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

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.

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

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

Here, f:[0,T]\times {\mathbb{R}}^2\to \mathbb{R} is a Carathéodory function, and \varphi :\mathbb{R}\to \mathbb{R} 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, [1–16] 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^{\prime }). A Nagumo growth condition was imposed on f. The case where f(t,u,u^{\prime }) is a Carathéodory map and \varphi (u^{\prime })=u^{\prime } 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:

with

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^{\prime }) 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 [4–6, 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

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.

In what follows, we denote I=[0,T]. The space of continuous functions \mathcal{C}(I), and the space of continuously derivable functions {\mathcal{C}}^1(I) are equipped with the usual norms {\parallel \cdot \parallel }_0 and {\parallel u\parallel }_1=max\{{\parallel u\parallel }_0,{\parallel u^{\prime }\parallel }_0\}, respectively. We denote the usual norm in L^p(I) by {\parallel u\parallel }_{L^p}, where 1\le p\le \mathrm{\infty }. We set

Definition 2.1 A map f:I\times {\mathbb{R}}^2\to \mathbb{R} is a Carathéodory function if

1. (i)

   f(t,\cdot ,\cdot ) is continuous for almost every t\in I;

2. (ii)

   f(\cdot ,x,y) is measurable for all (x,y)\in {\mathbb{R}}^2;

3. (iii)

   for all R>0, there exists h_R\in L^1(I) such that |f(t,x,y)|\le h_R(t) for all (x,y)\in {\mathbb{R}}^2 such that |x|\le R, |y|\le R, and for almost every t\in I.

Definition 2.2 We say that \alpha \in W(I) is a lower solution of (1.1) if

and, in addition,

1. (i)

   if ℬ denotes Dirichlet boundary condition (1.2), it satisfies

   \alpha (0)\le \mu \text{and}\alpha (T)\le \nu ;
2. (ii)

   if ℬ denotes periodic boundary condition (1.3), it satisfies

   \alpha (0)=\alpha (T)\text{and}\varphi ({\alpha }^{\prime }(0))\ge \varphi ({\alpha }^{\prime }(T));
3. (iii)

   if ℬ denotes Neumann boundary condition (1.4), it satisfies

   \varphi ({\alpha }^{\prime }(0))\ge \mu \text{and}\varphi ({\alpha }^{\prime }(T))\le \nu .

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 \alpha \in W(I) is a strict lower solution of (1.1) if for any t_0\in ]0,T[, there exist \epsilon >0 and {\mathcal{U}}_{t_0}, a neighborhood of t_0 in I, such that for almost every t\in {\mathcal{U}}_{t_0} and all (x,y)\in [\alpha (t),\alpha (t)+\epsilon ]\times [{\alpha }^{\prime }(t)-\epsilon ,{\alpha }^{\prime }(t)+\epsilon ],

and, in addition,

1. (i)

   if ℬ denotes Dirichlet boundary condition (1.2), it satisfies
2. (ii)

   if ℬ denotes periodic boundary condition (1.3), it satisfies

   \alpha (0)=\alpha (T)\text{and}\varphi ({\alpha }^{\prime }(0))>\varphi ({\alpha }^{\prime }(T));
3. (iii)

   if ℬ denotes Neumann boundary condition (1.4), it satisfies

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 \varphi :\mathbb{R}\to \mathbb{R} is a bijective increasing homeomorphism.

(H _f ) The map f:I\times {\mathbb{R}}^2\to \mathbb{R} is Carathéodory.

({\text{H}}_{\mathcal{B}}) There exist \alpha ,\beta \in W(I), respectively lower and upper solutions of (1.1), such that \alpha (t)\le \beta (t) for all t\in I.

(WN) There exist k>0, p\in ]1,\mathrm{\infty }], c\in L^p(I,[0,\mathrm{\infty }[), l\in L^1(I,[0,\mathrm{\infty }[) and \psi :[0,\mathrm{\infty }[\to [k,\mathrm{\infty }[ such that

and

with (p-1)/p=1 if p=\mathrm{\infty }.

In what follows, ({\text{H}}_{\mathcal{B}}) will be replaced by (H _D ), (H _N ) or (H _P ) if ℬ 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\times {\mathbb{R}}^2\to \mathbb{R} be a Carathéodory function and \sigma :I\times {\mathbb{R}}^2\to \mathbb{R} such that

1. (i)

   for any u\in {\mathcal{C}}^1(I), the map t\mapsto \sigma (t,u(t),u^{\prime }(t)) is measurable;

2. (ii)

   for any sequence \{u_n\} converging to u_0 in {\mathcal{C}}^1(I), there exists c>0 such that

   \parallel \sigma (t,u_n(t),u_n^{\prime }(t))\parallel \le c\mathit{\text{a.e. }}t\in I\mathit{\text{ and }}\mathrm{\forall }n\ge 0;

and

Then the operator N_g:{\mathcal{C}}^1(I)\to \mathcal{C}(I) defined by

with

is continuous and completely continuous. Moreover, for every u\in {\mathcal{C}}^1(I), N_g(u) is absolutely continuous and

Observe that the previous lemma holds with \sigma (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 \varphi :\mathbb{R}\to \mathbb{R} satisfy (H _ϕ ), and let \mathcal{G}:\mathbb{R}\times \mathcal{C}(I)\to \mathbb{R} be defined by

Then the following statements hold:

1. (1)

   For any c\in \mathbb{R} and any h\in \mathcal{C}(I), the equation

   \mathcal{G}(a,h)=c

has a unique solution {\overline{a}}_c(h).

1. (2)

   For any c\in \mathbb{R}, the function {\overline{a}}_c:\mathcal{C}(I)\to \mathbb{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 \varphi :\mathbb{R}\to \mathbb{R} satisfies (H _ϕ ). Let \mathrm{\Phi }:\mathbb{R}\times \mathcal{C}(I)\to {\mathcal{C}}^1(I) be defined by

Then Φ is continuous.

We will use the following maximum principle type result.

Lemma 2.7 Assume that (H _ϕ ) is satisfied. Let v,w\in W(I) be such that

Assume that one of the following conditions holds:

1. (i)

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

2. (ii)

   \varphi (v^{\prime }(0))\le \varphi (w^{\prime }(0)), \varphi (v^{\prime }(T))\ge \varphi (w^{\prime }(T));

3. (iii)

   v(0)=v(T), w(0)=w(T), \varphi (v^{\prime }(0))-\varphi (v^{\prime }(T))\le \varphi (w^{\prime }(0))-\varphi (w^{\prime }(T)).

Then v(t)\ge w(t) for all t\in I, or there exists c>0 such that v(t)=w(t)-c for all t\in I.

Proof Assume that

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

or

Indeed, if (2.5) does not hold, let \tau ,\rho \in I be such that

and

If \tau \in ]0,T[, then \varphi (v^{\prime }(\tau ))=\varphi (w^{\prime }(\tau )) and (2.3) or (2.4) hold.

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

Similarly, if (iii) holds, since

one has v^{\prime }(0)\ge w^{\prime }(0) and v^{\prime }(T)\le w^{\prime }(T). Using (iii) and the fact that ϕ is increasing, we get

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

Therefore, by assumption,

If (2.3) holds, for every t\in ]t_0,t_1],

Since ϕ is increasing, we deduce that v-w is nonincreasing in [t_0,t_1]. This is a contradiction. Similarly, we obtain a contradiction if (2.4) holds.

Therefore, v(t)\ge w(t) for all t\in I, or (2.5) is satisfied. □

Lemma 2.8 Assume (H _ϕ ) and (H _f ). Let \alpha ,\beta \in W(I) be respectively strict lower and upper solutions of (1.1) such that for all t\in I. If u\in W(I) is a solution of (1.1) such that \alpha (t)\le u(t)\le \beta (t) for all t\in I, then for all t\in I.

Proof Let u\in W(I) be a solution of (1.1) such that \alpha (t)\le u(t)\le \beta (t) for all t\in I. Assume that

First, we claim that 0,T\notin A. This is obviously the case if ℬ denotes Dirichlet boundary condition (1.2). If ℬ denotes periodic boundary condition (1.3), then u-\alpha attains a minimum at 0 and T. So, by (H _ϕ ) and Definition 2.3(ii),

a contradiction. If ℬ denotes Neumann boundary condition (1.4), then u-\alpha attains a minimum at 0 or at T. So, by (H _ϕ ) and Definition 2.3(iii),

a contradiction.

Let t_0=maxA\in ]0,T[. So, \varphi (u^{\prime }(t_0))=\varphi ({\alpha }^{\prime }(t_0)). By Definition 2.3, there exist \epsilon >0 and {\mathcal{U}}_{t_0}, a neighborhood of t_0, such that {(\varphi ({\alpha }^{\prime }(t)))}^{\prime }\ge f(t,x,y) a.e. t\in {\mathcal{U}}_{t_0} and all x, y such that \alpha (t)\le x\le \alpha (t)+\epsilon and |y-{\alpha }^{\prime }(t)|\le \epsilon. Since u\in {\mathcal{C}}^1(I), there exists t_1\in ]t_0,T[ such that u(t)\in ]\alpha (t),\alpha (t)+\epsilon ] and |u^{\prime }(t)-{\alpha }^{\prime }(t)|\le \epsilon for all t\in ]t_0,t_1]. Since t_0=maxA, there exists t\in ]t_0,t_1] such that u^{\prime }(t)>{\alpha }^{\prime }(t). Using the fact that ϕ is increasing, we deduce that

This is a contradiction. Therefore, for all t\in I.

Similarly, we show that \beta (t)>u(t) for all t\in I. □

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 \psi :[0,\mathrm{\infty }[\to [k,\mathrm{\infty }[ be such that

Then, for every d_0\ge 0, p\in ]1,\mathrm{\infty }], c_0\in L^p(I,[0,\mathrm{\infty }[), l_0\in L^1(I,[0,\mathrm{\infty }[), and B\subset \mathcal{C}(I) bounded, there exists M_0>d_0 such that for every

satisfying

one has .

Proof Let

Assumptions (H _ϕ ) and (2.7) imply that there exists M_0>d_0 such that

and

Assume that there exists

such that {\parallel u^{\prime }\parallel }_0\ge M_0. If max\{u^{\prime }(t):t\in I\}\ge M_0, there exist t_0,t_1\in I such that \varphi (u^{\prime }(t_0))=\varphi (d_0), \varphi (u^{\prime }(t_1))=\varphi (M_0), \varphi (u^{\prime }(t))\in ]\varphi (d_0),\varphi (M_0)[ and u^{\prime }(t)>0 for all t between t_0 and t_1. Without loss of generality, we assume that . Then, by assumption,

Integrating from t_0 to t_1 and using the Hölder inequality and the change of variable formula in an integral give us

This contradicts (2.8).

Similarly, if min\{u^{\prime }(t):t\in I\}\le -M_0, there exist t_0,t_1\in I such that \varphi (u^{\prime }(t_0))=\varphi (-d_0), \varphi (u^{\prime }(t_1))=\varphi (-M_0), \varphi (u^{\prime }(t))\in ]\varphi (-M_0),\varphi (-d_0)[ and for all t between t_0 and t_1. Arguing as above leads to a contradiction. □

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 \lambda \in [0,1]:

(3.1λ)

where \hat{f}:[0,1]\times {\mathbb{R}}^2\to \mathbb{R} is defined by

with \hat{g},\hat{h}\in L^1(I) chosen such that

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\{{\parallel \alpha \parallel }_0,{\parallel \beta \parallel }_0\} such that any solution u of (3.1 _λ ) satisfies .

Proof Fix M_1>0 such that

We claim that any solution u of (3.1 _λ ) is such that {\parallel u\parallel }_0\le M_1. Indeed, by (H _D ),

From the definition of \hat{f}, one has, almost everywhere on ,

Similarly,

It follows from Lemma 2.7 that -M_1\le u(t)\le M_1 for all t\in I.

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

Observe that, by (WN),

with

It follows from Lemma 2.9 applied with d_0=(\nu -\mu )/T that there exists M_0 such that any solution u of (3.1 _λ ) satisfies .

Finally, set M=max\{M_0,1+M_1\}. We have that 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 \lambda \in [0,1], problem (3.1 _λ ) has at least one solution.

Proof Let us define N_{\hat{f}}:{\mathcal{C}}^1(I)\to \mathcal{C}(I) and {\mathcal{S}}_D:[0,1]\times {\mathcal{C}}^1(I)\to \mathbb{R}\times \mathcal{C}(I) by

and

where {\overline{a}}_{\nu -\mu } is obtained in Lemma 2.5. Now, we define \mathcal{D}:[0,1]\times {\mathcal{C}}^1(I)\to {\mathcal{C}}^1(I) by

where Φ is defined in Lemma 2.6. We deduce that {\mathcal{S}}_D is continuous and completely continuous from Lemma 2.5 and from Lemma 2.4 applied with

This combined with Lemma 2.6 implies that \mathcal{D} is continuous and completely continuous.

Now, we study the fixed points of \mathcal{D}. Let u\in {\mathcal{C}}^1(I) and \lambda \in [0,1] be such that u=\mathcal{D}(\lambda ,u). One has, by Lemma 2.5,

Also, \varphi (u^{\prime })={\overline{a}}_{\nu -\mu }(\lambda N_{\hat{f}}(u))+\lambda N_{\hat{f}}(u). Hence, by Lemma 2.4, it is absolutely continuous and

So, fixed points of \mathcal{D} are solutions of (3.1 _λ ).

Let M>max\{{\parallel \alpha \parallel }_0,{\parallel \beta \parallel }_0\}\ge max\{\mu ,\nu \} be the constant obtained in Proposition 3.1 and set

Proposition 3.1 implies that u\ne \mathcal{D}(\lambda ,u) for all (\lambda ,u)\in [0,1]\times \partial \mathcal{U}. Observe that \mathcal{D}(0,u)=\mathrm{\Phi }(\mu ,{\overline{a}}_{\nu -\mu }(0))=u_0 with u_0(t)=\mu +(\nu -\mu )t/T. One has u_0\in \mathcal{U}. By the properties of the fixed point index (see [18] for more details),

Therefore, for every \lambda \in [0,1], \mathcal{D}(\lambda ,\cdot ) 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 u\in W(I) such that \alpha (t)\le u(t)\le \beta (t) for every t\in I.

Proof Proposition 3.2 insures the existence of u\in W(I), a solution of (3.1 _λ ) for \lambda =1. To conclude, we have to show that \alpha (t)\le u(t)\le \beta (t) for all t\in I since \hat{f}(t,x,y)=f(t,x,y) for x\in [\alpha (t),\beta (t)].

By (H _D ),

It follows from Lemma 2.7 that u(t)\ge \alpha (t) for all t\in I.

A similar argument yields u(t)\le \beta (t) for all t\in I. □

Remark 3.4 The hypothesis (WN) can be generalized by

with d_0=(\nu -\mu )/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:

where \varphi (s)=s^3, a\in C[0,T], g\in C(\mathbb{R}) and h\in L^1[0,T] with h(t)\ge 0 a.e. t\in [0,T]. Let m>0 be such that |a(t)|\le m for all t\in T. Then \alpha =-m and \beta =m are respectively lower and upper solutions of (3.10). Let \psi (s)=(|s{|}^3+1), c\ge max\{|g(x)|:|x|\le m\} and l=2mh. One has

and

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

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 \lambda \in [0,1]:

where \hat{f} 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\{{\parallel \alpha \parallel }_0,{\parallel \beta \parallel }_0\} such that any solution u of (4.1 _λ ) satisfies .

Proof Fix M_1>0 such that

We claim that any solution u of (4.1 _λ ) is such that {\parallel u\parallel }_0\le M_1. Observe that a solution of (4.1 _λ ) satisfies

So, it satisfies

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

It follows from Lemma 2.7 that {\parallel u\parallel }_0\le M_1, or there exists c>0 such that |u(t)|=M_1+c for all t\in I. If u\equiv M_1+c,

a contradiction. Similarly, one cannot have u\equiv -(M_1+c). Hence, {\parallel u\parallel }_0\le M_1.

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

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 .

Finally, set M=max\{M_0,1+M_1\}. We have that 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 \lambda \in [0,1], problem (4.1 _λ ) has at least one solution.

Proof Let N_{\hat{f}}:{\mathcal{C}}^1(I)\to \mathcal{C}(I) be defined in (3.5). Let us consider the operators \hat{L}:{\mathcal{C}}^1(I)\to \mathcal{C}(I) and {\mathcal{S}}_P:[0,1]\times {\mathcal{C}}^1(I)\to \mathbb{R}\times \mathcal{C}(I) defined by

and

where {\overline{a}}_0 is obtained in Lemma 2.5. Now, we define \mathcal{P}:[0,1]\times {\mathcal{C}}^1(I)\to {\mathcal{C}}^1(I) by

Again, we deduce from Lemmas 2.4, 2.5 and 2.6 that {\mathcal{S}}_P and \mathcal{P} are continuous and completely continuous.

Now, we study the fixed points of \mathcal{P}. Let u\in {\mathcal{C}}^1(I) and \lambda \in [0,1] be such that u=\mathcal{P}(\lambda ,u). One has, by Lemma 2.5,