Generalized $(f, \lambda)$-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces

1.   Introduction

In this paper, we consider the following Rao-Nakra sandwich beam with time-varying weights and frictional dampings in $(x, t) \in (0, L) \times (0, \infty)$,

in which $u$ and $v$ denote the longitudinal displacement and shear angle of the bottom and top layers, and $w$ represents the transverse displacement of the beam. The positive constants $\rho_i$, $h_i$, and $E_i$ $(i = 1, 3)$ are physical parameters representing, respectively, density, thickness, and Young's modulus of the $i$-th layer for $i = 1, 2, 3$ and $\rho h = \rho_1 h_1 + \rho_2 h_2 + \rho_3 h_3$. $E I = E_1 I_1 + E_3 I_3$, $\alpha = h_2 + \left(\frac{h_1 + h_3}{2}\right)$, $k = \frac{1}{h_2} \left(\frac{E_2}{2(1+\mu)} \right)$, where $\mu$ is the Poisson ratio $-1 < \mu < \frac{1}{2}$. The functions $\alpha_1(t)$, $\alpha_2(t)$, $\alpha_3(t)$ are the time dependent of the nonlinear frictional dampings $f_1\left(u_t\right)$, $f_2\left(v_t\right)$, $f_3\left(w_t\right)$, respectively. For the system (1.1), we consider the following Dirichlet-Neumann boundary conditions

and the initial conditions

The system (1.1)–(1.3) consists of one Euler-Bernoulli beam equation for the transversal displacement, and two wave equations for the longitudinal displacements of the top and bottom layers.

Our aim is to investigate the asymptotic behavior of solutions for the system (1.1)–(1.3). We study the effect of the three nonlinear dampings on the asymptotic behavior of the energy function. Under nonrestrictive on the growth assumption on the frictional damping terms, we establish exponential and general energy decay rates for this system by using the multiplier approach. The results generalize some earlier decay results on the Rao-Nakra sandwich beam equation.

First, let's review some previous findings about multilayered sandwich beam models. By applying the Riesz basis approach, Wang et al. ^[1] studied a sandwich beam system with a boundary control and established the exponential stability as well as the exact controllability and observability of the system. The author of ^[2] employed the multiplier approach to determine the precise controllability of a Rao-Nakra sandwich beam with boundary controls rather than the Riesz basis approach. Hansen and Imanuvilov ^[3,4] investigated a multilayer plate system with locally distributed control in the boundary and used Carleman estimations to determine the precise controllability results. Özer and Hansen ^[5,6] succeeded in obtaining, for a multilayer Rao-Nakra sandwich beam, boundary feedback stabilization and perfect controllability. One viscous damping effect on either the beam equation or one of the wave equations was all that was taken into account by Liu et al. ^[7] when they established the polynomial decay rate using the frequency domain technique. The semigroup created by the system is polynomially stable of order $1/2$, according to Wang ^[8], who analyzed a Rao-Nakra beam with boundary damping only on one end for two displacements using the same methodology. One can find additional results on the multilayer beam in ^{[9,10,11,12,13,14,15]}.

In this paper, we consider a Rao-Nakra sandwich beam with time-varying weights and frictional dampings, i.e., system (1.1)–(1.3). The main results are twofold:

(I) We establish an exponential decay of the system in the case of linear frictional dampings by using the multiplier approach, and the decay result depends on the time-varying weights $\alpha_i$.

(II) We establish more general decay of the system in the case of nonlinear frictional dampings by using the multiplier approach, and the decay result depends on the time-varying weights $\alpha_i$ and the frictional dampings $f_i$. To the best of our knowledge, there is no stability results on the Rao-Nakra sandwich beam with nonlinear frictional dampings. The remainder of the paper is as follows. In Section 2, we introduce some notations and preliminary results. In Section 3, we state the theorem of the stability and give a detailed proof.

2.   Preliminaries

In this section, we present some materials needed in the proof of our results. Throughout this paper, $c$ and $\varepsilon$ are used to denote generic positive constants. We consider the following assumptions:

$(H1)$ For $(i = 1, 2, 3)$, the functions $f_i:\mathbb{R} \rightarrow \mathbb{R}$ are $C^{0}$ nondecreasing satisfying, for $c_1, c_2 > 0$,

where $F_i: (0, \infty)\rightarrow (0, \infty)$ $(i = 1, 2, 3)$ are $C^{1}$ functions, which are linear or strictly increasing and strictly convex $C^{2}$ functions on $(0, r_i]$ with $F_i(0) = F_i^{\prime}(0) = 0$.

$(H2)$ For $(i = 1, 2, 3)$, the time dependent functions $\alpha_i: [0, \infty) \to (0, \infty)$ are $C^1$ functions satisfying $\int_{0}^{\infty}\alpha_i(t)dt = \infty$.

Remark 2.1. (1) Hypothesis $(H1)$ implies that $s f_i(s) > 0,$ for all $s\ne 0$. This condition $(H1)$ was introduced and employed by Lasiecka and Tataru ^[16]. It was shown there that the monotonicity and continuity of $f_i$ guarantee the existence of the function $F_i$ with the properties stated in $(H1)$.

(2) For more results on the convexity properties on the nonlinear frictional dampings and sharp energy decay rates, we refer to the works by Boussouira and her co-authors ^[17,18,19].

3.   Essential lemmas

The following lemmas will be of essential use in establishing our main results.

Lemma 3.1. ^[20] Let $E:\mathbb{R}^+\rightarrow \mathbb{R}^+$ be a nonincreasing function and $\gamma :\mathbb{R}^+\rightarrow \mathbb{R}^+$ be a strictly increasing $C^{1}$-function, with $\gamma (t)\rightarrow +\infty$ as $t\rightarrow +\infty.$ Assume that there exists $c > 0$ such that

then there exist positive constants $k$ and $\omega$ such that

Lemma 3.2. Let $E:\mathbb{R}^+\rightarrow \mathbb{R}^+$ be a differentiable and nonincreasing function and let $\chi: \mathbb{R}^+\rightarrow \mathbb{R}^+$ be a convex and increasing function such that $\chi(0) = 0$. Assume that

then $E$ satisfies the estimate

where $\psi(t) = \int_{t}^{1}\frac{1}{\chi(s)}ds$ for $t > 0$, and

Proof. Since $E'(t) \leq 0$, this implies $E(0) \leq E(t_0)$ for all $t \geq t_0 \geq 0$. If $E(t_0) = 0$ for $t_0 \geq 0$, then $E(t) = 0$ for all $t \geq t_0 \geq 0$, and there is nothing to prove in this case. As in ^[21], we assume that $E(t) > 0$ for $t \geq 0$ without loss of generality. Let

We have $L(s) \leq E(s), \; \forall\; s \geq 0$. The functional $L$ is positive, decreasing, and of class $C^1\left(0, \infty\right)$ satisfying

Since the functional $\chi$ is decreasing, we have

Integrating this differential equation over $(0, t)$, we get

Since $\chi$ is convex and $\chi(0) = 0$, we have

We find $\lim_{t \rightarrow 0} \psi(t) = \infty$ and $\left[\psi\left(E(0)\right), \infty)\right) \subset \text{Image}\left(\psi\right),$ then (3.3) imply that

Now, using the properties of $\chi$ and $E$, we have

Since $\lim_{t \rightarrow 0} \chi(t) = \infty, \; \chi(0) = 0$, and $\chi$ is increasing, (3.4) and (3.5) imply that

Now, let $t > \frac{E(0)}{\chi \left(E(0) \right)}$ and $J(s) = \dfrac{\psi^{-1} \left(t +\psi\left(E(0)\right)\right)}{t-s}, \; s\in[0, t)$.

The function $J$ is differentiable, then we have

Thus, $J'(s) = 0 \Leftrightarrow K(s) = t$ and $J'(s) < 0 \Leftrightarrow K(s) < t,$ where

Since $K(0) = \frac{E(0)}{\chi\left(E(0) \right)}$ and $K$ is increasing (because $\psi^{-1}$ is decreasing and $s \rightarrow \frac{s}{\chi(s)}, s > 0$ is nonincreasing thanks to the fact $\chi$ is convex), for $t > \frac{E(0)}{\chi\left(E(0) \right)}$, we have

Since $h$ satisfies $J' \left (h(t)\right) = 0$, we conclude from (3.6) our desired estimate (3.2). □

We define the energy associated to the problem (1.1)–(1.3) by the following formula

Lemma 3.3. The energy $E(t)$ satisfies

Proof. Multiplying (1.1)$_1$ by $u_t$, (1.1)$_2$ by $v_t$, (1.1)$_3$ by $w_t$ and integrating each of them by parts over $(0, L)$, we get the desired result. □

4.   Stability result

In this section, we state and prove our main result. For this purpose, we establish some lemmas. From now on, we denote by $c$ various positive constants, which may be different at different occurrences. For simplicity, we consider the case $\alpha(t) = \alpha_1(t) = \alpha_2(t) = \alpha_3(t)$.

Lemma 4.1. (Case: $F_i$ is linear) For $T > S \geq 0$, the energy functional of the system (1.1)–(1.3) satisfies

Proof. Multiplying (1.1)$_1$ by $\alpha u$ and integrating over $(S, T) \times (0, L)$, we obtain

Notice that

Using integration by parts and the boundary conditions, we get

Adding $2\int_S^T \alpha(t) \int_0^L \rho_1h_1 u_t^2 dx dt$ to both sides of the above equation, we have

Similarly, multiplying (1.1)$_2$ by $\alpha(t) v$ and (1.1)$_3$ by $\alpha(t) w$, and integrating each of them over $(S, T) \times (0, L)$, we obtain

and

Recalling the definition of $E$, and from (4.2)–(4.5), we get

Now, using Young's and Poincaré inequalities, we get for any $\varepsilon_i > 0 \; (i = 1, 2, 3)$,

which implies that

and

Using $(H1)$, the fact that $F_1$ is linear, Hölder and Poincaré inequalities, we obtain

Applying Young's inequality on the term $E^{\frac{1}{2}}(t) (-E'(t))^{\frac{1}{2}}$, we obtain for $\varepsilon > 0$

which implies that

In the same way, we have

and

Finally, using $(H1)$, and the fact that $F_1$ is linear, we find that

Similarly, we get

We combine the above estimates and take $\varepsilon$ small enough to get the estimate (4.1). □

Lemma 4.2. (Case: $F_i$ are nonlinear) For $T > S \geq 0$, the energy functional of the system (1.1)–(1.3) satisfies

where $\Lambda$ is convex, increasing, and of class $C^1[0, \infty)$ such that $\Lambda(0) = 0$.

Proof. We multiply $(1.1)_1$ by $\alpha(t) \frac{\Lambda(E)}{E}u$ and integrate over $(0, L) \times (S, T)$ to get

Also, we multiply $(1.1)_2$ by $\alpha(t) \frac{\Lambda(E)}{E}v$ and integrate over $(0, L) \times (S, T)$ to get

Similarly, we multiply $(1.1)_3$ by $\alpha(t) \frac{\Lambda(E)}{E}w$ and integrate over $(0, L) \times (S, T)$ to get

Integrating by parts in the first term of the righthand side of the Eq (4.16), we find that Eq (4.16) becomes

Using the fact that $\int_0^L uu_t dx \le cE(t)$, the properties of $\alpha(t)$, and the facts that the function $s\to \frac{\Lambda(s)}{s}$ is nondecreasing and $E$ is nonincreasing, we have

Similarly, we get

Combining (4.19)–(4.20), we have

Similarly, we have

and

Gathering all the above estimations by using (3.8), we obtain

This completes the proof of the estimate (4.15). □

In order to finalize the proof of our result, we let

where $F_i^*$ and $\Psi_i^*$ denote the dual functions of the convex functions $F_i$ and $\Psi_i$, respectively, in the sense of Young (see Arnold ^[22], pp. 64).

Lemma 4.3. Suppose $F_i (i = 1, 2, 3)$ are nonlinear, then the following estimates

and

hold.

Proof. We prove for $i = 1$, and the remaining are similar. Since $F_1^*$ and $\Psi_1^*$ are the dual functions of the convex functions $F_1$ and $\Psi_1$, respectively, then

and

Using (4.24) and the definition of $\Lambda$, we obtain (4.22).

For the proof of (4.23), we use (4.25) and the definitions of $\Psi_1$ and $\Lambda$ to obtain

□

Now, we state and prove our main decay results.

Theorem 4.4. Let $(u_0, u_1)\times (z_0, z_1) \in \left[H^2_{0}(0, L)\times L^2(0, L)\right]^2$. Assume that $(H1)$ and $(H2)$ hold, then there exist positive constants $k$ and $c$ such that, for $t$ large, the solution of the system (1.1)–(1.3) satisfies

where $\tilde{\alpha}(t) = \int_{0}^{t}\alpha(t)dt$, $\psi(t) = \int_{t}^{1}\frac{1}{\chi(s)}ds,$ $\chi(t) = c \Lambda(t)$, and

Proof. To establish (4.27), we use (4.1), and Lemma 3.1 for $\gamma(t) = \int_{0}^{t}\alpha(s)ds$. Consequently, the result follows.

For the proof of (4.28), we re-estimate the terms of (4.15) as follows:

We consider the following partition of the domain $(0, L)$:

So,

Using the definition of $\Omega_1$, condition $(H1)$, and (3.9), we have

After applying Young's inequality, and the condition $(H1)$, we obtain

The definition of $\Omega_1$, the condition $(H1)$, (3.8), (3.9), and (4.30) lead to

Using the definition of $\Lambda$, and the convexity of $F_1$, (4.31) becomes

Using Young's inequality, Jensen's inequality, condition $(H1)$, and (3.8), we get

We apply the generalized Young inequality

to the first term of (4.33) with $A = \frac{\Lambda(E)}{E}$ and $B = F_1^{-1}\left(\frac{1}{L}\int_0^L u_t f_1(u_t)dx\right)$ to get

We then apply it to the second term of (4.33) with $A = \frac{\Lambda(E)}{E} \sqrt{E}$ and

$B = \sqrt{L F_1^{-1}\left(\frac{1}{L}\int_0^L u_t f_1(u_t)dx\right)}$ to obtain

Combining (4.33)–(4.35), using (4.22), and (4.23), we arrive at

Using the fact that $s \to (F_1^{\prime})^{-1}(s)$ is nondecreasing, we deduce that, for $0 < \varepsilon_0\le \sqrt{E(0)}$,

Therefore, combining (4.15), (4.29), and (4.32), we find that

and similarly,

and

Combining all the above estimations with choosing $\varepsilon$ and $\varepsilon_0$ small enough, we arrive at

Using the facts that $E$ is nonincreasing and $s \to \frac{\Lambda(s)}{s}$ is nondecareasing, we can deduce that

Now, let $\tilde{E} = E\circ \tilde{\alpha}^{-1}$, where $\tilde{\alpha}(t) = \int_{0}^{t}\alpha(s)ds$, then we deduce from this inequality that

Using Lemma 3.1 for $\tilde{E}$ and $\chi(s) = c \Lambda(s)$, we deduce from (3.1) the following estimate

which, using the definition of $\tilde{E}$ and the change of variables, gives (4.28). □

Remark 4.1. The stability result (4.28) is a decay result. Indeed,

where $F = \min \{ F_i \}$ and $i = 1, 2, 3.$ Hence, $\lim_{t\to\infty}h^{-1}(t) = \infty$, which implies that $\lim_{t \to\infty}h(t) = \infty$. Using the convexity of $F$, we have

where $F = \min \{ F_i \}$ and $i = 1, 2, 3.$ Therefore, $\lim_{t\to 0^+}\psi(t) = \infty$, which leads to $\lim_{t\to \infty}\psi^{-1}(t) = 0$.

5.   Examples

Example 1. Let $f_i(s) = s^m$, $(i = 1, 2, 3)$, where $m\ge 1$, then the function $F$ ($F = \min \{ F_i \}$) is defined in the neighborhood of zero by

which gives, near zero,

So,

then in the neighborhood of $\infty$,

Using the fact that $h(t) = t$ as $t$ goes to infinity, we obtain from (4.27) and (4.28):

Example 2. Let $f_i(s) = s^m \sqrt{-\ln{s}}$, $(i = 1, 2, 3)$, where $m\ge 1$, then the function $F$ is defined in the neighborhood of zero by

which gives, near zero,

Therefore,

then in the neighborhood of $\infty$,

Using the fact that $h(t) = t$ as $t$ goes to infinity, we obtain

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

The authors would like to acknowledge the support provided by King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia. The support provided by the Interdisciplinary Research Center for Construction & Building Materials (IRC-CBM) at King Fahd University of Petroleum & Minerals (KFUPM), Saudi Arabia, for funding this work through Project No. INCB2402, is also greatly acknowledged.

Conflict of interest

The authors declare that there is no conflict of interest.