Open Access

Several kinds of oscillations in forced Liénard equations

Boundary Value Problems20132013:66

DOI: 10.1186/1687-2770-2013-66

Received: 5 November 2012

Accepted: 8 March 2013

Published: 29 March 2013

Abstract

Near an equilibrium we study the existence of asymptotically a.p. (almost periodic), asymptotically a.a. (almost automorphic), pseudo a.p., pseudo a.a., weighed pseudo a.p. and weighed pseudo a.a. solutions of Liénard differential equations in the form x ( t ) + f ( x ( t ) , p ) x ( t ) + g ( x ( t ) , p ) = e p ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq1_HTML.gif, where the forcing term possesses a similar nature, and where p is a parameter in a Banach space. We use a perturbation method around an equilibrium. We also study two special cases of the previous family of equations that are x ( t ) + f ( x ( t ) ) x ( t ) + g ( x ( t ) ) = e ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq2_HTML.gif and x ( t ) + f ( x ( t ) , q ) x ( t ) + g ( x ( t ) , q ) = e ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq3_HTML.gif.

MSC:34C27, 34C99, 47J07.

Keywords

asymptotically almost periodic functions asymptotically almost automorphic functions pseudo almost periodic functions pseudo almost automorphic functions weighted pseudo almost periodic functions weighted pseudo almost automorphic functions Liénard equations

1 Introduction

We consider the following family of forced Liénard equations:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equa_HTML.gif

where P is a Banach space, f : R × P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq4_HTML.gif and g : R × P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq5_HTML.gif are two functions for each p P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq6_HTML.gif and e p : R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq7_HTML.gif is a function.

When e p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq8_HTML.gif is almost periodic in the Bohr sense (respectively almost automorphic), in [1], Theorem 3.1 (respectively Theorem 3.2), we have proven the existence of an almost periodic (respectively almost automorphic) solution x p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq9_HTML.gif near an equilibrium, by using the perturbation method in the setting of Nonlinear Functional Analysis.

In the present paper, we extend this result to the frameworks of asymptotically almost periodic, asymptotically almost automorphic, pseudo almost periodic, pseudo almost automorphic, weighted pseudo almost periodic and weighted pseudo almost automorphic functions.

We also consider two special cases of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif), which are
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equb_HTML.gif

On the existence of such solutions for the two previous cases, we obtain similar results as the one obtained on ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif).

Martínez-Amores and Torres in [2], then Campos and Torres in [3] described the dynamics of equation ( F , e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq11_HTML.gif) in the periodic case, namely the forcing term e is periodic. Then Cieutat extended these results to the almost periodic case in [4], Ait Dads, Cieutat and Lhachimi did to the pseudo almost periodic case in [5], and Cieutat, Fatajou and N’Guérékata did to the almost automorphic case in [6].

For almost periodic solutions of second-order differential systems, results on the differentiable dependence were established in [7] by Blot, Cieutat and Mawhin.

Our approach of the problem is to transform it into a nonlinear equation with parameters in Banach function spaces, and to apply the implicit function theorem of the differential calculus in Banach spaces. To realize our aim, we use the Nemytskii operators (also called superposition operators) and state some properties on these operators. Then we establish, for a linear differential equation in a Banach space, a result on the existence and uniqueness of the solutions described above. We also extend the well-know result on the almost periodicity of the derivative of an almost periodic function to the weighted pseudo almost periodic and weighted pseudo almost automorphic cases.

Now we give a brief description of the contents of the paper. In Section 2, we fix our notation and we recall some definitions. In Section 3, we establish some results on the differentiation of weighted pseudo almost periodic functions and weighted pseudo almost automorphic functions, then on a linear differential equation and on the Nemytskii operators. In Section 4, we state the main theorem (Theorem 4.1) and we give the proof of this theorem. In Section 5, we establish two corollaries (Corollary 5.1 and Corollary 5.2) of the two special cases of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif).

2 Notation

When X and Y are Banach spaces, L ( X , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq12_HTML.gif stands for the space of all bounded linear operators from X into Y, C 0 ( X , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq13_HTML.gif (respectively C 1 ( X , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq14_HTML.gif, respectively C 2 ( X , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq15_HTML.gif) stands for the space of continuous (respectively Fréchet continuously differentiable, respectively twice Fréchet continuously differentiable) functions from X into Y.

B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq16_HTML.gif stands for the space of bounded continuous functions from into X. We also define B C 1 ( R , X ) : = { u C 1 ( X , Y ) : u , u B C 0 ( R , X ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq17_HTML.gif and B C 2 ( R , X ) : = { u C 2 ( X , Y ) : u , u , u B C 0 ( R , X ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq18_HTML.gif. Endowed with the norm u : = sup { u ( t ) : t R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq19_HTML.gif (respectively u B C 1 : = u + u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq20_HTML.gif, respectively u B C 2 : = u + u + u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq21_HTML.gif), B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq22_HTML.gif (respectively B C 1 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq23_HTML.gif, respectively B C 2 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq24_HTML.gif) is a Banach space.

A function u B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq25_HTML.gif is called an almost periodic function (in the Bohr sense) when it satisfies the following criterion (due to Bochner): { u ( + r ) : r R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq26_HTML.gif is relatively compact in B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq16_HTML.gif. A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq27_HTML.gif denotes the space of almost periodic functions from into X. Endowed with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq28_HTML.gif, A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq27_HTML.gif is a Banach space which is invariant by translation [8], that is to mean that [ t u ( t + τ ) ] A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq29_HTML.gif for all τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq30_HTML.gif, when u A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq31_HTML.gif.

A function u B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq25_HTML.gif is called an almost automorphic function (in the Bochner sense) when, for all real sequence ( s n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq32_HTML.gif, there exists a subsequence ( t n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq33_HTML.gif of ( s n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq32_HTML.gif such that for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq34_HTML.gif, lim n u ( t + t n ) = v ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq35_HTML.gif exists in X, and for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq34_HTML.gif, lim n v ( t t n ) = u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq36_HTML.gif exists. A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq37_HTML.gif denotes the space of almost automorphic functions from into X. Endowed with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq28_HTML.gif, A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq37_HTML.gif is a Banach space which is invariant by translation [9].

We also consider the following other function spaces which one can find in [10]:

  • C 0 ( X ) : = { u B C ( R , X ) : lim | t | u ( t ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq38_HTML.gif.

  • A A P 0 ( X ) : = A P 0 ( X ) C 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq39_HTML.gif the space of asymptotically almost periodic functions [8].

  • A A A 0 ( X ) : = A A 0 ( X ) C 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq40_HTML.gif the space of asymptotically almost automorphic functions [9].

  • P 0 ( X ) : = { u B C ( R , X ) : lim T 1 2 T T + T u ( s ) d s = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq41_HTML.gif.

  • P A P 0 ( X ) : = A P 0 ( X ) P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq42_HTML.gif the space of pseudo almost periodic functions [11, 12].

  • P A A 0 ( X ) : = A A 0 ( X ) P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq43_HTML.gif the space of pseudo almost automorphic functions [13, 14].

Endowed with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq28_HTML.gif, A A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq44_HTML.gif, A A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq45_HTML.gif, P A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq46_HTML.gif and P A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq47_HTML.gif are Banach spaces which are invariant by translation (cf. respectively [8, 9, 11] and [14]).

Let L loc 1 ( R , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq48_HTML.gif be the set of all functions ρ : R ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq49_HTML.gif which are positive and locally Lebesgue-integrable over . For a given r ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq50_HTML.gif and for each ρ L loc 1 ( R , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq51_HTML.gif, we set m ( r , ρ ) : = r + r ρ ( x ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq52_HTML.gif.

We define the following spaces:

  • U : = { ρ L loc 1 ( R , ( 0 , ) ) : lim r m ( r , ρ ) = } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq53_HTML.gif

  • U T : = { ρ U : ρ  satisfies (H) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq54_HTML.gif, where (H) is the following condition due to [15]:

  1. (H)

    For all τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq30_HTML.gif, there exist β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq55_HTML.gif and a bounded interval I such that ρ ( t + τ ) β ρ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq56_HTML.gif a.e. t R I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq57_HTML.gif,

     

which is equivalent to τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq58_HTML.gif, lim sup | t | ρ ( t + τ ) ρ ( t ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq59_HTML.gif (cf. Remark 3.4 in [15] or Remark 3.1 in [16]).

For ρ U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq60_HTML.gif, we consider the following spaces:

  • P 0 ( X , ρ ) : = { u B C ( R , X ) : lim r 1 m ( r , ρ ) r + r u ( s ) ρ ( s ) d s = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq61_HTML.gif.

  • W P A P 0 ( X , ρ ) : = A P 0 ( X ) P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq62_HTML.gif the space of weighted pseudo almost periodic functions [17, 18].

  • W P A A 0 ( X , ρ ) : = A A 0 ( X ) P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq63_HTML.gif the space of weighted pseudo almost automorphic functions [19].

Let μ be a positive measure on https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq64_HTML.gif ( https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq64_HTML.gif is a Lebesgue σ-field on ) satisfying μ ( R ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq65_HTML.gif and μ ( [ a , b ] ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq66_HTML.gif for all a b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq67_HTML.gif. A function u is called μ-pseudo almost periodic (respectively μ-pseudo almost automorphic) if u = g + ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq68_HTML.gif, where g A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq69_HTML.gif (respectively g A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq70_HTML.gif) and ϕ is a function satisfying
lim r 1 μ ( [ r , r ] ) [ r , r ] ϕ ( s ) d μ ( s ) d s = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equc_HTML.gif
where μ ( [ r , r ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq71_HTML.gif is the measure of the set [ r , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq72_HTML.gif. The set of such functions is denoted by P A P ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq73_HTML.gif (respectively P A A ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq74_HTML.gif) cf. [15] (respectively [16]). When the measure μ satisfies the following condition:
  1. (C)

    For all τ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq30_HTML.gif, there exist β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq75_HTML.gif and a bounded interval I such that μ ( { a + τ : a B } ) β μ ( B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq76_HTML.gif for all B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq77_HTML.gif satisfying B I = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq78_HTML.gif,

     

then the set P A P ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq73_HTML.gif (respectively P A A ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq74_HTML.gif) is a Banach space which is invariant by translation, cf. Corollary 2.31 and Theorem 3.3 in [15] (respectively Theorem 4.9 and Theorem 3.5 in [16]).

The space W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq79_HTML.gif (respectively W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq80_HTML.gif) is a special case of the space P A P ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq73_HTML.gif (respectively P A A ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq74_HTML.gif) in the following sense: W P A P 0 ( X , ρ ) = P A P ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq81_HTML.gif (respectively W P A A 0 ( X , ρ ) = P A A ( R , X , μ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq82_HTML.gif), where the measure μ is absolutely continuous with respect to the Lebesgue measure and its Radon-Nikodym derivative is ρ = d μ d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq83_HTML.gif. The function ρ satisfies hypothesis (H) if and only if the measure μ satisfies condition (C), cf. Remark 3.4 in [15] (respectively Remark 3.1 in [16]). Consequently, when ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq84_HTML.gif, the spaces W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq79_HTML.gif and W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq80_HTML.gif are Banach spaces which are invariant by translation.

E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif denotes one the following spaces: A A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq44_HTML.gif, A A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq45_HTML.gif, P A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq46_HTML.gif, P A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq47_HTML.gif, W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq86_HTML.gif, or W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq87_HTML.gif endowed with the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq88_HTML.gif. We have E 0 ( X ) B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq89_HTML.gif, and when ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif, E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif is a Banach space which is invariant by translation. E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq91_HTML.gif denotes the space of functions u B C 1 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq92_HTML.gif such that u , u E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq93_HTML.gif. Endowed with the norm B C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq94_HTML.gif, E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq95_HTML.gif is a Banach space. E 2 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq96_HTML.gif denotes the space of the functions u B C 2 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq97_HTML.gif such that u , u , u E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq98_HTML.gif. Endowed with the norm B C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq99_HTML.gif, E 2 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq96_HTML.gif is a Banach space.

3 Preliminary results

For the proof of the main result, we need the following lemmas.

Lemma 3.1 Let ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif and u C 1 ( R , X ) E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq100_HTML.gif. If the derivative u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq101_HTML.gif is uniformly continuous, then u E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq102_HTML.gif, thus u E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq103_HTML.gif.

Remark 3.2 Lemma 3.1 is well know for A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq27_HTML.gif, A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq37_HTML.gif. In the scalar case, for P A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq46_HTML.gif, this result is proved in [12], Corollary 5.6, p.59.

Proof Consider the function g n : R X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq104_HTML.gif defined by g n ( t ) : = n ( f ( t + 1 n ) f ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq105_HTML.gif for n N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq106_HTML.gif. Since E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif is a translation invariant vectorial space, then g n E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq107_HTML.gif. The equality g n ( t ) f ( t ) = n 0 1 n ( f ( t + s ) f ( t ) ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq108_HTML.gif shows that the uniform continuity of f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq109_HTML.gif implies that the sequence ( g n ) n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq110_HTML.gif with values in the Banach space E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif converges uniformly to f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq109_HTML.gif on . Then f E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq111_HTML.gif, and from the definition of E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq91_HTML.gif, we obtain f E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq112_HTML.gif. □

Lemma 3.3 Let ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif and A L ( X , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq113_HTML.gif. If the spectrum σ ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq114_HTML.gif of A does not intersect the imaginary axis, then for all h E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq115_HTML.gif, there exists a unique solution in E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif of the differential equation
u ( t ) = A u ( t ) + h ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ1_HTML.gif
(3.1)

Moreover, the solution u is in E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq91_HTML.gif.

Proof Applying Theorem 4.1, p.81 in [20] (or Theorem 4 in [21]), Equation (3.1) admits a unique bounded solution on which is given by the formula
u ( t ) = G ( t s ) h ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ2_HTML.gif
(3.2)

where G is the principal Green function for Equation (3.1). The Green function G : R L ( X , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq116_HTML.gif is continuous on R { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq117_HTML.gif, and there exist M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq118_HTML.gif and ω > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq119_HTML.gif such that G ( t ) M exp ( ω | t | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq120_HTML.gif for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq121_HTML.gif.

Now we prove that the bounded solution u defined by (3.2) belongs to E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif.

When E 0 ( X ) = A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq122_HTML.gif (respectively E 0 ( X ) = A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq123_HTML.gif), this result is a straightforward consequence of Theorem 3.8 in [15] (respectively Theorem 3.9 in [16]) and when E 0 ( X ) = C 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq124_HTML.gif, this result is proved in [21], Proposition 3. Then we deduce the result for E 0 ( X ) = A A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq125_HTML.gif (respectively E 0 ( X ) = A A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq126_HTML.gif).

Note that the case of the pseudo almost periodic (respectively pseudo almost automorphic) functions is a special case of the weighted pseudo almost periodic (respectively weighted pseudo almost automorphic) functions by taking ρ ( t ) : = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq127_HTML.gif for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq34_HTML.gif; remark that the associated measure is exactly the Lebesgue measure.

And so it suffices to prove the cases of weighted pseudo almost periodic functions and of weighted pseudo almost automorphic functions.

When E 0 ( X ) = W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq128_HTML.gif (respectively E 0 ( X ) = W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq129_HTML.gif), this result is a straightforward consequence of Theorem 3.8 in [15] (respectively Theorem 3.9 in [16]). Consequently, u E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq130_HTML.gif, and from the definition of E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif, we deduce that [ t A u ( t ) ] E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq131_HTML.gif. Since u satisfies Equation (3.1), then u E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq102_HTML.gif, and from the definition of E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq91_HTML.gif, we obtain u E 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq103_HTML.gif. □

Lemma 3.4 Let X and Y be two finite-dimensional Banach spaces, and let ϕ : X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq132_HTML.gif be a continuous mapping. Then the Nemytskii operator N ϕ : E 0 ( X ) E 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq133_HTML.gif, defined by N ϕ ( u ) : = [ t ϕ ( u ( t ) ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq134_HTML.gif, is continuous.

Remark 3.5 Contrary to the asymptotically almost periodic case and, in particular, for the almost periodic case, when the dimension of the Banach spaces X and Y is infinite, Lemma 3.4 does not hold for the pseudo almost periodic case, and thus for the weighted pseudo almost periodic case, without additional assumptions. This is due to the fact that the range of a pseudo almost periodic function is only bounded, but not relatively compact, contrary to the asymptotically almost periodic case. This last observation still holds when the word almost periodic is replaced by almost automorphic.

Proof When E 0 ( X ) = A A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq135_HTML.gif and E 0 ( Y ) = A A P 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq136_HTML.gif, replacing R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq137_HTML.gif by , this result is a variation of Theorem 8.4 in [22].

When E 0 ( X ) = A A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq138_HTML.gif and E 0 ( Y ) = A A A 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq139_HTML.gif, replacing R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq137_HTML.gif by , the inclusion N ϕ ( A A A 0 ( X ) ) A A A 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq140_HTML.gif is a variation of Theorem 2.15 in [9]. Moreover, using Lemma 1 in [23], we know that N ϕ : B C 0 ( R , X ) B C 0 ( R , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq141_HTML.gif is continuous, and so its restriction to A A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq45_HTML.gif is also continuous.

When E 0 ( X ) = P A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq142_HTML.gif and E 0 ( Y ) = P A P 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq143_HTML.gif (respectively E 0 ( X ) = P A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq144_HTML.gif and E 0 ( Y ) = P A A 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq145_HTML.gif), this result is a straightforward consequence of Theorem 4.1 (respectively Theorem 4.2) in [24].

When E 0 ( X ) = W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq146_HTML.gif and E 0 ( Y ) = W P A P 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq147_HTML.gif (respectively E 0 ( X ) = W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq148_HTML.gif and E 0 ( Y ) = W P A A 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq149_HTML.gif), using Corollary 4.12 in [15] (respectively Corollary 5.10 in [16]), we know that N ϕ ( W P A P 0 ( X , ρ ) ) W P A P 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq150_HTML.gif (respectively N ϕ ( W P A A 0 ( X , ρ ) ) W P A P 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq151_HTML.gif). Moreover, using Lemma 1 in [23], we know that N ϕ : B C 0 ( R , X ) B C 0 ( R , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq141_HTML.gif is continuous, and so its restriction to W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq79_HTML.gif (respectively W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq152_HTML.gif) is also continuous. □

Lemma 3.6 Let X and Y be two finite-dimensional Banach spaces, and let ϕ : X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq132_HTML.gif be a continuously Fréchet-differentiable mapping. Then the Nemytskii operator N ϕ : E 0 ( X ) E 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq133_HTML.gif is continuously Fréchet-differentiable on E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq85_HTML.gif, and we have D N ϕ ( u ) v = [ t D ϕ ( u ( t ) ) v ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq153_HTML.gif for all u , v E 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq154_HTML.gif.

Proof When E 0 ( X ) = A A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq135_HTML.gif and E 0 ( Y ) = A A P 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq136_HTML.gif, replacing R + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq137_HTML.gif by , this result is a variation of Theorem 8.5 in [22].

When E 0 ( X ) = A A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq138_HTML.gif and E 0 ( Y ) = A A A 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq139_HTML.gif, using Lemma 1 in [23], we know that N ϕ : B C 0 ( R , X ) B C 0 ( R , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq155_HTML.gif is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif and that we have D N ϕ ( u ) h = [ t D ϕ ( u ( t ) ) h ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq157_HTML.gif when u , h B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq158_HTML.gif. Now, using Theorem 2.15 in [9], we know that N ϕ ( A A A 0 ( X ) ) A A A 0 ( Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq159_HTML.gif and that D ϕ u A A A 0 ( L ( X , Y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq160_HTML.gif when u A A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq161_HTML.gif. And so N ϕ C 1 ( A A A 0 ( X ) , A A A 0 ( Y ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq162_HTML.gif and the announced formula for its Fréchet-differential is proven.

As in the proof of Lemma 3.3, the case of P A P 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq46_HTML.gif (respectively P A A 0 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq47_HTML.gif) is a corollary of the case W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq79_HTML.gif (respectively W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq80_HTML.gif). And so it suffices to prove the cases of the weighted pseudo almost periodic functions and of the weighted pseudo almost automorphic functions.

To prove the result in the case where E 0 ( X ) = W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq146_HTML.gif and E 0 ( Y ) = W P A P 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq147_HTML.gif (respectively E 0 ( X ) = W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq163_HTML.gif and E 0 ( Y ) = W P A A 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq164_HTML.gif), note that, using Lemma 1 in [23], we know that N ϕ : B C 0 ( R , X ) B C 0 ( R , Y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq141_HTML.gif is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif and that we have D N ϕ ( u ) h = [ t D ϕ ( u ( t ) ) h ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq165_HTML.gif when u , h B C 0 ( R , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq158_HTML.gif. Now, using Corollary 4.12 in [15] (respectively Corollary 5.10 in [16]), we know that N ϕ ( W P A P 0 ( X , ρ ) ) W P A P 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq166_HTML.gif (respectively N ϕ ( W P A A 0 ( X , ρ ) ) W P A A 0 ( Y , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq167_HTML.gif) and that D ϕ u W P A P 0 ( L ( X , Y ) , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq168_HTML.gif (respectively D ϕ u W P A A 0 ( L ( X , Y ) , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq169_HTML.gif) when u W P A P 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq170_HTML.gif (respectively W P A A 0 ( X , ρ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq80_HTML.gif).

Consequently, we obtain N ϕ C 1 ( W P A P 0 ( X , ρ ) , W P A P 0 ( Y , ρ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq171_HTML.gif (respectively N ϕ C 1 ( W P A A 0 ( X , ρ ) , W P A A 0 ( Y , ρ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq172_HTML.gif) and the announced formula for its Fréchet-differential is proven. □

Lemma 3.7 Let ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif, p P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq173_HTML.gif and e p E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq174_HTML.gif. If x is a solution of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif, that is, x C 2 ( R , R ) E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq176_HTML.gif and x satisfies ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif), then x E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq177_HTML.gif.

Proof Lemma 3.2 in [4] asserts that if x is a solution of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) in B C 0 ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq178_HTML.gif, then x B C 2 ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq179_HTML.gif, therefore the derivative x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq180_HTML.gif is uniformly continuous, and by help of Lemma 3.1, we obtain x E 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq181_HTML.gif. By using Lemma 3.4, the functions [ t f ( x ( t ) , p ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq182_HTML.gif and [ t g ( x ( t ) , p ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq183_HTML.gif are in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif. Applying again Lemma 3.4 to [ t ( f ( x ( t ) , p ) , x ( t ) ) ] E 0 ( R 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq184_HTML.gif and using the continuous function ϕ : R 2 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq185_HTML.gif defined by ϕ ( r , s ) : = r s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq186_HTML.gif, we obtain that [ t f ( x ( t ) , p ) x ( t ) ] E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq187_HTML.gif. Since x ( t ) = e p ( t ) f ( x ( t ) , p ) x ( t ) g ( x ( t ) , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq188_HTML.gif, then x E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq189_HTML.gif, and from the definition of E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif, we obtain that x E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq177_HTML.gif. □

4 The main result

First we announce the main result of the paper.

Theorem 4.1 Let ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif. Under the following assumptions:

(A1) f , g C 1 ( R × P , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq191_HTML.gif,

(A2) g ( 0 , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq192_HTML.gif,

(A3) p e p C 1 ( P , E 0 ( R ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq193_HTML.gif and e 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq194_HTML.gif,

(A4) f ( 0 , 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq195_HTML.gif when f ( 0 , 0 ) 2 < 4 g ( 0 , 0 ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq196_HTML.gif, and g ( 0 , 0 ) x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq197_HTML.gif when f ( 0 , 0 ) 2 4 g ( 0 , 0 ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq198_HTML.gif,

there exist a neighborhood https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq199_HTML.gif of 0 in E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif, a neighborhood https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq200_HTML.gif of 0 in P and a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif-mapping p x ̲ [ p ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq201_HTML.gif from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq200_HTML.gif into https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq199_HTML.gif which satisfies the following conditions:
  1. (i)

    x ̲ [ 0 ] = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq202_HTML.gif,

     
  2. (ii)

    for all p V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq203_HTML.gif, x ̲ [ p ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq204_HTML.gif is a solution of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif,

     
  3. (iii)

    if x U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq205_HTML.gif is a solution of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif with p V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq203_HTML.gif, then x = x ̲ [ p ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq206_HTML.gif.

     
To prove Theorem 4.1, we define the operator Φ : E 2 ( R ) × P E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq207_HTML.gif by setting
Φ ( x , p ) : = [ t x ( t ) + f ( x ( t ) , p ) x ( t ) + g ( x ( t ) , p ) e p ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ3_HTML.gif
(4.1)

when x E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq208_HTML.gif and p P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq209_HTML.gif.

Let p P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq173_HTML.gif. By using Lemma 3.7, we deduce that x E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq208_HTML.gif satisfies Φ ( x , p ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq210_HTML.gif if and only if x is a solution of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif.

Under (A2) and (A3), note that 0 is a solution of ( E , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq211_HTML.gif in E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif, and so the following equality holds:
Φ ( 0 , 0 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ4_HTML.gif
(4.2)
Lemma 4.2 Under (A1)-(A3), the operator Φ is well defined and it is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif on E 2 ( R ) × P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq212_HTML.gif. Moreover, the partial differential of Φ with respect to the first variable, at the point ( x , p ) = ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq213_HTML.gif, is given by
D x Φ ( 0 , 0 ) y = [ t y ( t ) + f ( 0 , 0 ) y ( t ) + g ( 0 , 0 ) x y ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equd_HTML.gif

when y E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq214_HTML.gif.

Proof First we introduce linear operators: d 2 d t 2 : E 2 ( R ) E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq215_HTML.gif defined by d 2 d t 2 x : = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq216_HTML.gif, d d t : E 1 ( R ) E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq217_HTML.gif defined by d d t x : = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq218_HTML.gif, in 1 : E 2 ( R ) E 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq219_HTML.gif defined by in 1 ( x ) : = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq220_HTML.gif and in 2 : E 2 ( R ) E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq221_HTML.gif defined by in 2 ( x ) : = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq222_HTML.gif. Since d 2 d t 2 x x B C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq223_HTML.gif, in 1 ( x ) B C 1 x B C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq224_HTML.gif and in 2 ( x ) x B C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq225_HTML.gif for all x E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq208_HTML.gif, and d d t x x B C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq226_HTML.gif for all x E 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq227_HTML.gif, these linear operators are continuous; and consequently, the following assertion holds:
d 2 d t 2 , d d t , in 1 , in 2  are of class  C 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Eque_HTML.gif
Now we define the Nemytskii operators build on the functions f and g: N f : E 0 ( R ) × E 0 ( P ) E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq228_HTML.gif defined by N f ( x , p ) : = [ t f ( x ( t ) , p ( t ) ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq229_HTML.gif and N g : E 0 ( R ) × E 0 ( P ) E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq230_HTML.gif defined by N g ( x , p ) : = [ t g ( x ( t ) , p ( t ) ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq231_HTML.gif. By using Lemma 3.6, N f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq232_HTML.gif and N g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq233_HTML.gif are of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif on E 0 ( R ) × E 0 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq234_HTML.gif assimilated to E 0 ( R × P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq235_HTML.gif, and Lemma 3.6 provides formulas for the differentials of these Nemytskii operators:
D x N f ( x , p ) y = [ t f ( x ( t ) , p ( t ) ) x y ( t ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ5_HTML.gif
(4.3)
D x N g ( x , p ) y = [ t g ( x ( t ) , p ( t ) ) x y ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ6_HTML.gif
(4.4)

for all x , y E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq236_HTML.gif.

We can assimilate a point p P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq209_HTML.gif to the constant function t p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq237_HTML.gif that belongs to E 0 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq238_HTML.gif, which permits us to look at P as a closed vector subspace of E 0 ( P ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq238_HTML.gif. Then we can consider the following restrictions of the operators N f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq232_HTML.gif and N g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq233_HTML.gif: S f : E 0 ( R ) × P E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq239_HTML.gif defined by S f ( x , p ) : = [ t f ( x ( t ) , p ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq240_HTML.gif and S g : E 0 ( R ) × P E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq241_HTML.gif defined by S g ( x , p ) : = [ t g ( x ( t ) , p ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq242_HTML.gif when x E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq243_HTML.gif and p P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq209_HTML.gif.

Since the restriction of a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif-mapping to a Banach subspace is also a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif-mapping, a straightforward consequence of the continuous differentiability of N f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq232_HTML.gif and of N g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq233_HTML.gif is that S f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq244_HTML.gif and S g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq245_HTML.gif are of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif, and the consequences of (4.3) and (4.4) are the following formulas:
D x S f ( x , p ) y = [ t f ( x ( t ) , p ) x y ( t ) ] , D x S g ( x , p ) y = [ t g ( x ( t ) , p ) x y ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equf_HTML.gif

for all x , y E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq236_HTML.gif for all p P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq209_HTML.gif.

Now we consider the following operators: π 1 : E 2 ( R ) × P E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq246_HTML.gif defined by π 1 ( x , p ) : = x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq247_HTML.gif, π 2 : E 2 ( R ) × P P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq248_HTML.gif defined by π 2 ( x , p ) : = p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq249_HTML.gif and B : R × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq250_HTML.gif defined by B ( r , s ) : = r s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq251_HTML.gif. We consider the Nemystkii operator build on B, N B : E 0 ( R ) × E 0 ( R ) E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq252_HTML.gif defined by N B ( u , v ) : = [ t u ( t ) v ( t ) = B ( u ( t ) , v ( t ) ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq253_HTML.gif and C : E 2 ( R ) × P E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq254_HTML.gif defined by C ( x , p ) : = e p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq255_HTML.gif.

Since π 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq256_HTML.gif and π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq257_HTML.gif are linear continuous, they are of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif. Since B is bilinear continuous, it is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif; and consequently, using Lemma 3.6, N B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq258_HTML.gif is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif on E 0 ( R ) × E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq259_HTML.gif. Denoting by ε : P E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq260_HTML.gif the mapping p e p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq261_HTML.gif, ε is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif after (A3), and C = ε π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq262_HTML.gif is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif as a composition of C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif-mappings. And so we can assert that π 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq256_HTML.gif, π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq263_HTML.gif, N B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq258_HTML.gif and C are of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif.

Now we note that the following equality holds:
Φ = d 2 d t 2 π 1 + N B ( S f ( in 2 π 1 , π 2 ) , d d t ( in 1 π 1 ) ) + S g ( in 2 π 1 , π 2 ) + C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ7_HTML.gif
(4.5)

Since all the mappings which are present in the previous formula are of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif, using the usual rules of the differential calculus in Banach spaces, we obtain that Φ is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif.

For all y E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq214_HTML.gif, by using the classical formulas of the differential calculus in Banach spaces and (4.5), we obtain
D x Φ ( 0 , 0 ) y = d 2 d t 2 y + N B ( f x ( 0 , 0 ) y , 0 ) + N B ( S f ( 0 , 0 ) , d d t ( in 1 ( y ) ) ) + D x S g ( 0 , 0 ) y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equg_HTML.gif
which implies, for all t R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq34_HTML.gif,
( D x Φ ( 0 , 0 ) y ) ( t ) = y ( t ) + f ( 0 , 0 ) y ( t ) + g ( 0 , 0 ) x y ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equh_HTML.gif

which is the announced formula. □

Lemma 4.3 Under (A1)-(A4), D x Φ ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq264_HTML.gif is a bijection from E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif onto E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif.

Proof Let b E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq265_HTML.gif. We want to prove that there exists a unique y E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq214_HTML.gif such that D x Φ ( 0 , 0 ) y = b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq266_HTML.gif. Using the formula provided by Lemma 4.2, this equation is equivalent to saying that y is a solution in E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif of the following second-order linear differential equation (which is a Duffing equation):
y ( t ) + f ( 0 , 0 ) y ( t ) + g ( 0 , 0 ) x y ( t ) = b ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ8_HTML.gif
(4.6)
Rewriting this second-order equation in the form of a first-order system, we obtain the following equivalent differential system:
X ( t ) = M X ( t ) + B ( t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equ9_HTML.gif
(4.7)

where X ( t ) : = [ y ( t ) y ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq267_HTML.gif, B ( t ) : = [ 0 b ( t ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq268_HTML.gif, and M : = [ 0 1 g ( 0 , 0 ) x f ( 0 , 0 ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq269_HTML.gif.

For ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif and with condition (A4), the assumptions of Lemma 3.3 are fulfilled, and we can assert that there exists a unique X E 1 ( R 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq270_HTML.gif, which is a solution of (4.7). Therefore the first coordinate of X, denoted by y, is the unique solution of (4.6) in E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif since y and y E 1 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq271_HTML.gif, and then y is the unique element of E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif which satisfies D x Φ ( 0 , 0 ) y = b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq272_HTML.gif. □

Proof of Theorem 4.1 By using (4.2), Lemma 4.2 and Lemma 4.3, we can use the implicit function theorem ([25], p.61) that permits us to say that there exist a neighborhood https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq199_HTML.gif of 0 in E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif, a neighborhood https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq200_HTML.gif of 0 in P and a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif-mapping p x ̲ [ p ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq201_HTML.gif from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq200_HTML.gif into https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq199_HTML.gif , which satisfies the following conditions:
  1. (a)

    x ̲ [ 0 ] = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq202_HTML.gif, that is, the condition (i) of Theorem 4.1.

     
  2. (b)

    Φ ( x ̲ [ p ] , p ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq273_HTML.gif for all p V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq203_HTML.gif, that ensures that x ̲ [ p ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq204_HTML.gif is a solution of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif for all p V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq203_HTML.gif, that is, the conclusion (ii) of Theorem 4.1.

     
  3. (c)

    { ( x , p ) U × V : Φ ( x , p ) = 0 } = { ( x ̲ [ p ] , p ) : p V } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq274_HTML.gif that implies the conclusion (iii) of Theorem 4.1.

     

And so Theorem 4.1 is proven. □

5 Special cases

We consider the equation
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equi_HTML.gif

which is a special case of ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) by taking f ( x , p ) = f 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq275_HTML.gif, g ( x , p ) = g 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq276_HTML.gif and p e p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq277_HTML.gif defined as the identity operator on P = E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq278_HTML.gif.

On the existence solution of ( F , e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq11_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq279_HTML.gif, we establish the following result.

Corollary 5.1 Let ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif. Under the following assumptions:

(A5) f 1 , g 1 C 1 ( R , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq280_HTML.gif,

(A6) g 1 ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq281_HTML.gif,

(A7) f 1 ( 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq282_HTML.gif when f 1 ( 0 ) 2 < 4 g 1 ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq283_HTML.gif, and g 1 ( 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq284_HTML.gif when f 1 ( 0 ) 2 4 g 1 ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq285_HTML.gif,

there exist a neighborhood https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq286_HTML.gif of 0 in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif, a neighborhood https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq199_HTML.gif of 0 in E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif and a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif-mapping e x ̲ [ e ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq287_HTML.gif from https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq286_HTML.gif into https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq199_HTML.gif , which satisfies the following conditions:
  1. (i)

    x ̲ [ 0 ] = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq202_HTML.gif,

     
  2. (ii)

    for all e W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq288_HTML.gif, x ̲ [ e ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq289_HTML.gif is a solution of ( F , e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq11_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq279_HTML.gif,

     
  3. (iii)

    if x U https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq205_HTML.gif is a solution of ( F , e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq11_HTML.gif) with e W https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq288_HTML.gif in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif, then we have x = x ̲ [ e ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq290_HTML.gif.

     
The second special case of the equation ( E , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq10_HTML.gif) is
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_Equj_HTML.gif

when q belongs to a Banach space Q, and by taking p = ( e , q ) P = E 0 ( R ) × Q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq291_HTML.gif, f ( x , e , q ) = f 2 ( x , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq292_HTML.gif, g ( x , e , q ) = g 2 ( x , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq293_HTML.gif and e ( e , q ) = e https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq294_HTML.gif. On the existence solution of ( G , e , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq295_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq279_HTML.gif, we establish the following result.

Corollary 5.2 Let ρ U T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq90_HTML.gif. Under the following assumptions:

(A8) f 2 , g 2 C 1 ( R × Q , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq296_HTML.gif,

(A9) g 2 ( 0 , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq297_HTML.gif,

(A10) f 2 ( 0 , 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq298_HTML.gif when f 2 ( 0 , 0 ) 2 < 4 g 2 ( 0 , 0 ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq299_HTML.gif, and g 2 ( 0 , 0 ) x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq300_HTML.gif when f 2 ( 0 , 0 ) 2 4 g 2 ( 0 , 0 ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq301_HTML.gif,

there exist a neighborhood W 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq302_HTML.gif of 0 in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif, a neighborhood U 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq303_HTML.gif of 0 in E 2 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq190_HTML.gif, a neighborhood V 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq304_HTML.gif of 0 in Q and a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq156_HTML.gif-mapping e x ̲ [ e , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq305_HTML.gif from W 2 × V 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq306_HTML.gif into U 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq303_HTML.gif which satisfies the following conditions:
  1. (i)

    x ̲ [ 0 , 0 ] = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq307_HTML.gif,

     
  2. (ii)

    for all e W 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq308_HTML.gif in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif and for all q V 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq309_HTML.gif, x ̲ [ e , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq310_HTML.gif is a solution of ( G , e , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq295_HTML.gif) in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif,

     
  3. (iii)

    if x U 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq311_HTML.gif is a solution of ( G , e , q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq295_HTML.gif) with e W 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq308_HTML.gif in E 0 ( R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq175_HTML.gif and q V 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq309_HTML.gif, then we have x = x ̲ [ e , q ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-66/MediaObjects/13661_2012_Article_418_IEq312_HTML.gif.

     

Declarations

Acknowledgements

The authors are very pleased to contribute to this special issue in honor of Jean Mawhin, an international expert in the field of nonlinear analysis and differential equations, whose opinion is ever very important and useful to us.

Authors’ Affiliations

(1)
Laboratoire SAMM EA 4543, Université Paris 1 Panthéon-Sorbonne, centre P.M.F.
(2)
Département de Mathématiques, Faculté des Sciences, Université de Skikda
(3)
Laboratoire de Mathématiques de Versailles, UMR-CNRS 8100, Université Versailles-Saint-Quentin-en-Yvelines

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