A Continuation Approach to the Periodic Boundary Value Problem for a Class of Nonlinear Coupled Oscillators with Potential Depending on Time

1 Introduction

In this paper, we are concerned with the 2⁢π-periodic boundary value problem for the second-order vector equation

u ′′ + g ⁢ ( t , u ) = p ⁢ ( t , u , u ′ ) , u ⁢ ( 2 ⁢ π ) - u ⁢ ( 0 ) = u ′ ⁢ ( 2 ⁢ π ) - u ′ ⁢ ( 0 ) = 0 ,

where g:ℝ×ℝd→ℝd and p:ℝ×ℝd×ℝd are all continuous functions and 2⁢π-periodic in the first variable.

The problems of the existence and the multiplicity of the periodic solutions for some second-order multidimensional nonlinear systems have been investigated in many papers for its applied background (only to mention some of them, see [2, 4, 5, 6, 8, 9] and the references therein).

In [4], Castro and Lazer obtained the existence of infinitely many periodic solutions for weakly coupled systems of the form

x i ′′ + g i ⁢ ( x i ) = p i ⁢ ( t , x i ) , i = 1 , 2 , … , d ,

when, for each i=1,2,…,d, gi⁢(x)/x→+∞ as |x|→∞, pi is bounded, and gi and pi are odd. They used shooting arguments and Miranda’s theorem.

In [2], Capietto, Mawhin and Zanolin considered the more general system

u i ′′ + g i ⁢ ( u i ) = p i ⁢ ( t , u , u ′ ) , i = 1 , 2 , … , d ,

without the oddness conditions and they proved that there is at least one 2⁢π-periodic solution to the equation. There, the relative a priori bounds of periodic solutions of the equations are found and so an approach for periodic solutions is proposed by a Leray–Schauder’s type continuation method.

In [8], Torres studied the problem of the existence of the T-periodic solutions for a class of lattices of Toda type and a singular type

x i ′′ + c ⁢ x i ′ = g i - 1 ⁢ ( x i - x i - 1 ) - g i ⁢ ( x i + 1 - x i ) + h i ⁢ ( t ) , i ∈ ℤ ,

where hi⁢(t) (i∈ℤ) are T-periodic. The strategy of proof is to find a priori bounds for periodic solutions of the homotopic equations of a related finite system. Then, the existence of a periodic solution of the finite system is ensured by standard properties of the coincidence degree. Furthermore, a limiting argument was applied to prove the existence of a periodic solution of the infinite systems.

In [9], Torres generalized the results of [8] and presented necessary and sufficient conditions for the existence of periodic solutions. Analogous results in the situation when the restoring forces are sub-linear functions of the distance between particles were given by Wang and Qian in [10] for the same equation. In the cases mentioned above, there are a priori bounds of periodic solutions or relative a priori bounds of periodic solutions for the equations.

A natural question is whether there is an analogous result for the equations that lack both a priori bounds and relative a priori bounds on periodic solutions.

In [7], Qian studied the problem of the existence of periodic solutions for a second-order scalar equation

x ′′ + g ⁢ ( t , x ) = 0

where the restoring force g is dependent on time and a non-resonant condition on the time-map of certain auxiliary equations is given. In this case, there are neither a priori bounds nor relative a priori bounds of periodic solutions. By using a continuation theorem developed initially in [3], Qian proved the existence of a periodic solution for the equation.

Our aim in this paper is to obtain an extension of the theorem in [7] to systems, dealing with a weakly-coupled system of the form

with g(t,u)=col(gi(t,ui))i=1,2,…,d, p(t,u,u′)=col(pi(t,u,u′))i=1,2,…,d and

1. |pi⁢(t,u,u′)|≤Pi∈R+ for i=1,2,…,d.

The 2⁢π-periodic boundary value condition can be rewritten as

For each i=1,…,d, consider the auxiliary equations

u i ′′ + g i , m ⁢ ( u i ) = 0 , u i ′′ + g i , M ⁢ ( u i ) = 0 ,

where gi,m and gi,M are continuous.

Now, assume that, for each i=1,2,…,d,

1. lim|x|→+∞⁡sgn⁡(x)⁡gi,m⁢(x)=+∞,

2. lim|x|→∞⁡inf⁡(gi⁢(t,x)/gi,m⁢(x))≥1 and lim|x|→∞⁡sup⁡(gi⁢(t,x)/gi,M⁢(x))≤1 uniformly for t∈[0,2⁢π].

Let

τ i , m ⁢ ( h ) = τ i , m ⁢ ( u i - , u i + ) = 2 ⁢ | ∫ u i - u i + d ⁢ s G i , m ⁢ ( u i + ) - G i , m ⁢ ( s ) |

be the minimal positive period of the solution on the orbit of

1 2 ⁢ v i 2 + G i , m ⁢ ( u i ) = h

with vi=ui′ and h an arbitrary large positive constant.

Denote

( τ i , m ) ± = lim sup x → ± ∞ ⁡ τ i , m ⁢ ( 0 , x ) , ( τ i , M ) ± = lim inf x → ± ∞ ⁡ τ i , M ⁢ ( 0 , x )

and suppose that

0 < ( τ i , M ) ± ≤ ( τ i , m ) ± < + ∞ ,

respectively. Moreover, we suppose the following:

1. For ni∈ℤ+, we have

   2 ⁢ π n i + 1 ≤ ( τ i , M ) - + ( τ i , M ) + ≤ ( τ i , m ) - + ( τ i , m ) + ≤ 2 ⁢ π n i .

2. There is a sequence {ui,l+} with ui,l+→+∞ as l→∞ such that

   2 ⁢ π n i + 1 + δ i ≤ ( τ i , M ) - + τ i , M ⁢ ( 0 , u i , l + ) ≤ ( τ i , m ) - + τ i , m ⁢ ( 0 , u i , l + ) ≤ 2 ⁢ π n i - δ i

   or there is a sequence {ui,l-} with ui,l-→-∞ as l→∞ such that

   2 ⁢ π n i + 1 + δ i ≤ ( τ i , M ) + + τ i , M ⁢ ( 0 , u i , l - ) ≤ ( τ i , m ) + + τ i , m ⁢ ( 0 , u i , l - ) ≤ 2 ⁢ π n i - δ i

   for some δi>0.

In this paper, we are concerned with the existence of periodic solutions to the equations (1.1) when the non-resonance conditions (τ0) and (τ1) are satisfied. In our case, on the one hand, few tools can be employed to develop a phase-plane analysis for the equivalent system

u i ′ = v i , v i = - g i ⁢ ( t , u i ) + p i ⁢ ( t , u , v ) , i = 1 , … , d ,

when d>1. On the other hand, because a priori bounds for periodic solutions under the condition (τ0) are lacking, the technique employed in [8, 9] is invalid. Furthermore, we note that because of appearance of u′, the classical variational techniques cannot be used as in [1] where a class of lattices of particles is concerned. As far as we know, no ready-made methods can be used directly to solve the problem.

To solve the problem, in this paper, we firstly extend the continuation theorem [3, Theorem 2.2] to the case of vector equations in Section 2. Subsequently, in Section 3, we apply the continuation theorem to prove the existence of periodic solutions of the equations (1.1). Our main result in this paper is:

2 Continuation Theorems

In this section we give an existence result (Theorem 2.1) for a coincidence equation in function spaces, whose application (Theorem 2.2) is used for the proof of Theorem 1.1 on the solvability of the periodic BVP (1.1)–(1.2). Now, we firstly introduce the following abstract setting.

Let X and Z be real Banach spaces with norm |⋅|X=∥⋅∥ and |⋅|Z, respectively. Let L:D⁢(L)⊂X→Z be a linear Fredholm mapping of index zero and let N:X×[0,1]→Z be an L-completely continuous operator. We denote by {e1,…,el} the canonical orthonormal basis of ℝl.

Consider the following operator equation:

We denote by

Σ = { ( x , λ ) ∈ D ⁢ ( L ) × [ 0 , 1 ] : L ⁢ x = N ⁢ ( x , λ ) }

the (possible empty) set of solutions of (2.1), and by

Σ λ = { x ∈ D ⁢ ( L ) : ( x , λ ) ∈ Σ } , 0 ≤ λ ≤ 1 ,

the section of Σ at λ. Clearly, Σ=⋃λ∈[0,1]Σλ. Let Xj (j=1,…,l) be closed linear subspaces of X such that X=⊕j=1lXj and suppose that linear continuous projections Πj:X→Xj and Πj⁢(x)=xj with ∥Πj∥=1 for j=1,…,l are selected, so that each x∈X can uniquely be expressed as x=∑j=1lxj. So, for each x∈X, we have

| x j | ≤ ∥ x ∥ ≤ ∑ j = 1 l ∥ x j ∥ .

We also define

Π ^ j : X × [ 0 , 1 ] → X j , Π ^ j ⁢ ( x , λ ) = Π j ⁢ ( x )

and

τ j : X × [ 0 , 1 ] → X j × [ 0 , 1 ] , τ j ⁢ ( x , λ ) = ( Π j ⁢ ( x ) , λ ) = ( x j , λ ) .

Let us consider the following assumptions:

1. Σ0 is bounded in X.

Define χ0:=|DL⁢(L-N⁢(⋅,0),X)|=|DL⁢(L-N⁢(⋅,0),Ω)|, where Ω⊃Σ0 is any open bounded subset of X and DL is the coincidence degree.

1. χ0≠0.

We introduce now 2⁢l continuous functionals ψ1,…,ψl,η1,…,ηl:X×[0,1]→ℝ.

1. ψj:Xj×[0,1]→ℝ (j=1,…,l) and there is an R0>0 such that

   ψ j ⁢ ( x , λ ) = ψ j ⁢ ( Π j ⁢ ( x ) , λ )

   for each (x,λ)∈Σ with ∥Πj⁢(x)∥≥R0, and

   | η j ⁢ ( x , λ ) - η j ⁢ ( Π j ⁢ ( x ) , λ ) | ≤ η ⁢ ( R 0 )   for ⁢ j = 1 , … , l ,

   for any (x,λ)∈Σ with ∥Πj⁢(x)∥≥R0, where η⁢(R0)→0 as R0→+∞.

2. Π^j⁢(ψj-1⁢(K)∩Σ) is bounded for each compact subset K⊂ℝ and each j∈{1,…,l}.

3. There exist constants R1>0,d>0 such that for each j,1≤j≤l,

   ψ j ⁢ ( x , λ ) ≥ - d , η j ⁢ ( x , λ ) ∈ ℤ

   for any (x,λ)∈Σ with ∥Πj⁢(x)∥≥R1.

The proof of Theorem 2.1 is the content of Section 4.

We consider now the second-order boundary value problem

where F:ℝ×ℝl×ℝl→ℝl is a continuous function which is T-periodic in the first variable.

We embed problem (2.3)–(2.4) into a one-parameter family of problems of the form

with f⁢(t,x,x′;λ):ℝ×ℝl×ℝl×[0,1]→ℝl continuous and T-periodic in t and such that

f ⁢ ( t , x , x ′ ; 1 ) = F ⁢ ( t , x , x ′ )   and   f ⁢ ( t , x , x ′ ; 0 ) = f 0 ⁢ ( x , x ′ ) .

For any solutions x⁢(⋅)=(x1⁢(⋅),x2⁢(⋅),…,xl⁢(⋅)) of the equations (2.5)–(2.6), we denote 𝐧⁢(xj)=Card⁡(Zxj), where Zxj:=xj-1⁢(0)∩[0,T).

3 Proof of the Main Result

In this section, let T=2⁢π and we will apply Theorem 2.2 to prove the existence of at least one solution of the finite system of boundedly-coupled second-order differential equations (1.1)–(1.2).

Now, we firstly embed problem (1.1)–(1.2) into a one-parameter family of problems of the form

where

f i ⁢ ( t , u , u ′ ; λ ) = λ ⁢ g i ⁢ ( t , u i ) + ( 1 - λ ) ⁢ ( g i , m ⁢ ( u i ) + η ⁢ ( u i ′ ) ) - λ ⁢ p i ⁢ ( t , u , u ′ )

such that

f i ⁢ ( t , u , u ′ ; 1 ) = g i ⁢ ( t , u i ) - p i ⁢ ( t , u , u ′ ) , f i ⁢ ( t , u , u ′ ; 0 ) = g i , m ⁢ ( u i ) + η ⁢ ( u i ′ ) ,

0≤λ≤1,η⁢(⋅) is a C0 function with a⁢η⁢(a)>0, for each a∈ℝ,a≠0, and |η⁢(a)|≤1.

Obviously, for each i=1,…,d, we can choose d0(i)≥0 suitably large such that

f i ⁢ ( t , u , u ′ ; λ ) ⋅ u i > 0 , i = 1 , … , d ,

and

| f i ⁢ ( t , u , u ′ ; λ ) - f i ⁢ ( t , Π i ⁢ ( u ) , Π i ⁢ ( u ′ ) ; λ ) | ≤ 2 ⁢ P i , i = 1 , … , d ,

for each t∈[0,2⁢π],λ∈[0,1] with |ui|≥d0(i). Thus, (2.7) and (2.8) are satisfied.

So, we now only need to verify all the conditions in Theorem 2.2.

Letting ui′=yi, i=1,…,d, (3.1)–(3.2) is equivalent to

and

u ⁢ ( 2 ⁢ π ) - u ⁢ ( 0 ) = y ⁢ ( 2 ⁢ π ) - y ⁢ ( 0 ) = 0 ,

where y=(y1,y2,…,yd).

In much the same way as in the proof of [2, Proposition 3], we can prove the following lemma.

We note that, for each i=1,…,d, there exists ρi>0 such that θi′⁢(t)<0 for any solution (u⁢(t),y⁢(t)) of (3.3) with |ui⁢(t)|+|yi⁢(t)|≥ρi, where θi⁢(t)=arg⁡(ui⁢(t)/yi⁢(t)). Choose ri>0 sufficiently large such that any 2⁢π-periodic solution (u⁢(t),y⁢(t)) of (3.3) with |ui⁢(0)|+|yi⁢(0)|≥ri satisfies, by Lemma 3.1,

| u i ⁢ ( t ) | + | y i ⁢ ( t ) | ≥ ρ i   for ⁢ t ∈ [ 0 , 2 ⁢ π ] ,

and then ui⁢(t) has only finite simple zeros in [0,2⁢π).

Without loss of generality, we assume that

g i , m ⁢ ( u i ) + ( P i + 1 ) ≤ g i ⁢ ( t , u i ) ≤ g i , M ⁢ ( u i ) - ( P i + 1 ) , i = 1 , … , d ,

for t∈[0,2⁢π] and all ui≥0 and

g i , M ⁢ ( u i ) + ( P i + 1 ) ≤ g i ⁢ ( t , u i ) ≤ g i , m ⁢ ( u i ) - ( P i + 1 ) , i = 1 , … , d ,

for t∈[0,2⁢π] and all ui≤0. These assumptions will be used when estimating the difference between any two consecutive zeros t1,t2 of ui⁢(t), the i-th component of a 2⁢π-periodic solution u⁢(t) of (3.3) with |ui⁢(t)|+|yi⁢(t)|≥ρi, ρi being large enough, t∈[t1,t2].