 3.1 and set

U= { u C 1 ( I ) : u 1 < M } .
(3.8)

Proposition 3.1 implies that uD(λ,u) for all (λ,u)[0,1]×U. Observe that D(0,u)=Φ(μ, a ¯ ν μ (0))= u 0 with u 0 (t)=μ+(νμ)t/T. One has u 0 U. By the properties of the fixed point index (see [18] for more details),

index ( D ( λ , ) , U ) =index ( D ( 0 , ) , U ) =index( u 0 ,U)=1λ[0,1].
(3.9)

Therefore, for every λ[0,1], D(λ,) has a fixed point, and hence (3.1 λ ) has a solution. □

Now, we can establish the existence of a solution to (1.1), (1.2).

Theorem 3.3 Assume that (H ϕ ), (H f ), (H D ) and (WN) hold. Then Dirichlet problem (1.1), (1.2) has a solution uW(I) such that α(t)u(t)β(t) for every tI.

Proof Proposition 3.2 insures the existence of uW(I), a solution of (3.1 λ ) for λ=1. To conclude, we have to show that α(t)u(t)β(t) for all tI since f ˆ (t,x,y)=f(t,x,y) for x[α(t),β(t)].

By (H D ),

( ϕ ( u ( t ) ) ) = f ( t , α ( t ) , α ( t ) ) + g ˆ ( t ) ( x ( t ) α ( t ) ) f ( t , α ( t ) , α ( t ) ) ( ϕ ( u ( t ) ) ) ( ϕ ( α ( t ) ) ) a.e.  t { t I : u ( t ) < α ( t ) } , u ( 0 ) = μ α ( 0 ) and u ( T ) = ν α ( T ) .

It follows from Lemma 2.7 that u(t)α(t) for all tI.

A similar argument yields u(t)β(t) for all tI. □

Remark 3.4 The hypothesis (WN) can be generalized by

min { ϕ ( d 0 ) d s ψ ( | ϕ 1 ( s ) | ) , ϕ ( d 0 ) d s ψ ( | ϕ 1 ( s ) | ) } >c,

with d 0 =(νμ)/T and c, a suitable constant which can be deduced from the proof of Lemma 2.9.

Example 3.5 Let us consider the following problem:

( ϕ ( u ( t ) ) ) = ( ( u ( t ) ) 3 + 1 ) ( h ( t ) ( u ( t ) + a ( t ) ) + g ( u ( t ) ) u ( t ) ) a.e.  t [ 0 , T ] , u ( 0 ) = 0 , u ( T ) = 0 ,
(3.10)

where ϕ(s)= s 3 , aC[0,T], gC(R) and h L 1 [0,T] with h(t)0 a.e. t[0,T]. Let m>0 be such that |a(t)|m for all tT. Then α=m and β=m are respectively lower and upper solutions of (3.10). Let ψ(s)=(|s | 3 +1), cmax{|g(x)|:|x|m} and l=2mh. One has

|f(t,x,y)|ψ ( | y | ) ( l ( t ) + c | y | ) x[m,m],yR, and a.e. t[0,T],

and

d s ψ ( | ϕ 1 ( s ) | ) = d s | s | + 1 =.

By Theorem 3.3, (3.10) has a solution u such that |u(t)|m for all t[0,T]. Notice that f does not satisfy a growth condition of Nagumo type.

4 The periodic problem

In this section, we consider problem (1.1) with the periodic boundary condition. In order to establish the existence of a solution to (1.1), (1.3), we consider the following family of problems defined for λ[0,1]:

where f ˆ is defined in (3.2).

We show that the solutions to these problems are a priori bounded.

Proposition 4.1 Assume that (H ϕ ), (H f ), (H P ) and (WN) hold. Then there exists M>max{ α 0 , β 0 } such that any solution u of (4.1 λ ) satisfies u 1 <M.

Proof Fix M 1 >0 such that

M 1 min t I α(t)1and M 1 max t I β(t)+1.

We claim that any solution u of (4.1 λ ) is such that u 0 M 1 . Observe that a solution of (4.1 λ ) satisfies

0 = ϕ ( u ( T ) ) ϕ ( u ( 0 ) ) = λ 0 T f ˆ ( t , u ( t ) , u ( t ) ) d t + ( 1 λ ) 0 T f ˆ ( t , u ( t ) , u ( t ) ) d t = 0 T f ˆ ( t , u ( t ) , u ( t ) ) d t .

So, it satisfies

( ϕ ( u ( t ) ) ) =λ f ˆ ( t , u ( t ) , u ( t ) ) a.e. tI,

with periodic boundary condition (1.3). Arguing as in the proof of Proposition 3.1, we obtain that

( ϕ ( u ( t ) ) ) 0 a.e. on  { t I : u ( t ) < M 1 } , ( ϕ ( u ( t ) ) ) 0 a.e. on  { t I : u ( t ) > M 1 } .

It follows from Lemma 2.7 that u 0 M 1 , or there exists c>0 such that |u(t)|= M 1 +c for all tI. If u M 1 +c,

0= 0 T f ˆ ( t , u ( t ) , u ( t ) ) dt= 0 T f ( t , β ( t ) , β ( t ) ) h ˆ (t) ( M 1 + c β ( t ) ) dt>0,

a contradiction. Similarly, one cannot have u( M 1 +c). Hence, u 0 M 1 .

As in the proof of Proposition 3.1, one has that any solution u of (4.1 λ ) satisfies

| ( ϕ ( u ( t ) ) ) |ψ ( | u ( t ) | ) ( l 0 ( t ) + c ( t ) | u ( t ) | ( p 1 ) / p ) a.e. tI,

where l 0 is defined in (3.4). It follows from Lemma 2.9 applied with d 0 =0 that there exists M 0 such that any solution u of (4.1 λ ) satisfies u 0 < M 0 .

Finally, set M=max{ M 0 ,1+ M 1 }. We have that u 1 <M for any solution u of (4.1 λ ). □

Proposition 4.2 Assume that (H ϕ ), (H f ), (H P ) and (WN) hold. Then, for every λ[0,1], problem (4.1 λ ) has at least one solution.

Proof Let N f ˆ : C 1 (I)C(I) be defined in (3.5). Let us consider the operators L ˆ : C 1 (I)C(I) and S P :[0,1]× C 1 (I)R×C(I) defined by

L ˆ (u)(t)= t T N f ˆ (u)(T),
(4.2)

and

S P (λ,u)= ( u ( 0 ) + 1 T N f ˆ ( u ) ( T ) , a ¯ 0 ( λ ( N f ˆ ( u ) L ˆ ( u ) ) ) + λ ( N f ˆ ( u ) L ˆ ( u ) ) ) ,
(4.3)

where a ¯ 0 is obtained in Lemma 2.5. Now, we define P:[0,1]× C 1 (I) C 1 (I) by

P=Φ S P .
(4.4)

Again, we deduce from Lemmas 2.4, 2.5 and 2.6 that S P and P are continuous and completely continuous.

Now, we study the fixed points of P. Let u C 1 (I) and λ[0,1] be such that u=P(λ,u). One has, by Lemma 2.5,

u ( 0 ) = u ( 0 ) + 1 T N f ˆ ( u ) ( T ) , u ( T ) = u ( 0 ) + 1 T N f ˆ ( u ) ( T ) + G ( a ¯ 0 ( λ ( N f ˆ ( u ) L ˆ ( u ) ) ) , λ ( N f ˆ ( u ) L ˆ ( u ) ) ) u ( T ) = u ( 0 ) .

Notice that ( N f ˆ (u)(T))/T=0. Also,

ϕ ( u ) = a ¯ 0 ( λ ( N f ˆ ( u ) L ˆ ( u ) ) ) +λ ( N f ˆ ( u ) L ˆ ( u ) ) .

Hence, by Lemma 2.4, it is absolutely continuous and

( ϕ ( u ( t ) ) ) = λ ( f ˆ ( t , u ( t ) , u ( t ) ) 1 T N f ˆ ( u ) ( T ) ) = λ f ˆ ( t , u ( t ) , u ( t ) ) + ( 1 λ ) T 0 T f ˆ ( t , u ( t ) , u ( t ) ) d t a.e.  t I .

Moreover,

ϕ ( u ( 0 ) ) = a ¯ 0 ( λ ( N f ˆ ( u ) L ˆ ( u ) ) ) + λ ( N f ˆ ( u ) ( 0 ) L ˆ ( u ) ( 0 ) ) , ϕ ( u ( T ) ) = a ¯ 0 ( λ ( N f ˆ ( u ) L ˆ ( u ) ) ) + λ ( N f ˆ ( u ) ( T ) L ˆ ( u ) ( T ) ) ϕ ( u ( T ) ) = ϕ ( u ( 0 ) ) .

So, fixed points of P are solutions of (4.1 λ ).

Let M>max{ α 0 , β 0 } be the constant obtained in Proposition 4.1 and set

U= { u C 1 ( I ) : u 1 < M } .
(4.5)

Proposition 4.1 implies that uP(λ,u) for all (λ,u)[0,1]×U. By the homotopy property of the fixed point index,

index ( P ( λ , ) , U ) =index ( P ( 0 , ) , U ) λ[0,1].

Observe that

P(0,u)=Φ ( u ( 0 ) + 1 T N f ˆ ( u ) ( T ) , a ¯ 0 ( 0 ) ) =u(0)+ 1 T N f ˆ (u)(T)R.

One has (UR)={M,M}, and

P ( 0 , M ) = M + 1 T 0 T f ˆ ( t , M , 0 ) d t = M + 1 T 0 T f ( t , β ( t ) , β ( t ) ) h ˆ ( t ) ( M β ( t ) ) d t > M ,

since M> β 0 and h ˆ (t)<min{0,f(t,β(t), β (t))}. Similarly,

P(0,M)<M.

By the contraction property of the fixed point index (see [18, Chapter 4, Section 12, Theorem 6.2]),

index ( P ( 0 , ) , U ) =index ( P ( 0 , ) , ( U R ) ) =1.

Therefore, for every λ[0,1],

index ( P ( λ , ) , U ) =1.
(4.6)

Thus, P(λ,) has a fixed point, and hence (4.1 λ ) has a solution. □

Now, we can establish the existence of a solution to (1.1), (1.3).

Theorem 4.3 Assume that (H ϕ ), (H f ), (H P ) and (WN) hold. Then periodic problem (1.1), (1.3) has a solution uW(I) such that α(t)u(t)β(t) for every tI.

Proof Proposition 4.2 insures the existence of uW(I), a solution of (4.1 λ ) for λ=1. To conclude, we have to show that α(t)u(t)β(t) for every tI, since f ˆ (t,x,y)=f(t,x,y) for x[α(t),β(t)].

Using (H P ), we obtain that

( ϕ ( u ( t ) ) ) < ( ϕ ( α ( t ) ) ) a.e.  t { t I : u ( t ) < α ( t ) } , ( ϕ ( u ( t ) ) ) > ( ϕ ( β ( t ) ) ) a.e.  t { t I : u ( t ) > β ( t ) } .

It follows from Lemma 2.7 that α(t)u(t)β(t) for all tI. □

5 The Neumann problem

In this section, we consider problem (1.1) with the Neumann boundary condition. In order to establish the existence of a solution to (1.1), (1.3), we consider the following family of problems defined for λ[0,1]:

where f ˜ :[0,1]× R 2 R is defined by

f ˜ (t,x,y)={ f ( t , β ( t ) , β ( t ) ) h ˜ ( t ) ( x β ( t ) ) if  x > β ( t ) , f ( t , x , y ) if  α ( t ) x β ( t ) , f ( t , α ( t ) , α ( t ) ) + g ˜ ( t ) ( x α ( t ) ) if  x < α ( t ) ;
(5.2)

with g ˜ , h ˜ L 1 (I) chosen such that

g ˜ ( t ) > max { 0 , f ( t , α ( t ) , α ( t ) ) 1 T ( ν μ ) } , h ˜ ( t ) < min { 0 , f ( t , β ( t ) , β ( t ) ) 1 T ( ν μ ) } .
(5.3)

We show that the solutions to these problems are a priori bounded. Let w C 1 (I) and vW(I) be defined by

w(t)=μ+ t T (νμ)andv=Φ(0,w).
(5.4)

Observe that

ϕ ( v ) =w.

Proposition 5.1 Assume that (H ϕ ), (H f ), (H N ) and (WN) hold. Then there exists M> v 0 such that any solution u of (5.1 λ ) satisfies u v 0 <M and u 0 <M. Moreover,

{ u C 1 ( I ) : α ( t ) u ( t ) β ( t ) t I } { u C 1 ( I ) : u v 0 < M } .

Proof Fix M 1 >0 such that

M 1 +v(t) min t I α(t)1and M 1 +v(t) max t I β(t)+1tI.

We claim that any solution u of (5.1 λ ) is such that u v 0 M 1 . For every uW(I), one has almost everywhere on {tI:u(t)< M 1 +v(t)},

f ˜ ( t , u ( t ) , u ( t ) ) = f ( t , α ( t ) , α ( t ) ) + g ˜ ( t ) ( u ( t ) α ( t ) ) < f ( t , α ( t ) , α ( t ) ) g ˜ ( t ) 1 T ( ν μ ) .
(5.5)

Similarly,

f ˜ ( t , u ( t ) , u ( t ) ) > 1 T (νμ)a.e. on  { t I : u ( t ) > M 1 + v ( t ) } .
(5.6)

Let u be a solution of (5.1 λ ). One has

νμ= 0 T ( ϕ ( u ( t ) ) ) dt= 0 T f ˜ ( t , u ( t ) , u ( t ) ) dt.
(5.7)

Combining (5.5), (5.6) and (5.7), we deduce that almost everywhere on {tI:u(t)< M 1 +v(t)},

( ϕ ( u ( t ) ) ) = λ f ˜ ( t , u ( t ) , u ( t ) ) + ( 1 λ ) T ( ν μ ) 1 T ( ν μ ) = ( ϕ ( v ( t ) ) ) .

Similarly,

( ϕ ( u ( t ) ) ) ( ϕ ( v ( t ) ) ) a.e. on  { t I : u ( t ) > M 1 + v ( t ) } .

Moreover,

ϕ ( u ( 0 ) ) =μ=ϕ ( v ( 0 ) ) andϕ ( u ( T ) ) =ν=ϕ ( v ( T ) ) .

It follows from Lemma 2.7 that u v 0 M 1 , or there exists c>0 such that |u(t)v(t)|= M 1 +c for all tI. If u(t)=v(t)+ M 1 +c for all tI, then

ν μ = 0 T f ˜ ( t , u ( t ) , u ( t ) ) d t = 0 T f ( t , β ( t ) , β ( t ) ) h ˜ ( t ) ( v ( t ) + M 1 + c β ( t ) ) d t > 0 T ( ϕ ( β ( t ) ) ) d t = ϕ ( β ( T ) ) ϕ ( β ( 0 ) ) ν μ ,

a contradiction. Similarly, one cannot have u(t)=v(t) M 1 c for all tI. Hence, u v 0 M 1 .

Let

l ˜ 1 (t)=max { | f ( t , α ( t ) , α ( t ) ) | + 2 M 1 | g ˜ ( t ) | , | f ( t , β ( t ) , β ( t ) ) | + 2 M 1 | h ˜ ( t ) | } .

Observe that, by (WN), one has that for any u solution of (5.1 λ ),

| ( ϕ ( u ( t ) ) ) | max { l ˜ 1 ( t ) , ψ ( | u ( t ) | ) ( l ( t ) + c ( t ) | u ( t ) | ( p 1 ) / p ) } ψ ( | u ( t ) | ) ( l ˜ 0 ( t ) + c ( t ) | u ( t ) | ( p 1 ) / p ) a.e.  t I ,

with

l ˜ 0 (t)=max { l ( t ) , l ˜ 1 ( t ) k } .
(5.8)

It follows from Lemma 2.9 applied with d 0 = ϕ 1 (μ) that there exists M 0 such that any solution u of (5.1 λ ) satisfies u 0 < M 0 .

Finally, set M=max{ M 0 ,1+ M 1 ,1+ v 0 }, we get the conclusion. □

Proposition 5.2 Assume that (H ϕ ), (H f ), (H N ) and (WN) hold. Then, for every λ[0,1], problem (5.1 λ ) has at least one solution.

Proof Let N f ˜ : C 1 (I)C(I), L ˜ : C 1 (I)C(I) and S N :[0,1]× C 1 (I)R×C(I) be defined respectively by

N f ˜ ( u ) ( t ) = 0 t f ˜ ( s , u ( s ) , u ( s ) ) d s , L ˜ ( u ) ( t ) = t T N f ˜ ( u ) ( T ) , S N ( λ , u ) = ( u ( 0 ) + 1 T ( μ ν + N f ˜ ( u ) ( T ) ) , w + λ ( N f ˜ ( u ) L ˜ ( u ) ) ) ,

with w C 1 (I) defined in (5.4). Now, we define N:[0,1]× C 1 (I) C 1 (I) by

N=Φ S N .
(5.9)

Again, we deduce from Lemmas 2.4, 2.5 and 2.6 that S N and N are continuous and completely continuous.

Now, we study the fixed points of N. Let u C 1 (I) and λ[0,1] be such that u=N(λ,u). One has

u(0)=u(0)+ 1 T ( μ ν + N f ˜ ( u ) ( T ) ) .

So,

νμ= N f ˜ (u)(T).

Also, ϕ( u )=w+λ( N f ˜ (u) L ˜ (u)). Hence, by Lemma 2.4, it is absolutely continuous and

( ϕ ( u ( t ) ) ) = λ ( f ˜ ( t , u ( t ) , u ( t ) ) 1 T N f ˜ ( u ) ( T ) ) + 1 T ( ν μ ) = λ f ˜ ( t , u ( t ) , u ( t ) ) + ( 1 λ ) T 0 T f ˜ ( t , u ( t ) , u ( t ) ) d t a.e.  t I .

Moreover,

ϕ ( u ( 0 ) ) = w ( 0 ) + λ ( N f ˜ ( u ) ( 0 ) L ˜ ( u ) ( 0 ) ) = μ , ϕ ( u ( T ) ) = w ( T ) + λ ( N f ˜ ( u ) ( T ) L ˜ ( u ) ( T ) ) = ν .

So, fixed points of N are solutions of (5.1 λ ).

Let M>0 be the constant obtained in Proposition 5.1 and v=Φ(0,w). We set

U= { u C 1 ( I ) : u v 0 < M , u 0 < M } .
(5.10)

Proposition 5.1 implies that uN(λ,u) for all (λ,u)[0,1]×U. By the homotopy property of the fixed point index,

index ( N ( λ , ) , U ) =index ( N ( 0 , ) , U ) λ[0,1].
(5.11)

Observe that

N(0,u)=u(0)+ 1 T ( μ ν + N f ˜ ( u ) ( T ) ) +Φ(0,w)R+v.

Let X={u=r+v:rR} C 1 (I). Notice that X is not a normed vectorial space if v0. Nevertheless, if v0, X is an affine space and hence it is an ANR. By the contraction property of the fixed point index (see [18, Chapter 4, Section 12, Theorem 6.2]),

index ( N ( 0 , ) , U ) =index ( N ( 0 , ) , U X ) .
(5.12)

One has (UX)={M+v,M+v}. Consider H:[M,M]R defined by

H(r)=N(0,r+v)v.

By (5.6),

H(M)=N(0,M+v)v=M+ 1 T ( μ ν + 0 T f ˜ ( t , M + v ( t ) , v ( t ) ) d t ) >M,

since M+v(t)>β(t) for every tI. Similarly,

H(M)<M.

By the commutativity property of the fixed point index (see [18, Chapter 4, Section 12, Theorem 6.2]),

index ( N ( 0 , ) , U X ) =index ( H , ] M , M [ ) =1.
(5.13)

Combining (5.11), (5.12) and (5.13), we deduce that for every λ[0,1],

index ( N ( λ , ) , U ) =1.
(5.14)

Thus, N(λ,) has a fixed point, and hence (5.1 λ ) has a solution. □

Now, we can establish the existence of a solution to (1.1), (1.4).

Theorem 5.3 Assume that (H ϕ ), (H f ), (H N ) and (WN) hold. Then Neumann problem (1.1), (1.4) has a solution uW(I) such that α(t)u(t)β(t) for every tI.

Proof Proposition 5.2 insures the existence of uW(I), a solution of (5.1 λ ) for λ=1. To conclude, we have to show that α(t)u(t)β(t) for every tI, since f ˜ (t,x,y)=f(t,x,y) for x[α(t),β(t)].

Using (H N ), we obtain that

( ϕ ( u ( t ) ) ) < ( ϕ ( α ( t ) ) ) a.e.  t { t I : u ( t ) < α ( t ) } , ( ϕ ( u ( t ) ) ) > ( ϕ ( β ( t ) ) ) a.e.  t { t I : u ( t ) > β ( t ) } .

It follows from Lemma 2.7 that α(t)u(t)β(t) for all tI. □

6 Multiplicity result

In this section, we establish the existence of at least three solutions to problem (1.1).

Theorem 6.1 Assume that (H ϕ ), (H f ) and the following conditions are satisfied:

( H B )′ For i=1,2, there exist α i , β i W(I), respectively strict lower and upper solutions of (1.1), such that α i (t)< β i (t), α 1 (t) α 2 (t), β 1 (t) β 2 (t) for all tI, and {tI: α 2 (t)> β 1 (t)}.

(WN)′ There exist k>0, p]1,], c L p (I,[0,[), l L 1 (I,[0,[) and ψ:[0,[[k,[ such that

1 ψ ϕ 1 L l o c 1 (R), 0 ± d s ψ ( | ϕ 1 ( s ) | ) =±,

and

| f ( t , x , y ) | ψ ( | y | ) ( l ( t ) + c ( t ) | y | ( p 1 ) / p ) x [ α 1 ( t ) , β 2 ( t ) ] , y R ,  and a.e.  t I ,

with (p1)/p=1 if p=.

Then problem (1.1) has at least three solutions u 1 , u 2 , u 3 such that

α 1 (t)< u 3 (t)< β 2 (t), α i (t)< u i (t)< β i (t)tI, and i=1,2,

and {tI: β 1 (t) u 3 (t) α 2 (t)}.

Proof Let denote Dirichlet boundary condition (1.2). Consider D 1 , D 2 and D 3 , the operators defined as in (3.7) and associated to the pairs of lower and upper solutions ( α 1 , β 1 ), ( α 2 , β 2 ) and ( α 1 , β 2 ), respectively. Let U i C 1 (I) be the open set defined as in (3.8) and associated to the operator D i for i=1,2,3. From equation (3.9), one has

index ( D i ( 1 , ) , U i ) =1for i=1,2,3.

From the proof of Theorem 3.3, we deduce that any fixed point u of D i (1,) is a solution of (1.1), (1.2) and is such that α i (t)u(t) β i (t) for all tI if i=1,2, and such that α 1 (t)u(t) β 2 (t) for all tI if i=3. Using the fact that α i and β i are respectively strict lower and upper solutions of (1.1), it follows from Lemma 2.8 that D i (1,) has no fixed points in U i V i ¯ , with

V i = { u C 1 ( I ) : α i ( t ) < u ( t ) < β i ( t ) t I } for  i = 1 , 2 , V 3 = { u C 1 ( I ) : α 1 ( t ) < u ( t ) < β 2 ( t ) t I } .

Hence, by the excision property of the fixed point index,

index ( D i ( 1 , ) , V i ) =1for i=1,2,3.

Since α 1 (t) β 1 (t) β 2 (t), α 1 (t) α 2 (t) β 2 (t) for all tI and {tI: α 2 (t)> β 1 (t)}, one has

V 1 V 2 V 3 ,and V 3 ( V 1 V 2 ¯ ),

and

D i (1,u)= D 3 (1,u)u V i ,i=1,2.

This combined with the additivity of the fixed point index implies that

index ( D 3 ( 1 , ) , V 3 ( V 1 V 2 ¯ ) ) = index ( D 3 ( 1 , ) , V 3 ) index ( D 3 ( 1 , ) , V 1 ) index ( D 3 ( 1 , ) , V 2 ) = 1 .

Therefore, problem (1.1) has at least three solutions u 1 V 1 , u 2 V 2 and u 3 V 3 ( V 1 V 2 ¯ ).

We argue similarly if denotes periodic boundary condition (1.3) (resp. Neumann boundary condition (1.4)) by using the results of Section 4 (resp. Section 5). □

References

  1. Ben-Naoum AK, De Coster C: On the existence and multiplicity of positive solutions of the p -Laplacian separated boundary value problem. Differ. Integral Equ. 1997, 10: 1093-1112.

    MathSciNet  MATH  Google Scholar 

  2. Bereanu C, Mawhin J: Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and ϕ -Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 2008, 15: 159-168. 10.1007/s00030-007-7004-x

    Article  MathSciNet  MATH  Google Scholar 

  3. Cabada A, Pouso RL:Existence result for the problem ( ϕ ( u ) ) =f(t,u, u ) with periodic and Neumann boundary conditions. Nonlinear Anal. 1997, 30: 1733-1742. 10.1016/S0362-546X(97)00249-6

    Article  MathSciNet  MATH  Google Scholar 

  4. Cabada A, Pouso RL:Existence results for the problem ( ϕ ( u ) ) =f(t,u, u ) with nonlinear boundary conditions. Nonlinear Anal. 1999, 35: 221-231. 10.1016/S0362-546X(98)00009-1

    Article  MathSciNet  MATH  Google Scholar 

  5. Cabada A, Pouso RL: Existence theory for functional p -Laplacian equations with variable exponents. Nonlinear Anal. 2003, 52: 557-572. 10.1016/S0362-546X(02)00122-0

    Article  MathSciNet  MATH  Google Scholar 

  6. Cabada A, O’Regan D, Pouso RL: Second order problems with functional conditions including Sturm-Liouville and multipoint conditions. Math. Nachr. 2008, 281: 1254-1263. 10.1002/mana.200510675

    Article  MathSciNet  MATH  Google Scholar 

  7. De Coster C: On pairs of positive solutions for the one dimensional p -Laplacian. Nonlinear Anal. 1994, 23: 669-681. 10.1016/0362-546X(94)90245-3

    Article  MathSciNet  MATH  Google Scholar 

  8. El Khattabi N: Problèmes périodiques du second ordre à croissance au plus linéaire. Topol. Methods Nonlinear Anal. 1995, 5: 365-383.

    MathSciNet  MATH  Google Scholar 

  9. Ferracuti L, Papalini F: Boundary-value problems for strongly non-linear multivalued equations involving different ϕ -Laplacians. Adv. Differ. Equ. 2009, 14: 541-566.

    MathSciNet  MATH  Google Scholar 

  10. Frigon M: Théorèmes d’existence de solutions d’inclusions différentielles. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 472. In Topological Methods in Differential Equations and Inclusions. Kluwer Academic, Dordrecht; 1995:51-87. Montreal 1994

    Chapter  Google Scholar 

  11. Goudreau K: A multiplicity result for a nonlinear boundary value problem. J. Math. Anal. Appl. 1998, 218: 395-408. 10.1006/jmaa.1997.5799

    Article  MathSciNet  MATH  Google Scholar 

  12. Mawhin J, Thompson HB: Nagumo conditions and second-order quasilinear equations with compatible nonlinear functional boundary conditions. Rocky Mt. J. Math. 2011, 41: 573-596. 10.1216/RMJ-2011-41-2-573

    Article  MathSciNet  MATH  Google Scholar 

  13. O’Regan D:Some general existence principles and results for ( ϕ ( y ) ) =q(t)f(t,y, y ), 0<t<1. SIAM J. Math. Anal. 1993, 24: 648-668. 10.1137/0524040

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang J, Gao W, Lin Z: Boundary value problems for general second order equations and similarity solutions to the Rayleigh problem. Tohoku Math. J. 1995, 45: 327-344.

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang J, Gao W: Existence of solutions to boundary value problems for a nonlinear second order equation with weak Carathéodory functions. Differ. Equ. Dyn. Syst. 1997, 5: 175-185.

    MATH  Google Scholar 

  16. Zhang JJ, Liu WB, Ni JB, Chen TY: Multiple periodic solutions of p -Laplacian equation with one-side Nagumo condition. J. Korean Math. Soc. 2008, 45: 1549-1559. 10.4134/JKMS.2008.45.6.1549

    Article  MathSciNet  MATH  Google Scholar 

  17. Manásevich R, Mawhin J: Periodic solutions for nonlinear systems with p -Laplacian-like operators. J. Differ. Equ. 1998, 145: 367-393. 10.1006/jdeq.1998.3425

    Article  MathSciNet  MATH  Google Scholar 

  18. Granas A, Dugundji J Springer Monographs in Mathematics. In Fixed Point Theory. Springer, New York; 2003.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marlene Frigon.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Authors contributed equally. They read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

El Khattabi, N., Frigon, M. & Ayyadi, N. Multiple solutions of boundary value problems with ϕ-Laplacian operators and under a Wintner-Nagumo growth condition. Bound Value Probl 2013, 236 (2013). https://doi.org/10.1186/1687-2770-2013-236

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-2770-2013-236

Keywords