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Small-amplitude nonlinear periodic oscillations in a suspension bridge system

Boundary Value Problems20182018:141

https://doi.org/10.1186/s13661-018-1063-2

  • Received: 21 June 2018
  • Accepted: 11 September 2018
  • Published:

Abstract

This paper deals with a coupled nonlinear beam-wave system, proposed by Lazer and McKenna, modeling a suspension bridge. By constructing a suitable Nash–Moser-type iteration scheme, the existence of small-amplitude nonlinear periodic oscillations, i.e., the existence of small-amplitude periodic solutions for the suspension bridge system, is obtained, and so is their local uniqueness.

Keywords

  • Suspension bridge system
  • Small divisors
  • Nash–Moser iteration scheme

1 Introduction

In this paper, we consider the following suspension bridge system:
$$\begin{aligned}& m_{b}u_{tt}-Qu_{xx}-K(w-u)^{+}=m_{b}g+ \epsilon h_{1}(x,\omega t), \end{aligned}$$
(1.1)
$$\begin{aligned}& m_{c}w_{tt}+EIw_{xxxx}+K(w-u)^{+}=m_{c}g+ \epsilon h_{2}(x,\omega t), \end{aligned}$$
(1.2)
with the boundary condition
$$ \begin{aligned} &u(0,t)=u(L,t)=0, \\ &w(0,t)=w(L,t)=0,\qquad w_{xx}(0,t)=w_{xx}(L,t)=0, \end{aligned} $$
(1.3)
where \(x\in(0,\pi)\), \(t\in\mathbb{R}\), \(m_{b}\), \(m_{c}\), Q and EI are positive constants, the parameter \(\epsilon\geq0\), \((w-u)^{+}=\max\{w-u,0\}\), \(h_{i}\) (\(i=1,2\)) is a time-periodic external force with period \(2\pi/\omega\) and amplitude ϵ.
This model is first proposed by Lazer and McKenna [11] through the observation of the fundamental nonlinearity in suspension bridges of which the stays connecting the supporting cables and the roadbed resist expansion but do not resist compression. The roadbed of length L is modeled by a horizontal vibrating beam with both ends being simply supported; the supporting cable of length L is modeled by a horizontal vibrating string with both ends being fixed; the vertical stays connecting the roadbed to the supporting cable are modeled by one-sided springs with resist expansion but not resist compression; In the model (1.1)–(1.2), \(u(x,t)\) and \(w(x,t)\) denote the downward deflections of the cable and the roadbed, respectively; \(m_{b}\) and \(m_{c}\) denote the mass densities of the cable and the roadbed, respectively; Q denotes the coefficient of cable tensile strength; EI denotes the roadbed flexural rigidity; K is Hooke’s constant of the stays; \(h_{1}\) and \(h_{2}\) denote the external aerodynamic forces. In fact, we can write system (1.1) as a Hamiltonian system. Introducing the variables \(Z=(q_{1},q_{2},p_{1},p_{2})\), with \(q_{1}(x,t)=u(x,t)\), \(q_{2}(x,t)=w(x,t)\), \(p_{1}(x,t)=u_{t}(x,t)\) and \(p_{2}(x,t)=w_{t}(x,t)\), and the Hamiltonian
$$\begin{aligned} H(Z) =&\frac{1}{2} \int_{0}^{\pi}\bigl[m_{b}q_{1}^{2}+m_{c}q_{2}^{2}+Q( \partial _{x}q_{1})^{2}+EI(\partial_{xx}q_{2})^{2}+ \bigl(K(q_{2}-q_{1})^{+}\bigr)^{2}\bigr]\,dx \\ &{}- \int_{0}^{\pi}\bigl[F_{1}(q_{1},p_{1})+F_{2}(q_{2},p_{2}) \bigr]\,dx, \end{aligned}$$
the system (1.1) can be written in the form
$$ \frac{\partial}{\partial_{t}}Z=J\nabla_{Z}H(Z), $$
where \(F_{1}(q_{1},p_{1})=m_{b}gq_{1}+\epsilon h_{1}(x,t)q_{1}\), \(F_{2}(q_{2},p_{2})=m_{b}gq_{2}+\epsilon h_{1}(x,t)q_{2}\) and J is a \(4\times4\) symplectic matrix.
Theoretically and numerically of periodic oscillations in for suspension bridge model have attracted much attention [5, 911]. McKenna and Walter [12] first studied large-amplitude periodic oscillations for a single suspension bridge equation (a beam equation) via Leray–Schauder degree theory. By considering the motion of the cable in suspension bridge, Lazer and MaKenna [11] considered the model (1.1)–(1.2) under boundary condition (1.3). By using a standard IMSL subroutine on a mainframe using high precision, they showed that the approximation system of (1.1)–(1.2) with damped term
$$\begin{aligned}& u''+\delta_{1}u'+a_{1}u-K(w-u)^{+}= \epsilon g(t), \\& w''+\delta_{2}w'+a_{2}w+K(w-u)^{+}=W_{0}, \end{aligned}$$
has large and small periodic solutions. Recently, when \(\omega=1\), by using Leray–Schauder degree theory, Ding [7] proved that system (1.1)–(1.2) under boundary condition (1.3) has at least two large-amplitude π-periodic solutions, under the assumption that
$$\begin{aligned}& Q\leq m_{c}, \qquad EI\leq m_{b}, \\& \lambda_{m,n}=Q(2n+1)^{2}-4m_{c}m^{2} \neq0,\qquad \mu _{m,n}=EI(2n+1)^{4}-4m_{b}m^{2} \neq0; \\& \lambda_{m,n}+\mu_{m,n}\neq0,\quad \mbox{for }m\geq1, n \geq1, \end{aligned}$$
where \(\lambda_{m,n}\) and \(\mu_{m,n}\) denote the eigenvalue of wave operator \(m_{b}u_{tt}-Qu_{xx}\) and beam operator \(m_{c}w_{tt}+EIw_{xxxx}\), respectively. Both \(\sqrt{\frac{Q}{m_{c}}}\) and \(\sqrt{\frac{Q}{m_{b}}}\) are rational numbers.

In the present paper, we show the existence of small-amplitude periodic oscillations in problem (1.1)–(1.3). Here, one of our main results is the following.

Theorem 1.1

Let \(0<\bar{\sigma}<\sigma<\tilde{\sigma}<1\), \((m_{b}g,m_{c}g)\in[0,\chi_{0}]\times[0,\chi_{0}]\). Assume that \(Q,EI>K\), \((h_{1},h_{2})\in X_{\bar{\sigma}}\times Y_{\bar{\sigma}}\). Then there exist \(\epsilon_{0}>0\) and \(K_{0}>0\) sufficiently small and a Cantor set \(\mathcal{X}(\nu)=\{\omega:(\nu,\omega)\in\mathcal{D}_{\gamma}\}\), \(\mathcal{D}_{\gamma}\) to be specified in (4.17), such that, for \(\epsilon\in[0,\epsilon_{0}]\) and \(K\in[0,K_{0}]\), and every \((\nu,\omega)\in\mathcal{X}(\nu)\) there exists a solution of system (1.1)(1.2) under boundary condition (1.3),
$$\begin{aligned}& \bigl(u(\nu,\omega),w(\nu,\omega)\bigr) \\& \quad =\bigl(\bar{u}+\tilde{u}(\nu,\omega),\bar{w}+\tilde{w}(\nu,\omega)\bigr) \in\mathbb{V}_{1}\times\mathbb{V}_{2}\oplus( \mathbb{W}_{1}\cap X_{\bar{\sigma},s})\times(\mathbb{W}_{2}\cap Y_{\bar{\sigma},s}), \end{aligned}$$
where the spaces \(\mathbb{V}_{i}\), \(\mathbb{W}_{i}\), \(X_{\bar{\sigma},s}\) and \(Y_{\bar{\sigma},s}\) are to be specified in Sect. 2, \(\chi_{0}\) is a sufficiently small constant.
Moreover, for every \(0<\omega_{1}<\omega_{2}<\infty\), there exists a constant C depending on Q such that in the rectangular region \(\mathcal{Y}=(\nu_{1},\nu_{2})\times(\omega_{1},\omega_{2})\) we have
$$\begin{aligned} \frac{|\mathcal{X}(\nu)\cap\mathcal{Y}|}{|\mathcal{Y}|}\geq1-C\gamma. \end{aligned}$$
For dissipative system
$$\begin{aligned}& m_{b}u_{tt}-Qu_{xx}+ \delta_{1} u_{t}-K(w-u)^{+}=m_{b}g+\epsilon h_{1}(x,\omega t), \end{aligned}$$
(1.4)
$$\begin{aligned}& m_{c}w_{tt}+EIw_{xxxx}+ \delta_{2}w_{t}+K(w-u)^{+}=m_{c}g+\epsilon h_{2}(x,\omega t), \end{aligned}$$
(1.5)
where \(\delta_{1}, \delta_{2}>0\), we have the same result.

Theorem 1.2

Let \(0<\bar{\sigma}<\sigma<\tilde{\sigma}<1\), \((m_{b}g,m_{c}g)\in[0,\chi_{0}]\times[0,\chi_{0}]\). Assume that \(Q,EI>K\), \(\delta_{1},\delta_{2}>0\), \((h_{1},h_{2})\in X_{\bar{\sigma}}\times Y_{\bar{\sigma}}\). Then there exist \(\epsilon_{0}>0\) and \(K_{0}>0\) sufficiently small and a Cantor set \(\mathcal{X}(\nu)=\{\omega:(\nu,\omega)\in\mathcal{D}_{\gamma}\}\), \(\mathcal{D}_{\gamma}\) to be specified in (4.17), such that, for \(\epsilon\in[0,\epsilon_{0}]\) and \(K\in[0,K_{0}]\), and every \((\nu,\omega)\in\mathcal{X}(\nu)\) there exists a solution of system (1.4)(1.5) under boundary condition (1.3)
$$\begin{aligned}& \bigl(u(\nu,\omega),w(\nu,\omega)\bigr) \\& \quad = \bigl(\bar{u}+\tilde{u}(\nu,\omega),\bar{w}+\tilde{w}(\nu,\omega)\bigr) \in\mathbb{V}_{1}\times\mathbb{V}_{2}\oplus( \mathbb{W}_{1}\cap X_{\bar{\sigma},s})\times(\mathbb{W}_{2}\cap Y_{\bar{\sigma},s}), \end{aligned}$$
where spaces \(\mathbb{V}_{i}\), \(\mathbb{W}_{i}\), \(X_{\bar{\sigma},s}\) and \(Y_{\bar{\sigma},s}\) are to be specified in Sect. 2, \(\chi_{0}\) is a sufficiently small constant.
Moreover, for every \(0<\omega_{1}<\omega_{2}<\infty\), there exists a constant C depending on Q such that in the rectangular region \(\mathcal{Y}=(\nu_{1},\nu_{2})\times(\omega_{1},\omega_{2})\) we have
$$ \frac{|\mathcal{X}(\nu)\cap\mathcal{Y}|}{|\mathcal{Y}|}\geq1-C\gamma. $$

The main difficulty in dealing with problems (1.1)–(1.3) and (1.4)–(1.5) is in the presence of the term \((w-u)^{+}\), which is piecewise linear. So, we cannot simply apply the contraction map theorem in solving the range equation which is obtained by Lyapunov–Schmidt decomposition. To obtain a \(2\pi/\omega\)-time-periodic solution, a possible method is to construct a rapidly convergent Nash–Moser iteration. The first pioneering work is due to Moser [1315]. Rabinowitz [16] showed that a class of nonlinear wave equations with damped term has periodic solutions via constructing a rapidly convergent Nash–Moser iteration. Celletti and Chierchia [4] established a dissipative Nash–Moser theorem via a rapidly convergent Newton iteration, and proved KAM tori smoothly bifurcating into quasi-periodic attractors in dissipative mechanical models. Inspired by the work of [14, 6, 8, 1618], we will construct a suitable rapidly convergent Nash–Moser iteration scheme to prove our results.

We organize the paper as follows. In Sect. 2, we recall some basic notations, splitting the problem into the bifurcation equation and the range equation. Section 3, the bifurcation equation is solved via direct calculus of variations. Section 4 is divided into two subsections. The first subsection is to give the strategy of the Nash–Moser algorithm and some KAM estimates. The last subsection is to prove the convergence of the Nash–Moser algorithm and local uniqueness of solutions. In the Appendix, the measure of the Cantor set \(\mathcal{D}_{\gamma}\) is also estimated.

2 Some notations

We start this section by introducing some notations. Consider the following space:
$$\begin{aligned} X_{\sigma,s} :=&\biggl\{ u(t,x):=\sum_{l\in\mathbb{Z}}u_{l}(x)e^{ilt} \Bigm| u_{0}\in\mathbb{H}^{1}_{0}\bigl((0,\pi), \mathbb{R}\bigr), u_{l}\in\mathbb{C}\bigl((0,\pi ),\mathbb{R}\bigr)\ (l\neq0),u_{-l}=u^{*}_{l}, \\ &\|u\|^{2}_{\sigma,s} :=|u_{0}|_{\mathbb{H}^{1}_{0}}^{2}+ \sum_{|l|\geq1}\Bigl(\max_{x\in(0,\pi )}u_{l}(x) \Bigr)^{2}|l|^{2s}e^{2\sigma|l|}< \infty\biggr\} , \end{aligned}$$
and
$$\begin{aligned} Y_{\sigma,s} :=&\biggl\{ w(t,x):=\sum_{l\in\mathbb{Z}}w_{l}(x)e^{ilt} \Bigm| w_{0}\in\mathbb{H}^{2}_{0}\bigl((0,\pi), \mathbb{R}\bigr),w_{l}\in\mathbb{C}\bigl((0,\pi ),\mathbb{R}\bigr)\ (l \neq0), w_{-l}=w^{*}_{l}, \\ &\|w\|^{2}_{\sigma,s} :=|w_{0}|_{\mathbb{H}^{2}_{0}}^{2}+ \sum_{|l|\geq1}\Bigl(\max_{x\in(0,\pi )}w_{l}(x) \Bigr)^{2}|l|^{2s}e^{2\sigma|l|}< \infty\biggr\} , \end{aligned}$$
where \(u_{l}\) and \(w_{l}\) denote the lthe Fourier coefficients.
Obviously, for a nested family of Banach spaces \(\{X_{\sigma,s}: \sigma\geq0, s\geq0\}\) and \(\{Y_{\sigma,s}: \sigma\geq0, s\geq0\}\), we have
$$ X_{\sigma_{2},s}\subset X_{\sigma_{1},s}, \qquad Y_{\sigma_{2},s}\subset Y_{\sigma_{1},s}, $$
and
$$\begin{aligned}& \|u\|_{\sigma_{1},s}\leq\|u\|_{\sigma_{2},s}, \quad \forall 0\leq \sigma_{1}\leq\sigma_{2}, u\in X_{\sigma,s}, \\& \|w\|_{\sigma_{1},s}\leq\|w\|_{\sigma_{2},s}, \quad \forall 0\leq \sigma_{1}\leq\sigma_{2}, w\in Y_{\sigma,s}. \end{aligned}$$
For \(\sigma\geq0\), \(s\geq0\), the space \(X_{\sigma,s}\) is a Banach algebra with respect to multiplication of functions, i.e., if \(u_{1},u_{2}\in X_{\sigma,s}\), then \(u_{1}u_{2}\in X_{\sigma,s}\) and there exists a positive constant C, such that
$$ \|u_{1}u_{2}\|_{\sigma,s}\leq C\|u_{1} \|_{\sigma,s}\|u_{2}\|_{\sigma,s}. $$
The space \(Y_{\sigma,s}\) is a Banach algebra with respect to multiplication of functions. It is obvious that each function in \(X_{\sigma,s}\) or (\(Y_{\sigma,s}\)) has bounded analytic extension in the complex multi-strip \(|Im\vartheta_{i}|<\sigma\), where \(\vartheta_{i}\in C\), \(i=1,2\). By the definition of the space \(X_{\sigma,s}\), the following inequality holds:
$$ \bigl\Vert \partial_{\vartheta}^{h} u \bigr\Vert _{\sigma,s}\leq \bigl\Vert \partial_{\vartheta}^{k} u \bigr\Vert _{\sigma,s}, \quad \forall h, k\in N^{2}: h_{i} \leq k_{i}. $$
For more details, see [1].
After a time rescaling, we look for 2π periodic solutions of
$$\begin{aligned}& \omega^{2}m_{b}u_{tt}-Qu_{xx}-K(w-u)^{+}=m_{b}g+ \epsilon h_{1}(x,t), \end{aligned}$$
(2.1)
$$\begin{aligned}& \omega^{2}m_{c}w_{tt}+EIw_{xxxx}+K(w-u)^{+}=m_{c}g+ \epsilon h_{2}(x,t), \end{aligned}$$
(2.2)
with the boundary condition
$$ \begin{aligned} &u(0,t)=u(\pi,t)=0, \\ &w(0,t)=w(\pi,t)=0,\qquad w_{xx}(0,t)=w_{xx}(\pi,t)=0. \end{aligned} $$
(2.3)
Denote the wave operator (d’Alembertian operator) \(\mathcal{L}_{\omega}\) by
$$ \mathcal{L}_{\omega}u=\omega^{2}m_{b}u_{tt}-Qu_{xx}, $$
with the Dirichlet boundary condition
$$ u(0,t)=u(\pi,t)=0, $$
and the beam operator \(\mathcal{J}_{\omega}\) by
$$ \mathcal{J}_{\omega}w=\omega^{2}m_{c}w_{tt}+EIw_{xxxx}, $$
with the hinged boundary condition
$$ w(0,t)=w(\pi,t)=0,\qquad w_{xx}(0,t)=w_{xx}(\pi,t)=0. $$
Let \(\lambda_{l,j}\) denote the eigenvalues of the wave operator \(\mathcal{L}_{\omega}\) and \(\mu_{l,j}\) denote the eigenvalues of the beam operator \(\mathcal{J}_{\omega}\). Then it follows from a direct calculation that
$$ \lambda_{l,j}=-\omega^{2}m_{b}l^{2}+Qj^{2}, \qquad \mu_{l,j}=-\omega^{2}m_{c}l^{2}+EIj^{4}, $$
where \(l\in\mathbb{Z}\) and \(j=1,2,\ldots\) .
To find the solution of system (2.1)–(2.2) under boundary condition (2.3), we will introduce the Lyapunov–Schmidt reduction according to the decomposition
$$ X_{\sigma,s}\times Y_{\sigma,s}=(\mathbb{V}_{1}\times \mathbb{V}_{2})\oplus\bigl((\mathbb{W}_{1}\cap X_{\sigma,s})\times(\mathbb{W}_{2}\cap Y_{\sigma,s})\bigr), $$
where
$$\begin{aligned}& \mathbb{V}_{1}:=\mathbb{H}_{0}^{1}(0,\pi), \qquad \mathbb{W}_{1}:=\biggl\{ u=\sum_{l\neq 0}u_{l}(x)e^{ilt} \in X_{\sigma,s}\biggr\} , \\& \mathbb{V}_{2}:=\mathbb{H}_{0}^{2}(0,\pi), \qquad \mathbb{W}_{2}:=\biggl\{ w=\sum_{l\neq 0}w_{l}(x)e^{ilt} \in Y_{\sigma,s}\biggr\} . \end{aligned}$$
Then the solution \((u,w)\in X_{\sigma,s}\times Y_{\sigma,s}\) of system (2.1)–(2.2) can be written as
$$ u(x,t)=u_{0}(x)+\sum_{l\neq0}u_{l}(x)e^{ilt}, \qquad w(x,t)=w_{0}(x)+\sum_{l\neq 0}w_{l}(x)e^{ilt}. $$
Note that \(h_{1}(x,t)\) and \(h_{2}(x,t)\) are 2π time-periodic forcing terms. So
$$ h_{1}(x,t)=\bar{h}_{1}(x)+\sum_{l\neq0}h_{1l}(x)e^{ilt}, \qquad h_{2}(x,t)=\bar {h}_{2}(x)+\sum _{l\neq0}h_{2l}(x)e^{ilt}. $$
Set
$$ u(x,t)=\bar{u}(x)+\tilde{u}(x,t), \qquad w(x,t)=\bar{w}(x)+\tilde{w}(x,t) $$
and
$$ h_{1}(x,t)=\bar{h}_{1}(x)+\tilde{h}_{1}(x,t), \qquad h_{2}(x,t)=\bar{h}_{2}(x)+\tilde{h}_{2}(x,t). $$
Projecting system (2.1)–(2.2) by \(P_{V}\) and \(P_{W}\), the bifurcation equation and the range equation are obtained:
$$ \begin{aligned} &{-}Q\bar{u}''(x)=KP_{V}( \bar{w}-\bar{u}+\tilde{w}-\tilde{u})^{+}+\epsilon \bar{h}_{1}(x), \\ &EI\bar{w}''''(x)=-KP_{V}( \bar{w}-\bar{u}+\tilde{w}-\tilde{u})^{+}+\epsilon \bar{h}_{2}(x), \end{aligned} $$
(2.4)
and
$$ \begin{aligned} &\mathcal{L}_{\omega} \tilde{u}=KP_{W}(\tilde{w}-\tilde{u}+\bar{w}-\bar {u})^{+}+m_{c}g+ \epsilon\tilde{h}_{1}(x,t), \\ &\mathcal{J}_{\omega}\tilde{w}=-KP_{W}(\tilde{w}-\tilde{u}+ \bar{w}-\bar {u})^{+}+m_{b}g+\epsilon\tilde{h}_{2}(x,t). \end{aligned} $$
(2.5)

Remark 2.1

To obtain more relaxation conditions on Q and EI, both constants \(m_{b}g\) and \(m_{c}g\) are put in the range equation. In doing so, \(EI>K\) is sufficient to solve the bifurcation equation. On the other hand, if we get the bifurcation equation in time t (not x) by splitting system (2.1)–(2.2), \(m_{b}g\) and \(m_{c}g\) will appear in the range equation.

Remark 2.2

We shall find solutions of (2.4)–(2.5) when the ratio \(\frac{\nu}{\omega}\) is small (see (4.17)). In this limit and ũ tends to 0 and the bifurcation equation reduces to the time-independent equation
$$\begin{aligned}& -Q\bar{u}''(x)=K(\bar{w}-\bar{u})^{+}+\epsilon \bar{h}_{1}(x), \\& EI\bar{w}''''(x)=-K( \bar{w}-\bar{u})^{+}+\epsilon\bar{h}_{2}(x). \end{aligned}$$

Remark 2.3

For dissipative system (1.4)–(1.5), by Lyapunov–Schmidt reduction, the bifurcation equation is the same as (2.4)–(2.6), and the range equation is
$$\begin{aligned}& \mathcal{L}'_{\omega}\tilde{u}=K(\tilde{w}- \tilde{u})^{+}+m_{b}g+\epsilon \tilde{h}_{1}(x,t), \end{aligned}$$
(2.6)
$$\begin{aligned}& \mathcal{J}'_{\omega}\tilde{w}=-K(\tilde{w}- \tilde{u})^{+}+m_{c}g+\epsilon \tilde{h}_{2}(x,t), \end{aligned}$$
(2.7)
where \(\mathcal{L}'_{\omega}u=\omega^{2}m_{b}u_{tt}-Qu_{xx}+\delta_{1}u_{t}\) and \(\mathcal{J}'_{\omega}w=\omega^{2}m_{c}w_{tt}+EIw_{xxxx}+\delta_{2}w_{t}\).

3 The bifurcation equation

For convenience, we denote \(u(x)=\bar{u}(x)\) and \(w(x)=\bar{w}(x)\). Consider the following coupled ODEs:
$$\begin{aligned}& -Q u''(x)=K(w-u)^{+}+\epsilon h_{1}(x), \end{aligned}$$
(3.1)
$$\begin{aligned}& EI w''''(x)=-K(w-u)^{+}+ \epsilon h_{2}(x), \end{aligned}$$
(3.2)
with the boundary condition
$$\begin{aligned}& u(0)=u(\pi)=0, \\& w(0)=w(\pi), \qquad w''(0)=w''( \pi). \end{aligned}$$
Define an action functional
$$\begin{aligned} I(u,w) =&\frac{1}{2} \int_{0}^{\pi}Q \bigl\vert u' \bigr\vert ^{2}+EI \bigl\vert w'' \bigr\vert ^{2}+K\bigl[(w-u)^{+}\bigr]^{2}\,dx \\ &{}- \int _{0}^{\pi}\bar{F}_{1}(u,w)+ \bar{F}_{2}(u,w)\,dx, \end{aligned}$$
where \(I:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\),
$$ \bar{F}_{1}(u,w)=\epsilon h_{1}(x)u, \qquad \bar{F}_{2}(u,w)=\epsilon h_{2}(x)w. $$

Theorem 3.1

Assume that \(Q,EI>K\). Then system (3.1)(3.2) admits a solution \((u,w)\in\bar{\mathbb{V}}=\mathbb{H}_{0}^{1}(0,\pi)\times\mathbb {H}_{0}^{2}(0,\pi)\).

Proof

By the direct method of the calculus of variations, it suffices to prove that the action functional \(I(u,w)\) is weakly lower semi-continuous and coercive.

We first prove the weakly lower semi-continuity of \(I(u,w)\). Note that the first part of \(I(u,w)\), \(Q|u'|^{2}+EI|w''|^{2}\) is convex, we only need to show that \(\frac{1}{2}\int_{0}^{\pi}K[(w-u)^{+}]^{2}\,dx-\int_{0}^{\pi}\bar {F}_{1}(u,w)+\bar{F}_{1}(u,w)\,dx\) is weakly lower semi-continuous. Take a sequence \(\{(u_{j},w_{j})\}_{j\in\mathbb{Z}}\in\bar{\mathbb{V}}\) such that \((u_{j},w_{j})\longrightarrow(u,w)\) weakly. By the Sobolev embedding \(\bar{\mathbb{V}}\hookrightarrow\bar{\mathbb{L}}:=\mathbb{L}^{2}(0,\pi )\times\mathbb{L}^{2}(0,\pi)\), we know that \(\{(u_{j},w_{j})\}_{j\in\mathbb{Z}}\) is strongly convergent in \(\bar{\mathbb{L}}\). We derive
$$ \frac{1}{2} \int_{0}^{\pi}K\bigl[(w_{j}-u_{j})^{+} \bigr]^{2}\,dx\longrightarrow\frac {1}{2} \int_{0}^{\pi}K\bigl[(w-u)^{+}\bigr]^{2} \,dx. $$
Note that operators \(\bar{F}_{1}(u,w)\) and \(\bar{F}_{2}(u,w)\) are linear on \((u,w)\). So we know that \(\frac{1}{2}\int_{0}^{\pi}K[(w-u)^{+}]^{2}\,dx-\int_{0}^{\pi}\bar {F}_{1}(u,w)+\bar{F}_{1}(u,w)\,dx\) is weakly semi-continuous.
Next, we prove that the action functional \(I(u,w)\) is coercive, i.e.
$$ I(u,w)\longrightarrow\infty, \quad \mbox{as } \bigl\Vert (u,w) \bigr\Vert _{\bar{V}}\longrightarrow \infty. $$
(3.3)
By assumption, there exist a sufficiently small constant \(\epsilon_{0}>0\) and positive constants \(C_{0}:=C_{0}(Q,K,\epsilon_{0})\) and \(C_{1}:=C_{1}(EI,K,\epsilon_{0})\) such that
$$\begin{aligned} I(u,w) =&\frac{1}{2} \int_{0}^{\pi}Q \bigl\vert u' \bigr\vert ^{2}+EI \bigl\vert w'' \bigr\vert ^{2}+K\bigl[(w-u)^{+}\bigr]^{2}\,dx- \int _{0}^{\pi}\bar{F}_{1}(u,w)+ \bar{F}_{1}(u,w)\,dx \\ \geq&\frac{1}{2}\bigl[(Q-K) \Vert u \Vert ^{2}_{\mathbb{H}^{1}_{0}}+(EI-K) \Vert w \Vert ^{2}_{\mathbb {H}^{2}_{0}}\bigr]-\epsilon \Vert h_{1} \Vert _{\mathbb{H}^{1}_{0}} \Vert u \Vert ^{2}_{\mathbb{H}^{1}_{0}} -\epsilon \Vert h_{2} \Vert _{\mathbb{H}^{2}_{0}} \Vert w \Vert ^{2}_{\mathbb{H}^{2}_{0}} \\ =&\frac{1}{2}\bigl(Q-K-2\epsilon \Vert h_{1} \Vert _{\mathbb{H}^{1}_{0}}\bigr) \Vert u \Vert ^{2}_{\mathbb {H}^{1}_{0}}+ \frac{1}{2}\bigl(EI-K-2\epsilon \Vert h_{2} \Vert _{\mathbb{H}^{2}_{0}}\bigr) \Vert w \Vert ^{2}_{\mathbb{H}^{2}_{0}} \\ \geq&C_{0} \Vert u \Vert ^{2}_{\mathbb{H}^{1}_{0}}+C_{1} \Vert w \Vert ^{2}_{\mathbb{H}^{2}_{0}}, \end{aligned}$$
which implies that (3.3) holds.

Therefore, we conclude that the action functional \(I(u,w)\) is weakly lower semi-continuous and coercive on the Hilbert space \(\bar{\mathbb{V}}\), and it possesses the minimum point \((u^{*},w^{*})\in\bar{\mathbb{V}}\) which is a solution of system (3.1)–(3.2). □

4 The range equation

This section is devoted to solving the range equation (2.5) for system (1.1)–(1.2). Note that the method of solving the range equation (2.6)–(2.7) for system (1.4)–(1.5) is similar, so, we mainly discuss the range equation for the conservation case.

4.1 The Nash–Moser algorithm

For convenience, we denote \(u(x,t)=\tilde{u}(x,t)\), \(w(x,t)=\tilde{w}(x,t)\) and \(P_{W}(\tilde{w}-\tilde{u}+\bar{w}-\bar{u})^{+}=(\tilde{w}-\tilde{u})^{+}\). Consider the range equation
$$\begin{aligned}& \mathcal{L}_{\omega}u=K(w-u)^{+}+m_{b}g+\epsilon \tilde{h}_{1}(x,t), \end{aligned}$$
(4.1)
$$\begin{aligned}& \mathcal{J}_{\omega}w=-K(w-u)^{+}+m_{c}g+ \epsilon\tilde{h}_{2}(x,t), \end{aligned}$$
(4.2)
with the boundary condition
$$\begin{aligned}& u(0,t)=u(\pi,t)=0, \\& w(0,t)=w(\pi,t)=0,\qquad w_{xx}(0,t)=w_{xx}(\pi,t)=0. \end{aligned}$$
Define a sequence of subspaces
$$\begin{aligned}& W^{(i)}_{1}=\biggl\{ u=\sum_{1\leq|l|\leq N_{i}}u_{l,j} \varphi_{j}(x)e^{ilt}\biggr\} ,\qquad W^{(i)}_{2}= \biggl\{ w=\sum_{1\leq|l|\leq N_{i}}w_{l,j} \psi_{j}(x)e^{ilt}\biggr\} , \\& \bigl(W^{(i)}_{1}\bigr)^{\perp}=\biggl\{ u=\sum _{|l|\geq N_{i}}u_{l,j}\varphi_{j}(x)e^{ilt} \biggr\} , \qquad \bigl(W^{(i)}_{2}\bigr)^{\perp}=\biggl\{ w=\sum_{|l|\geq N_{i}}w_{l,j}\psi_{j}(x)e^{ilt} \biggr\} . \end{aligned}$$
Then
$$ W_{1}=W^{(i)}_{1}\otimes\bigl(W^{(i)}_{1} \bigr)^{\perp},\qquad W_{2}=W^{(i)}_{2}\otimes \bigl(W^{(i)}_{2}\bigr)^{\perp}, \quad i\in\mathbb{N}. $$
Let \(S_{i}\) denote the smooth projections on \(W^{(i)}_{1}\) and \(W^{(i)}_{2}\). For all \(\sigma, \sigma'\geq0\), the following smoothing properties hold:
$$ \begin{aligned} &\|S_{i}u\|_{\sigma+\sigma'}\leq N_{i}^{\sigma'}\|u\|_{\sigma}, \\ &N_{i}^{\sigma'} \bigl\Vert (I-S_{i})u \bigr\Vert _{\sigma}\leq\|u\|_{\sigma+\sigma'}, \end{aligned} $$
(4.3)
where θ satisfies assumption (4.53) below and \(N_{i}=e^{\theta^{i}}\) for all \(i\in\mathbb{N}\).
For convenience, we write in short
$$\begin{aligned}& X_{\sigma}=X_{\sigma,s}, \qquad \|u\|_{\sigma}=\|u \|_{\sigma,s}, \\& Y_{\sigma}=Y_{\sigma,s},\qquad \|w\|_{\sigma}=\|w \|_{\sigma,s}. \end{aligned}$$
Note that \((\mathcal{L}_{\omega}+C)^{-1}1=\frac{1}{C}\) (or for \(\mathcal{J}_{\omega}\)) holds in a suitable big space, for a constant \(C\neq0\). We introduce a fixed positive constant \(\eta_{i}\) (\(i\in\mathbb{N}\)) and operators
$$\begin{aligned}& \tilde{\mathcal{L}}_{\omega}=\omega^{2}m_{b} \partial_{tt}-Q\partial _{xx}+\eta_{i}, \\& \tilde{\mathcal{J}}_{\omega}=\omega^{2}m_{c} \partial_{tt}+EI\partial _{xxxx}+\eta_{i}. \end{aligned}$$

Remark 4.1

Another reason of introducing the parameters \(\eta_{i}\) is to get a more exact domain where \(m_{c}g\) and \(m_{b}g\) exist. Moreover, the optimal value of parameters \(\eta_{i}\) is obtained from the estimate in (4.57).

The range equations (4.1)–(4.2) become
$$\begin{aligned}& \tilde{\mathcal{L}}_{\omega}u=K(w-u)^{+}+\eta_{i} u+m_{c}g+\epsilon\tilde {h}_{1}(x,t), \end{aligned}$$
(4.4)
$$\begin{aligned}& \tilde{\mathcal{J}}_{\omega}w=-K(w-u)^{+}+\eta_{i} w+m_{b}g+\epsilon\tilde{h}_{2}(x,t). \end{aligned}$$
(4.5)
Summing up (4.4)–(4.5) yields
$$ \tilde{\mathcal{L}}_{\omega}u +\tilde{\mathcal{J}}_{\omega}w =\eta_{i}(u+w)+(m_{b}+m_{c})g+\epsilon( \tilde{h}_{1}+\tilde{h}_{2}). $$
(4.6)
Set
$$ A_{1}=\tilde{\mathcal{J}}_{\omega}^{-1}u, \qquad A_{2}=\tilde{\mathcal{L}}_{\omega}^{-1}w,\qquad A_{3}=\tilde{\mathcal{L}}_{\omega}^{-1}A_{1}+ \tilde{\mathcal{J}}_{\omega}^{-1}A_{2}. $$
(4.7)
By Eq. (4.6), it follows that
$$ A_{1}+A_{2}=\eta_{i}A_{3}+ \frac{1}{\eta_{i}^{2}}(m_{b}+m_{c})g+\epsilon \tilde{ \mathcal{L}}_{\omega}^{-1}\tilde{\mathcal{J}}_{\omega }^{-1}( \tilde{h}_{1}+\tilde{h}_{2}). $$
(4.8)
Combining the second equation in (4.2) with (4.8), we get
$$ \mathcal{J}_{\omega}\tilde{\mathcal{L}}_{\omega}A_{2} =-K(f_{1})^{+}+m_{b}g+\epsilon\tilde{h}_{2}(x,t), $$
(4.9)
where
$$ f_{1}=\biggl((\tilde{\mathcal{L}}_{\omega}+\tilde{ \mathcal{J}}_{\omega})A_{2}-\eta _{i}\tilde{ \mathcal{J}}_{\omega}A_{3}-\frac{1}{\eta_{i}}(m_{b}+m_{c})g -\epsilon\tilde{\mathcal{L}}_{\omega}^{-1}(\tilde{h}_{1}+ \tilde{h}_{2})\biggr)^{+}. $$
Let
$$ \Lambda=\mathcal{J}_{\omega}\tilde{\mathcal{L}}_{\omega}( \tilde{\mathcal {L}}_{\omega}+\tilde{\mathcal{J}}_{\omega})^{-1}, $$
(4.10)
and
$$ A_{4}=(\tilde{\mathcal{L}}_{\omega}+\tilde{ \mathcal{J}}_{\omega})A_{2}. $$
(4.11)
Then we rewrite (4.9) as
$$ \Lambda A_{4}+K(A_{4}-f_{2})^{+}=m_{b}g+ \epsilon\tilde{h}_{2}(x,t), $$
(4.12)
where
$$ f_{2}=\eta_{i}\tilde{\mathcal{J}}_{\omega}A_{3}+ \frac{1}{\eta_{i}}(m_{b}+m_{c})g +\epsilon\tilde{ \mathcal{L}}_{\omega}^{-1}(\tilde{h}_{1}+ \tilde{h}_{2}). $$
For convenience, we rewrite (4.12) as
$$ \Lambda a+K(a-f_{2})^{+}=m_{c}g+\epsilon \tilde{h}_{2}(x,t). $$
(4.13)
Then from
$$ w-u=\tilde{\mathcal{L}}_{\omega}w-\tilde{ \mathcal{J}}_{\omega}u=a-f_{2}, $$
(4.14)
the solution of system (4.1)–(4.2) can be written as
$$\begin{aligned}& u=\mathcal{L}_{\omega}^{-1}\bigl[K(a-f_{2})^{+}+m_{b}g+ \epsilon \tilde{h}_{1}\bigr], \end{aligned}$$
(4.15)
$$\begin{aligned}& w=\mathcal{J}_{\omega}^{-1}\bigl[-K(a-f_{2})^{+}+m_{c}g+ \epsilon \tilde{h}_{2}\bigr]. \end{aligned}$$
(4.16)

Remark 4.2

Outline of the strategy of the Nash–Moser algorithm:

Our target is to construct the approximation solution \(a_{\infty}= \sum_{i=0}^{\infty}a_{i}\) and the approximation parameters \((m_{c}g)_{\infty}= \sum_{i=0}^{\infty}(m_{c}g)_{i}\) and \((m_{b}g)_{\infty}= \sum_{i=0}^{\infty}(m_{b}g)_{i}\) of Eq. (4.13) by the Nash–Moser algorithm. Then, by (4.15)–(4.16), the solution in the range equations (4.1)–(4.2) is
$$\begin{aligned}& u_{\infty}=\mathcal{L}_{\omega}^{-1}\bigl[K \bigl(a_{\infty}-f_{2}^{(\infty )}\bigr)^{+}+(m_{b}g)_{\infty}+ \epsilon \tilde{h}_{1}\bigr], \\& w_{\infty}=\mathcal{J}_{\omega}^{-1}\bigl[-K \bigl(a_{\infty}-f_{2}^{(\infty )}\bigr)^{+}+(m_{c}g)_{\infty}+ \epsilon \tilde{h}_{2}\bigr], \end{aligned}$$
where
$$ f_{2}^{(\infty)}=\frac{K}{2}+\epsilon\tilde{L}_{\omega}( \tilde{h}_{1}+\tilde{h}_{2}). $$
In fact, if we choose suitable initial approximation \((u_{0},w_{0})\), then, by (4.7) and (4.11), the initial approximation \(a_{0}\) of (4.13) can be obtained. Furthermore, by (4.15)–(4.16), the corresponding first step approximation solution of (4.1)–(4.2) is
$$\begin{aligned}& u_{1}=\mathcal{L}_{\omega }^{-1}\bigl[K \bigl(a_{0}+a_{1}-f_{2}^{(0)} \bigr)^{+}+(m_{b}g)_{0}+(m_{b}g)_{1}+ \epsilon \tilde{h}_{1}\bigr]\in W_{1}^{(1)}, \\& w_{1}=\mathcal{J}_{\omega }^{-1}\bigl[-K \bigl(a_{0}+a_{1}-f_{2}^{(0)} \bigr)^{+}+(m_{c}g)_{0}+(m_{c}g)_{1}+ \epsilon \tilde{h}_{2}\bigr]\in W_{2}^{(1)}. \end{aligned}$$
To obtain the ith step approximation solution \((u_{i},w_{i})\), we first need to get the ith step approximation solution \(\sum_{k=0}^{i}a_{k}\) and the ith step approximation parameters \(\sum_{k=0}^{i}(m_{b}g)_{k}\) and \(\sum_{k=0}^{i}(m_{c}g)_{k}\). Lemmas 4.34.4 show how to get the ith approximation step \(a_{i}\). In the process of proving convergence of Nash–Moser algorithm, the optimal ith approximation step value of parameters \((m_{b}g)_{i}\) and \((m_{c}g)_{i}\) can be determined in Lemmas 4.54.6. Then, by (4.15)–(4.16), the corresponding ith step approximation solution of (4.1)–(4.2) is
$$\begin{aligned}& u_{i}=\mathcal{L}_{\omega}^{-1}\Biggl[K\Biggl(\sum _{k=0}^{i}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}+\sum_{k=0}^{i}(m_{c}g)_{k}+ \epsilon S_{i-1}\tilde{h}_{1}\Biggr]\in W_{1}^{(i)}, \\& w_{i}=\mathcal{J}_{\omega}^{-1}\Biggl[-K\Biggl(\sum _{k=0}^{i}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}+\sum_{k=0}^{i}(m_{b}g)_{k}+ \epsilon S_{i-1}\tilde{h}_{2}\Biggr]\in W_{2}^{(i)}, \end{aligned}$$
where
$$ f_{2}^{(i-1)}=\eta_{i-1}\tilde{\mathcal{J}}_{\omega}A_{3}^{(i-1)}+ \frac {(m_{b}g)_{i-1}+(m_{c}g)_{i-1}}{\eta_{i-1}} +\epsilon\tilde{\mathcal{L}}_{\omega}^{-1}S_{i-1}( \tilde{h}_{1}+\tilde{h}_{2}). $$
Since there are errors (denoted by \(E_{i}\)) in constructing each approximation step, the convergence of Nash–Moser algorithm remains to be treated. We will prove it in Lemma 4.6.
Fix the following “nonresonant” set:
$$\begin{aligned} \mathcal{D}_{\gamma} :=&\biggl\{ (\nu,\omega)\in\bigl(\nu', \nu''\bigr)\times(\gamma ,+\infty): \bigl\vert \omega\sqrt{m_{b}+m_{c}}l-\sqrt{EIj^{4}+Qj^{2}} \bigr\vert \geq\frac{\gamma}{ \vert l \vert ^{\kappa +1}}, \\ & \biggl\vert 1+K\biggl(\frac{1}{2\omega\sqrt{m_{b}}l(\omega\sqrt{m_{b}}l-\sqrt{Q}j)}+\frac {1}{2\omega\sqrt{m_{c}}l(\omega\sqrt{m_{c}}l-\sqrt{EI}j^{2})}\biggr) \biggr\vert \geq\frac{\gamma }{ \vert l \vert ^{\kappa+1}}, \\ & \vert \omega\sqrt{m_{b}}l-\sqrt{Q}j \vert \geq \frac{\gamma}{ \vert l \vert ^{\kappa+1}}, \bigl\vert \omega\sqrt{m_{c}}l- \sqrt{EI}j^{2} \bigr\vert \geq\frac{\gamma}{ \vert l \vert ^{\kappa+1}}, \frac{\nu}{\omega} \leq C\gamma^{2}, l,j\geq1\biggr\} , \end{aligned}$$
(4.17)
where \(\kappa\in(0,+\infty)\), \((\nu',\nu'')\) denotes a neighborhood of \(\nu_{0}\), for some \(\nu_{0}\in[0,\nu''']\), and
$$ \nu=\max\biggl\{ \frac{1}{\sqrt{m_{b}}},\frac{1}{\sqrt{m_{c}}}\biggr\} . $$

Remark 4.3

In what follows, for each iteration step \(i\in\mathbb{N}\), the nonresonant conditions
$$\begin{aligned}& \bigl\vert \sqrt{\omega^{2}(m_{b}+m_{c})l^{2}-2 \eta_{i}}-\sqrt{EIj^{4}+Qj^{2}} \bigr\vert \geq \frac {\gamma}{ \vert l \vert ^{\kappa+1}}, \\& \bigl\vert \sqrt{\omega^{2}m_{b}l^{2}- \eta_{i}}-\sqrt{Q}j \bigr\vert \geq\frac{\gamma}{ \vert l \vert ^{\kappa+1}},\qquad \bigl\vert \sqrt{\omega^{2}m_{c}l^{2}- \eta_{i}}-\sqrt{Q}j^{2} \bigr\vert \geq\frac{\gamma }{ \vert l \vert ^{\kappa+1}}, \end{aligned}$$
and
$$\begin{aligned}& \biggl\vert 1+K\biggl[\frac{1}{-\omega^{2}m_{b}l+Qj^{2}+\eta_{i}}+\frac{1}{-\omega ^{2}m_{c}l^{2}+EIj^{4}}+\frac{\eta_{i}}{(-\omega^{2}m_{b}l+Qj^{2}+\eta_{i})(-\omega ^{2}m_{c}l^{2}+EIj^{4})} \biggr] \biggr\vert \\& \quad \geq\frac{\gamma}{|l|^{\kappa+1}} \end{aligned}$$
are also needed. But we find that if \(\omega\in[\gamma,+\infty]\backslash\mathcal{D}_{\gamma}\), then, for \(i\in\mathbb{N}\), \(|l|< N_{i}\) and \(\eta_{i}< e^{-N_{i}}\), one derives
$$\begin{aligned}& \bigl\vert \sqrt{\omega^{2}(m_{b}+m_{c})l^{2}-2 \eta_{i}}-\sqrt{EIj^{4}+Qj^{2}} \bigr\vert \\& \quad = \bigl\vert \omega\sqrt{m_{b}+m_{c}}l- \sqrt{EIj^{4}+Qj^{2}}+\sqrt{\omega ^{2}(m_{b}+m_{c})l^{2}-2 \eta_{i}}-\omega\sqrt{m_{b}+m_{c}}l \bigr\vert \\& \quad \geq \bigl\vert \omega\sqrt{m_{b}+m_{c}}l- \sqrt{EIj^{4}+Qj^{2}} \bigr\vert -\frac{2\eta_{i}}{\sqrt {\omega^{2}(m_{b}+m_{c})l^{2}-2\eta_{i}}+\omega\sqrt{m_{b}+m_{c}}l} \\& \quad \geq \frac{\gamma}{ \vert l \vert ^{\kappa+1}}-\frac{C(\gamma)\eta_{i}}{2 \vert l \vert } \\& \quad \geq \frac{\gamma}{2 \vert l \vert ^{\kappa+1}}. \end{aligned}$$
In a similar manner, we obtain
$$ \bigl\vert \sqrt{\omega^{2}m_{b}l^{2}- \eta_{i}}-\sqrt{Q}j \bigr\vert \geq\frac{\gamma}{ \vert l \vert ^{\kappa+1}},\qquad \bigl\vert \sqrt{\omega^{2}m_{c}l^{2}- \eta_{i}}-\sqrt{Q}j^{2} \bigr\vert \geq\frac{\gamma}{ \vert l \vert ^{\kappa+1}}. $$
Finally, for \(i\in\mathbb{N}\), \(|l|< N_{i}\) and \(\eta_{i}< e^{-N_{i}}\), we have
$$\begin{aligned}& \biggl\vert 1+K\biggl[\frac{1}{-\omega^{2}m_{b}l+Qj^{2}+\eta_{i}}+\frac{1}{-\omega ^{2}m_{c}l^{2}+EIj^{4}}+\frac{\eta_{i}}{(-\omega^{2}m_{b}l+Qj^{2}+\eta_{i})(-\omega ^{2}m_{c}l^{2}+EIj^{4})} \biggr] \biggr\vert \\& \quad \geq \biggl\vert 1+K\biggl(\frac{1}{-\omega^{2}m_{b}l+Qj^{2}}+\frac{1}{-\omega ^{2}m_{c}l^{2}+EIj^{4}}\biggr) \biggr\vert \\& \qquad {}-K \biggl\vert \frac{1}{-\omega^{2}m_{b}l+Qj^{2}+\eta_{i}}-\frac{1}{-\omega ^{2}m_{b}l+Qj^{2}} \biggr\vert \\& \qquad {} - \biggl\vert \frac{K\eta_{i}}{(-\omega^{2}m_{b}l+Qj^{2}+\eta_{i})(-\omega ^{2}m_{c}l^{2}+EIj^{4})} \biggr\vert \\& \quad \geq \biggl\vert 1+K\biggl(\frac{1}{2\omega\sqrt{m_{b}}l(\omega\sqrt{m_{b}}l-\sqrt {Q}j)}+\frac{1}{2\omega\sqrt{m_{c}}l(\omega\sqrt{m_{c}}l-\sqrt {EI}j^{2})}\biggr) \biggr\vert \\& \qquad {}-2K\eta_{i}\gamma^{2} \vert l \vert ^{2\kappa+2} \\& \quad \geq\frac{\gamma}{2 \vert l \vert ^{\kappa+1}}. \end{aligned}$$
Therefore, the nonresonant condition is sufficient to keep the operators \(\mathcal{L}_{\omega}\), \(\tilde{\mathcal{L}}_{\omega}\), \(\mathcal{J}_{\omega}\), \(\tilde{\mathcal{J}}_{\omega}\), Λ and \(1+K\Lambda^{-1}\) invertible in a bigger space.

Lemma 4.1

Let \(\omega\in\mathcal{X}(\nu)\) and \(\bar{\sigma}>\tilde{\sigma}\geq0\). Then the “diagonal” operators \(\mathcal{L}_{\omega}\), \(\mathcal{J}_{\omega}\), \(\tilde{\mathcal{L}}_{\omega}\) and \(\tilde{\mathcal{J}}_{\omega}\) satisfy the following:

(1) For any \((u,w)\in X_{\sigma}\times Y_{\sigma}\),
$$\begin{aligned}& \mathcal{L}_{\omega}u = \mathcal{L}_{\omega}\biggl( \sum _{(l,j)\in \mathbb{Z}^{2}}u_{l,j}\varphi_{j}(x)e^{ilt} \biggr)=\lambda_{l,j} u, \\& \mathcal{J}_{\omega}w = \mathcal{J}_{\omega}\biggl( \sum _{(l,j)\in \mathbb{Z}^{2}}w_{l,j}\psi_{j}(x)e^{ilt} \biggr)=\mu_{l,j} w, \\& \tilde{\mathcal{L}}_{\omega}u = \tilde{\mathcal{L}}_{\omega } \biggl( \sum_{(l,j)\in\mathbb{Z}^{2}}u_{l,j}\varphi _{j}(x)e^{ilt}\biggr)=\tilde{\lambda}_{l,j} u, \\& \tilde{\mathcal{J}}_{\omega}w = \tilde{\mathcal{J}}_{\omega } \biggl( \sum_{(l,j)\in\mathbb{Z}^{2}}w_{l,j}\psi _{j}(x)e^{ilt}\biggr)=\tilde{\mu}_{l,j}w, \end{aligned}$$
where
$$ \begin{aligned} &\lambda_{l,j}=- \omega^{2}m_{b}l^{2}+Qj^{2},\qquad \mu_{l,j}=-\omega ^{2}m_{c}l^{2}+EIj^{4}, \\ &\tilde{\lambda}_{l,j}=-\omega^{2}m_{b}l^{2}+Qj^{2}+ \eta_{i},\qquad \tilde{\mu }_{l,j}=-\omega^{2}m_{c}l^{2}+EIj^{4}+ \eta_{i}. \end{aligned} $$
(4.18)
Operators \(\mathcal{L}_{\omega}\), \(\tilde{\mathcal{L}}_{\omega}\), \(\mathcal{J}_{\omega}\) and \(\tilde{\mathcal{J}}_{\omega}\) are invertible and map the spaces \(X_{\bar{\sigma},s}\) and \(Y_{\bar{\sigma},s}\) onto space \(X_{\tilde{\sigma},s}\) and \(Y_{\bar{\sigma},s}\), respectively, and
$$\begin{aligned}& \mathcal{L}_{\omega}^{-1}u=\mathcal{L}_{\omega}^{-1} \biggl(\sum_{(l,j)\in \mathbb{Z}^{2}}u_{l,j} \varphi_{j}(x)e^{ilt}\biggr)=\lambda_{l,j}^{-1} u, \\& \mathcal{J}_{\omega}^{-1}w=\mathcal{J}_{\omega}^{-1} \biggl(\sum_{(l,j)\in \mathbb{Z}^{2}}w_{l,j} \psi_{j}(x)e^{ilt}\biggr)=\mu_{l,j}^{-1} w, \\& \tilde{\mathcal{L}}_{\omega}^{-1}u=\tilde{\mathcal{L}}_{\omega }^{-1} \biggl(\sum_{(l,j)\in\mathbb{Z}^{2}}u_{l,j} \varphi_{j}(x)e^{ilt}\biggr)=\tilde {\lambda}_{l,j}^{-1} u, \\& \tilde{\mathcal{J}}_{\omega}^{-1}w=\tilde{\mathcal{J}}_{\omega }^{-1} \biggl(\sum_{(l,j)\in\mathbb{Z}^{2}}w_{l,j} \psi_{j}(x)e^{ilt}\biggr)=\tilde{\mu }_{l,j}^{-1}w, \\& \Lambda^{-1}u=\biggl(\frac{1}{\mathcal{J}_{\omega}}+\frac{1}{\tilde{\mathcal {L}}_{\omega}}+ \frac{\eta_{i}}{\mathcal{J}_{\omega}\tilde{\mathcal {L}}}_{\omega}\biggr) \biggl(\sum _{(l,j)\in\mathbb{Z}^{2}}u_{l,j}\varphi_{j}(x)e^{ilt} \biggr) =\bigl(\mu_{l,j}^{-1}+\tilde{\lambda}_{l,j}^{-1}+ \eta_{i}\mu_{l,j}^{-1}\tilde {\lambda}_{l,j}^{-1} \bigr)u, \end{aligned}$$
where \(\lambda_{l,j}\), \(\mu_{l,j}\), \(\tilde{\lambda}_{l,j}\) and \(\tilde{\mu}_{l,j}\) are defined in (4.18).
(2) Set
$$\begin{aligned}& \Sigma_{1}(\varpi):=\sup_{(l,j)\in\mathbb{Z}^{2}/\{0\}}\bigl( \bigl\vert \omega ^{2}m_{b}l^{2}-Qj^{2} \bigr\vert ^{-1}e^{-\varpi \vert l \vert }\bigr), \\& \Sigma_{2}(\varpi):=\sup_{(l,j)\in\mathbb{Z}^{2}/\{0\}}\bigl( \bigl\vert \omega ^{2}m_{c}l^{2}-EIj^{4} \bigr\vert ^{-1}e^{-\varpi \vert l \vert }\bigr), \\& \Sigma_{3}(\varpi):=\sup_{(l,j)\in\mathbb{Z}^{2}/\{0\}}\bigl( \bigl\vert \omega ^{2}m_{b}l^{2}-Qj^{2}+ \eta_{i} \bigr\vert ^{-1}e^{-\varpi \vert l \vert }\bigr), \\& \Sigma_{4}(\varpi):=\sup_{(l,j)\in\mathbb{Z}^{2}/\{0\}}\bigl( \bigl\vert \omega ^{2}m_{c}l^{2}-EIj^{4}+ \eta_{i} \bigr\vert ^{-1}e^{-\varpi \vert l \vert }\bigr), \\& \Sigma_{5}(\varpi):=\sup_{(l,j)\in\mathbb{Z}^{2}/\{0\}}\bigl( \eta_{i} \bigl\vert \omega ^{2}m_{c}l^{2}-EIj^{4} \bigr\vert ^{-1} \bigl\vert \omega^{2}m_{b}l^{2}-Qj^{2}+ \eta_{i} \bigr\vert ^{-1}e^{-2\varpi \vert l \vert }\bigr). \end{aligned}$$
Then we have
$$\begin{aligned}& \bigl\Vert \mathcal{L}_{\omega}^{-1}u \bigr\Vert _{\tilde{\sigma}}\leq\Sigma_{1}(\bar {\sigma}-\tilde{\sigma}) \Vert u \Vert _{\bar{\sigma}},\qquad \bigl\Vert \mathcal{J}_{\omega }^{-1}w \bigr\Vert _{\tilde{\sigma}}\leq\Sigma_{2}(\bar{\sigma}-\tilde{ \sigma}) \Vert w \Vert _{\bar{\sigma}}, \\& \bigl\Vert \tilde{\mathcal{L}}_{\omega}^{-1}u \bigr\Vert _{\tilde{\sigma}}\leq\Sigma _{3}(\bar{\sigma}-\tilde{\sigma}) \Vert u \Vert _{\bar{\sigma}}, \qquad \bigl\Vert \tilde{\mathcal {J}}_{\omega}^{-1}w \bigr\Vert _{\tilde{\sigma}}\leq\Sigma_{4}(\bar{\sigma}-\tilde { \sigma}) \Vert w \Vert _{\bar{\sigma}}, \\& \bigl\Vert \Lambda^{-1}u \bigr\Vert _{\tilde{\sigma}}\leq\bigl( \Sigma_{2}(\bar{\sigma}-\tilde {\sigma})+\Sigma_{3}(\bar{ \sigma}-\tilde{\sigma})+\Sigma_{5}(\bar{\sigma }-\tilde{\sigma})\bigr) \Vert u \Vert _{\bar{\sigma}}, \end{aligned}$$
and
$$ \Sigma_{1}(\varpi),\Sigma_{2}(\varpi),\Sigma_{3}( \varpi),\Sigma_{4}(\varpi)\leq \frac{C\gamma}{\varpi^{\kappa}}\biggl( \frac{\kappa}{e}\biggr)^{\kappa},\qquad \Sigma_{5}(\varpi)\leq \frac{C^{2}\gamma^{2}}{\varpi^{2\kappa+2}}\biggl(\frac{2\kappa +2}{e}\biggr)^{2\kappa+2}, $$
where C is a positive constant.

Proof

The property of the operators \(\mathcal{L}_{\omega}\), \(\mathcal{J}_{\omega}\), \(\tilde{\mathcal{L}}_{\omega}\) and \(\tilde{\mathcal{J}}_{\omega}\) is obvious. Now we verify the property of the operators \(\mathcal{L}_{\omega}^{-1}\), \(\mathcal{J}_{\omega}^{-1}\), \(\tilde{\mathcal{L}}_{\omega}^{-1}\) and \(\tilde{\mathcal{J}}_{\omega}^{-1}\). We have
$$\begin{aligned} \bigl\Vert \mathcal{L}_{\omega}^{-1}u \bigr\Vert _{\tilde{\sigma}} =&\sum_{l\in\mathbb{Z}}\bigl( \bigl\vert \omega^{2}m_{b}l^{2}-Qj^{2} \bigr\vert ^{-1}e^{-(\bar{\sigma }-\tilde{\sigma}) \vert l \vert }\bigr) \vert u_{l,j} \vert \bigl\vert \varphi_{j}(x) \bigr\vert e^{ \vert l \vert \bar{\sigma}} \\ \leq&\Sigma_{1}(\bar{\sigma}-\tilde{\sigma})\|u\|_{\bar{\sigma}}. \end{aligned}$$
Since \(\omega\in\mathcal{D}_{\gamma}\), we have
$$\begin{aligned} \bigl\vert \omega^{2}m_{b}l^{2}-Qj^{2} \bigr\vert ^{-1} =& \vert \omega\sqrt{m_{b}}l-\sqrt{Q}j \vert ^{-1} \vert \omega \sqrt{m_{b}}l+\sqrt{Q}j \vert ^{-1} \\ \leq&\frac{|l|^{\kappa+1}}{\gamma}\frac{\nu|l|}{\omega}=|l|^{\kappa }\gamma. \end{aligned}$$
Then, from \(\sup_{x>0}(x^{y}e^{-x})=(\frac{y}{e})^{y}\), \(\forall y\geq0\), we obtain
$$ \Sigma_{1}(\varpi)\leq\frac{\gamma}{\varpi^{\kappa}}\biggl(\frac{\kappa }{e} \biggr)^{\kappa}. $$
In a same manner, we can get the property of the operators \(\mathcal{J}_{\omega}^{-1}\), \(\tilde{\mathcal{L}}_{\omega}^{-1}\), \(\tilde{\mathcal{J}}_{\omega}^{-1}\) and \(\Lambda^{-1}\). □
To solve (4.13), introduce the function spaces
$$ W^{(i)}_{3}:=\biggl\{ a=\sum_{1\leq|l|\leq N_{i}}a_{l,j} \phi_{j}(x)e^{ilt}\biggr\} , $$
where \(\{\phi_{j}(x)=\sin(jx)\}\) is the complete orthonormal system of the eigenfunctions of the operator Λ.

Lemma 4.2

Let \(\omega\in\mathcal{X}(\nu)\). Then, for a constant \(K>0\), equation
$$ \Lambda a_{i}+K(a_{i})^{+}+E_{i-1}=0 $$
(4.19)
has a unique solution \(a_{i}\in W^{(i)}_{3}\). Especially, equation
$$ \Lambda a_{i}+K(a_{i})^{+}=0 $$
has a unique solution \(a_{i}=0\). Furthermore,
$$ \|a_{i}\|_{\sigma}\leq C\Sigma_{5}( \bar{\sigma}-\sigma)\|E_{i-1}\|_{\bar{\sigma}}, $$
(4.20)
where \(i\in\mathbb{N}\), \(\bar{\sigma}>\sigma\), \(E_{i-1}\) is periodic in time t and does not depend on \(a_{i}\).

Proof

For convenience, we denote \(a=a_{i}\) and \(E=E_{i-1}\), \(i\in\mathbb{N}\). From the definition of operator Λ in (4.10), \(a= \sum_{1\leq|l|\leq N_{i}}a_{l,j}\phi_{j}(x)e^{ilt}\) and \(E= \sum_{1\leq|l|\leq N_{i}}E_{l,j}\phi_{j}(x)e^{ilt}\), Eq. (4.19) can be written as
$$\begin{aligned} &\sum_{1\leq|l|\leq N_{i}}\frac{\tilde{\lambda}_{l,j}\mu_{l,j}}{\tilde{\lambda}_{l,j}+\tilde {\mu}_{l,j}}a_{l,j} \phi_{j}(x)e^{ilt}+K\biggl(\sum_{1\leq|l|\leq N_{i}}a_{l,j} \phi_{j}(x)e^{ilt}\biggr)^{+} \\ &\quad {}+\sum_{1\leq|l|\leq N_{i}}E_{l,j} \phi_{j}(x)e^{ilt}=0. \end{aligned}$$
(4.21)
Denote the domain \(\Omega^{+}:=\{(x,t)\mid a= \sum_{1\leq|l|\leq N_{i}}a_{l,j}\phi_{j}(x)e^{ilt}\geq0\}\). Then, comparing the coefficients of the above equation in \(\Omega^{+}\), we get
$$ \biggl(\frac{\tilde{\lambda}_{l,j}\mu_{l,j}}{\tilde{\lambda}_{l,j}+\tilde{\mu }_{l,j}}+K\biggr)a_{l,j}=-E_{l,j}, $$
which implies that
$$ a_{l,j}=\frac{-\tilde{\lambda}_{l,j}-\tilde{\mu}_{l,j}}{\tilde{\lambda }_{l,j}\mu_{l,j}+K\tilde{\lambda}_{l,j}+K\tilde{\mu}_{l,j}}E_{l,j}. $$
(4.22)
Denote the domain \(\Omega^{-}:=\{(x,t)\mid a= \sum_{1\leq|l|\leq N_{i}}a_{l,j}\phi_{j}(x)e^{ilt}\leq0\}\). Then, from (4.21), it follows
$$ \sum_{1\leq|l|\leq N_{i}}\frac{\tilde{\lambda}_{l,j}\mu_{l,j}}{\tilde{\lambda}_{l,j}+\tilde {\mu}_{l,j}}a_{l,j} \phi_{j}(x)e^{ilt}+\sum_{1\leq|l|\leq N_{i}}E_{l,j} \phi_{j}(x)e^{ilt}=0. $$
Comparing the coefficients of the above equation, one derives
$$ a_{l,j}=\frac{-\tilde{\lambda}_{l,j}-\tilde{\mu}_{l,j}}{\tilde{\lambda }_{l,j}\mu_{l,j}}E_{l,j}. $$
(4.23)
Note that \(\phi_{j}(x)=\sin(jx)\), \(j\in\mathbb{Z}\). So, we have
$$\begin{aligned} a\biggl(t,\frac{\pi}{j}\biggr) =& \sum_{1\leq|l|\leq N_{i}} \frac{-\tilde{\lambda}_{l,j}-\tilde{\mu}_{l,j}}{\tilde{\lambda }_{l,j}\mu_{l,j}+K\tilde{\lambda}_{l,j}+K\tilde{\mu}_{l,j}}E_{l,j}\phi _{j}\biggl(\frac{\pi}{j} \biggr)e^{ilt} \\ =& \sum_{1\leq|l|\leq N_{i}}\frac{-\tilde{\lambda}_{l,j}-\tilde{\mu}_{l,j}}{\tilde{\lambda }_{l,j}\mu_{l,j}}E_{l,j} \phi_{j}\biggl(\frac{\pi}{j}\biggr)e^{ilt}=0. \end{aligned}$$
(4.24)
Combining (4.22)–(4.23) with (4.24), there exists a unique solution \(a_{i}\in W_{3}^{(i)}\). Here, the solution \(a_{i}\) is in \(\mathbb{C}((0,\pi),\mathbb{R})\). Especially, when \(E_{i-1}=0\), there exists a unique zero solution.
Due to nonresonant condition and Remark 4.3, we have
$$\begin{aligned} \biggl\vert \frac{1}{\mu_{l,j}}+\frac{1}{\tilde{\lambda}_{l,j}}+\frac{\eta _{i}}{\tilde{\lambda}_{l,j}\mu_{l,j}} \biggr\vert \leq& \biggl\vert \frac{1}{\mu_{l,j}} \biggr\vert + \biggl\vert \frac {1}{\tilde{\lambda}_{l,j}} \biggr\vert +\eta_{i} \biggl\vert \frac{1}{\tilde{\lambda}_{l,j}\mu _{l,j}} \biggr\vert \\ \leq& \vert l \vert ^{\kappa+1}\gamma\bigl(2+\eta_{i} \vert l \vert ^{\kappa+1}\gamma\bigr). \end{aligned}$$
(4.25)
Furthermore, by (4.22)–(4.25), Remark 4.3 and \(\sup_{x>0}(x^{y}e^{-x})=(\frac{y}{e})^{y}\), one derives
$$\begin{aligned} \|a\|_{\sigma} \leq&\max\biggl\{ \sum_{1\leq \vert l \vert , \vert j \vert \leq N_{i}} \biggl\vert \frac{\tilde{\lambda}_{l,j}+\tilde{\mu}_{l,j}}{\tilde{\lambda }_{l,j}\mu_{l,j}+K\tilde{\lambda}_{l,j}+K\tilde{\mu }_{l,j}} \biggr\vert \vert E_{l,j} \vert \bigl\vert \phi_{j}(x) \bigr\vert e^{ \vert l \vert \sigma}, \\ &\sum_{1\leq \vert l \vert , \vert j \vert \leq N_{i}} \biggl\vert \frac{\tilde{\lambda}_{l,j}+\tilde{\mu }_{l,j}}{\tilde{\lambda}_{l,j}\mu_{l,j}} \biggr\vert \vert E_{l,j} \vert \bigl\vert \phi _{j}(x) \bigr\vert e^{ \vert l \vert \sigma}\biggr\} \\ \leq&\max\biggl\{ \sum_{1\leq \vert l \vert , \vert j \vert \leq N_{i}} \biggl\vert \frac{1}{\mu_{l,j}}+\frac{1}{\tilde{\lambda}_{l,j}}+\frac{\eta _{i}}{\tilde{\lambda}_{l,j}\mu_{l,j}} \biggr\vert \biggl\vert 1+K\biggl(\frac{1}{\mu_{l,j}}+\frac{1}{\tilde{\lambda}_{l,j}}+\frac{\eta _{i}}{\tilde{\lambda}_{l,j}\mu_{l,j}}\biggr) \biggr\vert \\ &{}\times \vert E_{l,j} \vert \bigl\vert \phi_{j}(x) \bigr\vert e^{ \vert l \vert \sigma }, \\ &\sum_{1\leq \vert l \vert , \vert j \vert \leq N_{i}} \biggl\vert \frac{1}{\mu_{l,j}}+ \frac{1}{\tilde {\lambda}_{l,j}}+\frac{\eta_{i}}{\tilde{\lambda}_{l,j}\mu _{l,j}} \biggr\vert \vert E_{l,j} \vert \bigl\vert \phi_{j}(x) \bigr\vert e^{ \vert l \vert \sigma}\biggr\} \\ \leq&\max\biggl\{ \sum_{1\leq \vert l \vert , \vert j \vert \leq N_{i}} \vert l \vert ^{2\kappa+2}\bigl(2+\eta_{i} \vert l \vert ^{\kappa+1}\gamma \bigr) \vert E_{l,j} \vert \bigl\vert \phi _{j}(x) \bigr\vert e^{ \vert l \vert \sigma}, \\ &\gamma\sum_{1\leq \vert l \vert , \vert j \vert \leq N_{i}} \vert l \vert ^{\kappa+1}\bigl(2+\eta_{i} \vert l \vert ^{\kappa+1}\gamma \bigr) \vert E_{l,j} \vert \bigl\vert \phi _{j}(x) \bigr\vert e^{ \vert l \vert \sigma}\biggr\} \\ \leq&\max\bigl\{ C\Sigma_{5}(\bar{\sigma}-\sigma)\|E \|_{\bar{\sigma}},\gamma \Sigma_{1}(\bar{\sigma}-\sigma)\|E \|_{\bar{\sigma}}\bigr\} \\ \leq&C\Sigma_{5}(\bar{\sigma}-\sigma)\|E\|_{\bar{\sigma}}, \end{aligned}$$
where C is a constant, \(\Sigma_{1}(\bar{\sigma}-\sigma)\) and \(\Sigma_{5}(\bar{\sigma}-\sigma)\) are defined in Lemma 4.1. This completes the proof. □
Define
$$ \mathcal{M}(a,m_{b}g)=\Lambda a+K(a-f_{2})^{+}-m_{b}g- \epsilon\tilde{h}_{2}(x,t)=0. $$
(4.26)

Lemma 4.3

Let \(\omega\in\mathcal{X}(\nu)\). Then Eq. (4.26) possesses the first step approximation \(a_{1}\in W_{3}^{(1)}\) satisfying (4.20) for \(i=1\). For the range equation (4.1)(4.2), we obtain the corresponding approximation solution
$$\begin{aligned}& u_{1}=\mathcal{L}_{\omega }^{-1} \bigl[K_{0}\bigl(a_{0}+a_{1}-f_{2}^{(0)} \bigr)^{+}+(m_{c}g)_{0}+(m_{c}g)_{1}+ \epsilon S_{1}\tilde{h}_{1}\bigr]\in W_{1}^{(1)}, \end{aligned}$$
(4.27)
$$\begin{aligned}& w_{1}=\mathcal{J}_{\omega }^{-1} \bigl[-K_{0}\bigl(a_{0}+a_{1}-f_{2}^{(0)} \bigr)^{+}+(m_{b}g)_{0}+(m_{b}g)_{1}+ \epsilon S_{1}\tilde{h}_{2}\bigr]\in W_{2}^{(1)}, \end{aligned}$$
(4.28)
where
$$\begin{aligned}& f_{2}^{(0)}=\eta_{0}\tilde{\mathcal{J}}_{\omega}A_{3}^{(0)}+ \frac {(m_{b}g)_{0}+(m_{c}g)_{0}}{\eta_{0}} +\epsilon\tilde{\mathcal{L}}_{\omega}^{-1}( \tilde{h}_{1}+\tilde{h}_{2}), \\& A_{1}^{(0)}=\tilde{\mathcal{J}}_{\omega}^{-1}u_{0}, \qquad A_{2}^{(0)}=\tilde {\mathcal{L}}_{\omega}^{-1}w_{0}, \qquad A_{3}^{(0)}=\tilde{\mathcal{L}}_{\omega}^{-1}A_{1}^{(0)}+ \tilde{\mathcal {J}}_{\omega}^{-1}A_{2}^{(0)}, \\& E_{0}=\Lambda a_{0}+K\bigl(a_{0}-f_{2}^{(0)} \bigr)^{+}-m_{b}g-\epsilon S_{0}\tilde{h}_{2}(x,t). \end{aligned}$$

Proof

Define
$$\begin{aligned} R_{0} =&K\bigl(a_{0}+a_{1}-f_{2}^{(1)} \bigr)^{+}-K\bigl(a_{0}-f_{2}^{(0)} \bigr)^{+}-K(a_{1})^{+}-(m_{b}g)_{1} \\ &{}+\epsilon(S_{0}- S_{1})\tilde{h}_{2}(x,t). \end{aligned}$$
Then we have
$$\begin{aligned}& \mathcal{M}\bigl(a_{0}+a_{1},(m_{b}g)_{0}+(m_{b}g)_{1} \bigr) \\& \quad = \Lambda a_{0}+\Lambda a_{1}+K\bigl(a_{0}+a_{1}-f_{2}^{(1)} \bigr)^{+}-(m_{b}g)_{0}-(m_{b}g)_{1}- \epsilon S_{1}\tilde{h}_{2}(x,t) \\& \quad = \Lambda a_{0}+K\bigl(a_{0}-f_{2}^{(0)} \bigr)^{+}-(m_{b}g)_{0}-\epsilon S_{0} \tilde{h}_{2}(x,t)+\Lambda a_{1}+K(a_{1})^{+} \\& \qquad {} +K\bigl(a_{0}+a_{1}-f_{2}^{(1)} \bigr)^{+}-K\bigl(a_{0}-f_{2}^{(0)} \bigr)^{+}-K(a_{1})^{+}-(m_{b}g)_{1}+\epsilon (S_{0}-S_{1})\tilde{h}_{2}(x,t) \\& \quad = E_{0}+\Lambda a_{1}+K(a_{1})^{+}+R_{0}. \end{aligned}$$
On the basis of our approximation method, we need to solve the following equation:
$$ \Lambda a_{1}+K(a_{1})^{+}+E_{0}=0. $$
(4.29)
Lemma 4.2 shows that Eq. (4.29) has a unique solution \(a_{1}\in W_{3}^{(1)}\) which satisfies
$$ \|a_{1}\|_{\sigma}\leq C(\bar{\sigma}-\sigma) \|E_{0}\|_{\bar{\sigma}}, $$
where \(C(\bar{\sigma}-\sigma)=\Sigma_{1}(\bar{\sigma}-\sigma)+\Sigma_{2}(\bar {\sigma}-\sigma)+\Sigma_{5}(\bar{\sigma}-\sigma)\).
By (4.14)–(4.16), we obtain
$$\begin{aligned}& u_{1}=\mathcal{L}_{\omega }^{-1}\bigl[K \bigl(a_{1}+a_{0}-f_{2}^{(0)} \bigr)^{+}+(m_{c}g)_{0}+(m_{c}g)_{1}+ \epsilon S_{1}\tilde{h}_{1}\bigr], \\& w_{1}=\mathcal{J}_{\omega }^{-1}\bigl[-K \bigl(a_{1}+a_{0}-f_{2}^{(0)} \bigr)^{+}+(m_{b}g)_{0}+(m_{b}g)_{1}+ \epsilon S_{1}\tilde{h}_{2}\bigr]. \end{aligned}$$
This completes the proof. □

Using the same method in Lemma 4.3, the following result holds.

Lemma 4.4

Let \(\omega\in\mathcal{X}(\nu)\). Then (4.26) possesses the ith step approximation \(a_{i}\in W_{3}^{(i)}\) satisfying (4.20). For the range equation (4.1)(4.2), we obtain the corresponding approximation solution
$$\begin{aligned}& u_{i}=\mathcal{L}_{\omega}^{-1}\Biggl[K \Biggl(\sum_{k=0}^{i}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}+\sum_{k=0}^{i}(m_{c}g)_{k}+ \epsilon S_{i-1}\tilde{h}_{1}\Biggr], \end{aligned}$$
(4.30)
$$\begin{aligned}& w_{i}=\mathcal{J}_{\omega}^{-1} \Biggl[-K\Biggl(\sum_{k=0}^{i}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}+\sum_{k=0}^{i}(m_{b}g)_{k}+ \epsilon S_{i-1}\tilde{h}_{2}\Biggr], \end{aligned}$$
(4.31)
where
$$\begin{aligned}& f_{2}^{(i-1)}=\eta_{i-1}\tilde{ \mathcal{J}}_{\omega}A_{3}^{(i-1)}+\frac {[(m_{b}g)_{i-1}+(m_{c}g)_{i-1}]}{\eta_{i-1}} + \epsilon\tilde{\mathcal{L}}_{\omega}^{-1}S_{i-1}( \tilde{h}_{1}+\tilde {h}_{2}), \end{aligned}$$
(4.32)
$$\begin{aligned}& A_{1}^{(i-1)}=\tilde{\mathcal{J}}_{\omega }^{-1}u_{i-1}, \qquad A_{2}^{(i-1)}=\tilde{\mathcal{L}}_{\omega}^{-1}w_{i-1}, \qquad A_{3}^{(i-1)}=\tilde{\mathcal{L}}_{\omega}^{-1}A_{1}^{(i-1)}+ \tilde {\mathcal{J}}_{\omega}^{-1}A_{2}^{(i-1)}, \\& R_{i-1}=K\Biggl(\sum_{k=0}^{i}a_{k}-f_{2}^{(i)} \Biggr)^{+}-K\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}-K(a_{i})^{+}+(m_{b}g)_{i} \\& \hphantom{R_{i-1}={}}{}+\epsilon(S_{i-1}-S_{i}) \tilde{h}_{2}, \\& E_{i}=\sum_{k=0}^{i}\Lambda a_{k}+K\Biggl(\sum_{k=0}^{i}a_{k}-f_{2}^{(i)} \Biggr)^{+}-m_{b}g-\epsilon S_{i}\tilde{h}_{2}. \end{aligned}$$
(4.33)
In order to prove the convergence of Nash–Moser algorithm, we need the following KAM estimates. For convenience, we choose the initial step \((u_{0},w_{0})=(0,0)\) and parameters \((m_{b}g)_{0}=(m_{c}g)_{0}=0\). Then, by (4.11), it follows that \(a_{0}=0\). Set
$$\begin{aligned}& E_{0} = K\bigl(-f_{2}^{(0)} \bigr)^{+}-\epsilon S_{0}\tilde{h}_{2}, \end{aligned}$$
(4.34)
$$\begin{aligned}& E_{1} = R_{0} =K\bigl(a_{1}-f_{2}^{(1)}\bigr)^{+}-K \bigl(-f_{2}^{(0)}\bigr)^{+}-K(a_{1})^{+}-(m_{b}g)_{1} +\epsilon(S_{0}-S_{1}) \tilde{h}_{2}(x,t), \end{aligned}$$
(4.35)
$$\begin{aligned}& f_{2}^{(0)} = \epsilon\tilde{ \mathcal{L}}_{\omega}^{-1}S_{0}(\tilde {h}_{1}+\tilde{h}_{2}), \end{aligned}$$
(4.36)
$$\begin{aligned}& f_{2}^{(1)} = \eta_{1}\tilde{ \mathcal{J}}_{\omega}A_{3}^{(1)}+\frac {[(m_{b}g)_{1}+(m_{c}g)_{1}]}{\eta_{1}} + \epsilon\tilde{\mathcal{L}}_{\omega}^{-1}S_{1}( \tilde{h}_{1}+\tilde {h}_{2}), \\& A_{1}^{(1)} = \tilde{\mathcal{J}}_{\omega}^{-1}u_{1}, \qquad A_{2}^{(1)}=\tilde {\mathcal{L}}_{\omega}^{-1}w_{1}, \qquad A_{3}^{(1)}=\tilde{\mathcal{L}}_{\omega}^{-1}A_{1}^{(1)}+ \tilde{\mathcal {J}}_{\omega}^{-1}A_{2}^{(1)} . \end{aligned}$$
(4.37)

Lemma 4.5

(KAM estimates)

Let \(\omega\in\mathcal{X}(\nu)\). Then, for any \(0<\alpha<\sigma\), the following estimates hold:
$$\begin{aligned}& \|E_{0}\|_{\sigma}\leq K\epsilon C(\alpha)N_{0}^{\alpha} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma}+\| \tilde{h}_{2}\|_{\sigma }\bigr)+\epsilon N_{0}^{\alpha} \|\tilde{h}_{2}\|_{\sigma}, \\& \|a_{1}\|_{\sigma-\frac{\alpha}{3}}\leq\biggl(2C\biggl(\frac{\alpha}{3} \biggr)+C'\biggl(\frac {\alpha}{3}\biggr)\biggr)\|E_{0} \|_{\sigma}, \\& \|u_{1}\|_{\sigma-\frac{2\alpha}{3}}\leq C(K) \biggl(C\biggl(\frac{\alpha}{3} \biggr)C'\biggl(\frac{\alpha}{3}\biggr)+ C\biggl( \frac{2\alpha}{3}\biggr)\biggr)\|E_{0}\|_{\sigma}, \\& \|w_{1}\|_{\sigma-\frac{2\alpha}{3}}\leq C(K) \biggl(C\biggl(\frac{\alpha}{3} \biggr)C'\biggl(\frac{\alpha}{3}\biggr)+ C\biggl( \frac{2\alpha}{3}\biggr)\biggr)\|E_{0}\|_{\sigma}, \\& \|E_{1}\|_{\sigma-\frac{\alpha}{3}}\leq C(\eta_{0},K,\alpha) \|E_{0}\|^{2}_{\sigma}+\epsilon C\biggl( \frac{\alpha}{3}\biggr)N_{0}^{-\frac{\alpha}{3}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma}+\| \tilde{h}_{2} \|_{\sigma}\bigr) \\& \hphantom{\|E_{1}\|_{\sigma-\frac{\alpha}{3}}\leq{}}{}+\epsilon N_{0}^{-\frac{2\alpha}{3}} \bigl\Vert \tilde{h}_{2}(x,t) \bigr\Vert _{\sigma}+(m_{b}g)_{1}, \end{aligned}$$
where \(C(K)\) and \(C(\eta_{0},K,\alpha)\) are constants, \(C(\alpha)\) and \(C'(\alpha)\) are defined in (4.38)(4.39).

Proof

Denote
$$\begin{aligned}& C(\alpha) = \frac{\gamma}{\alpha^{\kappa}}\biggl(\frac{\kappa}{e} \biggr)^{\kappa }=C(\kappa,\gamma)\alpha^{-\kappa}, \end{aligned}$$
(4.38)
$$\begin{aligned}& C'(\alpha) = \frac{C^{2}\gamma^{2}}{\alpha^{2\kappa+2}}\biggl( \frac{2\kappa +2}{e}\biggr)^{2\kappa+2}\leq C'(\kappa,\gamma) \alpha^{-2\kappa-2}. \end{aligned}$$
(4.39)
From the definition of \(E_{0}\) in (4.34) and \(f_{2}^{(0)}\) in (4.36), it follows that
$$\begin{aligned} \bigl\Vert f_{2}^{(0)} \bigr\Vert _{\sigma} =& \bigl\Vert \epsilon\tilde{\mathcal{L}}_{\omega }^{-1}S_{0}( \tilde{h}_{1}+\tilde{h}_{2}) \bigr\Vert _{\sigma}\leq \epsilon \bigl\Vert \tilde {\mathcal{L}}_{\omega}^{-1}S_{0}( \tilde{h}_{1}+\tilde{h}_{2}) \bigr\Vert _{\sigma } \\ \leq&\epsilon C(\alpha)N_{0}^{\alpha}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma}+ \Vert \tilde{h}_{2} \Vert _{\sigma}\bigr) \end{aligned}$$
(4.40)
and
$$\begin{aligned} \Vert E_{0} \Vert _{\sigma} =& \bigl\Vert K \bigl(-f_{2}^{(0)}\bigr)^{+}-\epsilon S_{0} \tilde{h}_{2} \bigr\Vert _{\sigma} \\ \leq&K \bigl\Vert f_{2}^{(0)} \bigr\Vert _{\sigma}+\epsilon \Vert S_{0}\tilde{h}_{2} \Vert _{\sigma} \\ \leq&K\epsilon C(\alpha)N_{0}^{\alpha}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma}+ \Vert \tilde{h}_{2} \Vert _{\sigma }\bigr)+\epsilon N_{0}^{\alpha} \Vert \tilde{h}_{2} \Vert _{\sigma}. \end{aligned}$$
(4.41)
By (4.20), for \(i=1\),
$$ \|a_{1}\|_{\sigma-\frac{\alpha}{3}}\leq C'\biggl( \frac{\alpha}{3}\biggr)\|E_{0}\|_{\sigma}. $$
(4.42)
Then, from the property of operator Λ in Lemma 4.1, (4.40) and (4.42), it follows that
$$\begin{aligned} \Vert u_{1} \Vert _{\sigma-\frac{2\alpha}{3}} =& \bigl\Vert \mathcal{L}_{\omega }^{-1}\bigl[K\bigl(a_{1}-f_{2}^{(0)} \bigr)^{+}+\epsilon S_{0}\tilde{h}_{1}\bigr] \bigr\Vert _{\sigma-\frac{2\alpha}{3}} \\ \leq& \bigl\Vert K\mathcal{L}_{\omega}^{-1}a_{1}+ \mathcal{L}_{\omega }^{-1}\bigl[\bigl(-f_{2}^{(0)} \bigr)^{+}+\epsilon S_{0}\tilde{h}_{1}\bigr] \bigr\Vert _{\sigma-\frac{2\alpha}{3}} \\ \leq&KC\biggl(\frac{\alpha}{3}\biggr) \Vert a_{1} \Vert _{\sigma-\frac{\alpha}{3}}+C\biggl(\frac {2\alpha}{3}\biggr) \bigl\Vert K \bigl(-f_{2}^{(0)}\bigr)^{+}+\epsilon S_{0} \tilde{h}_{1} \bigr\Vert _{\sigma} \\ \leq&C(K) \biggl(C\biggl(\frac{\alpha}{3}\biggr)C'\biggl( \frac{\alpha}{3}\biggr)+ C\biggl(\frac{2\alpha}{3}\biggr)\biggr) \Vert E_{0} \Vert _{\sigma} \end{aligned}$$
(4.43)
and
$$\begin{aligned} \Vert w_{1} \Vert _{\sigma-\frac{2\alpha}{3}} =& \bigl\Vert \mathcal{L}_{\omega }^{-1}\bigl[K\bigl(a_{1}-f_{2}^{(0)} \bigr)^{+}-\epsilon S_{0}\tilde{h}_{1}\bigr] \bigr\Vert _{\sigma-\frac{2\alpha}{3}} \\ \leq& \bigl\Vert K\mathcal{L}_{\omega}^{-1}a_{1}+ \mathcal{L}_{\omega }^{-1}\bigl[\bigl(-f_{2}^{(0)} \bigr)^{+}-\epsilon S_{0}\tilde{h}_{1}\bigr] \bigr\Vert _{\sigma-\frac{2\alpha}{3}} \\ \leq&KC\biggl(\frac{\alpha}{3}\biggr) \Vert a_{1} \Vert _{\sigma-\frac{\alpha}{3}}+C\biggl(\frac {2\alpha}{3}\biggr) \bigl\Vert K \bigl(-f_{2}^{(0)}\bigr)^{+}-\epsilon S_{0} \tilde{h}_{1} \bigr\Vert _{\sigma} \\ \leq&C(K) \biggl(C\biggl(\frac{\alpha}{3}\biggr)C'\biggl( \frac{\alpha}{3}\biggr)+ C\biggl(\frac{2\alpha}{3}\biggr)\biggr) \Vert E_{0} \Vert _{\sigma}, \end{aligned}$$
(4.44)
where \(C(K)\) is a constant depending on K.
Denote
$$ \tilde{f}_{3}^{(1)} = \tilde{L}^{-1}( \eta_{1}u_{1}-\eta_{0}u_{0}),\qquad \tilde{f}_{4}^{(1)}=\tilde{L}^{-1}( \eta_{1}w_{1}-\eta_{0}w_{0}). $$
Then
$$ f_{2}^{(1)}-f_{2}^{(0)}= \tilde{f}_{3}^{(1)}+\tilde{f}_{4}^{(1)}+ \epsilon\tilde {L}_{\omega}^{-1}(S_{1}-S_{0}) (\tilde{h}_{1}+\tilde{h}_{2}). $$
(4.45)
By (4.43)–(4.44), we derive
$$\begin{aligned} \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma-\alpha} \leq& \bigl\Vert \tilde{L}^{-1}(\eta _{1}u_{1}-\eta_{0}u_{0}) \bigr\Vert _{\sigma-\alpha}\leq \eta_{1}C\biggl(\frac{\alpha}{3}\biggr) \|u_{1}\|_{\sigma-\frac{2\alpha}{3}} \\ \leq&\eta_{1}C(K)C\biggl(\frac{\alpha}{3}\biggr) \biggl(C\biggl( \frac{\alpha}{3}\biggr)C'\biggl(\frac{\alpha}{3}\biggr)+ C \biggl(\frac{2\alpha}{3}\biggr)\biggr)\|E_{0}\|_{\sigma} \\ =&C_{1}(\eta_{1},K,\alpha)\|E_{0} \|_{\sigma} \end{aligned}$$
(4.46)
and
$$\begin{aligned} \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma-\alpha} \leq& \bigl\Vert \tilde{L}^{-1}(\eta _{1}w_{1}-\eta_{1}w_{0}) \bigr\Vert _{\sigma-\alpha}\leq \eta_{1}C\biggl(\frac{\alpha}{3}\biggr) \|w_{1}\|_{\sigma-\alpha} \\ \leq&\eta_{1}C(K)C\biggl(\frac{\alpha}{3}\biggr) \biggl(C\biggl( \frac{\alpha}{3}\biggr)C'\biggl(\frac{\alpha}{3}\biggr)+ C \biggl(\frac{2\alpha}{3}\biggr)\biggr)\|E_{0}\|_{\sigma} \\ =&C_{2}(\eta_{1},K,\alpha)\|E_{0} \|_{\sigma}, \end{aligned}$$
(4.47)
where \(C_{1}(\eta_{1},K,\alpha)\) and \(C_{2}(\eta_{1},K,\alpha)\) are constants.
By the definition of \(E_{1}\) in (4.35) and (4.45), we derive
$$\begin{aligned} \Vert E_{1} \Vert _{\sigma-\alpha} =& \bigl\Vert K \bigl(a_{1}-f_{2}^{(1)}\bigr)^{+}-K \bigl(-f_{2}^{(0)}\bigr)^{+}-K(a_{1})^{+}-(m_{b}g)_{1} \\ &{}+\epsilon(S_{0}- S_{1})\tilde{h}_{2}(x,t) \bigr\Vert _{\sigma-\alpha} \\ \leq& \bigl\Vert K\bigl(a_{1}-f_{2}^{(1)}+f_{2}^{(0)} \bigr)^{+}-K(a_{1})^{+}+K\bigl(-f_{2}^{(0)}\bigr)^{+}-K \bigl(-f_{2}^{(0)}\bigr)^{+} \bigr\Vert _{\sigma-\alpha} \\ &{}+(m_{b}g)_{1}+\epsilon \bigl\Vert (S_{0}- S_{1})\tilde{h}_{2}(x,t) \bigr\Vert _{\sigma-\frac{2\alpha}{3}} \\ \leq& \bigl\Vert K\bigl(\tilde{f}_{3}^{(1)}+ \tilde{f}_{4}^{(1)}\bigr)^{+} \bigr\Vert _{\sigma-\alpha }+ \epsilon \bigl\Vert \tilde{L}_{\omega}^{-1}(S_{0}-S_{1}) (\tilde{h}_{1}+\tilde{h}_{2}) \bigr\Vert _{\sigma-\alpha} \\ &{}+(m_{b}g)_{1}+\epsilon \bigl\Vert (S_{0}- S_{1})\tilde{h}_{2}(x,t) \bigr\Vert _{\sigma-\alpha} \\ \leq&K\bigl[ \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma-\alpha}+ \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma-\alpha}\bigr] +\epsilon C\biggl(\frac{\alpha}{3} \biggr)N_{0}^{-\frac{2\alpha}{3}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma}+ \Vert \tilde{h}_{2} \Vert _{\sigma} \bigr) \\ &{}+\epsilon N_{0}^{-\alpha} \bigl\Vert \tilde{h}_{2}(x,t) \bigr\Vert _{\sigma}+(m_{b}g)_{1}, \end{aligned}$$
which, together with (4.42), (4.46) and (4.47), yields
$$\begin{aligned} \Vert E_{1} \Vert _{\sigma-\alpha} \leq&C(\eta_{0},K, \alpha) \Vert E_{0} \Vert ^{2}_{\sigma }+\epsilon C\biggl(\frac{\alpha}{3}\biggr)N_{0}^{-\frac{\alpha}{3}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma}+ \Vert \tilde{h}_{2} \Vert _{\sigma}\bigr) \\ &{}+\epsilon N_{0}^{-\frac{2\alpha}{3}} \bigl\Vert \tilde{h}_{2}(x,t) \bigr\Vert _{\sigma}+(m_{b}g)_{1}, \end{aligned}$$
where \(C(\eta_{0},K,\alpha)\) is a constant. This completes the proof. □

4.2 Convergence of Nash–Moser algorithm, and local uniqueness

In the following, we will give a sufficient condition on the convergence of Newton algorithm. For \(i\in\mathbb{N}\) and \(0<\bar{\sigma}<\sigma<\tilde{\sigma}\), set
$$\begin{aligned}& \sigma_{i}:=\bar{\sigma}+\frac{\sigma-\bar{\sigma}}{2^{i}}, \end{aligned}$$
(4.48)
$$\begin{aligned}& \alpha_{i+1}:=\sigma_{i}-\sigma_{i+1}= \frac{\sigma-\bar{\sigma}}{2^{i+1}}. \end{aligned}$$
(4.49)
By (4.48), we have
$$ \sigma_{0}>\sigma_{1}>\cdots>\sigma_{i}> \sigma_{i+1}>\cdots,\quad i\in\mathbb{N}. $$
For the convergence of the Nash–Moser algorithm, we need to choose
$$ \bigl[(m_{b}g)_{i}+(m_{c}g)_{i} \bigr]=e^{-\frac{N_{i+1}}{2^{i-1}}},\qquad \eta_{i}=\bigl[(m_{b}g)_{i}+(m_{c}g)_{i} \bigr]=e^{-\frac{N_{i+1}}{2^{i-1}}}, \quad i\in\mathbb{N}. $$
(4.50)
Furthermore, for convenience, choose
$$ (m_{b}g)_{i}=(m_{c}g)_{i}= \frac{1}{2}e^{-\frac{N_{i+1}}{2^{i-1}}}, \quad i\in\mathbb{N}. $$
(4.51)

Remark 4.4

The choice of \(\eta_{i}\) in (4.50) depends on making
$$ \eta_{i-1}\|a_{i}\|_{\sigma_{i-2}}\leq\eta_{i-1}e^{\frac {N_{i}}{2^{i-1}}+\frac{N_{i}}{2^{i-2}}} \|a_{i}\|_{\sigma_{i}}\leq\|a_{i}\| _{\sigma_{i}} $$
holds. Moreover, to make \(\lim_{i\rightarrow\infty}\frac{[(m_{b}g)_{i}+(m_{c}g)_{i}]}{\eta_{i}}=C\) (C is a constant), the choice of \((m_{b}g)_{i}+(m_{c}g)_{i}\) in (4.50) is determined by \(\eta_{i}\). For convenience, the values of \((m_{b}g)_{i}\) and \((m_{c}g)_{i}\) are chosen in (4.51). It is important to prove the convergence of the Nash–Moser algorithm.
We assume there exist sufficiently small K and ϵ such that
$$\begin{aligned}& \Xi\leq2^{-24\kappa}C^{-2}(\kappa,\gamma,\sigma,\bar{ \sigma})\rho ^{-1}, \end{aligned}$$
(4.52)
$$\begin{aligned}& \theta\geq2\theta_{1}>2\ln\Xi^{-1}, \end{aligned}$$
(4.53)
where Ξ is to be defined in (4.75) and depends on K and ϵ.

Lemma 4.6

(Convergence of Nash–Moser algorithm)

Let \(\omega\in\mathcal{X}(\nu)\). Assume that (4.52)(4.53) holds. Then there exist \(\sum_{i=0}^{\infty}a_{i}\) and \(\sum_{i=0}^{\infty}(m_{b}g)_{i}\) such that
$$ \mathcal{M}\Biggl(\sum_{i=0}^{\infty}a_{i}, \sum_{i=0}^{\infty}(m_{b}g)_{i} \Biggr)=0. $$

Proof

The target is to prove that the convergence of the error \(E_{i}\) of the ith iterative step, i.e.,
$$\lim_{i\rightarrow\infty}\|E_{i}\|_{\sigma_{i}}=0. $$
Denote
$$ \tilde{f}_{3}^{(i)} = \tilde{L}_{\omega}^{-1}( \eta_{i}u_{i}-\eta_{i-1}u_{i-1}),\qquad \tilde{f}_{4}^{(i)}=\tilde{L}_{\omega}^{-1}( \eta_{i}w_{i}-\eta_{i-1}w_{i-1}). $$
(4.54)
Then, by (4.54), we have
$$ f_{2}^{(i)}-f_{2}^{(i-1)}= \tilde{f}_{3}^{(i)}+\tilde{f}_{4}^{(i)}+ \epsilon \mathcal{L}_{\omega}^{-1}(S_{i-1}-S_{i-2}) (\tilde{h}_{1}+\tilde{h}_{2}). $$
(4.55)
From (4.33) and (4.55), it follows
$$\begin{aligned} \Vert E_{i} \Vert _{\sigma_{i}} =& \Vert R_{i-1} \Vert _{\sigma_{i}} \\ =& \Biggl\Vert K\Biggl(\sum_{k=0}^{i}a_{k}-f_{2}^{(i)} \Biggr)^{+}-K\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}-K(a_{i})^{+} \\ &{}+\epsilon(S_{i-1}-S_{i})\tilde {h}_{2}+(m_{b}g)_{i} \Biggr\Vert _{\sigma_{i}} \\ \leq& \Biggl\Vert K\bigl(a_{i}+f_{2}^{(i-1)}-f^{(i)}_{2} \bigr)^{+}-K(a_{i})^{+}+K\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}-K\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-1)} \Biggr)^{+} \\ &{} +\epsilon(S_{i-1}-S_{i})\tilde{h}_{2} \Biggr\Vert _{\sigma_{i}}+(m_{b}g)_{i} \\ \leq& \bigl\Vert K\bigl(\tilde{f}_{3}^{(i)}+ \tilde{f}_{4}^{(i-1)}\bigr)^{+} \bigr\Vert _{\sigma _{i}}+K \epsilon C(\alpha_{i}) e^{-(\frac{\theta}{2})^{i}(\sigma-\bar{\sigma})}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma _{i}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i}}\bigr) \\ &{} +\epsilon \bigl\Vert (S_{i-1}-S_{i}) \tilde{h}_{2} \bigr\Vert _{\sigma_{i}}+(m_{b}g)_{i} \\ \leq& K\bigl( \bigl\Vert \tilde{f}_{3}^{(i)} \bigr\Vert _{\sigma_{i}}+ \bigl\Vert \tilde{f}_{4}^{(i)} \bigr\Vert _{\sigma _{i}}\bigr)+K\epsilon C(\alpha_{i})N_{i}^{-\alpha_{i}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i}}\bigr) \\ &{}+\epsilon N_{i}^{-\alpha_{i}} \Vert \tilde{h}_{2} \Vert _{\sigma _{i}}+e^{-\theta_{1}^{i}} \\ \leq& K\bigl( \bigl\Vert \tilde{f}_{3}^{(i)} \bigr\Vert _{\sigma_{i}}+ \bigl\Vert \tilde{f}_{4}^{(i)} \bigr\Vert _{\sigma _{i}}\bigr)+K\epsilon C(\alpha_{i}) e^{-\theta_{1}^{i}(\sigma-\bar{\sigma})} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i}}+ \Vert \tilde {h}_{2} \Vert _{\sigma_{i}}\bigr) \\ &{} +\epsilon e^{-\theta_{1}^{i}(\sigma-\bar{\sigma})} \Vert \tilde{h}_{2} \Vert _{\sigma _{i}}+e^{-\theta_{1}^{i}}. \end{aligned}$$
(4.56)
In what follows, we need to estimate \(\|\tilde{f}_{3}^{(i)}\|_{\sigma_{i}}\) and \(\|\tilde{f}_{4}^{(i)}\|_{\sigma_{i}}\). From the definition of \(\tilde{f}_{3}^{(i)}\) in (4.54) and \(\eta_{i}\) in (4.50), we derive
$$\begin{aligned} \bigl\Vert \tilde{f}_{3}^{(i)} \bigr\Vert _{\sigma_{i}} =& \bigl\Vert \tilde{L}_{\omega}^{-1}(\eta _{i}u_{i}-\eta_{i-1}u_{i-1}) \bigr\Vert _{\sigma_{i}} \\ \leq&\eta_{i-1} \Biggl\Vert \tilde{L}_{\omega}^{-1}L_{\omega}^{-1}K \Biggl(\Biggl(\sum_{k=0}^{i}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}-\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-2)} \Biggr)^{+}\Biggr) \Biggr\Vert _{\sigma_{i}} \\ &{}+\epsilon\eta_{i-1} \bigl\Vert \mathcal{L}_{\omega}^{-1}(S_{i-1}-S_{i-2}) \tilde {h}_{1} \bigr\Vert _{\sigma_{i}} \\ \leq&\eta_{i-1}C(\alpha_{i})C(\alpha_{i-1}) \bigl\Vert K\bigl[a_{i}+f_{2}^{(i-2)}-f_{2}^{(i-1)} \bigr]^{+} \bigr\Vert _{\sigma_{i-2}} \\ &{}+\epsilon\eta_{i-1} \bigl\Vert \mathcal{L}_{\omega}^{-1}(S_{i-1}-S_{i-2}) \tilde {h}_{1} \bigr\Vert _{\sigma_{i}} \\ \leq&\frac{\eta_{i-1}}{2}C(\alpha_{i})C(\alpha_{i-1}) \Vert a_{i} \Vert _{\sigma_{i-2}}^{2} +\frac{\eta_{i-1}}{2}C( \alpha_{i})C(\alpha_{i-1}) \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-2}}^{2} \\ &{}+\frac{\eta_{i-1}}{2}C(\alpha_{i})C(\alpha_{i-1}) \bigl\Vert \tilde {f}_{4}^{(i-1)} \bigr\Vert _{\sigma_{i-2}}^{2} \\ &{}+\epsilon K\eta_{i-1} C(\alpha_{i})C( \alpha_{i-1})C(\alpha_{i-2})N_{i-2}^{-\alpha_{i-2}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-2}}\bigr) \\ &{}+\epsilon\eta_{i-1} C(\alpha_{i})N_{i-1}^{-\alpha_{i-1}} \Vert h_{1} \Vert _{\sigma_{i}}+2\eta_{i-1}C(\alpha _{i})C(\alpha_{i-1})K^{2} \\ \leq&\eta_{i-1}e^{\frac{2N_{i}}{2^{i}}}e^{\frac {2N_{i-1}}{2^{i-1}}}C(\alpha_{i})C( \alpha_{i-1}) \Vert a_{i} \Vert _{\sigma_{i}}^{2} +\eta_{i-1}e^{\frac{2N_{i-1}}{2^{i-1}}}C(\alpha_{i})C( \alpha_{i-1}) \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2} \\ &{}+\eta_{i-1}e^{\frac{2N_{i-1}}{2^{i-1}}}C(\alpha_{i})C( \alpha_{i-1}) \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2} \\ &{}+\epsilon K\eta_{i-1} C(\alpha_{i})C( \alpha_{i-1})C(\alpha_{i-2})N_{i-2}^{-\alpha_{i-2}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-2}}\bigr) \\ &{}+\epsilon\eta_{i-1} C(\alpha_{i})N_{i-1}^{-\alpha_{i-1}} \Vert \tilde{h}_{1} \Vert _{\sigma_{i}}+2\eta _{i-1}C( \alpha_{i})C(\alpha_{i-1})K^{2} \\ \leq& \Vert a_{i} \Vert _{\sigma_{i}}^{2} +C( \alpha_{i})C(\alpha_{i-1}) \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma _{i-1}}^{2} \\ &{}+C(\alpha_{i})C(\alpha_{i-1}) \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma _{i-1}}^{2} \\ &{}+\epsilon K\eta_{i-1} C(\alpha_{i})C( \alpha_{i-1})C(\alpha_{i-2})N_{i-2}^{-\alpha_{i-2}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-2}}\bigr) \\ &{}+\epsilon\eta_{i-1} C(\alpha_{i})N_{i-1}^{-\alpha_{i-1}} \Vert \tilde{h}_{1} \Vert _{\sigma_{i}}+2\eta _{i-1}C( \alpha_{i})C(\alpha_{i-1})K^{2}. \end{aligned}$$
(4.57)
In a similar manner, we get
$$\begin{aligned} \bigl\Vert \tilde{f}_{4}^{(i)} \bigr\Vert _{\sigma_{i}} =& \bigl\Vert \tilde{L}_{\omega}^{-1}(\eta _{i}w_{i}-\eta_{i-1}w_{i-1}) \bigr\Vert _{\sigma_{i}} \\ \leq& \Vert a_{i} \Vert _{\sigma_{i}}^{2} +C( \alpha_{i})C(\alpha_{i-1}) \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma _{i-1}}^{2} \\ &{}+C(\alpha_{i})C(\alpha_{i-1}) \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma _{i-1}}^{2} \\ &{}+\epsilon K\eta_{i-1} C(\alpha_{i})C( \alpha_{i-1})C(\alpha_{i-2})N_{i-2}^{-\alpha_{i-2}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-2}}\bigr) \\ &{}+\epsilon\eta_{i-1} C(\alpha_{i})N_{i-1}^{-\alpha_{i-1}} \Vert \tilde{h}_{2} \Vert _{\sigma_{i}}+2\eta _{i-1}C( \alpha_{i})C(\alpha_{i-1})K^{2}. \end{aligned}$$
(4.58)
Hence, by (4.57)–(4.58), it follows that
$$\begin{aligned}& \bigl\Vert \tilde{f}_{3}^{(i)} \bigr\Vert _{\sigma_{i}}+ \bigl\Vert \tilde{f}_{4}^{(i)} \bigr\Vert _{\sigma_{i}} \\& \quad \leq 2 \Vert a_{i} \Vert ^{2}_{\sigma_{i}} +2C(\alpha_{i})C(\alpha_{i-1}) \bigl( \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2}+ \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2}\bigr) \\& \qquad {}+2\epsilon K\eta_{i-1} C(\alpha_{i})C( \alpha_{i-1})C(\alpha_{i-2})N_{i-2}^{-\alpha_{i-2}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-2}}\bigr) \\& \qquad {}+\epsilon\eta_{i-1} C(\alpha_{i})N_{i-1}^{-\alpha_{i-1}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i}}+ \Vert \tilde {h}_{2} \Vert _{\sigma_{i}}\bigr)+4\eta_{i-1}C( \alpha_{i})C(\alpha_{i-1})K^{2}. \end{aligned}$$
(4.59)
Furthermore, by using the Young inequality, we get
$$\begin{aligned} \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2} \leq& 6C^{2}(\alpha _{i-1})C^{2}( \alpha_{i-2}) \Vert a_{i-1} \Vert _{\sigma_{i-1}}^{2^{2}} +6C^{2}(\alpha_{i-1})C^{2}(\alpha_{i-2}) \bigl\Vert \tilde{f}_{3}^{(i-2)} \bigr\Vert _{\sigma _{i-2}}^{2^{2}} \\ & {}+6C^{2}(\alpha_{i-1})C^{2}( \alpha_{i-2}) \bigl\Vert \tilde{f}_{4}^{(i-2)} \bigr\Vert _{\sigma _{i-2}}^{2^{2}} \\ & {}+6\epsilon^{2}K^{2}\eta_{i-2}^{2} C^{2}(\alpha_{i-1})C^{2}(\alpha_{i-2})C^{2}( \alpha_{i-3})N_{i-3}^{-2\alpha _{i-3}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-3}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-3}}\bigr)^{2} \\ &{}+6\epsilon^{2}\eta_{i-2}^{2}C^{2}( \alpha_{i-1})N_{i-2}^{-2\alpha_{i-2}} \Vert \tilde{h}_{1} \Vert _{\sigma_{i-1}}^{2} \\ &{}+6 \times2^{2}\eta_{i-2}^{2}C^{2}(\alpha _{i-1})C^{2}(\alpha_{i-2})K^{2^{2}}. \end{aligned}$$
(4.60)
Also,
$$\begin{aligned} \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2} \leq&6C^{2}(\alpha _{i-1})C^{2}( \alpha_{i-2}) \Vert a_{i-1} \Vert _{\sigma_{i-1}}^{2^{2}} +6C^{2}(\alpha_{i-1})C^{2}(\alpha_{i-2}) \bigl\Vert \tilde{f}_{3}^{(i-2)} \bigr\Vert _{\sigma _{i-2}}^{2^{2}} \\ &{}+6C^{2}(\alpha_{i-1})C^{2}( \alpha_{i-2}) \bigl\Vert \tilde{f}_{4}^{(i-2)} \bigr\Vert _{\sigma _{i-2}}^{2^{2}} \\ &{}+6\epsilon^{2}K^{2}\eta_{i-2}^{2} C^{2}(\alpha_{i-1})C^{2}(\alpha_{i-2})C^{2}( \alpha_{i-3})N_{i-3}^{-2\alpha _{i-3}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-3}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-3}}\bigr)^{2} \\ &{}+6\epsilon^{2}\eta_{i-2}^{2}C^{2}( \alpha_{i-1})N_{i-2}^{-2\alpha_{i-2}} \Vert \tilde{h}_{2} \Vert _{\sigma_{i-1}}^{2} \\ &{}+6 \times2^{2}\eta_{i-2}^{2}C^{2}(\alpha _{i-1})C^{2}(\alpha_{i-2})K^{2^{2}}. \end{aligned}$$
(4.61)
This, combining (4.60)–(4.61), shows that
$$\begin{aligned}& \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2}+ \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2} \\& \quad \leq 6\times2C^{2}( \alpha_{i-1})C^{2}(\alpha_{i-2}) \Vert a_{i-1} \Vert _{\sigma _{i-1}}^{2^{2}} \\& \qquad {}+6\times2C^{2}(\alpha_{i-1})C^{2}( \alpha_{i-2}) \bigl\Vert \tilde{f}_{3}^{(i-2)} \bigr\Vert _{\sigma_{i-2}}^{2^{2}} \\& \qquad {}+6\times2C^{2}(\alpha_{i-1})C^{2}( \alpha_{i-2}) \bigl\Vert \tilde{f}_{4}^{(i-2)} \bigr\Vert _{\sigma_{i-2}}^{2^{2}} \\& \qquad {}+6\times2\epsilon^{2}K^{2} C^{2}( \alpha_{i-1})C^{2}(\alpha_{i-2})C^{2}( \alpha_{i-3})N_{i-3}^{-2\alpha _{i-3}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-3}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-3}}\bigr)^{2} \\& \qquad {}+6\epsilon^{2}\eta_{i-2}^{2}C^{2}( \alpha_{i-1})N_{i-2}^{-2\alpha_{i-2}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-1}}^{2}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-1}}^{2}\bigr) \\& \qquad {}+6\times2^{2^{2}-1}\eta_{i-2}^{2}C^{2}( \alpha_{i-1})C^{2}(\alpha_{i-2})K^{2^{2}}. \end{aligned}$$
(4.62)
Then, applying the Young inequality to (4.60)–(4.61), we have
$$\begin{aligned}& \bigl\Vert \tilde{f}_{3}^{(i-2)} \bigr\Vert _{\sigma_{i-2}}^{2^{2}} + \bigl\Vert \tilde{f}_{4}^{(i-2)} \bigr\Vert _{\sigma_{i-2}}^{2^{2}} \\& \quad \leq 6^{2^{2}-1}\times2^{2^{2}-1}C^{2^{2}}( \alpha_{i-2})C^{2^{2}}(\alpha _{i-3}) \Vert a_{i-2} \Vert _{\sigma_{i-2}}^{2^{3}} \\& \qquad {} +6^{2^{2}-1}\times2^{2^{2}-1}C^{2^{2}}( \alpha_{i-2})C^{2^{2}}(\alpha_{i-3}) \bigl\Vert \tilde{f}_{3}^{(i-3)} \bigr\Vert _{\sigma_{i-3}}^{2^{3}} \\& \qquad {} +6^{2^{2}-1}\times2^{2^{2}-1}C^{2^{2}}( \alpha_{i-2})C^{2^{2}}(\alpha_{i-3}) \bigl\Vert \tilde{f}_{4}^{(i-3)} \bigr\Vert _{\sigma_{i-3}}^{2^{3}} \\& \qquad {} +6^{2^{2}-1}\times2^{2^{2}-1}\epsilon^{2^{2}}K^{2^{2}} C^{2^{2}}(\alpha_{i-2})C^{2^{2}}(\alpha_{i-3})C^{2^{2}}( \alpha _{i-4})N_{i-4}^{-2^{2}\alpha_{i-4}} \\& \qquad {} \times\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-4}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-4}}\bigr)^{2^{2}} \\& \qquad {} +6^{2^{2}-1}\epsilon^{2^{2}}\eta_{i-3}^{2^{2}}C^{2^{2}}( \alpha _{i-2})N_{i-3}^{-2^{2}\alpha_{i-3}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma _{i-2}}^{2^{2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-2}}^{2^{2}}\bigr) \\& \qquad {} +6^{2^{2}-1}\times2^{2^{3}-1}\eta_{i-3}^{2^{2}}C^{2^{2}}( \alpha _{i-2})C^{2^{2}}(\alpha_{i-3})K^{2^{3}}, \\& \ldots, \\& \bigl\Vert \tilde{f}_{3}^{(2)} \bigr\Vert _{\sigma_{2}}^{2^{i-2}} + \bigl\Vert \tilde{f}_{4}^{(2)} \bigr\Vert _{\sigma_{2}}^{2^{i-2}} \\& \quad \leq 6^{2^{i-2}-1}\times2^{2^{i-2}-1}C^{2^{i-2}}(\alpha _{2})C^{2^{i-2}}(\alpha_{1}) \Vert a_{2} \Vert _{\sigma_{2}}^{2^{i-1}} \\& \qquad {} +6^{2^{i-2}-1}\times2^{2^{i-2}-1}C^{2^{i-2}}(\alpha _{2})C^{2^{i-2}}(\alpha_{1}) \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma _{1}}^{2^{i-1}} \\& \qquad {} +6^{2^{i-2}-1}\times2^{2^{i-2}-1}C^{2^{i-2}}(\alpha _{2})C^{2^{i-2}}(\alpha_{1}) \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma _{1}}^{2^{i-1}} \\& \qquad {} +6^{2^{i-2}-1}\times2^{2^{i-2}-1}\epsilon^{2^{i-2}}K^{2^{i-2}} C^{2^{i-2}}(\alpha_{2})C^{2^{i-2}}(\alpha_{1})C^{2^{i-2}}( \alpha _{0})N_{0}^{-{2^{i-2}}\alpha_{0}} \\& \qquad {} \times\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{0}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{0}}\bigr)^{2^{i-2}} \\& \qquad {} +6^{2^{i-2}-1}\epsilon^{2^{i-2}}C^{2^{i-2}}(\alpha _{2})N_{0}^{-{2^{i-2}}\alpha_{0}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma _{2}}^{2^{i-2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{2}}^{2^{i-2}}\bigr) \\& \qquad {} +6^{2^{i-2}-1}\times2^{2^{i-1}-1}\eta_{0}^{2^{i-2}}C^{2^{i-2}}( \alpha _{2})C^{2^{i-2}}(\alpha_{1})K^{2^{i-1}}. \end{aligned}$$
Iterating the above estimates one by one, we obtain
$$ \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}^{2}+ \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma _{i-1}}^{2}\leq R_{1}+R_{2}+R_{3}+R_{4}+R_{5}, $$
(4.63)
where
$$\begin{aligned}& R_{1} = \sum_{k=2}^{i-1} \Vert a_{i-k+1} \Vert _{\sigma_{i-k+1}}^{2^{k}}\prod _{j=2}^{k}12^{2^{j-1}}C^{2^{j-1}}( \alpha_{i-j+1})C^{2^{j-1}}(\alpha _{i-j}), \\& R_{2} = \bigl( \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma_{1}}^{2^{i-1}}+ \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma_{1}}^{2^{i-1}}\bigr)\prod _{k=2}^{i-1}12^{2^{k-1}}C^{2^{k-1}}(\alpha _{i-k+1})C^{2^{k-1}}(\alpha_{i-k}), \\& R_{3} = \sum_{k=2}^{i-2}(K \epsilon)^{2^{k+1}}\bigl(C(\alpha_{i-k+1})C(\alpha _{i-k})C(\alpha_{i-k-1})\bigr)^{2^{k}} N_{i-k-1}^{-2^{k}\alpha_{i-k-1}} \\& \hphantom{R_{3} ={}}{} \times\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-k-1}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-k-1}} \bigr)^{2^{k}}\prod_{j=2}^{k}12^{2^{j-1}}C^{2^{j-1}}( \alpha _{i-j+1})C^{2^{j-1}}(\alpha_{i-j}), \\& R_{4} = \sum_{k=2}^{i-2} \epsilon^{2^{k}}C^{2^{k}}(\alpha_{i-k}) N_{i-k}^{-2^{k-1}\alpha_{i-k}}\bigl( \Vert \tilde{h}_{1} \Vert ^{2^{k}}_{\sigma_{i-k}}+ \Vert \tilde{h}_{2} \Vert ^{2^{k}}_{\sigma_{i-k}}\bigr) \\& \hphantom{R_{4} ={}}{} \times\prod_{j=2}^{k}6^{2^{j-1}}C^{2^{j-1}}( \alpha _{i-j+1})C^{2^{j-1}}(\alpha_{i-j}), \\& R_{5} = \sum_{k=2}^{i-1} \eta_{i-k-2}^{2^{k}}K^{2^{k}}\prod _{j=2}^{k}12^{2^{j}}C^{2^{j-1}}( \alpha_{i-j+1})C^{2^{j-1}}(\alpha_{i-j}). \end{aligned}$$
By (4.48) and (4.49), there exist positive constants \(C(\kappa,\gamma,\sigma,\bar{\sigma})\) and \(C_{1}(\kappa,\gamma,\sigma,\bar{\sigma})\), depending on κ, σ, σ̄ such that
$$ C(\alpha_{i})\leq C(\kappa,\gamma,\sigma,\bar{ \sigma})2^{i\kappa},\qquad C_{1}(\alpha_{i})\leq C_{1}(\kappa,\gamma,\sigma,\bar{\sigma})2^{2i\kappa}, $$
(4.64)
where \(C(\alpha)\) and \(C'(\alpha)\) are defined in (4.38)–(4.39), \(C_{1}(\alpha)=2C(\alpha)+C'(\alpha)\).
On the other hand, from \(N_{n}=e^{\theta^{n}}\), \(\theta>2\theta_{1}\) and \(k\geq1\), it follows that
$$\begin{aligned}& N_{i-k-1}^{-2^{k}\alpha_{i-k-1}} = e^{-(\frac{\theta}{2})^{i-1}}e^{-(\frac {4}{\theta})^{k}(\sigma-\bar{\sigma})} \leq e^{-\theta_{1}^{i-1}}, \end{aligned}$$
(4.65)
$$\begin{aligned}& N_{i-k}^{-2^{k-1}\alpha_{i-k}} = e^{-(\frac{\theta}{2})^{i}}e^{-2(\frac {4}{\theta})^{k}(\sigma-\bar{\sigma})} \leq e^{-\theta_{1}^{i-1}}. \end{aligned}$$
(4.66)
Note that \(N_{i-k-1}=e^{\theta^{i-k-1}}>\theta^{i-k-1}\). We have
$$ \eta_{i-k-2}^{2^{k}} = e^{-N_{i-k-1}2^{2k-i+3}}\leq e^{-\theta_{1}^{i-1}}. $$
(4.67)
So, by (4.64)–(4.67), we estimate \(R_{1}\), \(R_{2}\), \(R_{3}\) and \(R_{4}\), having
$$\begin{aligned}& \begin{aligned}[b] R_{1} &= \sum _{k=2}^{i-1} \Vert a_{i-k+1} \Vert _{\sigma_{i-k+1}}^{2^{k}}\prod_{j=2}^{k}12^{2^{j-1}}C^{2^{j-1}}( \alpha_{i-j+1})C^{2^{j-1}}(\alpha _{i-j}) \\ &\leq \sum_{k=2}^{i-1}12^{2^{k}} \times2^{[2^{k+1}(i-k+2)-8(i-1)]\kappa} C^{2^{k+1}}(\kappa,\gamma,\sigma,\bar{\sigma}) \Vert a_{i-k+1} \Vert _{\sigma_{i-k+1}}^{2^{k}} \\ &\leq \sum_{k=2}^{i-1}\bigl(12C^{2}( \kappa,\gamma,\sigma,\bar{\sigma })\bigr)^{2^{k}}\times2^{[2^{k+1}(i-k+2)-8i+8]\kappa} \Vert a_{i-k+1} \Vert _{\sigma_{i-k+1}}^{2^{k}}, \end{aligned} \end{aligned}$$
(4.68)
$$\begin{aligned}& \begin{aligned}[b] R_{2} &\leq 12^{2^{i-1}} \times2^{(2^{i+2}-4i)\kappa}C^{2^{i}}(\kappa,\gamma ,\sigma,\bar{\sigma}) \bigl( \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma_{1}}^{2^{i-1}}+ \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma_{1}}^{2^{i-1}}\bigr) \\ &\leq \bigl[12\times2^{8\kappa} C^{2}(\kappa,\gamma,\sigma, \bar{\sigma}) \bigl( \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma _{1}}+ \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma_{1}}\bigr)\bigr]^{2^{i-1}}, \end{aligned} \end{aligned}$$
(4.69)
$$\begin{aligned}& \begin{aligned}[b] R_{3} &\leq e^{-\theta_{1}^{i-1}}\sum _{k=2}^{i-2}2^{[2^{k}(5i-5k+2)-8i+8]\kappa} \bigl(12K \epsilon C^{5}\bigr)^{2^{k}}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-k-2}}+ \Vert \tilde {h}_{2} \Vert _{\sigma_{i-k-2}}\bigr)^{2^{k+1}} \\ &\leq e^{-\theta_{1}^{i-1}}\sum_{k=2}^{i-2}2^{[2^{k}(5i-5k+2)-8i+8]\kappa } \bigl[12K\epsilon C^{5}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{0}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{0}} \bigr)^{2}\bigr]^{2^{k}}, \end{aligned} \end{aligned}$$
(4.70)
$$\begin{aligned}& \begin{aligned}[b] R_{4} &\leq e^{-\theta_{1}^{i-1}}\sum _{k=2}^{i-2}2^{[2^{k}(5i-5k+2)-8i+8]\kappa}\bigl(6\epsilon C^{2}(\kappa,\gamma,\sigma,\bar{\sigma})\bigr)^{2^{k}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-k}}^{2^{k}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-k}}^{2^{k}}\bigr) \\ &\leq e^{-\theta_{1}^{i-1}}\sum_{k=2}^{i-2}2^{[2^{k}(5i-5k+2)-8i+8]\kappa } \bigl[6\epsilon C^{2}(\kappa,\gamma,\sigma,\bar{\sigma}) \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{0}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{0}}\bigr)\bigr]^{2^{k}}, \end{aligned} \end{aligned}$$
(4.71)
and
$$ R_{5}\leq e^{-\theta_{1}^{i-1}}\sum _{k=2}^{i-1}\bigl(12KC^{2}(\kappa,\gamma, \sigma,\bar {\sigma})\bigr)^{2^{k}}\times2^{[2^{k+1}(i-k+2)-8i+8]\kappa}. $$
(4.72)
Inserting (4.59), (4.63), (4.68)–(4.72) into (4.56), then, by (4.20), the relation between \(\|E_{i}\|_{\sigma_{i}}\) and \(\|E_{i-1}\|_{\sigma_{i-1}}\) is
$$\begin{aligned} \|E_{i}\|_{\sigma_{i}} \leq&2KC_{1}^{2}( \alpha_{i})\|E_{i-1}\|^{2}_{\sigma _{i-1}}+K\sum _{k=2}^{i-1}\rho^{2^{k}}2^{[2^{k+2}(i-k+2)-4i]\kappa} \| E_{i-k}\|_{\sigma_{i-k}}^{2^{k}} \\ &{}+\Upsilon_{i}^{(1)}+\Upsilon_{i}^{(2)}+ \Upsilon_{i}^{(3)} +\Upsilon_{i}^{(4)}+ \Upsilon_{i}^{(5)} \\ \leq&2KC_{1}^{2}(\alpha_{i})\|E_{i-1} \|^{2}_{\sigma_{i-1}}+K\sum_{k=1}^{i-1} \rho^{2^{k}}2^{[2^{k+2}(i-k+2)-4i]\kappa}\|E_{i-k}\|_{\sigma _{i-k}}^{2^{k+1}} \\ &{}+\Upsilon_{i}^{(1)}+\Upsilon_{i}^{(2)}+ \Upsilon_{i}^{(3)} +\Upsilon_{i}^{(4)}+ \Upsilon_{i}^{(5)}, \end{aligned}$$
(4.73)
where
$$\begin{aligned}& \rho = 12 C^{2}(\kappa,\gamma,\sigma,\bar{\sigma})C^{2}_{1}( \kappa,\gamma,\sigma,\bar {\sigma}), \\ & \Upsilon_{i}^{(1)} = C^{2}(\kappa,\gamma,\sigma, \bar{\sigma})\bigl[12\times 2^{8\kappa} C^{2}(\kappa,\gamma, \sigma,\bar{\sigma}) \bigl( \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma _{1}}+ \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma_{1}}\bigr)\bigr]^{2^{i-1}}, \\ & \Upsilon_{i}^{(2)} = e^{-\theta_{1}^{i-1}}C^{2}( \kappa,\gamma,\sigma ,\tilde{\sigma})\sum_{k=2}^{i-2}2^{[2^{k}(5i-5k+2)-4i]\kappa } \bigl[12K\epsilon C^{5}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{0}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{0}} \bigr)^{2}\bigr]^{2^{k}}, \\ & \Upsilon_{i}^{(3)} = e^{-\theta_{1}^{i-1}}\sum _{k=2}^{i-2}2^{[2^{k}(5i-5k+2)-4i]\kappa}\bigl[6\epsilon C^{2}(\kappa,\gamma,\sigma,\bar{\sigma}) \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{0}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{0}}\bigr)\bigr]^{2^{k}}, \\ & \Upsilon_{i}^{(4)} = e^{-\theta_{1}^{i}}C^{2}( \kappa,\gamma,\sigma,\tilde {\sigma})\sum_{k=1}^{i-1} \bigl(12KC^{2}(\kappa,\gamma,\sigma,\bar{\sigma })\bigr)^{2^{k}} \times2^{[2^{k+1}(i-k+2)-4i]\kappa}, \\ & \Upsilon_{i}^{(5)} = 3K\epsilon C(\alpha_{i}) e^{-\theta_{1}^{i}(\sigma-\bar{\sigma})}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-2}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-2}}\bigr)+\epsilon e^{-\theta_{1}^{i}(\sigma-\bar{\sigma})} \Vert \tilde{h}_{2} \Vert _{\sigma_{i}} \\ & \hphantom{\Upsilon_{i}^{(5)} ={}}{} +4e^{-\theta_{1}^{i}}C(\alpha_{i})C( \alpha_{i-1})K^{2}+e^{-\theta_{1}^{i}}. \end{aligned}$$
By (4.53), it follows that
$$ \lim_{i\rightarrow+\infty}e^{-\theta_{1}^{i}}\sum_{k=1}^{i}2^{(i-k)2^{k}}=0. $$
Hence, for small K and ϵ, by (4.46) and assumptions (4.52)–(4.53), it is easy to check that
$$ \lim_{i\rightarrow\infty}\Upsilon_{i}^{(j)}=0,\quad j=1,2,3,4,5. $$
In what follows, we will prove \(\lim_{i\rightarrow\infty}E_{i}=0\) by induction.
When \(i=2\), by (4.73), we have
$$\begin{aligned} \|E_{2}\|_{\sigma_{2}} \leq&2KC_{1}^{2}( \alpha_{2})\|E_{1}\|_{\sigma_{1}}^{2}+\rho ^{2}2^{2^{4}\kappa}\|E_{1}\|^{2^{2}}_{\sigma_{1}} +\Upsilon_{2}^{(1)}+\Upsilon_{2}^{(2)}+ \Upsilon_{2}^{(3)}+\Upsilon _{2}^{(4)}+ \Upsilon_{2}^{(5)}, \\ \leq&\Xi^{2}C_{1}^{2}(\alpha_{2}) \bigl(2Kb^{2}+\rho^{2}2^{2^{4}\kappa}C_{1}^{-2}( \alpha _{2})\Xi^{2}b^{2^{2}}+\Phi_{2}\bigr) \\ \leq&\Xi^{2}C_{1}^{2}(\alpha_{2}) \Theta_{2}, \end{aligned}$$
(4.74)
where
$$\begin{aligned}& b\Xi = \|E_{1}\|_{\sigma_{1}}, \\& \Xi^{2}\Phi_{2} = \Upsilon_{2}^{(1)}+ \Upsilon_{2}^{(2)}+\Upsilon _{2}^{(3)}+ \Upsilon_{2}^{(4)}+\Upsilon_{2}^{(5)}, \\& \Theta_{2} = \bigl(2Kb^{2}+\rho^{2}2^{2^{4}\kappa}C_{1}^{-4}( \alpha_{2})\Xi ^{2}b^{2^{2}}+\Phi_{2}\bigr) . \end{aligned}$$
(4.75)
By assumptions (4.52)–(4.53), for a small \(b>0\), it is easy to check that
$$ \Theta_{2}< 1. $$
When \(i=3\), by (4.73), we derive
$$\begin{aligned} \|E_{3}\|_{\sigma_{3}} \leq&2KC_{1}^{2}( \alpha_{3})\|E_{2}\|_{\sigma_{2}}^{2}+\rho ^{2}2^{2^{5}\kappa}\|E_{2}\|^{2^{2}}_{\sigma_{2}} +\rho^{2^{2}}2^{2^{6}\kappa}\|E_{1}\|^{2^{3}}_{\sigma_{1}} \\ &{}+\Upsilon_{3}^{(1)}+\Upsilon_{3}^{(2)}+ \Upsilon_{3}^{(3)}+\Upsilon _{3}^{(4)}+ \Upsilon_{3}^{(5)}, \\ \leq&2KC_{1}^{2}(\alpha_{3})C_{1}^{2^{2}}( \alpha_{2})\Xi^{2^{2}}\Theta_{2}^{2}+\rho ^{2}2^{2^{5}\kappa}C_{1}^{2^{3}}( \alpha_{2})\Xi^{2^{3}}\Theta_{2}^{2} \\ &{}+\rho^{2^{2}}2^{2^{6}\kappa}b^{2^{3}}\Xi^{2^{3}}+ \Upsilon_{3}^{(1)}+\Upsilon _{3}^{(2)}+ \Upsilon_{3}^{(3)}+\Upsilon_{3}^{(4)}+ \Upsilon_{3}^{(5)} \\ \leq&\Xi^{2^{2}}C_{1}^{2}(\alpha_{3})C_{1}^{2^{2}}( \alpha_{2})\Theta_{3}, \end{aligned}$$
where
$$\begin{aligned}& \Xi^{2}\Phi_{3} = \Upsilon_{3}^{(1)}+ \Upsilon_{3}^{(2)}+\Upsilon _{3}^{(3)}+ \Upsilon_{3}^{(4)}+\Upsilon_{3}^{(5)}, \\& \Theta_{3} = 2K\Theta_{2}^{2}+ \rho^{2}2^{2^{5}\kappa}C_{1}^{-2}(\alpha _{3})C_{1}^{2^{2}}(\alpha_{2}) \Xi^{2^{2}}\Theta_{2}^{2^{2}} \\& \hphantom{\Theta_{3} ={}}{} +C_{1}^{-2}(\alpha_{3})C_{1}^{-2^{2}}( \alpha_{2})\rho^{2^{2}}2^{2^{6}\kappa }b^{2^{3}} \Xi^{2^{2}}+C_{1}^{-2}(\alpha_{3})C_{1}^{-2^{2}}( \alpha_{2})\Phi_{3}. \end{aligned}$$
By assumptions (4.52)–(4.53), for a small \(b>0\), it is easy to check
$$ \Theta_{3}< 1. $$
For \(2\leq k\leq i-1\), assume that the following estimate holds:
$$ \|E_{k}\|_{\sigma_{k}}\leq\Xi^{2^{k-1}} C_{1}^{2}( \alpha_{k})C_{1}^{2^{2}}(\alpha_{k-1})\cdots C_{1}^{2^{k-1}}(\alpha_{2})\Theta_{k}, $$
where
$$\begin{aligned}& \Xi^{2^{k-1}}\Phi_{k} = \Upsilon_{k}^{(1)}+ \Upsilon_{k}^{(2)}+\Upsilon _{k}^{(3)}+ \Upsilon_{k}^{(4)}+\Upsilon_{k}^{(5)}, \\& \Theta_{k} = 2K\Theta^{2}_{k-1}+ \rho^{2}2^{(2^{4}-4)i\kappa}C_{1}^{-2}(\alpha _{k})C_{1}^{2^{2}}(\alpha_{k-1}) \Xi^{2^{k-1}}\Theta^{2^{2}}_{k-1} \\& \hphantom{\Theta_{k} ={}}{} +\rho^{2^{2}}2^{(2^{5}(i-1)-4i)\kappa}C_{1}^{-2}( \alpha_{k})C_{1}^{2^{2}}(\alpha _{k-1})\cdots C_{1}^{2^{i-1}}(\alpha_{2})\Xi^{2^{k-1}} \Theta_{i-2}^{2^{3}} \\& \hphantom{\Theta_{k} ={}}{} +\cdots \\& \hphantom{\Theta_{k} ={}}{} +C_{1}^{-2}(\alpha_{k})C_{1}^{-2^{2}}( \alpha_{k-1})\cdots C_{1}^{-2^{k-1}}( \alpha_{2})\rho^{2^{k-1}}2^{(2^{k+3}(i-k+1)-4i)\kappa}\Xi ^{2^{k-1}}b^{2^{k}} \\& \hphantom{\Theta_{k} ={}}{} +C_{1}^{-2}(\alpha_{k})C_{1}^{-2^{2}}( \alpha_{k-1})\cdots C_{1}^{-2^{k-1}}( \alpha_{2})\Xi^{-2^{k-2}}K^{2^{k}} \\& \hphantom{\Theta_{k} ={}}{} +C_{1}^{-2}(\alpha_{i})C_{1}^{-2^{2}}( \alpha_{i-1})\cdots C_{1}^{-2^{k-1}}(\alpha_{2}) \Phi_{k}, \end{aligned}$$
and
$$ \Theta_{k}< 1. $$
By (4.73), we get
$$\begin{aligned} \|E_{i}\|_{\sigma_{i}} \leq&C_{1}^{2}( \alpha_{i})\|E_{i-1}\|_{\sigma_{i-1}}^{2}+\sum _{k=1}^{i-1}\rho^{2^{k}}2^{[2^{k+2}(i-k+2)-4i]\kappa} \|E_{i-k}\|_{\sigma _{i-k}}^{2^{k+1}} \\ &{}+\Upsilon_{i}^{(1)}+\Upsilon_{i}^{(2)}+ \Upsilon_{i}^{(3)}+\Upsilon _{i}^{(4)} \\ \leq&C_{1}^{2}(\alpha_{i})C_{1}^{2^{2}}( \alpha_{i-1})\cdots C_{1}^{2^{i-1}}(\alpha_{2}) \Xi^{2^{i-1}}\Theta_{i-1}^{2} \\ &{}+\rho^{2}2^{(2^{4}-4)i\kappa}C_{1}^{2^{3}}( \alpha_{i-1})\cdots C_{1}^{2^{i}}(\alpha_{2}) \Xi^{2^{i}}\Theta^{2^{2}}_{i-1} \\ &{}+\rho^{2^{2}}2^{(2^{5}(i-1)-4i)\kappa}C_{1}^{2^{4}}( \alpha_{i-2})\cdots C_{1}^{2^{i}}(\alpha_{2}) \Xi^{2^{i}}\Theta_{i-2}^{2^{3}} \\ &{}+\cdots+\rho^{2^{i-1}}2^{(2^{i+3}-4i)\kappa}b^{2^{i}} \Xi^{2^{i}} +\Upsilon_{i}^{(1)}+\Upsilon_{i}^{(2)}+ \Upsilon_{i}^{(3)}+\Upsilon _{i}^{(4)}+ \Upsilon_{i}^{(5)} \\ \leq&C_{1}^{2}(\alpha_{i})C_{1}^{2^{2}}( \alpha_{i-1})\cdots C_{1}^{2^{i-1}}(\alpha_{2}) \Xi^{2^{i-1}}\Theta_{i}, \end{aligned}$$
(4.76)
where
$$\begin{aligned}& \Xi^{2^{i-1}}\Phi_{i} = \Upsilon_{i}^{(1)}+ \Upsilon_{i}^{(2)}+\Upsilon _{i}^{(3)}+ \Upsilon_{i}^{(4)}+\Upsilon_{i}^{(5)}, \\& \Theta_{i} = 2K\Theta^{2}_{i-1}+ \rho^{2}2^{(2^{4}-4)i\kappa}C_{1}^{-2}(\alpha _{i})C_{1}^{2^{2}}(\alpha_{i-1}) \Xi^{2^{i-1}}\Theta^{2^{2}}_{i-1} \\& \hphantom{\Theta_{i} ={}}{} +\rho^{2^{2}}2^{(2^{5}(i-1)-4i)\kappa}C_{1}^{-2}( \alpha_{i})C_{1}^{2^{2}}(\alpha _{i-1})\cdots C_{1}^{2^{i-1}}(\alpha_{2})\Xi^{2^{i-1}} \Theta_{i-2}^{2^{3}} \\& \hphantom{\Theta_{i} ={}}{} +\cdots \\& \hphantom{\Theta_{i} ={}}{} +C_{1}^{-2}(\alpha_{i})C_{1}^{-2^{2}}( \alpha_{i-1})\cdots C_{1}^{-2^{i-1}}(\alpha_{2}) \rho^{2^{i-1}}2^{(2^{i+3}-4i)\kappa}\Xi ^{2^{i-1}}b^{2^{i}} \\& \hphantom{\Theta_{i} ={}}{} +C_{1}^{-2}(\alpha_{i})C_{1}^{-2^{2}}( \alpha_{i-1})\cdots C_{1}^{-2^{i-1}}(\alpha_{2}) \Phi_{i}. \end{aligned}$$
Note that
$$ C_{1}^{2}(\alpha_{i})C_{1}^{2^{2}}( \alpha_{i-1})\cdots C_{1}^{2^{i-1}}(\alpha_{2}) \leq\bigl(2^{8\kappa}C_{1}^{2}(\kappa,\gamma,\sigma, \bar {\sigma})\bigr)^{2^{i-1}}. $$
Hence, by (4.76), it shows that
$$ \|E_{i}\|_{\sigma_{i}}\leq\bigl(2^{8\kappa}C_{1}^{2}( \kappa,\gamma,\sigma,\bar {\sigma})\Xi\bigr)^{2^{i-1}}\Theta_{i} $$
(4.77)
and
$$\begin{aligned} \Theta_{i} \leq&2K\Theta^{2}_{i-1}+ \rho^{2}C_{1}^{2}(\kappa,\gamma,\sigma ,\bar{ \sigma})\Xi^{2^{i-1}}\Theta^{2^{2}}_{i-1} + \rho^{2^{2}}\bigl(2^{8\kappa}C_{1}(\kappa,\gamma,\sigma, \bar{\sigma})\Xi \bigr)^{2^{i-1}}\Theta_{i-2}^{2^{3}} \\ &{}+\cdots+\bigl(2^{24\kappa}C_{1}^{2}(\kappa, \gamma,\sigma,\bar{\sigma})\Xi\rho b^{2}\bigr)^{2^{i-1}}+\bigl(4 \times2^{8\kappa}C_{1}^{2}(\kappa,\gamma,\sigma,\bar { \sigma})\bigr)^{-2^{i-1}}\Phi_{i}. \end{aligned}$$
By assumptions (4.52)–(4.53), for small \(b>0\), it is also easy to check
$$ \Theta_{i}< \sum_{i=1}^{\infty}2^{-i}=1. $$
Hence, we have
$$ 0\leq\lim_{i\rightarrow\infty}\|E_{i}\|_{\sigma_{i}}\leq\lim _{i\rightarrow \infty}\bigl(2^{8\kappa}C_{1}^{2}( \kappa,\gamma,\sigma,\bar{\sigma})\Xi \bigr)^{2^{i-1}}\Theta_{i} \rightarrow0, $$
which implies that
$$ \lim_{i\rightarrow\infty}\|E_{i}\|_{\sigma_{i}}=0. $$
This means that Eq. (4.26) has a solution \(a_{\infty}=\sum_{k=1}^{\infty}a_{k}\). This completes the proof. □

Lemma 4.7

(Existence of solution)

Let \(\omega\in\mathcal{X}(\nu)\). Assume (4.52)(4.53) hold. Then system (4.1)(4.2) has a solution \((u_{\infty},w_{\infty})\in X_{\bar{\sigma}}\times Y_{\bar{\sigma}}\) to be defined in (4.82)(4.83).

Proof

This result is to prove the existence of solution for system (4.1)–(4.2), i.e., \(\lim_{j\rightarrow\infty}\|u_{j}\|_{\sigma_{j}}\) and \(\lim_{j\rightarrow\infty}\|w_{j}\|_{\sigma_{j}}\) exist. Since the methods for the two are the same, we only prove the former.

From the definition of \(u_{i}\) in (4.30), for \(j\geq i\), we have
$$\begin{aligned}& \Vert u_{i}-u_{i-1} \Vert _{\sigma_{j+1}} \\& \quad \leq \Vert u_{i}-u_{i-1} \Vert _{\sigma _{i+1}} \\& \quad \leq \Biggl\Vert L_{\omega}^{-1}K\Biggl(\Biggl(\sum _{k=0}^{i}a_{k}-f_{2}^{(i-1)} \Biggr)^{+}-\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-2)} \Biggr)^{+}\Biggr) \Biggr\Vert _{\sigma_{i+1}} \\& \qquad {}+\epsilon \bigl\Vert \mathcal {L}_{\omega}^{-1}(S_{i-1}-S_{i-2}) \tilde{h}_{1} \bigr\Vert _{\sigma_{i+1}} \\& \quad \leq C(\alpha_{i+1}) \Biggl\Vert K\bigl(a_{i}+f_{2}^{(i-2)}-f_{2}^{(i-1)} \bigr)^{+}+\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-2)} \Biggr)^{+}-\Biggl(\sum_{k=0}^{i-1}a_{k}-f_{2}^{(i-2)} \Biggr)^{+} \Biggr\Vert _{\sigma_{i}} \\& \qquad {}+\epsilon \bigl\Vert \mathcal{L}_{\omega}^{-1}(S_{i-1}-S_{i-2}) \tilde{h}_{1} \bigr\Vert _{\sigma_{i+1}} \\& \quad = C(\alpha_{i+1}) \bigl\Vert K\bigl(a_{i}+f_{2}^{(i-2)}-f_{2}^{(i-1)} \bigr)^{+} \bigr\Vert _{\sigma_{i}} +\epsilon \bigl\Vert \mathcal{L}_{\omega}^{-1}(S_{i-1}-S_{i-2}) \tilde{h}_{1} \bigr\Vert _{\sigma_{i+1}} \\& \quad \leq C(\alpha_{i+1}) \bigl( \Vert a_{i} \Vert _{\sigma_{i}} + \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}} + \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}\bigr)+K\epsilon C(\alpha_{i})C( \alpha_{i-1})N_{i-1}^{-\alpha_{i-1}} \\& \qquad {}\times\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-1}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{i-1}}\bigr)+\epsilon C( \alpha_{i})N_{i-1}^{-\alpha_{i-1}} \Vert h_{1} \Vert _{\sigma_{i}}. \end{aligned}$$
(4.78)
In what follows, we estimate the term \(\|\tilde{f}_{3}^{(i-1)}\|_{\sigma_{i-1}}\) and \(\|\tilde{f}_{4}^{(i-1)}\|_{\sigma_{i-1}}\). By (4.59), (4.63), (4.68)–(4.71), it follows that
$$\begin{aligned}& \bigl\Vert \tilde{f}_{3}^{(i-1)} \bigr\Vert _{\sigma_{i-1}}+ \bigl\Vert \tilde{f}_{4}^{(i-1)} \bigr\Vert _{\sigma _{i-1}} \\& \quad \leq2\|a_{i-1}\|^{2}_{\sigma_{i-1}} +2C(\kappa,\gamma, \sigma,\bar{\sigma})\sum_{k=2}^{i-2}\rho ^{2^{k}}2^{[2^{k+1}(i-k+1)-4i+4]}\|a_{i-k}\|_{\sigma_{i-k}}^{2^{k}} \\& \qquad {}+\Upsilon_{i-1}^{(1)}+\Upsilon_{i-1}^{(2)}+ \Upsilon _{i-1}^{(3)}+\Upsilon_{i-1}^{(4)}+R_{6}, \end{aligned}$$
(4.79)
where
$$\begin{aligned}& \Upsilon_{i-1}^{(1)} = C^{2}(\kappa,\gamma,\sigma, \bar{\sigma})\bigl[12\times 2^{8\kappa} C^{2}(\kappa,\gamma, \sigma,\bar{\sigma}) \bigl( \bigl\Vert \tilde{f}_{3}^{(1)} \bigr\Vert _{\sigma _{1}}+ \bigl\Vert \tilde{f}_{4}^{(1)} \bigr\Vert _{\sigma_{1}}\bigr)\bigr]^{2^{i-2}}, \\& \Upsilon_{i-1}^{(2)} = e^{-\theta_{1}^{i-2}}C^{2}( \kappa,\gamma,\sigma ,\tilde{\sigma})\sum_{k=2}^{i-3}2^{[2^{k}(5i-5k-3)-4i+4]\kappa } \bigl[12K\epsilon C^{5}\bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{0}}+ \Vert \tilde{h}_{2} \Vert _{\sigma _{0}} \bigr)^{2}\bigr]^{2^{k}}, \\& \Upsilon_{i-1}^{(3)} = e^{-\theta_{1}^{i-2}}\sum _{k=2}^{i-3}2^{[2^{k}(5i-5k-3)-4i+4]\kappa}\bigl[6\epsilon C^{2}(\kappa,\gamma,\sigma,\bar{\sigma}) \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{0}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{0}}\bigr)\bigr]^{2^{k}}, \\& \Upsilon_{i-1}^{(4)} = e^{-\theta_{1}^{i-1}}C^{2}( \kappa,\gamma,\sigma ,\tilde{\sigma})\sum_{k=1}^{i-2} \bigl(12KC^{2}(\kappa,\gamma,\sigma,\bar{\sigma })\bigr)^{2^{k}} \times2^{[2^{k+1}(i-k+1)-4i+4]\kappa}, \\& R_{6} = 2\epsilon K\eta_{i-2} C(\alpha_{i-1})C( \alpha_{i-2})C(\alpha_{i-3})N_{i-3}^{-\alpha_{i-3}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-3}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-3}}\bigr) \\& \hphantom{R_{6} = {}}{} +\epsilon\eta_{i-2} C(\alpha_{i-1})N_{i-2}^{-\alpha_{i-2}} \bigl( \Vert \tilde{h}_{1} \Vert _{\sigma_{i-1}}+ \Vert \tilde{h}_{2} \Vert _{\sigma_{i-1}}\bigr)+4\eta_{i-2}C( \alpha_{i-1})C(\alpha _{i-2})K^{2}. \end{aligned}$$
Combining (4.78) with (4.79), we get
$$\begin{aligned}& \Vert u_{i}-u_{i-1} \Vert _{\sigma_{j+1}} \\& \quad \leq C( \alpha_{i+1})C_{1}(\alpha_{i}) \Vert E_{i-1} \Vert _{\sigma_{i-1}}+2C(\alpha_{i+1})C( \alpha_{i-1}) \Vert E_{i-2} \Vert ^{2}_{\sigma_{i-2}} \\& \qquad {}+2C^{2}(\kappa,\gamma,\sigma,\bar{\sigma})\sum _{k=2}^{i-2}\bigl(C(\kappa ,\gamma,\sigma,\bar{ \sigma})\rho\bigr)^{2^{k}}2^{2^{k+2}(i-k+1)-2i} \Vert E_{i-k-1} \Vert _{\sigma_{i-k-1}}^{2^{k}} \\& \qquad {}+C(\alpha_{i+1}) \bigl(\Upsilon_{i-1}^{(1)}+ \Upsilon_{i-1}^{(2)}+\Upsilon _{i-1}^{(3)}+ \Upsilon_{i-1}^{(4)}+R_{6}\bigr)+R_{7}, \end{aligned}$$
(4.80)
where
$$\begin{aligned}& \rho = 12C^{2}(\kappa,\gamma,\sigma,\bar{\sigma})C^{2}_{1}( \kappa,\gamma ,\sigma,\bar{\sigma}), \\& R_{7} = K\epsilon C^{2}(\kappa,\gamma,\sigma,\bar{ \sigma})2^{(2i-1)\kappa}e^{-\theta _{1}^{i-1}(\sigma-\bar{\sigma})}(\|\tilde{h}_{1} \|_{\sigma_{i-1}}+\|\tilde {h}_{2}\|_{\sigma_{i-1}}) \\& \hphantom{R_{7} ={}}{}+\epsilon C(\kappa,\gamma,\sigma,\bar{\sigma})2^{(i)\kappa}e^{-\theta _{1}^{i-1}(\sigma-\bar{\sigma})} \|h_{1}\|_{\sigma_{i}}. \end{aligned}$$
By (4.77), for \(\forall1\leq k\leq i\), it follows that
$$ \|E_{i-k}\|_{\sigma_{i-k}}\leq\bigl(2^{8\kappa}C_{1}^{2}( \kappa,\gamma,\sigma ,\bar{\sigma})\Xi\bigr)^{2^{i-k-1}}\Theta_{i-k}, $$
(4.81)
where
$$\begin{aligned} \Theta_{i} \leq&2K\Theta^{2}_{i-1}+ \rho^{2}C_{1}^{2}(\kappa,\gamma,\sigma ,\bar{ \sigma})\Xi^{2^{i-1}}\Theta^{2^{2}}_{i-1} + \rho^{2^{2}}\bigl(2^{8\kappa}C_{1}(\kappa,\gamma,\sigma, \bar{\sigma})\Xi \bigr)^{2^{i-1}}\Theta_{i-2}^{2^{3}} \\ &{}+\cdots+\bigl(2^{24\kappa}C_{1}^{2}(\kappa, \gamma,\sigma,\bar{\sigma})\Xi\rho b^{2}\bigr)^{2^{i-1}}+\bigl(4 \times2^{8\kappa}C_{1}^{2}(\kappa,\gamma,\sigma,\bar { \sigma})\bigr)^{-2^{i-1}}\Phi_{i} \\ < &1. \end{aligned}$$
Inserting (4.81) into (4.80), we derive
$$\begin{aligned} \|u_{i}-u_{i-1}\|_{\sigma_{j+1}} \leq&C( \alpha_{i+1})C_{1}(\alpha _{i}) \bigl(2^{8\kappa}C_{1}^{2}(\kappa,\gamma,\sigma,\bar{ \sigma})\Xi \bigr)^{2^{i-2}}\Theta_{i-1} \\ &{}+2C(\alpha_{i+1})C(\alpha_{i-1}) \bigl(2^{8\kappa}C_{1}^{2}( \kappa,\gamma,\sigma ,\bar{\sigma})\Xi\bigr)^{2^{i-2}}\Theta_{i-2}^{2} \\ &{}+2C^{2}(\kappa,\gamma,\sigma,\bar{\sigma}) \bigl(2^{8\kappa}C_{1}^{2}( \kappa ,\gamma,\sigma,\bar{\sigma})\Xi\bigr)^{2^{i-1}} \\ &{}\times\sum_{k=2}^{i-2}\bigl(C(\kappa, \gamma,\sigma,\bar{\sigma})\rho \bigr)^{2^{k}}2^{2^{k+2}(i-k+1)-2i} \Theta_{i-2}^{2^{k}} \\ &{}+C(\alpha_{i+1}) \bigl(\Upsilon_{i-1}^{(1)}+ \Upsilon_{i-1}^{(2)}+\Upsilon _{i-1}^{(3)}+ \Upsilon_{i-1}^{(4)}+R_{6}\bigr)+R_{7}. \end{aligned}$$
By (4.52)–(4.53), the above estimate implies that \(i\rightarrow\infty\) as \(j\rightarrow\infty\). Then
$$ \|u_{i}-u_{i-1}\|_{\sigma_{j}}\rightarrow0. $$
Therefore, by the relation \(\|u_{j}\|_{\sigma_{j}}=\|u_{0}+\sum_{i=1}^{j}(u_{i}-u_{i-1})\|_{\sigma_{j}}\leq\sum_{i=1}^{j}\|u_{i}-u_{i-1}\|_{\sigma_{i}}\), we find that \(\lim_{j\rightarrow\infty}\|u_{j}\|_{\sigma_{j}}\) exists.
Thus, the solution of system (4.1)–(4.2) is
$$\begin{aligned}& \tilde{u}_{\infty}=\mathcal{L}_{\omega}^{-1} \Biggl[K\Biggl(\sum_{k=1}^{\infty }a_{k}-f_{2}^{(\infty)} \Biggr)^{+}+\sum_{k=1}^{\infty}(m_{b}g)_{k}+ \epsilon \tilde{h}_{1}\Biggr], \end{aligned}$$
(4.82)
$$\begin{aligned}& \tilde{w}_{\infty}=\mathcal{J}_{\omega}^{-1} \Biggl[-K\Biggl(\sum_{k=1}^{\infty }a_{k}-f_{2}^{(\infty)} \Biggr)^{+}+\sum_{k=1}^{\infty}(m_{c}g)_{k}+ \epsilon \tilde{h}_{2}\Biggr], \end{aligned}$$
(4.83)
where
$$ f_{2}^{(\infty)}=\frac{K}{2}+\epsilon \tilde{L}_{\omega}(\tilde{h}_{1}+\tilde{h}_{2}). $$
(4.84)
 □

Lemma 4.8

(Local uniqueness of solution)

System (4.1)(4.2) has a unique solution \((u_{\infty},w_{\infty})\in X_{\bar{\sigma}}\times Y_{\bar{\sigma}}\), which is obtained in (4.82)(4.83).

Proof

Suppose that there exist two solutions \((u,w)\) and \((u',w')\) of system (4.1)–(4.2). Then we denote by ā and ã the corresponding solutions of Eq. (4.26), respectively. Let \((h^{(1)},h^{(2)})=(u-u',w-w')\) and \(h^{(3)}=\bar{a}-\tilde{a}\). Then, by (4.26), we have
$$\begin{aligned}& \mathcal{M}\bigl(h^{(3)},(m_{b}g)-(m_{b}g)' \bigr) \\& \quad = \Lambda h^{(3)}+K\bigl(h^{(3)}\bigr)^{+}+K( \bar{a}-f_{2})^{+}-K(\tilde {a}-f_{2})^{+}-K\bigl(h^{(3)} \bigr)^{+}-(m_{b}g)+(m_{b}g)' \\& \quad = \Psi_{1}\bigl(h^{(3)}\bigr)+\Psi_{2} \bigl(h^{(3)}\bigr)=0, \end{aligned}$$
(4.85)
where \(f_{2}\) is defined in (4.84), and
$$\begin{aligned}& \Psi_{1}\bigl(h^{(3)}\bigr) = \Lambda h^{(3)}+K \bigl(h^{(3)}\bigr)^{+}, \\& \Psi_{2}\bigl(h^{(3)}\bigr) = K(\bar{a}-f_{2})^{+}-K( \tilde {a}-f_{2})^{+}-K\bigl(h^{(3)}\bigr)^{+}-(m_{b}g)+(m_{b}g)'. \end{aligned}$$
Take \((m_{b}g)=(m_{b}g)'\). Then, by the definition of \(\Psi_{2}(h^{(3)})\), we derive
$$\begin{aligned} \bigl\Vert \Psi_{2}\bigl(h^{(3)}\bigr) \bigr\Vert _{0} =& \bigl\Vert K(\bar{a}-f_{2})^{+}-K(\tilde {a}-f_{2})^{+}-K\bigl(h^{(3)}\bigr)^{+}-(m_{b}g)+(m_{b}g)' \bigr\Vert _{0} \\ \leq& \bigl\Vert K\bigl(h^{(3)}\bigr)^{+}+K(\tilde{a}-f_{2})^{+}-K( \tilde {a}-f_{2})^{+}-K\bigl(h^{(3)}\bigr)^{+} \bigr\Vert _{0} \\ \leq&0, \end{aligned}$$
which means \(\Psi_{2}(h^{(3)})=0\). Then, from (4.85), we get \(\Psi_{1}(h^{(3)})=\Lambda h^{(3)}+K(h^{(3)})^{+}=0\). Therefore, by Lemma 4.2, \(\bar{a}=\tilde{a}\). Then, from the definition \((u,w)\) and \((u',w')\) in (4.82)–(4.83), we obtain \(u=u'\) and \(w=w'\). This completes the proof. □

Declarations

Acknowledgements

The authors express sincere thanks to the anonymous referees for very careful reading and for providing many valuable comments and suggestions, which led to improvement of this paper. The second author is supported by program for innovative research team of Huizhou University (IRTHZU).

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Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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Authors’ Affiliations

(1)
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou, P.R. China
(2)
School of Mathematics and Big Data, Huizhou University, Huizhou, P.R. China

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