Existence and Multiplicity Results for Systems of First-Order Differential Equations via the Method of Solution-Regions

Marlène Frigon

doi:10.1515/ans-2017-6049

Article Open Access

Existence and Multiplicity Results for Systems of First-Order Differential Equations via the Method of Solution-Regions

Marlène Frigon

Published/Copyright: February 7, 2018

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From the journal Advanced Nonlinear Studies Volume 18 Issue 3

Abstract

In this paper, we establish existence and multiplicity results for systems of first-order differential equations. To this end, we introduce the method of solution-regions. It generalizes the method of upper and lower solutions and the method of solution-tubes. Our results can also be seen as viability results since we obtain solutions remaining in suitable regions. We give conditions insuring the existence of at least three viable solutions of a system of first-order differential equations. Many examples are presented to show that a large variety of sets can be solution-regions.

Keywords: System of First Order Differential Equations; Multiple Solutions; Initial Value and Periodic Boundary Value Problems; Fixed Point Index; Solution-Region; Upper and Lower Solutions

MSC 2010: 34B15; 34A34; 34A12

1 Introduction

In this paper, we establish existence and multiplicity results for systems of first-order differential equations of the form

(1.1) u ′ ⁢ ( t ) = f ⁢ ( t , u ⁢ ( t ) ) for a.e. ⁢ t ∈ [ 0 , T ] , u ∈ ℬ ,

where f:[0,T]×ℝN→ℝN is a Carathéodory function and ℬ denotes the initial value or the periodic boundary value conditions

(1.2) u ⁢ ( 0 ) = r ,

(1.3) u ⁢ ( 0 ) = u ⁢ ( T ) .

No growth conditions will be imposed on f. Even though this problem is widely treated in the literature, few multiplicity results were obtained. In particular, there are almost no articles in the literature establishing the existence of more than one solution of systems of differential equations.

To our knowledge, the method of upper and lower solutions was first used to obtain existence results for the periodic problem in the case of one differential equation (N=1) by Moretto [20] who considered a locally Lipschitzian right-hand side. This method was extended to continuous maps by Mawhin [18] and then to Carathéodory functions by Nkashama [22] who considered absolutely continuous lower and upper solutions. Other generalizations of the method of lower and upper solutions were obtained for example by Pouso [25] and by Frigon and O’Regan [9] who allowed the lower and upper solutions to be functions of bounded variation.

In 1962, Knobloch [15] introduced the notion of component-wise lower and upper solutions which he used to obtain an existence result for the periodic problem for a system of differential equation in the case where f satisfies a locally Lipschitz condition. Inspired by [4], an other extension of the method of lower and upper solutions to systems of differential equations is due to Mirandette [19] who introduced the notion of tube-solutions of (1.1) (see also [6, 10]).

In all the previous results, the method of lower and upper solutions and the method of solution-tubes give information on the location of the solution by insuring the existence of a solution u staying between a lower and an upper solution (α⁢(t)≤u⁢(t)≤β⁢(t)) or staying inside a tube (∥u⁢(t)-v⁢(t)∥≤M⁢(t)).

On the other hand, given K⊂ℝN, a solution of (1.1) remaining in K is called a viable solution (i.e. u⁢(t)∈K for all t). To our knowledge, the first viability result was obtained in 1942 by Nagumo [21]. Since then, his result has been generalized to differential inclusions by many authors. The interested reader is referred to [1] and the references therein. In particular, viability results were obtained by Plaskacz [24] and Bielawski, Górniewicz and Plaskacz [2] in the case where K is a proximate retact and the right-hand side satisfies a tangential condition (see also [7, 11] for other viability results in proximate retracts).

It is interesting to realize that solutions obtained via the method of lower and upper solutions or the method of solution-tubes can be seen as viable solutions since they remain in a given region.

The fixed point theory and topological methods were used to establish multiplicity results for problem (1.1). Many multiplicity results which can be found in the literature rely on fixed point theorems on cones due to Krasnosel’skiĭ or due to Leggett and Williams, see for example [14, 16, 17, 23] and the references therein. Starting with the pioneering work of Mawhin [18], an other approach relies on the method of strict lower and upper solutions in the case of one differential equation (N=1). Other multiplicity results relying on the method of strict lower and upper solutions can be found in [12, 22].

As mentioned before, few multiplicity results were obtained in the case where system (1.1) has more than one equation (N>1). In [8], Frigon and Lotfipour introduced the notion of strict solution-tubes on which rely their multiplicity results. This method was used in [3] to obtain multiplicity results for systems of differential equations with a nonlinear differential operator.

In this paper, we introduce the method of solution-regions to establish existence and multiplicity results for system (1.1). A solution-region will be a suitable set R⊂[0,T]×ℝN for which we will deduce that it contains the graph of viable solutions. As particular cases, a solution-region can be the set between ordered upper and lower solutions or the set given by a solution-tube. However, it could be more general regions. In particular, it does not have to be a proximate retract.

The paper is organized as follows. In Sections 3 and 4, we introduce respectively the notions of admissible regions and solution-regions. Existence results with the method of solution-regions are obtained in Section 5. In the last section, we introduce the notion of strict solution-regions and we give conditions insuring the existence of at least three viable solutions of (1.1). Many nontrivial examples are presented throughout the text to show that the method of solution-regions is a powerful tool to establish the existence of solutions of systems of differential equations. For sake of completeness, in Section 2, we recall some definitions and preliminary results.

2 Preliminaries

In what follows, we denote I=[0,T]. For h:I×ℝN→ℝ having partial derivatives with respect to its last N variables at some point (t^,x^)∈I×ℝN, we denote

∇ x ⁡ h ⁢ ( t ^ , x ^ ) = ( ∂ ⁡ h ∂ ⁡ x 1 ⁢ ( t ^ , x ^ ) , … , ∂ ⁡ h ∂ ⁡ x N ⁢ ( t ^ , x ^ ) ) .

For J⊂I, we consider the following set of locally absolutely continuous maps:

W loc 1 , 1 ( J , ℝ ) = { u : J → ℝ ∣ u ∈ W 1 , 1 ( [ t 0 , t 1 ] , ℝ ) for every [ t 0 , t 1 ] ⊂ J } .

We introduce the notion of locally Carathéodory functions.

Definition 2.1.

Let D⊂I×ℝN. A map f:D→ℝm is a Carathéodory function if

f ⁢ ( t , ⋅ ) is continuous on Dt={x:(t,x)∈D} for almost every t∈I,
f ⁢ ( ⋅ , x ) is measurable for all x∈⋃t∈IDt,
for all k>0, there exists ψk∈L1⁢(I,ℝ) such that ∥f⁢(t,x)∥≤ψk⁢(t) for a.e. t and every x such that ∥x∥≤k and (t,x)∈D.

A map f:D→ℝm is locally Carathéodory if f:A→ℝm is a Carathéodory function for every compact set A⊂D.

It is well known that a completely continuous operator is associated to a Carathéodory map. The reader is referred to [5] for the proof.

Lemma 2.2.

Let g:I×RN→RN be a Carathéodory function. Then the operator Ng:C⁢(I,RN)→C⁢(I,RN) defined by

(2.1) N g ⁢ ( u ) ⁢ ( t ) = ∫ 0 t g ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s

is continuous and completely continuous.

The following comparison result will be useful to obtain viability results.

Lemma 2.3.

Let w:[a,b]→R be a continuous map and J={t∈[a,b]:0<w⁢(t)}. Assume that

w ∈ W loc 1 , 1 ⁢ ( J , ℝ ) ,
w ′ ⁢ ( t ) ≤ 0 a.e. t ∈ J ,
one of the following conditions holds:
1. w ⁢ ( a ) ≤ 0 ,
2. w ⁢ ( a ) ≤ w ⁢ ( b ) .

Then w⁢(t)≤0 for all t∈[a,b] or there exists k>0 such that w⁢(t)=k for all t∈[a,b].

Proof.

Assume that J≠∅. Let

J 0 = { τ ∈ [ a , b ] : w ⁢ ( τ ) = max t ∈ [ a , b ] ⁡ w ⁢ ( t ) } .

Observe that ∅≠J0⊂J.

Case 1: J0=[a,b]. In this case, there exists k>0 such that

w ⁢ ( t ) = k for every ⁢ t ∈ [ a , b ] .

Case 2: J0≠[a,b]. Let t^=max⁡J0. We claim that

(2.2) t ^ > a and there exists ρ ∈ ] a , t ^ ] ∩ J ∖ J 0 .

Indeed, if (iii.1) is satisfied, then a∉J. So t^>a. The intermediate value theorem insures that there exists ρ∈]a,t^[ such that 0<w⁢(ρ)<w⁢(t^).

If (iii.2) is satisfied, w⁢(a)≤w⁢(b). So, if a∈J0, then t^=b. Since J0≠[a,b], [a,t^]∖J0=[a,b]∖J0≠∅. On the other hand, if a∉J0, then t^>a. In both cases, there exists ρ∈]a,t^[ such that 0<w⁢(ρ)<w⁢(t^).

Let ρ be given in (2.2). Let

t 1 = min ⁡ ( [ ρ , b ] ∩ J 0 ) and t 0 = max ⁡ { t ∈ [ ρ , t 1 ] : w ⁢ ( t ) = w ⁢ ( ρ ) } .

So,

t 1 ∈ J 0 , [ t 0 , t 1 ] ⊂ J and w ⁢ ( t 0 ) < w ⁢ ( t 1 ) .

By (i) and (ii), w∈W1,1⁢([t0,t1],ℝ) and w′⁢(t)≤0 almost everywhere on [t0,t1]. Therefore,

0 ≥ ∫ t 0 t 1 w ′ ⁢ ( t ) ⁢ 𝑑 t = w ⁢ ( t 1 ) - w ⁢ ( t 0 ) > 0 .

This is a contradiction. So, J=∅ and hence, w⁢(t)≤0 for all t∈[a,b]. ∎

For sake of completeness, we recall the notions of lower and upper solutions and the notion of solution-tubes of (1.1), see [8, 18, 19, 22].

Definition 2.4.

Let N=1 and β∈W1,1⁢(I,ℝ).

The map β is called an upper solution of (1.1) if
1. f ⁢ ( t , β ⁢ ( t ) ) ≤ β ′ ⁢ ( t ) for almost every t∈I,
2. β ⁢ ( 0 ) ≥ r if ℬ denotes (1.2), β⁢(T)≤β⁢(0) if ℬ denotes (1.3).
It is called a lower solution of (1.1) if it satisfies conditions (i) and (ii) with the reverse inequalities.
The map β is called a strict upper solution of (1.1) (resp. strict lower solution of (1.1)) if there exists ε>0 such that
1. f ⁢ ( t , x ) ≤ β ′ ⁢ ( t ) for almost every t∈I and every x∈(β⁢(t)-ε,β⁢(t)] (resp. f⁢(t,x)≥β′⁢(t) for a.e. t∈I and every x∈[β⁢(t),β⁢(t)+ε)),
2. β ⁢ ( 0 ) > r (resp. β⁢(0)<r) if ℬ denotes (1.2), β⁢(T)<β⁢(0) (resp. β⁢(T)>β⁢(0)) if ℬ denotes (1.3).

Definition 2.5.

Let v∈W1,1⁢(I,ℝN) and M∈W1,1⁢(I,[0,∞)).

The pair (v,M) is called a solution-tube of (1.1) if
1. 〈 x - v ⁢ ( t ) , f ⁢ ( t , x ) - v ′ ⁢ ( t ) 〉 ≤ M ⁢ ( t ) ⁢ M ′ ⁢ ( t ) for almost every t∈I and every x such that ∥x-v⁢(t)∥=M⁢(t),
2. v ′ ⁢ ( t ) = f ⁢ ( t , v ⁢ ( t ) ) a.e. t∈{t∈I:M⁢(t)=0},
3. ∥ v ( 0 ) - r ∥ ≤ M ( 0 ) if ℬ denotes (1.2), ∥v⁢(0)-v⁢(T)∥≤M⁢(0)-M⁢(T) if ℬ denotes (1.3).
The pair (v,M) is called a strict solution-tube of (1.1) if
1. M ⁢ ( t ) > 0 for every t∈I,
2. there exists ε>0 such that, for almost every t∈I and every x with ∥x-v⁢(t)∥∈(M⁢(t)-ε,M⁢(t)],
  
  〈 x - v ⁢ ( t ) , f ⁢ ( t , x ) - v ′ ⁢ ( t ) 〉 ≤ M ′ ⁢ ( t ) ⁢ ∥ x - v ⁢ ( t ) ∥ ,
3. ∥ v ( 0 ) - r ∥ < M ( 0 ) if ℬ denotes (1.2), ∥v⁢(0)-v⁢(T)∥<M⁢(0)-M⁢(T) if ℬ denotes (1.3).

We recall also the notion of proximate retract introduced by Bielawski, Górniewicz and Plaskacz [2, 24].

Definition 2.6.

A compact subset K⊂ℝN is called a proximate retract if there exists an open neighborhood U of K and a continuous function ρ:U→K such that ∥ρ⁢(x)-x∥=dist⁡(x,K) for every x∈U. The map ρ is called a metric retraction.

3 Admissible Regions

We introduce the notion of admissible regions in I×ℝN. Those regions will play a key role in our existence results.

Definition 3.1.

We say that a set R⊂I×ℝN is an admissible region if there exist two continuous maps h:I×ℝN→ℝ and p=(p1,p2):I×ℝN→I×ℝN satisfying the following conditions:

R = { ( t , x ) : h ⁢ ( t , x ) ≤ 0 } is bounded and, for every t∈I,

R t = { x ∈ ℝ N : ( t , x ) ∈ R } ≠ ∅ .
The map h has partial derivatives at (t,x) for almost every t and every x with (t,x)∈Rc=(I×ℝN)∖R, and ∂⁡h∂⁡t, ∇x⁡h are locally Carathéodory maps on Rc.
p is bounded and such that p⁢(t,x)=(t,x) for every (t,x)∈R and

〈 ∇ x ⁡ h ⁢ ( t , x ) , p 2 ⁢ ( t , x ) - x 〉 < 0 for a.e. t and every x with ( t , x ) ∈ R c .

We call (h,p) an admissible pair associated to R.

We present many examples to show that the notion of admissible regions covers a large variety of sets R even in the one-dimensional case N=1.

Example 3.2.

Let α,β∈W1,1⁢(I,ℝ) be such that α⁢(t)≤β⁢(t) for every t∈I. It is easy to verify that the set R={(t,x)∈I×ℝ:α⁢(t)≤x≤β⁢(t)} is an admissible region.

Here is an example in [0,T]×ℝ which shows that the boundary of an admissible region may have corners.

Example 3.3.

Let T=2. The set R=[0,1]×[-2,2]∪[1,2]×[-1,1] is an admissible region. Indeed, we define the continuous maps h:[0,2]×ℝ→ℝ and p:[0,2]×ℝ→[0,2]×ℝ by

h ⁢ ( t , x ) = { 0 if ( t , x ) ∈ R , 3 2 ⁢ ( | x | - 2 ) 2 if 0 ≤ t ≤ 1 , 2 < | x | < ∞ , ( | x | + t - 2 ) ⁢ ( | x | - 1 ) if 1 ≤ t ≤ 2 , 1 < | x | ≤ t , ( t - 1 ) ⁢ ( 3 ⁢ | x | - t - 2 ) if 1 ≤ t ≤ 2 , t < | x | ≤ 1 + t , 1 2 ⁢ ( 6 ⁢ ( t - 1 ) + ( t - 1 ) 2 + 3 ⁢ ( | x | - 2 ) 2 ) if 1 ≤ t ≤ 2 , 1 + t < | x | ,

and

p ⁢ ( t , x ) = { ( t , x ) if ( t , x ) ∈ R , ( t , 2 ⁢ x | x | ) if 0 ≤ t ≤ 1 , 2 < | x | , ( 1 + t - | x | , x | x | ) if 1 ≤ t ≤ 2 , 1 < | x | ≤ t , ( 1 , x + x ⁢ ( 1 - t ) | x | ) if 1 ≤ t ≤ 2 , t < | x | ≤ 1 + t , ( 1 , 2 ⁢ x | x | ) if 1 ≤ t ≤ 2 , 1 + t < | x | .

One can verify that ∂⁡h∂⁡t and ∂⁡h∂⁡x are Carathéodory on Rc, and

∂ ⁡ h ∂ ⁡ x ⁢ ( t , x ) ⁢ ( p 2 ⁢ ( t , x ) - x ) < 0 for ( t , x ) ∈ R c .

Therefore, (h,p) is an admissible pair associated to the admissible region R. It is worthwhile to notice that

R = { ( t , x ) ∈ [ 0 , 2 ] × ℝ : α ⁢ ( t ) ≤ x ≤ β ⁢ ( t ) ⁢ for all ⁢ t ∈ [ 0 , 2 ] } ,

with

β ⁢ ( t ) = { 2 if t ∈ [ 0 , 1 ] , 1 if t ∈ ( 1 , 2 ] , α ⁢ ( t ) = { - 2 if t ∈ [ 0 , 1 ] , - 1 if t ∈ ( 1 , 2 ] .

However, α and β cannot be respectively lower and upper solutions in the sense of Definition 2.4 (1) since they are not continuous.

Now, we present some examples of admissible regions in I×ℝN with N>1.

Example 3.4.

Let M∈W1,1⁢(I,[0,∞)), v∈W1,1⁢(I,ℝN) and a∈W1,1(I,]0,∞[N). The set

R = { ( t , x ) ∈ I × ℝ N : ∑ i = 1 N a i ⁢ ( t ) ⁢ ( x i - v i ⁢ ( t ) ) 2 ≤ M 2 ⁢ ( t ) }

is an admissible region with the associated admissible pair (h,p) given by

h ⁢ ( t , x ) = ( ∑ i = 1 N a i ⁢ ( t ) ⁢ ( x i - v i ⁢ ( t ) ) 2 ) 1 2 - M ⁢ ( t ) ,

and p⁢(t,x)=(t,p2⁢(t,x)) with p2⁢(t,x) the projection of x on Rt.

Here is an example which shows that ∂⁡Rt the boundary of an admissible region R does not need to be smooth.

Example 3.5.

For i=1,…,N, let ai≤bi. It is easy to verify that the set I×[a1,b1]×⋯×[aN,bN] is an admissible region in I×ℝN.

In the next example, we show that there are admissible regions which are not proximate retracts.

Example 3.6.

The region R=I×R0⊂I×ℝ2 is admissible, where

R 0 = { ( x 1 , x 2 ) ∈ ℝ 2 : ( 1 - | x 1 | ) 2 + x 2 2 ≤ 1 } .

Indeed, let us define p:I×ℝ2→I×ℝ2 and h:I×ℝ2→ℝ by

h ⁢ ( t , x 1 , x 2 ) = { ( 1 - | x 1 | ) 2 + x 2 2 - 1 if ( t , x 1 , x 2 ) ∈ R , ( ( 1 - | x 1 | ) 2 + x 2 2 ) 2 - 1 if | x 1 | > 1 , ( 1 - | x 1 | ) 2 + x 2 2 > 1 , x 2 4 - x 1 2 ⁢ ( 2 - | x 1 | ) 2 if x 1 ∈ [ - 1 , 1 ] , ( 1 - | x 1 | ) 2 + x 2 2 > 1 ,

and p⁢(t,x)=(t,p2⁢(t,x)) given by

p 2 ( t , x 1 , x 2 ) = { ( x 1 , x 2 ) if ( t , x 1 , x 2 ) ∈ R , ( x 1 | x 1 | ⁢ ( 1 - 1 - | x 1 | ( 1 - | x 1 | ) 2 + x 2 2 ) , x 2 ( 1 - | x 1 | ) 2 + x 2 2 ) if | x 1 | > 1 and ( 1 - | x 1 | ) 2 + x 2 2 > 1 , ( x 1 , x 2 ⁢ 1 - ( 1 - | x 1 | ) 2 | x 2 | ) if x 1 ∈ [ - 1 , 1 ] and ( 1 - | x 1 | ) 2 + x 2 2 > 1 .

The maps p and h are continuous and h⁢(t,x)≤0 if and only if (t,x)∈R. Also, ∂⁡h∂⁡t and ∇x⁡h are continuous on Rc. Moreover,

〈 ∇ x ⁡ h ⁢ ( t , x ) , p 2 ⁢ ( t , x ) - x 〉 < 0 for every ( t , x ) ∈ R c .

So, R is an admissible region with the admissible pair (h,p). Observe that R0 is not a proximate retract (see Definition 2.6).

4 Solution-Regions

We will be interested to give conditions insuring the existence of a solution of (1.1) in a suitable admissible region. To this aim, we introduce the notion of solution-regions of (1.1).

Definition 4.1.

A set R⊂I×ℝN is called a solution-region of (1.1) if it is an admissible region with an associated admissible pair (h,p) satisfying the following conditions:

For almost every t and every x with (t,x)∉R, one has

∂ ⁡ h ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h ⁢ ( t , x ) , f ⁢ ( p ⁢ ( t , x ) ) 〉 ≤ 0 .
1. If ℬ denotes the initial value condition (1.2), it satisfies h⁢(0,r)≤0.
2. If ℬ denotes the periodic boundary value condition (1.3), it satisfies h⁢(0,x)≤h⁢(T,x) for every x such that (0,x)∉R.

Example 4.2.

Let α,β∈W1,1⁢(I,ℝ) be respectively lower and upper solutions of (1.1) such that α⁢(t)≤β⁢(t) for every t∈I (see Definition 2.4(1)). It can easily be seen that R={(t,x)∈I×ℝ:α⁢(t)≤x≤β⁢(t)} is a solution-region of (1.1).

Example 4.3.

Let (v,M)∈W1,1⁢(I,ℝN)×W1,1⁢(I,[0,∞)) be a solution-tube of (1.1) (see Definition 2.5 (1)). Then

R = { ( t , x ) ∈ I × ℝ N : ∥ x - v ⁢ ( t ) ∥ ≤ M ⁢ ( t ) }

is a solution-region of (1.1). Indeed, R is an admissible region with (h,p) the associated admissible pair given in Example 3.4 with a=(1,…,1). It is easy to check that (h,p) satisfies conditions (i) and (ii) of Definition 4.1.

5 Existence Results Relying on the Method of Solution-Regions

In this section, we establish the following existence result relying on what we call the method of solution-regions.

Theorem 5.1.

Let f:I×RN→RN be a Carathéodory function. Assume that there exists R a solution-region of (1.1). Then problem (1.1) has a solution u∈W1,1⁢(I,RN) such that (t,u⁢(t))∈R for every t∈I.

To prove this result, we modify our problem (1.1). Let (h,p) be an admissible pair associated to the solution-region R. For λ∈[0,1], we consider the following families of problems:

(5.1${{}_{\lambda}}$)

u ′ ⁢ ( t ) = λ ⁢ f R ⁢ ( t , u ⁢ ( t ) ) for a.e. ⁢ t ∈ I ,

u ⁢ ( 0 ) = r ,

if ℬ in (1.1) denotes (1.2), and if it denotes (1.3), we consider

(5.2${{}_{\lambda}}$)

u ′ ⁢ ( t ) = λ ⁢ f R ⁢ ( t , u ⁢ ( t ) ) + 1 - λ T ⁢ ∫ 0 T f R ⁢ ( t , u ⁢ ( t ) ) ⁢ 𝑑 t for a.e. ⁢ t ∈ I ,

u ⁢ ( 0 ) = u ⁢ ( T ) ,

where fR:I×ℝN→ℝN is defined by

(5.3) f R ⁢ ( t , x ) = { f ⁢ ( t , x ) if ( t , x ) ∈ R , f ⁢ ( p ⁢ ( t , x ) ) - c ⁢ ( t ) ⁢ ( x - p 2 ⁢ ( t , x ) ) otherwise,

with c∈L1⁢(I,ℝ) chosen such that

(5.4) c ⁢ ( t ) > ∥ f ⁢ ( p ⁢ ( t , x ) ) ∥ for a.e. t ∈ I and every x ∈ ℝ N .

Observe that such a function c exists since f is Carathéodory and p is a bounded map.

Let us consider the operators ℐ,𝒫:[0,1]×C⁢(I,ℝN)→C⁢(I,ℝN) defined by

ℐ ⁢ ( λ , u ) = r + λ ⁢ N f R ⁢ ( u )

and

(5.5) 𝒫 ⁢ ( λ , u ) ⁢ ( t ) = u ⁢ ( 0 ) - ( 1 + λ ⁢ t ) T ⁢ N f R ⁢ ( u ) ⁢ ( T ) + λ ⁢ N f R ⁢ ( u ) ⁢ ( t ) ,

where NfR is defined in (2.1).

We first establish an existence result for the family of problems ((5.2${{}_{\lambda}}$)).

Proposition 5.2.

Let f:I×RN→RN be a Carathéodory function and (h,p) an admissible pair associated to R a solution-region of (1.1), (1.3). Then, for every λ∈[0,1], ((5.2${{}_{\lambda}}$)) has at least one solution. Moreover, there exists

M > max ⁡ { ∥ x ∥ : ( t , x ) ∈ R ⁢ for some ⁢ t ∈ I }

such that the fixed point index

index ⁡ ( 𝒫 ⁢ ( λ , ⋅ ) , 𝒰 ) = ( - 1 ) N for every λ ∈ [ 0 , 1 ]

with U={u∈C⁢(I,RN):∥u∥0<M}.

Proof.

The fact that f is Carathéodory and condition (iii) of Definition 3.1 imply that fR is also a Carathéodory function. It follows from Lemma 2.2 that 𝒫 is continuous and completely continuous.

We claim that the fixed points of 𝒫 are solutions of ((5.2${{}_{\lambda}}$)). Indeed, if u=𝒫⁢(λ,u),

(5.6) u ⁢ ( t ) = u ⁢ ( 0 ) - ( 1 + λ ⁢ t ) T ⁢ ∫ 0 T f R ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s + λ ⁢ ∫ 0 t f R ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s for every ⁢ t ∈ I .

In particular, for t=0 and for t=T, one has

u ⁢ ( 0 ) = u ⁢ ( 0 ) - 1 T ⁢ ∫ 0 T f R ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s = u ⁢ ( T ) ,

and hence,

(5.7) N f R ⁢ ( u ) ⁢ ( T ) = ∫ 0 T f R ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s = 0 .

From (5.6) and (5.7), one deduces that, for almost every t∈I,

u ′ ⁢ ( t ) = λ ⁢ ( f R ⁢ ( t , u ⁢ ( t ) ) - 1 T ⁢ ∫ 0 T f R ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s ) = λ ⁢ f R ⁢ ( t , u ⁢ ( t ) ) + 1 - λ T ⁢ ∫ 0 T f R ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s .

Thus, u is a solution of ((5.2${{}_{\lambda}}$)).

Fix M>0 such that

(5.8) M > 1 + max ⁡ { ∥ p 2 ⁢ ( t , x ) ∥ : ( t , x ) ∈ I × ℝ N } .

We show that

(5.9) ∥ u ∥ 0 < M for any solution u of (5.2 λ ) .

Indeed, choose m such that

(5.10) M > m > 1 + max ⁡ { ∥ p 2 ⁢ ( t , x ) ∥ : ( t , x ) ∈ I × ℝ N } .

Assume that u is a solution of ((5.2${{}_{\lambda}}$)) for some λ∈[0,1]. If λ=0 and

{ t ∈ I : ∥ u ⁢ ( t ) ∥ > m } ≠ ∅ ,

then, by (5.3), (5.7) and (5.10), u≡k with ∥k∥>m and

0 = ∫ 0 T f R ⁢ ( t , u ⁢ ( t ) ) ⁢ 𝑑 t = ∫ 0 T f ⁢ ( p ⁢ ( t , k ) ) + c ⁢ ( t ) ⁢ ( p 2 ⁢ ( t , k ) - k ) ⁢ d ⁢ t .

This inequality combined with (5.4) and (5.10) imply that

∥ k ∥ ⁢ ∥ c ∥ L 1 ≤ ∫ 0 T c ⁢ ( t ) ⁢ ( 1 + ∥ p 2 ⁢ ( t , k ) ∥ ) ⁢ 𝑑 t ⁢ < m ∥ ⁢ c ∥ L 1 .

This is a contradiction.

On the other hand, if λ∈(0,1], combining (5.3), (5.4), (5.7) and (5.10) permits to deduce that, almost everywhere on {t∈I:∥u⁢(t)∥>m},

∥ u ⁢ ( t ) ∥ ′ = 〈 u ⁢ ( t ) ∥ u ⁢ ( t ) ∥ , u ′ ⁢ ( t ) 〉

= 〈 u ⁢ ( t ) ∥ u ⁢ ( t ) ∥ , λ ⁢ f R ⁢ ( t , u ⁢ ( t ) ) 〉

= λ ⁢ 〈 u ⁢ ( t ) ∥ u ⁢ ( t ) ∥ , f ⁢ ( p ⁢ ( t , u ⁢ ( t ) ) ) + c ⁢ ( t ) ⁢ ( p 2 ⁢ ( t , u ⁢ ( t ) ) - u ⁢ ( t ) ) 〉

≤ λ ⁢ c ⁢ ( t ) ⁢ ( 1 + ∥ p 2 ⁢ ( t , u ⁢ ( t ) ) ∥ - ∥ u ⁢ ( t ) ∥ )

< 0 .

It follows from Lemma 2.3 that u⁢(t)≤m for every t∈I.

Let

𝒰 = { u ∈ C ⁢ ( I , ℝ N ) : ∥ u ∥ 0 < M } .

It follows from (5.9) and the homotopy property of the fixed point index that

(5.11) index ⁡ ( 𝒫 ⁢ ( λ , ⋅ ) , 𝒰 ) = index ⁡ ( 𝒫 ⁢ ( 0 , ⋅ ) , 𝒰 ) for every ⁢ λ ∈ [ 0 , 1 ] .

Observe that

𝒫 ⁢ ( 0 , u ) = u ⁢ ( 0 ) - 1 T ⁢ ∫ 0 T f R ⁢ ( s , u ⁢ ( s ) ) ⁢ 𝑑 s ∈ ℝ N

and

𝒰 ∩ ℝ N = B ℝ N ⁢ ( 0 , M ) .

By the contraction property of the fixed point index (see [13, Chapter 4, Section 12, Theorem 6.2]), one has

(5.12) index ⁡ ( 𝒫 ⁢ ( 0 , ⋅ ) , 𝒰 ) = index ⁡ ( 𝒫 ⁢ ( 0 , ⋅ ) , B ℝ N ⁢ ( 0 , M ) ) .

Let us define P0:[0,1]×BℝN⁢(0,M)¯→ℝN by

P 0 ⁢ ( λ , x ) = λ ⁢ 𝒫 ⁢ ( 0 , x ) + 2 ⁢ ( 1 - λ ) ⁢ x .

We claim that x≠P0⁢(λ,x) for every (λ,x)∈[0,1]×∂⁡BℝN⁢(0,M). Indeed, assume that there exists x∈ℝN such that ∥x∥=M and x=P0⁢(λ,x) for some λ∈(0,1]. By (5.3), (5.4) and (5.8), it satisfies

x = P 0 ⁢ ( λ , x ) = 2 ⁢ ( 1 - λ ) ⁢ x + λ ⁢ ( x - 1 T ⁢ ∫ 0 T f R ⁢ ( s , x ) ⁢ 𝑑 s )

= 2 ⁢ ( 1 - λ ) ⁢ x + λ ⁢ ( x - 1 T ⁢ ∫ 0 T f ⁢ ( p ⁢ ( s , x ) ) + c ⁢ ( t ) ⁢ ( p 2 ⁢ ( s , x ) - x ) ⁢ d ⁢ s )

= ( 2 - λ + λ ⁢ ∥ c ∥ L 1 T ) ⁢ x - λ T ⁢ ∫ 0 T f ⁢ ( p ⁢ ( s , x ) ) + c ⁢ ( t ) ⁢ p 2 ⁢ ( s , x ) ⁢ d ⁢ s .

So,

M ⁢ ( 1 - λ + λ ⁢ ∥ c ∥ L 1 T ) = ( 1 - λ + λ ⁢ ∥ c ∥ L 1 T ) ⁢ ∥ x ∥

= ∥ λ T ⁢ ∫ 0 T f ⁢ ( p ⁢ ( s , x ) ) + c ⁢ ( t ) ⁢ p 2 ⁢ ( s , x ) ⁢ d ⁢ s ∥

< λ T ⁢ ∫ 0 T c ⁢ ( t ) + c ⁢ ( t ) ⁢ ( M - 1 ) ⁢ d ⁢ s

= M ⁢ λ ⁢ ∥ c ∥ L 1 T .

This is a contradiction.

The homotopy property of the fixed point index implies that

(5.13) index ⁡ ( P 0 ⁢ ( 1 , ⋅ ) , B ℝ N ⁢ ( 0 , M ) ) = index ⁡ ( P 0 ⁢ ( 0 , ⋅ ) , B ℝ N ⁢ ( 0 , M ) ) = index ⁡ ( 2 ⁢ id , B ℝ N ⁢ ( 0 , M ) ) = ( - 1 ) N .

Combining (5.11), (5.12) and (5.13), one obtains

index ⁡ ( 𝒫 ⁢ ( λ , ⋅ ) , 𝒰 ) = ( - 1 ) N for every ⁢ λ ∈ [ 0 , 1 ] .

Therefore, for every λ∈[0,1], 𝒫⁢(λ,⋅) has a fixed point, and hence ((5.2${{}_{\lambda}}$)) has a solution. ∎

We obtain an analogous result for the family of problems ((5.1${{}_{\lambda}}$)) but, in this case, the proof is simpler.

Proposition 5.3.

Let f:I×RN→RN be a Carathéodory function and (h,p) an admissible pair associated to a solution-region R of (1.1), (1.2). Then ((5.1${{}_{\lambda}}$)) has at least one solution for every λ∈[0,1]. Moreover, there exists

M > max ⁡ { ∥ x ∥ : ( t , x ) ∈ R ⁢ for some ⁢ t ∈ I }

such that the fixed point index

index ⁡ ( ℐ ⁢ ( λ , ⋅ ) , 𝒰 ) = 1 for every λ ∈ [ 0 , 1 ]

with U={u∈C⁢(I,RN):∥u∥0<M}.

Proof.

It follows from the assumptions and Lemma 2.2 that ℐ is continuous and completely continuous. It is easy to verify that the fixed points of ℐ are solutions of ((5.1${{}_{\lambda}}$)).

Fix M>0 such that

M > 1 + max ⁡ { ∥ p 2 ⁢ ( t , x ) ∥ : ( t , x ) ∈ I × ℝ N }

and let

𝒰 = { u ∈ C ⁢ ( I , ℝ N ) : ∥ u ∥ 0 < M } .

Arguing as in the proof of Proposition 5.2, one can show that u≠ℐ⁢(λ,u) for all (λ,u)∈[0,1]×∂⁡𝒰. The properties of the fixed point index and the fact that the constant map r∈𝒰 imply that

index ⁡ ( ℐ ⁢ ( λ , ⋅ ) , 𝒰 ) = index ⁡ ( ℐ ⁢ ( 0 , ⋅ ) , 𝒰 ) = index ⁡ ( r , 𝒰 ) = 1 for every ⁢ λ ∈ [ 0 , 1 ] .

Therefore, for every λ∈[0,1], ℐ⁢(λ,⋅) has a fixed point, and hence ((5.1${{}_{\lambda}}$)) has a solution. ∎

We are ready to prove Theorem 5.1.

Proof of Theorem 5.1.

We first consider the case where ℬ denotes the periodic boundary value condition (1.3). Proposition 5.2 insures the existence of u a solution of ((5.2${{}_{\lambda}}$)) with λ=1.

It follows from (5.3) and Definitions 3.1 and 4.1 that, almost everywhere on {t:h⁢(t,u⁢(t))>0}, one has

d ⁢ h d ⁢ t ⁢ ( t , u ⁢ ( t ) ) = ∂ ⁡ h ∂ ⁡ t ⁢ ( t , u ⁢ ( t ) ) + 〈 ∇ x ⁡ h ⁢ ( t , u ⁢ ( t ) ) , u ′ ⁢ ( t ) 〉

= ∂ ⁡ h ∂ ⁡ t ( t , u ( t ) ) + 〈 ∇ x h ( t , u ( t ) ) , f ( p ( t , u ( t ) ) ) + c ( t ) ( p 2 ( t , u ( t ) ) - u ( t ) 〉

< 0 .

Since u⁢(0)=u⁢(T), the previous inequality combined with Lemma 2.3 and Definition 4.1 (ii) imply that (t,u⁢(t))∈R for every t∈I. Therefore, u is a solution of (1.1) since p=id on R.

The case where ℬ denotes the initial value condition (1.2) is analogous. ∎

We present some examples of applications of Theorem 5.1.

Example 5.4.

We consider the following problem:

(5.14)

u ′ ⁢ ( t ) = f ⁢ ( t , u ⁢ ( t ) ) for a.e. ⁢ t ∈ [ 0 , 3 ] ,

u ⁢ ( 0 ) = u ⁢ ( 3 ) ,

where

f ⁢ ( t , x ) = e 6 ⁢ x ⁢ ( 1 - t ) ⁢ ( x - 1 t 2 3 ) ⁢ ( 4 ⁢ x - 5 ⁢ sin 2 ⁡ ( t ⁢ π 3 ) ) - 3 ⁢ ( x 5 - | x | 5 ) .

We claim that there exists u a solution of this problem such that (t,u⁢(t))∈R for every t∈[0,3], where R={(t,x)∈[0,3]×ℝ:-1≤x≤b⁢(t)}, with

b ⁢ ( t ) = { 1 if t ∈ [ 0 , 1 ) , 2 if t ∈ [ 1 , 2 ] , 5 - 3 ⁢ t 2 if t ∈ ( 2 , 3 ] .

First of all, we show that R is an admissible region. We define the continuous maps h:[0,3]×ℝ→ℝ and p:[0,3]×ℝ→[0,3]×ℝ by

h ⁢ ( t , x ) = { 0 if ( t , x ) ∈ R , - 1 - x if x ≤ - 1 , ( x - t ) ⁢ ( x - 1 ) if t ∈ [ 0 , 1 ] , 1 < x ≤ 2 - t , ( t + 3 ⁢ x - 4 ) ⁢ ( 1 - t ) if t ∈ [ 0 , 1 ] , 2 - t < x ≤ 3 - t , 1 2 ⁢ ( 6 ⁢ ( 1 - t ) + ( 1 - t ) 2 + 3 ⁢ ( x - 2 ) 2 ) if t ∈ [ 0 , 1 ] , x > 3 - t , 3 2 ⁢ ( x - 2 ) 2 if t ∈ ( 1 , 2 ] , x > 2 , 3 8 ⁢ ( 2 ⁢ x + 3 ⁢ t - 10 ) 2 if t ∈ ( 2 , 3 ] , x > 5 - 3 ⁢ t 2 ,

and

p ⁢ ( t , x ) = { ( t , x ) if ( t , x ) ∈ R , ( t , - 1 ) if x ≤ - 1 , ( x + t - 1 , 1 ) if t ∈ [ 0 , 1 ] , 1 < x ≤ 2 - t , ( 1 , x + t - 1 ) if t ∈ [ 0 , 1 ] , 2 - t < x ≤ 3 - t , ( 1 , 2 ) if t ∈ [ 0 , 1 ] , x > 3 - t , ( t , 2 ) if t ∈ ( 1 , 2 ] , x > 2 , ( t , 5 - 3 ⁢ t 2 ) if t ∈ ( 2 , 3 ] , x > 5 - 3 ⁢ t 2 .

It is easy to check that ∂⁡h∂⁡t and ∂⁡h∂⁡x are Carathéodory functions on ([0,3]×ℝ)∖R and

( p 2 ⁢ ( t , x ) - x ) ⁢ ∂ ⁡ h ∂ ⁡ x ⁢ ( t , x ) < 0 for a.e. t and every x such that ( t , x ) ∉ R .

So R is an admissible region.

To show that R is a solution-region of (5.14), we first check that h satisfies the boundary condition. One has for (0,x)∉R,

h ⁢ ( 0 , x ) = { - 1 - x if x < - 1 , x ⁢ ( x - 1 ) if 1 < x ≤ 2 , 3 ⁢ x - 4 if 2 < x ≤ 3 , 7 2 + 3 2 ⁢ ( x - 2 ) 2 if x > 3 ,

≤ { - 1 - x if x ≤ - 1 , 3 8 ⁢ ( 2 ⁢ x - 1 ) 2 if x > 1 ,

= h ⁢ ( 3 , x ) .

Finally, for almost every t and every x such that (t,x)∉R,

∂ ⁡ h ∂ ⁡ t ⁢ ( t , x ) + ∂ ⁡ h ∂ ⁡ x ⁢ ( t , x ) ⁢ f ⁢ ( p ⁢ ( t , x ) ) = { - f ⁢ ( t , - 1 ) if x ≤ - 1 , ( 1 - x ) + ( 2 ⁢ x - t - 1 ) ⁢ f ⁢ ( x + t - 1 , 1 ) if t ∈ [ 0 , 1 ] , 1 < x ≤ 2 - t , ( 5 - 3 ⁢ x - 2 ⁢ t ) + 3 ⁢ ( 1 - t ) ⁢ f ⁢ ( 1 , x + t - 1 ) if t ∈ [ 0 , 1 ] , 2 - t < x ≤ 3 - t , ( t - 4 ) + 3 ⁢ ( x - 2 ) ⁢ f ⁢ ( 1 , 2 ) if t ∈ [ 0 , 1 ] , x > 3 - t , 3 ⁢ ( x - 2 ) ⁢ f ⁢ ( t , 2 ) if t ∈ ( 1 , 2 ] , x > 2 , 3 2 ⁢ ( 2 ⁢ x + 3 ⁢ t - 10 ) ⁢ ( 3 2 + f ⁢ ( t , 5 - 3 ⁢ t 2 ) ) if t ∈ ( 2 , 3 ] , x > 5 - 3 ⁢ t 2 ,

< 0 .

We conclude that R is a solution-region of (5.14). Theorem 5.1 implies that (5.14) has a solution whose graph is included in R. It is worthwhile to notice that b is not an upper solution of (5.14) in the sense of Definition 2.4 (1).

Here is an example for a system of two differential equations.

Example 5.5.

We consider the following problem:

(5.15)

u ′ ⁢ ( t ) = ( u ⁢ ( t ) ) 1 3 ⁢ ( 1 - u ⁢ ( t ) 2 ) ⁢ e t + 5 ⁢ v ⁢ ( t ) ,

v ′ ⁢ ( t ) = - t 1 2 ⁢ v ⁢ ( t ) ⁢ e t ⁢ u ⁢ ( t ) for a.e. ⁢ t ∈ I = [ 0 , T ] ,

( u ⁢ ( 0 ) , v ⁢ ( 0 ) ) = ( u 0 , v 0 ) ,

where (1-|u0|)2+v02≤1. We claim that there exists u a solution of this problem such that

( t , u ⁢ ( t ) ) ∈ R = I × R 0 for every ⁢ t ∈ I ,

where

R 0 = { ( x 1 , x 2 ) ∈ ℝ 2 : ( 1 - | x 1 | ) 2 + x 2 2 ≤ 1 } .

We have seen that R is an admissible region with the admissible pair (h,p) given in Example 3.6. It is easy to verify that h⁢(0,u0,v0)≤0 and

∂ ⁡ h ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h ⁢ ( t , x ) , f ⁢ ( p ⁢ ( t , x ) ) 〉 = 〈 ∇ x ⁡ h ⁢ ( t , x ) , f ⁢ ( p ⁢ ( t , x ) ) 〉 ≤ 0 for every ⁢ ( t , x ) ∉ R .

So, R is a solution-region of (5.15). Theorem 5.1 insures the existence of a solution of (5.15) whose graph is included in R. We recall that, in this example, R0 is not a proximate retract.

6 Multiplicity Results

In this section, we establish multiplicity results for problem (1.1). To this end, we introduce the following notion of strict solution-regions.

Definition 6.1.

We say that R⊂I×ℝN is a strict solution-region of (1.1) if R is a solution-region with an associated admissible pair (h,p) satisfying the following conditions:

For every t∈I, int⁡(Rt)≠∅, where Rt={x∈ℝN:(t,x)∈R}.
There exists ε>0 such that, for almost every t and every x with h⁢(t,x)∈(-ε,0), the map h has partial derivatives at (t,x) and

∂ ⁡ h ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h ⁢ ( t , x ) , f ⁢ ( t , x ) 〉 ≤ 0 ,

and ∂⁡h∂⁡t,∇x⁡h are locally Carathéodory maps on h-1⁢((-ε,0)).
If ℬ denotes the initial value condition (1.2), it satisfies h⁢(0,r)<0. If ℬ denotes the periodic boundary value condition (1.3), it satisfies h⁢(0,x)<h⁢(T,x) for every x such that h⁢(0,x)=0.

Example 6.2.

It can easily be seen that R={(t,x)∈I×ℝ:α⁢(t)≤x≤β⁢(t)} is a strict solution-region of (1.1) if α,β∈W1,1⁢(I,ℝ) are respectively strict lower and upper solutions of (1.1) such that α⁢(t)<β⁢(t) for every t∈I (see Definition 2.4 (2)). Similarly, for (v,M)∈W1,1⁢(I,ℝN)×W1,1⁢(I,[0,∞)) a strict solution-tube of (1.1) (see Definition 2.5 (2)), it is easy to verify that

R = { ( t , x ) ∈ I × ℝ N : ∥ x - v ⁢ ( t ) ∥ ≤ M ⁢ ( t ) }

is a strict solution-region of (1.1).

A strict solution-region R has a nice property. Indeed, a solution of (1.1) remaining in R must stay in its interior.

Proposition 6.3.

Let f:I×RN→RN be a Carathéodory function and R a strict solution-region of (1.1). Assume that u is a solution of (1.1) such that (t,u⁢(t))∈R for every t∈I. Then

u ⁢ ( t ) ∈ int ⁡ ( R t ) for every t ∈ I .

More precisely, h⁢(t,u⁢(t))<0 for every t∈I, where (h,p) is an admissible pair associated to R satisfying Definition 6.1.

Proof.

Let (h,p) be an admissible pair associated to R satisfying Definition 6.1. By assumption,

( t , u ⁢ ( t ) ) ∈ R = { ( t , x ) ∈ I × ℝ N : h ⁢ ( t , x ) ≤ 0 } for every t ∈ I .

Assume that

J = { t ∈ I : h ⁢ ( t , u ⁢ ( t ) ) = 0 } ≠ ∅ .

We claim that

(6.1) 0 ∉ J .

This is obviously the case if ℬ denotes the initial value condition (1.2) by Definition 6.1 (iii). If 0∈J and ℬ denotes the periodic boundary value condition (1.3), one has by Definition 6.1 (iii) that

0 = h ⁢ ( 0 , u ⁢ ( 0 ) ) < h ⁢ ( T , u ⁢ ( 0 ) ) = h ⁢ ( T , u ⁢ ( T ) ) ≤ 0 .

This is a contradiction.

Let tm=min⁡J∈(0,T]. So, by (6.1),

h ⁢ ( t m , u ⁢ ( t m ) ) = 0 and h ⁢ ( t , u ⁢ ( t ) ) < 0 for every ⁢ t ∈ [ 0 , t m ) .

Let ε>0 be given by Definition 6.1 (ii). Since u and h are continuous, there exist t0 and t1 such that 0≤t0<t1<tm,

h ⁢ ( t 0 , u ⁢ ( t 0 ) ) < h ⁢ ( t 1 , u ⁢ ( t 1 ) ) and h ⁢ ( t , u ⁢ ( t ) ) ∈ ( - ε , 0 ) for all ⁢ t ∈ [ t 0 , t 1 ] .

From Definition 6.1, we deduce that h⁢(⋅,u⁢(⋅))∈W1,1⁢([t0,t1],ℝ) and

0 < h ⁢ ( t 1 , u ⁢ ( t 1 ) ) - h ⁢ ( t 0 , u ⁢ ( t 0 ) ) = ∫ t 0 t 1 d d ⁢ t ⁢ h ⁢ ( t , u ⁢ ( t ) ) ⁢ 𝑑 t = ∫ t 0 t 1 ∂ ⁡ h ∂ ⁡ t ⁢ ( t , u ⁢ ( t ) ) + 〈 ∇ x ⁡ h ⁢ ( t , u ⁢ ( t ) ) , f ⁢ ( t , u ⁢ ( t ) ) 〉 ⁢ d ⁢ t ≤ 0 .

This is a contradiction. Therefore, J=∅ and hence h⁢(t,u⁢(t))<0 for every t∈I. ∎

Using the notions of solution-regions and strict solution-regions, we establish a multiplicity result for problem (1.1).

Theorem 6.4.

Let f:I×RN→RN be a Carathéodory function. Assume that there exist R1 and R2 two strict solution-regions and R3 a solution-region of (1.1) such that

R 1 ∪ R 2 ⊂ R 3 𝑎𝑛𝑑 R t 1 ∩ R t 2 = ∅ for some t ∈ I .

Then problem (1.1) has at least three distinct solutions u1,u2,u3 such that

( t , u i ⁢ ( t ) ) ∈ R i for every t ∈ I and i = 1 , 2 , 3 ,

and {t∈I:(t,u3⁢(t))∈R3∖(R1∪R2)}≠∅.

Proof.

First, we consider the case where ℬ denotes (1.3). For i=1,2,3, let (hi,pi) be an admissible pair associated to Ri and insured by Definition 4.1 and Definition 6.1, respectively. Consider 𝒫i the operator defined as in (5.5) and associated to the admissible pair (hi,pi). Proposition 5.2 implies that there exists an open set 𝒰i⊂C⁢(I,ℝN) such that

index ⁡ ( 𝒫 i ⁢ ( 1 , ⋅ ) , 𝒰 i ) = ( - 1 ) N ,

and

{ u ∈ C ⁢ ( I , ℝ N ) : ( t , u ⁢ ( t ) ) ∈ R i ⁢ for all ⁢ t ∈ I } ⊂ 𝒰 i .

From the proof of Theorem 5.1, we deduce that any fixed point u of 𝒫i⁢(1,⋅) is a solution of (1.1), (1.3) and is such that

(6.2) ( t , u ⁢ ( t ) ) ∈ R i for every t ∈ I .

For i=2,3, it follows from (6.2), Proposition 6.3 and the fact that Ri is a strict solution-region of problem (1.1), (1.3), that 𝒫i⁢(1,⋅) has no fixed points in 𝒰i∖𝒱¯i, with

𝒱 i = { u ∈ C ⁢ ( I , ℝ N ) : h i ⁢ ( t , u ⁢ ( t ) ) < 0 ⁢ for all ⁢ t ∈ I } .

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

(6.3) index ⁡ ( 𝒫 i ⁢ ( 1 , ⋅ ) , 𝒱 i ) = ( - 1 ) N for ⁢ i = 2 , 3 .

Since R1∪R2⊂R3 and Rt1∩Rt2=∅ for some t∈I, one has

𝒱 1 ∩ 𝒱 2 = ∅ , 𝒱 1 ∪ 𝒱 2 ⊂ 𝒰 3 , 𝒰 3 ∖ ( 𝒱 1 ∪ 𝒱 2 ¯ ) ≠ ∅ ,

and

𝒫 i ⁢ ( 1 , u ) = 𝒫 3 ⁢ ( 1 , u ) for every u ∈ 𝒱 i and i = 2 , 3 .

This combined with (6.3) and the additivity property of the fixed point index imply that

index ⁡ ( 𝒫 3 ⁢ ( 1 , ⋅ ) , 𝒰 3 ∖ ( 𝒱 1 ∪ 𝒱 2 ¯ ) ) = index ⁡ ( 𝒫 3 ⁢ ( 1 , ⋅ ) , 𝒰 3 ) - index ⁡ ( 𝒫 3 ⁢ ( 1 , ⋅ ) , 𝒱 1 ) - index ⁡ ( 𝒫 3 ⁢ ( 1 , ⋅ ) , 𝒱 2 )

= index ⁡ ( 𝒫 3 ⁢ ( 1 , ⋅ ) , 𝒰 3 ) - index ⁡ ( 𝒫 1 ⁢ ( 1 , ⋅ ) , 𝒱 1 ) - index ⁡ ( 𝒫 2 ⁢ ( 1 , ⋅ ) , 𝒱 2 )

= ( - 1 ) N + 1 .

Therefore, problem (1.1), (1.3) has at least three solutions u1∈𝒱1, u2∈𝒱2 and u3∈𝒰3∖(𝒱1∪𝒱2¯) such that (t,u3⁢(t))∈R3 for every t∈I.

The case where ℬ denotes (1.2) is analogous. ∎

Here is an example of a system of differential equations for which we can insure the existence of at least three solutions.

Example 6.5.

We consider the following system of differential equations with the periodic boundary value condition:

(6.4)

u ′ ⁢ ( t ) = ( u 2 ⁢ ( t ) - t 2 + 10 ⁢ t - 25 ) ⁢ ( sin ⁡ ( t + v ⁢ ( t ) ) - 2 ⁢ u ⁢ ( t ) ) ,

v ′ ⁢ ( t ) = t 3 - v ⁢ ( t ) ⁢ e 1 + u 2 ⁢ ( t ) for a.e. ⁢ t ∈ [ 0 , 1 ] ,

u ⁢ ( 0 ) = u ⁢ ( 1 ) , v ⁢ ( 0 ) = v ⁢ ( 1 ) .

We claim that this problem has at least three solutions. Let us consider the following regions in [0,1]×ℝ2:

R 1 = { ( t , x 1 , x 2 ) ∈ [ 0 , 1 ] × ℝ 2 : ( x 1 + t - 5 ) 2 + x 2 2 ≤ ( 4 - 2 ⁢ t ) 2 } ,

R 2 = { ( t , x 1 , x 2 ) ∈ [ 0 , 1 ] × ℝ 2 : ( x 1 - t + 5 ) 2 + x 2 2 ≤ ( 4 - 2 ⁢ t ) 2 } ,

R 3 = { ( t , x 1 , x 2 ) ∈ [ 0 , 1 ] × ℝ 2 : t - 5 ≤ x 1 ≤ 5 - t , | x 2 | ≤ ( 4 - 2 ⁢ t ) }

∪ { ( t , x 1 , x 2 ) ∈ [ 0 , 1 ] × ℝ 2 : x 1 > 5 - t , ( x 1 + t - 5 ) 2 + x 2 2 ≤ ( 4 - 2 ⁢ t ) 2 }

∪ { ( t , x 1 , x 2 ) ∈ [ 0 , 1 ] × ℝ 2 : x 1 < t - 5 , ( x 1 + t - 5 ) 2 + x 2 2 ≤ ( 4 - 2 ⁢ t ) 2 } .

They are admissible regions. Indeed, for i=1,2,3, we define the continuous maps hi:[0,1]×ℝ2→ℝ and pi:[0,1]×ℝ2→[0,1]×ℝ2 by

(6.5)

h 1 ⁢ ( t , x ) = 2 ⁢ t - 4 + ( x 1 + t - 5 ) 2 + x 2 2 ,

h 2 ⁢ ( t , x ) = 2 ⁢ t - 4 + ( x 1 - t + 5 ) 2 + x 2 2 ,

h 3 ⁢ ( t , x ) = { h 1 ⁢ ( t , x 1 , x 2 ) if x 1 > 5 - t , h 2 ⁢ ( t , x 1 , x 2 ) if x 1 < t - 5 , 2 ⁢ t - 4 + | x 2 | if | x 1 | ≤ 5 - t ,

and

(6.6)

p 1 ( t , x ) = { ( t , x ) if ( t , x ) ∈ R 1 , ( t , 5 - t + ( x 1 + t - 5 ) ⁢ ( 4 - 2 ⁢ t ) ( x 1 + t - 5 ) 2 + x 2 2 , x 2 ⁢ ( 4 - 2 ⁢ t ) ( x 1 + t - 5 ) 2 + x 2 2 ) if ( t , x ) ∉ R 1 ,

p 2 ( t , x ) = { ( t , x ) if ( t , x ) ∈ R 2 , ( t , t - 5 + ( x 1 - t + 5 ) ⁢ ( 4 - 2 ⁢ t ) ( x 1 - t + 5 ) 2 + x 2 2 , x 2 ⁢ ( 4 - 2 ⁢ t ) ( x 1 - t + 5 ) 2 + x 2 2 ) if ( t , x ) ∉ R 2 ,

p 3 ⁢ ( t , x ) = { ( t , x ) if ( t , x ) ∈ R 3 , p 1 ⁢ ( t , x ) if ( t , x ) ∉ R 3 , x 1 > 5 - t , p 2 ⁢ ( t , x ) if ( t , x ) ∉ R 3 , x 1 < t - 5 , ( t , x 1 , 4 - 2 ⁢ t ) if ( t , x ) ∉ R 3 , x 1 ∈ [ t - 5 , 5 - t ] , x 2 > 4 - 2 ⁢ t , ( t , x 1 , 2 ⁢ t - 4 ) if ( t , x ) ∉ R 3 , x 1 ∈ [ t - 5 , 5 - t ] , x 2 < 2 ⁢ t - 4 .

It is easy to check that ∂⁡h3∂⁡t, ∇x⁡h3 are Carathéodory on {(t,x):h3⁢(t,x)>0}, and, for i=1,2, ∂⁡hi∂⁡t and ∇x⁡hi are Carathéodory on {(t,x):hi⁢(t,x)≥-1}. Also, for i=1,2,3,

〈 ∇ x ⁡ h i ⁢ ( t , x ) , p 2 i ⁢ ( t , x ) - x 〉 < 0 for a.e. t and all x with ( t , x ) ∉ R i .

So, R1,R2 and R3 are admissible regions.

Now, we show that R1, R2 and R3 are solution-regions of (6.4). For (t,x)∉R1, let us denote

p 1 ⁢ ( t , x ) = ( t , 5 - t + a ⁢ c , b ⁢ c ) ,

where

a = x 1 + t - 5 ( x 1 + t - 5 ) 2 + x 2 2 , b = x 2 ( x 1 + t - 5 ) 2 + x 2 2 and c = 4 - 2 ⁢ t .

Using the fact that a2+b2=1 and c∈[2,4], we deduce that

∂ ⁡ h 1 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 1 ⁢ ( t , x ) , f ⁢ ( p 1 ⁢ ( t , x ) ) 〉

= 2 + a + 〈 ( a , b ) , f ⁢ ( p 1 ⁢ ( t , x ) ) 〉

= 2 + a - c ⁢ a 2 ⁢ ( ( 10 - 2 ⁢ t + a ⁢ c ) ⁢ ( 10 - 2 ⁢ t + 2 ⁢ a ⁢ c - sin ⁡ ( t + b ⁢ c ) ) ) - c ⁢ b 2 ⁢ e ( 1 + ( 5 - t + a ⁢ c ) 2 ) + b ⁢ t 3

≤ 3 + a - 6 ⁢ c ⁢ a 2 - e ⁢ c ⁢ b 2

≤ 4 - e ⁢ c .

Thus, for every (t,x)∉R1,

(6.7) ∂ ⁡ h 1 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 1 ⁢ ( t , x ) , f ⁢ ( p 1 ⁢ ( t , x ) ) 〉 ≤ 4 - 2 ⁢ e < 0 .

Similarly, for every (t,x)∉R2,

(6.8) ∂ ⁡ h 2 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 2 ⁢ ( t , x ) , f ⁢ ( p 2 ⁢ ( t , x ) ) 〉 < 0 .

Now, combining (6.5), (6.6), (6.7) and (6.8), we have for (t,x)∉R3,

∂ ⁡ h 3 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 3 ⁢ ( t , x ) , f ⁢ ( p 3 ⁢ ( t , x ) ) 〉 = { ∂ ⁡ h 1 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 1 ⁢ ( t , x ) , f ⁢ ( p 1 ⁢ ( t , x ) ) 〉 if x 1 > 5 - t , ∂ ⁡ h 2 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 2 ⁢ ( t , x ) , f ⁢ ( p 2 ⁢ ( t , x ) ) 〉 if x 1 < t - 5 , 2 - ( 4 - 2 ⁢ t ) ⁢ e 1 + x 1 2 + t 3 ⁢ x 2 | x 2 | if x 1 ∈ [ t - 5 , 5 - t ] .

So, for every (t,x)∉R3,

(6.9) ∂ ⁡ h 3 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 3 ⁢ ( t , x ) , f ⁢ ( p 3 ⁢ ( t , x ) ) 〉 < 0 .

We have to verify that hi satisfies the boundary condition. Notice that the set B01={x∈ℝ2:(0,x)∈R1} is the closed ball in ℝ2 centered in (5,0) of radius 4 while B11={x∈ℝ2:(1,x)∈R1} is the closed ball centered in (4,0) of radius 2. So, B11⊂B01. For (0,x)∉R1,

(6.10) h 1 ⁢ ( 0 , x ) = dist ⁡ ( x , B 0 1 ) < dist ⁡ ( x , B 1 1 ) = h 1 ⁢ ( 1 , x ) .

Similarly,

(6.11) h 2 ⁢ ( 0 , x ) < h 2 ⁢ ( 1 , x ) for every ⁢ ( 0 , x ) ∉ R 2 .

Combining (6.5), (6.10) and (6.11), we obtain for (0,x)∉R3,

h 3 ⁢ ( 0 , x ) = { h 1 ⁢ ( 0 , x 1 , x 2 ) if x 1 > 5 - t , h 2 ⁢ ( 0 , x 1 , x 2 ) if x 1 < t - 5 , | x 2 | - 4 if x 1 ∈ [ t - 5 , t + 5 ] ,

< { h 1 ⁢ ( 1 , x 1 , x 2 ) if x 1 > 5 - t , h 2 ⁢ ( 1 , x 1 , x 2 ) if x 1 < t - 5 , | x 2 | - 2 if x 1 ∈ [ t - 5 , t + 5 ] ,

= h 3 ⁢ ( 1 , x ) .

Thus, R1, R2 and R3 are solution-regions of (6.4).

We verify that R1 and R2 are strict solution-regions of (6.4). Let ε>0 which will be fixed later. Again, if we denote

a = x 1 + t - 5 ( x 1 + t - 5 ) 2 + x 2 2 , b = x 2 ( x 1 + t - 5 ) 2 + x 2 2 ,

we deduce that, for (t,x) such that -ε<h1⁢(t,x)<0, one has x>1 and

∂ ⁡ h 1 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 1 ⁢ ( t , x ) , f ⁢ ( t , x ) 〉 = 2 + a + 〈 ( a , b ) , f ⁢ ( t , x ) 〉

= 2 + a - a ⁢ ( x 1 + t - 5 ) ⁢ ( ( 5 - t + x 1 ) ⁢ ( 2 ⁢ x 1 - sin ⁡ ( t + x 2 ) ) ) - b ⁢ x 2 ⁢ e 1 + x 1 2 + b ⁢ t 3

≤ 3 + a - 5 ⁢ a ⁢ ( x 1 + t - 5 ) - e ⁢ b ⁢ x 2

≤ 3 + a - e ⁢ ( x 1 + t - 5 ) 2 + x 2 2

≤ 4 - e ⁢ ( 4 - 2 ⁢ t - ε )

≤ 4 - e ⁢ ( 2 - ε ) .

Thus, we can choose ε sufficiently small such that, for every (t,x) satisfying h1⁢(t,x)∈(-ε,0), one has

∂ ⁡ h 1 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 1 ⁢ ( t , x ) , f ⁢ ( t , x ) 〉 ≤ 4 - e ⁢ ( 2 - ε ) ≤ 0 .

Similarly, for every (t,x) satisfying h2⁢(t,x)∈(-ε,0), one has

∂ ⁡ h 2 ∂ ⁡ t ⁢ ( t , x ) + 〈 ∇ x ⁡ h 2 ⁢ ( t , x ) , f ⁢ ( t , x ) 〉 ≤ 0 .

Arguing as we did to obtain (6.10) and (6.11) permit us to deduce that

if h i ⁢ ( 0 , x ) = 0 , then h i ⁢ ( 0 , x ) < h i ⁢ ( 1 , x ) for i = 1 , 2 .

Therefore, R1 and R2 are two strict solution-regions and R3 is a solution-region of (6.4) such that

R 1 ∪ R 2 ⊂ R 3 and R 3 ∖ ( R 1 ∪ R 2 ) ≠ ∅ .

It follows from Theorem 6.4 that there exist (u1,v1), (u2,v2) and (u3,v3) three solutions of (6.4) such that

( t , u i ⁢ ( t ) , v i ⁢ ( t ) ) ∈ R i for every t ∈ [ 0 , 1 ] for ⁢ i = 1 , 2 , 3 ,

and

{ t ∈ [ 0 , 1 ] : ( t , u 3 ⁢ ( t ) , v 3 ⁢ ( t ) ) ∉ R 1 ∪ R 2 } ≠ ∅ .

Communicated by Ken Palmer

Funding source: Natural Sciences and Engineering Research Council of Canada

Award Identifier / Grant number: 170452-2013

Funding statement: This work was partially supported by the Natural Sciences and Engineering Research Council of Canada–NSERC (Discovery Grant 170452-2013).

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Received: 2017-01-30

Revised: 2018-01-11

Accepted: 2018-01-16

Published Online: 2018-02-07

Published in Print: 2018-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2017-6049

Keywords for this article

System of First Order Differential Equations; Multiple Solutions; Initial Value and Periodic Boundary Value Problems; Fixed Point Index; Solution-Region; Upper and Lower Solutions

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