**2. Problem Formulation**

In this work, we consider the equation of the form

$$D\_{0t}^{\mathfrak{a}}u(\mathbf{x},t) + (-\Delta)^{\mathbf{v}}u(\mathbf{x},t) = f(\mathbf{x},t), \ (\mathbf{x},t) \in \Pi \times (0,\infty), \ l - 1 < \mathfrak{a} \le l, \ l, \mathbf{v} \in N \tag{4}$$

with initial conditions

$$\lim\_{t \to 0} D\_{0t}^{a-k} u(\mathbf{x}, t) = \varphi\_k(\mathbf{x}), \quad k = 1, 2, \dots, l \tag{5}$$

and boundary conditions

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *<sup>α</sup><sup>j</sup>* · (−Δ)*<sup>i</sup> <sup>u</sup>*(*x*1, ..., *xj*, ..., *xN*, *<sup>t</sup>*) <sup>|</sup>*xj*=<sup>0</sup> <sup>+</sup>*β<sup>j</sup>* · (−Δ)*<sup>i</sup> u*(*x*1, ..., *xj*, ..., *xN*, *t*) |*xj*=*π*= 0, 1 ≤ *j* ≤ *p*, *β<sup>j</sup>* · *<sup>∂</sup>*(−Δ)*<sup>i</sup> u*(*x*1,...,*xj*,...,*xN*,*t*) *<sup>∂</sup>xj* |*xj*=<sup>0</sup> +*α<sup>j</sup>* · *<sup>∂</sup>*(−Δ)*<sup>i</sup> u*(*x*1,...,*xj*,...,*xN*,*t*) *<sup>∂</sup>xj* |*xj*=*π*= 0, 1 ≤ *<sup>j</sup>* ≤ *<sup>p</sup>*, (−Δ)*<sup>i</sup> <sup>u</sup>*(*x*1, ..., *xj*, ..., *xN*, *<sup>t</sup>*) <sup>|</sup>*xj*=0= (−Δ)*<sup>i</sup> u*(*x*1, ..., *xj*, ..., *xN*, *t*) |*xj*=*π*, *p* + 1 ≤ *j* ≤ *q*, *<sup>∂</sup>*(−Δ)*<sup>i</sup> u*(*x*1,...,*xj*,...,*xN*,*t*) *<sup>∂</sup>xj* <sup>|</sup>*xj*=0<sup>=</sup> *<sup>∂</sup>*(−Δ)*<sup>i</sup> u*(*x*1,...,*xj*,...,*xN*,*t*) *<sup>∂</sup>xj* |*xj*=*π*, *<sup>p</sup>* + <sup>1</sup> ≤ *<sup>j</sup>* ≤ *<sup>q</sup>*, (−Δ)*<sup>i</sup> u*(*x*1, ..., *xj*, ..., *xN*, *t*) |*xj*=0= 0, *q* + 1 ≤ *j* ≤ *N*, (−Δ)*<sup>i</sup> u*(*x*1, ..., *xj*, ..., *xN*, *t*) |*xj*=*π*= 0, *q* + 1 ≤ *j* ≤ *N*, 1 ≤ *p* ≤ *q* ≤ *N*, *i* = 0, 1, . . . , *ν* − 1, (6)

where (*x*, *t*)=(*x*1, ... , *xj*, ... , *xN*, *t*) ∈ Π × (0, ∞), Π = (0, *π*) ×···× (0, *π*), *α<sup>j</sup>* = *const*, *β<sup>j</sup>* = *const*, and *f*(*x*, *t*), *ϕk*(*x*), *k* = 1, 2, ... , *l* are functions that can be expanded in terms of the system of eigenfunctions {*vn*(*x*), *<sup>n</sup>* <sup>∈</sup> *<sup>Z</sup>N*} of the spectral problem:

$$
\mu(-\Delta)^{\nu}v(\mathbf{x}) = \mu v(\mathbf{x}),
\tag{7}
$$

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *<sup>α</sup><sup>j</sup>* · (−Δ)*<sup>i</sup> <sup>v</sup>*(*x*1,..., *xj*,..., *xN*) <sup>|</sup>*xj*=<sup>0</sup> <sup>+</sup>*β<sup>j</sup>* · (−Δ)*<sup>i</sup> v*(*x*1,..., *xj*,..., *xN*) |*xj*=*π*= 0, 1 ≤ *j* ≤ *p*, *β<sup>j</sup>* · *<sup>∂</sup>*(−Δ)*<sup>i</sup> v*(*x*1,...,*xj*,...,*xN*) *<sup>∂</sup>xj* |*xj*=<sup>0</sup> +*α<sup>j</sup>* · *<sup>∂</sup>*(−Δ)*<sup>i</sup> v*(*x*1,...,*xj*,...,*xN*) *<sup>∂</sup>xj* |*xj*=*π*= 0, 1 ≤ *<sup>j</sup>* ≤ *<sup>p</sup>*, (−Δ)*<sup>i</sup> <sup>v</sup>*(*x*1,..., *xj*,..., *xN*) <sup>|</sup>*xj*=0= (−Δ)*<sup>i</sup> v*(*x*1,..., *xj*,..., *xN*) |*xj*=*π*, *p* + 1 ≤ *j* ≤ *q*, *<sup>∂</sup>*(−Δ)*<sup>i</sup> v*(*x*1,...,*xj*,...,*xN*) *<sup>∂</sup>xj* <sup>|</sup>*xj*=0<sup>=</sup> *<sup>∂</sup>*(−Δ)*<sup>i</sup> v*(*x*1,...,*xj*,...,*xN*) *<sup>∂</sup>xj* |*xj*=*π*, *<sup>p</sup>* + <sup>1</sup> ≤ *<sup>j</sup>* ≤ *<sup>q</sup>*, (−Δ)*<sup>i</sup> v*(*x*1,..., *xj*,..., *xN*) |*xj*=0= 0, *q* + 1 ≤ *j* ≤ *N*, (−Δ)*<sup>i</sup> v*(*x*1,..., *xj*,..., *xN*) |*xj*=*π*= 0, *q* + 1 ≤ *j* ≤ *N*, 1 ≤ *p* ≤ *q* ≤ *N*, *i* = 0, 1, . . . , *ν* − 1. (8)

Here, for *α* < 0,, fractional integral *D<sup>α</sup>* has the form

$$D\_{at}^{\alpha}u(x,t) = \frac{\operatorname{sign}(t-a)}{\Gamma(-\alpha)} \int\_{a}^{t} \frac{u(x,\tau) \cdot d\tau}{|t-\tau|^{\alpha+1}} d\tau$$

*D<sup>α</sup> atu*(*x*, *t*) = *u*(*x*, *t*) for *α* = 0, and for *l* − 1 < *α* ≤ *l*, *l* ∈ *N*, the fractional derivative has the form

$$\begin{aligned} D\_{at}^{\alpha}u(\mathbf{x},t) &= \operatorname{sign}^l(t-a) \frac{d^l}{dt^l} D\_{at}^{\alpha-l} u(\mathbf{x},t) = 0 \\ &= \frac{\operatorname{sign}^{l+1}(t-a)}{\Gamma(l-a)} \frac{d^l}{dt^l} \int\_a^t \frac{u(\mathbf{x},\tau) \cdot d\tau}{|t-\tau|^{\alpha-l+1}}. \end{aligned}$$

In [17], Problems (4)–(6) and, accordingly, spectral Problems (7) and (8) in the case *ν* = 1, were considered.

#### **3. Preliminaries**

More detailed information for this section can be found in [17]. We look for eigenfunctions of spectral Problems (7) and (8) in the form of the product *v*(*x*) = *y*1(*x*1) ····· *yN*(*xN*). Then, we obtain, instead of spectral Problems (7) and (8), the following spectral problem:

$$-y''(\mathbf{x}) = \mu y(\mathbf{x}), \mu = \lambda^2 \tag{9}$$

$$\begin{cases} \arg(0) + \beta y(\pi) = 0, \\ \beta y'(0) + \alpha y'(\pi) = 0. \end{cases} \tag{10}$$

In the case of |*α*| = |*β*|, i.e., with boundary conditions *y*(0) = *y*(*π*), *y* (0) = *y* (*π*) or *y*(0) = −*y*(*π*), *y* (0) = −*y* (*π*), spectral Problems (7) and (8) were investigated by many authors (see, for example, [35–41]). In order to simplify calculations, we confined ourselves to the case of |*α*| = |*β*|, *α* = 0, *β* = 0. It is not difficult to see that *μ* = 0 is not an eigenvalue of Problems (9) and (10). In fact, if *μ* = 0 is the eigenvalue, then *y* = 0, *y* = *ax* + *b*, *αb* + *β*(*aπ* + *b*) = 0, *βa* + *αa* = 0. We obtained from here *a* = 0, *b* = 0, i.e., *y* ≡ 0. Similarly, for *μ* < 0, Problems (9) and (10) have no nontrivial solutions.

For *μ* > 0, the general solution of Problem (9) has the form

$$y(\mathbf{x}) = A\cos\lambda\,\mathbf{x} + B\sin\lambda\,\mathbf{x}.$$

From boundary conditions, we have:

$$
\alpha y(0) + \beta y(\pi) = \alpha A + \beta (A \cos \lambda \pi + B \sin \lambda \pi) = 0,
$$

$$\begin{aligned} \beta y'(0) + \alpha y'(\pi) &= \beta(\lambda B) + \alpha(\lambda B \cos \lambda \pi - \lambda A \sin \lambda \pi) = 0, \\\\ \begin{cases} (\alpha + \beta \cos \lambda \pi)A + \beta \sin \lambda \pi B = 0, \\\ a \sin \lambda \pi A - (\beta + \alpha \cos \lambda \pi)B = 0. \end{cases} \end{aligned}$$

i.e.,

$$\text{Hence, the nontrivial solutions of Problems (9) and (10) are only possible in the case of } \frac{1}{2}$$

$$(\alpha + \beta \cos \lambda \pi)(-\beta - \alpha \cos \lambda \pi) - \alpha \beta \sin^2(\lambda \pi) = 0.$$

Furthermore,

$$-a\beta - a^2 \cos \lambda \pi - \beta^2 \cos \lambda \pi - a\beta \cos^2 \lambda \pi - a\beta \sin^2 \lambda \pi = 0,$$

i.e., <sup>−</sup>(*α*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*2)*cosλπ* <sup>=</sup> <sup>2</sup>*αβ* or *cosλπ* <sup>=</sup> <sup>−</sup>2*αβ <sup>α</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> .

Therefore, *λπ* <sup>=</sup> arccos <sup>−</sup>2*αβ <sup>α</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> or

$$
\lambda \pi = \pm \arccos \frac{-2a\beta}{\pi^2 + \beta^2} + 2n\pi, \quad n \in Z.
$$

Further,

$$\mu\_n^{\pm} = (2n + \varepsilon\_n \varrho)^2 = (-2n - \varepsilon\_n \varrho)^2 = \mu\_{-n'}^{\mp}, \ \varepsilon\_n = \pm 1, \ \varrho = \frac{1}{\pi} \arccos \frac{-2a\beta}{a^2 + \beta^2}, n \in \mathbb{Z}.$$

That's why *μ*± *<sup>n</sup>* = *μ*<sup>±</sup> <sup>−</sup>*<sup>n</sup>* means that *<sup>ε</sup>*−*<sup>n</sup>* <sup>=</sup> <sup>−</sup>*εn*, i.e., *<sup>ε</sup>*−*<sup>n</sup>* <sup>=</sup> *<sup>ε</sup>n*, *<sup>n</sup>* <sup>∈</sup> *<sup>Z</sup>*. Thus, the eigenvalues and eigenfunctions of Problems (9) and (10) are

$$\mu\_n = \lambda\_n^2 = (2n + \varepsilon\_n \varrho)^2, \ \varrho = \frac{1}{\pi} \arccos \frac{-2a\beta}{a^2 + \beta^2}, \ \varepsilon\_n = \pm 1, \varepsilon\_{-n} = \varepsilon\_n, \ n \in Z\_+$$

and

$$y\_n(\mathbf{x}) = B\_n \left( \frac{\beta + \kappa \cos \lambda\_n \pi}{\kappa \sin \lambda\_n \pi} \cos \lambda\_n \mathbf{x} + \sin \lambda\_n \mathbf{x} \right),$$

respectively, where

$$\frac{\beta + a \cos \lambda\_n \pi}{a \sin \lambda\_n \pi} = \frac{\beta - \frac{2a^2 \beta}{a^2 + \beta^2}}{\varepsilon\_n a \sqrt{1 - \frac{4a^2 \beta^2}{(a^2 + \beta^2)^2}}} = \frac{\beta(\beta^2 - a^2)}{\varepsilon\_n a \left| \ \beta^2 - a^2 \right|} = \varepsilon\_n \operatorname{sign}(\beta^2 - a^2) \frac{\beta}{a} \sqrt{1 - \frac{4a^2 \beta}{a^2 + \beta^2}}$$

hence, *yn*(*x*) = *Bn <sup>ε</sup>nsign*(*β*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*2) *<sup>β</sup> <sup>α</sup>* cos *λnx* + *sinλnx* . Choosing

$$B\_{\rm ll} = \varepsilon\_{\rm n} \text{sign}(\beta^2 - \alpha^2) \frac{\alpha}{\sqrt{\alpha^2 + \beta^2}} \sqrt{\frac{2}{\pi}} \frac{1}{\sqrt{1 + (2n)^{2s}}}$$

we obtain

$$y\_n(\mathbf{x}) = \sqrt{\frac{2}{\pi}} \frac{1}{\sqrt{a^2 + \beta^2}} \frac{1}{\sqrt{1 + (2n)^{2s}}} \left(\beta \cos \lambda\_n \mathbf{x} + \varepsilon\_n \operatorname{sign}(\beta^2 - a^2) a \sin \lambda\_n \mathbf{x}\right).$$

Denote *ω<sup>n</sup>* = <sup>2</sup> *<sup>π</sup>* <sup>√</sup> <sup>1</sup> *<sup>α</sup>*2+*β*<sup>2</sup> <sup>√</sup> <sup>1</sup> <sup>1</sup>+(2*n*)2*<sup>s</sup>* . Then, *yn*(*x*) = *ω<sup>n</sup> <sup>β</sup>* cos *<sup>λ</sup>nx* <sup>+</sup> *<sup>ε</sup>nsign*(*β*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*2)*<sup>α</sup>* sin *<sup>λ</sup>nx* .

The norm in space *W<sup>s</sup>* <sup>2</sup>(0, *π*) is introduced as follows:

$$\left\| f \right\|\_{\mathcal{W}\_{\underline{\mathbb{Q}}}(0,\pi)}^2 = \left\| f \right\|\_{L\_2(0,\pi)}^2 + \left\| D^s f \right\|\_{L\_2(0,\pi)}^2.$$

Let *ε<sup>n</sup>* = *ε*−*n*. Then, system of vectors

$$z\_{\mathfrak{n}}(\mathbf{x}) = \omega\_{\mathfrak{n}} \left( \beta \cos 2n\mathbf{x} + \varepsilon\_{\mathfrak{n}} \operatorname{sign}(\beta^2 - a^2) \alpha \sin 2n\mathbf{x} \right),$$

forms the complete orthonormal system in *W<sup>s</sup>* <sup>2</sup>(0, *π*). The following lemma holds.

**Lemma 1.** *Let* {*an*} *be a finite system of complex numbers. Then, inequalities*

$$\left\| \sum\_{-N}^{N} a\_{\mathbb{H}}(y\_{\mathbb{H}}(\mathbf{x}) - z\_{\mathbb{H}}(\mathbf{x})) \right\|\_{L\_{2}(0,\pi)} \leq \sqrt{2} \cdot \max\_{\mathbf{x} \in [0,\pi]} \left| e^{iq\chi} - 1 \right| \cdot \sqrt{\sum\_{-N}^{N} |\!\!/ \! a\_{\mathbb{H}} \cdot \!\!/ \! a\_{\mathbb{H}}} |^{2}$$

*are valid where*

$$\mathbf{c}\_{\mathrm{fl}} = \frac{1}{\sqrt{1 + (2n)^{2s}}}, \text{ s} = 1, 2, 3, \dots$$

**Proof.** Calculating the difference of *yn*(*x*) − *zn*(*x*), we obtain

$$y\_{\mathbb{M}}(\mathbf{x}) - z\_{\mathbb{M}}(\mathbf{x}) = $$

$$\omega\_{\mathbb{M}}[\beta(\cos \lambda\_{\mathbb{M}} \mathbf{x} - \cos 2n\mathbf{x}) + \varepsilon\_{\mathbb{M}} \operatorname{sign}(\beta^2 - a^2)a(\sin \lambda\_{\mathbb{M}} \mathbf{x} - \sin 2n\mathbf{x})] = $$

$$= \omega\_{\mathbb{M}}[(\varepsilon\_{\mathbb{M}} \operatorname{sign}(\beta^2 - a^2)a + \beta i)\frac{e^{i\varepsilon\_{\mathbb{M}}\operatorname{sgn}x} - 1}{2i}e^{2\pi i x} + $$

$$+ (\varepsilon\_{\mathbb{M}} \operatorname{sign}(\beta^2 - a^2)a - \beta i)\frac{1 - e^{-i\varepsilon\_{\mathbb{M}}\operatorname{sgn}x}}{2i}e^{-2\pi i x}].$$

Then,

$$\sum\_{-N}^{N} a\_{n}(y\_{n} - z\_{n}) = \sum\_{-N}^{N} a\_{n} \omega\_{n} \left[ (\varepsilon\_{n} \text{sign}(\beta^{2} - a^{2})a + \beta i) \frac{\varepsilon^{ix\_{n}\eta x} - 1}{2i} \varepsilon^{2nix} + \dots \right]$$

$$+ (\varepsilon\_{n} \text{sign}(\beta^{2} - a^{2})a - \beta i) \frac{1 - e^{-ix\_{n}\eta x}}{2i} \varepsilon^{-2nix} \left] \text{.}$$

Using properties of the norm, we have

$$\left\|\sum\_{-N}^{N} a\_{\hbar} (y\_n - z\_n) \right\|\_{L\_2(0, \pi)} = $$

$$= \left\| \frac{\operatorname{sign}(\beta^2 - a^2)\alpha + \beta i}{2i} (e^{iqx} - 1) \sum\_{-N, \varepsilon\_n = 1}^{N} a\_{\hbar} \omega\_n e^{2\pi i x} + 1 \right\|\_{L\_2(0, \pi)}$$

$$+ \frac{-\operatorname{sign}(\beta^2 - a^2)\alpha + \beta i}{2i} (e^{-iqx} - 1) \sum\_{-N, \varepsilon\_n = -1}^{N} a\_{\hbar} \omega\_n e^{2\pi i x} + 1$$

<sup>+</sup>*sign*(*β*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*2)*<sup>α</sup>* <sup>−</sup> *<sup>β</sup><sup>i</sup>* <sup>2</sup>*<sup>i</sup>* (<sup>1</sup> <sup>−</sup> *<sup>e</sup>* <sup>−</sup>*iϕx*) *N* ∑−*N*,*εn*=1 *anωne* <sup>−</sup>2*nix*+ <sup>+</sup>−*sign*(*β*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*2)*<sup>α</sup>* <sup>−</sup> *<sup>β</sup><sup>i</sup>* <sup>2</sup>*<sup>i</sup>* (<sup>1</sup> <sup>−</sup> *<sup>e</sup> <sup>i</sup>ϕx*) *N* ∑ −*N*,*εn*=−1 *anωne* −2*nix* 4 4 4 4 4 *L*2(0,*π*) = = 4 4 4 4 *sign*(*β*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*2)*<sup>α</sup>* <sup>+</sup> *<sup>β</sup><sup>i</sup>* <sup>2</sup>*<sup>i</sup>* (*<sup>e</sup> <sup>i</sup>ϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup>)<sup>×</sup> × *N* ∑−*N*,*εn*=1 *anωne* <sup>2</sup>*nix* + *N* ∑ −*N*,*εn*=−1 *anωne* −2*nix* + <sup>+</sup>−*sign*(*β*<sup>2</sup> <sup>−</sup> *<sup>α</sup>*2)*<sup>α</sup>* <sup>+</sup> *<sup>β</sup><sup>i</sup>* <sup>2</sup>*<sup>i</sup>* (*<sup>e</sup>* <sup>−</sup>*iϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup>)<sup>×</sup> × *N* ∑ −*N*,*εn*=−1 *anωne* <sup>2</sup>*nix* + *N* ∑−*N*,*εn*=1 *anωne* −2*nix*4 4 4 4 4 *L*2(0,*π*) ≤ ≤ *α*<sup>2</sup> + *β*<sup>2</sup> <sup>2</sup> · max *x*∈[0,*π*] *e <sup>i</sup>ϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup> × × ⎛ ⎝ 4 4 4 4 4 *N* ∑−*N*,*εn*=1 *anωne* <sup>2</sup>*nix* + *N* ∑ −*N*,*εn*=−1 *anωne* −2*nix* 4 4 4 4 4 *L*2(0,*π*) + + 4 4 4 4 4 *N* ∑ −*N*,*εn*=−1 *anωne* <sup>2</sup>*nix* + *N* ∑−*N*,*εn*=1 *anωne* −2*nix* 4 4 4 4 4 *L*2(0,*π*) ⎞ ⎠ = = *α*<sup>2</sup> + *β*<sup>2</sup> <sup>2</sup> max *x*∈[0,*π*] *e <sup>i</sup>ϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup> ⎛ ⎝ ! *<sup>N</sup>* ∑−*N*,*εn*=1 | *anω<sup>n</sup>* |<sup>2</sup> + *N* ∑ −*N*,*εn*=−1 | *anω<sup>n</sup>* |2+ + ! *<sup>N</sup>* ∑ −*N*,*εn*=−1 | *anω<sup>n</sup>* |<sup>2</sup> + *N* ∑−*N*,*εn*=1 | *anω<sup>n</sup>* |<sup>2</sup> ⎞ ⎠ · <sup>√</sup>*<sup>π</sup>* <sup>=</sup> = *α*<sup>2</sup> + *β*<sup>2</sup> · max *x*∈[0,*π*] *e <sup>i</sup>ϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup> · <sup>√</sup>*<sup>π</sup>* · ! *N* ∑ −*N* | *anω<sup>n</sup>* |2. Thus, denoting *cn* = √ <sup>1</sup> <sup>1</sup>+(2*n*)2*<sup>s</sup>* , we obtain

$$\left\| \left| \sum\_{-N}^{N} a\_{\boldsymbol{n}} (y\_{\boldsymbol{n}}(\mathbf{x}) - z\_{\boldsymbol{n}}(\mathbf{x})) \right| \right\|\_{L\_{2}(0,\pi)} \leq \sqrt{2} \cdot \max\_{\boldsymbol{x} \in [0,\pi]} \left| e^{i\boldsymbol{q}\cdot\mathbf{x}} - 1 \right| \cdot \sqrt{\sum\_{-N}^{N} |a\_{\boldsymbol{n}} \cdot c\_{\boldsymbol{n}}|^{2}}.$$

**Lemma 2.** *Let* {*an*} *be a finite system of complex numbers. Then, inequalities*

$$\left\| D^s \sum\_{-N}^N a\_\hbar (y\_\hbar(\mathbf{x}) - z\_\hbar(\mathbf{x})) \right\|\_{L\_2(0, \pi)} \le$$

$$\frac{1}{2} \le \sqrt{2} \left[ \max\_{\mathbf{x} \in [0, \pi]} \left| e^{i\boldsymbol{\rho}\mathbf{x}} - 1 \right| + (\boldsymbol{\varrho} + 1)^s - 1 \right] \cdot \sqrt{\sum\_{-N}^N |a\_\hbar \cdot c\_\hbar \cdot (2n)^s|^2}$$

*are valid at s* = 1, 2, 3, . . . .

**Proof.** Denote

$$\theta = \sqrt{2} \cdot \max\_{x \in [0, \pi]} |e^{i\varphi x} - 1| \text{.} $$

since *<sup>N</sup>*

$$\sum\_{-N}^{N} a\_{\boldsymbol{n}} (y\_{\boldsymbol{n}} - z\_{\boldsymbol{n}}) = \frac{\operatorname{sign}(\beta^2 - \alpha^2)\alpha + \beta\dot{\boldsymbol{t}}}{2\dot{\boldsymbol{i}}} \cdot (\boldsymbol{e}^{iq\boldsymbol{x}} - 1).$$

$$\cdot \left(\sum\_{-N,\boldsymbol{x}\_{\boldsymbol{n}}=1}^{N} a\_{\boldsymbol{n}} \cdot \omega\_{\boldsymbol{n}} \cdot \boldsymbol{e}^{2\text{mix}} + \sum\_{-N,\boldsymbol{x}\_{\boldsymbol{n}}=-1}^{N} a\_{\boldsymbol{n}} \cdot \omega\_{\boldsymbol{n}} \cdot \boldsymbol{e}^{-2\text{mix}}\right) + \cdots$$

$$+ \frac{-\operatorname{sign}(\beta^2 - \alpha^2)\alpha + \beta\dot{\boldsymbol{t}}}{2\dot{\boldsymbol{i}}} \cdot \left(\boldsymbol{e}^{-iq\boldsymbol{x}} - 1\right).$$

$$\cdot \cdot \left(\sum\_{-N,\boldsymbol{x}\_{\boldsymbol{n}}=-1}^{N} a\_{\boldsymbol{n}} \cdot \omega\_{\boldsymbol{n}} \cdot \boldsymbol{e}^{2\text{mix}} + \sum\_{-N,\boldsymbol{x}\_{\boldsymbol{n}}=1}^{N} a\_{\boldsymbol{n}} \cdot \omega\_{\boldsymbol{n}} \cdot \boldsymbol{e}^{-2\text{mix}}\right),$$

using properties of the norm, we have

$$\begin{split} \left\| D^{s} \sum\_{i=1}^{N} a\_{\mathfrak{a}} (y\_{\mathfrak{a}} - z\_{\mathfrak{a}}) \right\|\_{L\_{2}(0,\varpi)} \leq & \frac{\sqrt{\varpi^{2} + B^{2}}}{\sum\_{i=1}^{N} a\_{\mathfrak{a}} \cdot \left( \left\| \sum\_{k=0}^{z} C\_{k}^{k} \cdot D^{k} (e^{iq\pi} - 1) \cdot \right. \\\\ \left. \cdot D^{s-k} \left( \sum\_{-N\le s-1}^{N} a\_{\mathfrak{a}} \cdot \omega\_{n} \cdot e^{2izx} + \sum\_{-N\le s-1}^{N} a\_{\mathfrak{a}} \cdot \omega\_{n} \cdot e^{-2izx} \right) \right\|\_{L\_{2}(0,\varpi)} + \\ & + \left\| \sum\_{k=0}^{z} C\_{k}^{k} \cdot D^{k} (e^{-iq\pi} - 1) \cdot \right. \\\\ \left. \cdot D^{s-k} \left( \sum\_{-N\le s-1}^{N} a\_{\mathfrak{a}} \cdot \omega\_{n} \cdot e^{-2izx} + \sum\_{-N\le s-1}^{N} a\_{\mathfrak{a}} \cdot \omega\_{n} \cdot e^{-2izx} \right) \right\|\_{L\_{2}(0,\varpi)} \leq \\ & \left. \leq \frac{\sqrt{\varpi^{2} + B^{2}}}{2} \cdot \left( \max\_{-N\le s-1} \left| e^{iq\pi} - 1 \right| \right. \\\\ \left. \times \left\| \sum\_{-N\le s-1}^{N} a\_{\mathfrak{a}} \cdot \omega\_{n} \cdot (2n) e^{2izx} + \sum\_{-N\le s-1}^{N} a\_{\mathfrak{a}} \cdot a\_{\mathfrak{a}} \cdot (-2n)^{s} e^{-2izx} \right| \right. \\ & + \left$$

$$\begin{split} \lambda &\times \left( \max\_{x \in [0,\pi]} \left| e^{iqx} - 1 \right| \sqrt{\sum\_{-N}^{N} |a\_{\mathbb{R}} a\_{\mathbb{R}} (2n)^{s}|^{2}} + \sum\_{k=1}^{s} C\_{s}^{k} \Phi^{k} \sqrt{\sum\_{-N}^{N} |a\_{\mathbb{R}} a\_{\mathbb{R}} (2n)^{s-k}|^{2}} \right) \times \\ & \quad \times \sqrt{\pi} = \sqrt{2} \left( \max\_{x \in [0,\pi]} \left| e^{iqx} - 1 \right| \cdot \sqrt{\sum\_{-N}^{N} |a\_{\mathbb{R}} \cdot c\_{\mathbb{R}} \cdot (2n)^{s}|^{2}} + \\ & \quad \quad + \sum\_{k=1}^{s} C\_{s}^{k} \cdot \Phi^{k} \cdot \sqrt{\sum\_{-N}^{N} |a\_{\mathbb{R}} \cdot c\_{\mathbb{R}} \cdot (2n)^{s-k}|^{2}} \right) \le \\ & \le \sqrt{2} \left[ \max\_{x \in [0,\pi]} \left| e^{iqx} - 1 \right| + (\varrho + 1)^{s} - 1 \right] \cdot \sqrt{\sum\_{-N}^{N} |a\_{\mathbb{R}} \cdot c\_{\mathbb{R}} \cdot (2n)^{s}|^{2}}. \end{split}$$

Thus, inequalities

$$\left\| D^s \sum\_{-N}^N a\_n(y\_n(\mathbf{x}) - z\_n(\mathbf{x})) \right\|\_{L\_2(0, \pi)} \le$$

$$\frac{1}{2} \le \sqrt{2} \left[ \max\_{\mathbf{x} \in [0, \pi]} \left| e^{i\varphi \mathbf{x}} - 1 \right| + (\varrho + 1)^s - 1 \right] \cdot \sqrt{\sum\_{-N}^N |a\_n \cdot c\_n \cdot (2n)^s|^2}$$

hold at *s* = 1, 2, 3, . . . .

Using Lemmas 1 and 2, we obtain

**Lemma 3.** *Let* {*an*} *be a finite system of complex numbers. Then the following inequality*

$$\begin{aligned} & \left\| \sum\_{-N}^{N} a\_{\hbar} (y\_{\hbar}(\mathbf{x}) - z\_{\hbar}(\mathbf{x})) \right\|\_{W\_{2}^{0}(0,\pi)} \leq \\ & \leq \sqrt{\theta^{2} + 2 \left( \frac{\theta}{\sqrt{2}} + (\varrho + 1)^{s} - 1 \right)^{2}} \cdot \sigma(\mathbf{s}) \cdot \sqrt{\sum\_{-N}^{N} |a\_{\hbar}|^{2}} \\ & \leq \frac{1}{\tau^{2}}, \sigma(\mathbf{s}) = 1 \text{ at } \mathbf{s} > 0. \end{aligned}$$

*is valid where <sup>σ</sup>*(0) = <sup>1</sup> √2 , *σ*(*s*) = 1 *at s* > 0.

**Lemma 4.** *Let α* = 0, *β* = 0, |*α*| = |*β*| *be real numbers, and*

$$\rho = \sqrt{\theta^2 + 2\left(\frac{\theta}{\sqrt{2}} + (\varrho + 1)^s - 1\right)^2} \cdot \sigma(s) < 1$$

*where <sup>σ</sup>*(0) = <sup>1</sup> √2 , *<sup>σ</sup>*(*s*) = <sup>1</sup> *at <sup>s</sup>* <sup>&</sup>gt; 0, *<sup>θ</sup>* <sup>=</sup> <sup>√</sup><sup>2</sup> · max *x*∈[0,*π*] *eiϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup> , *<sup>λ</sup><sup>n</sup>* <sup>=</sup> <sup>2</sup>*<sup>n</sup>* <sup>+</sup> *<sup>ε</sup><sup>n</sup>* · *<sup>ϕ</sup>*, *<sup>ϕ</sup>* <sup>=</sup> <sup>1</sup> *<sup>π</sup>* arccos <sup>−</sup>2*αβ <sup>α</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> , *ε<sup>n</sup>* = *ε*−*<sup>n</sup>* = ±1 *at n* ∈ *Z*.

*Then, eigenfunction system*

$$y\_n(\mathbf{x}) = \sqrt{\frac{2}{\pi}} \cdot \frac{\not{\otimes} \cos \lambda\_n \mathbf{x} + \varepsilon\_n \cdot \operatorname{sign}(\beta^2 - \alpha^2) \cdot \alpha \sin \lambda\_n \mathbf{x}}{\sqrt{\alpha^2 + \beta^2} \cdot \sqrt{1 + (2n)^{2s}}}, \quad n \in \mathbb{Z}\_+$$

*of spectral Problems (9) and (10) forms the Riesz basis in the space W<sup>s</sup>* <sup>2</sup>(0, *π*).

**Proof.** Vector system

$$z\_n(\mathbf{x}) = \sqrt{\frac{2}{\pi}} \cdot \frac{\beta \cos 2nx + \varepsilon\_n \cdot \operatorname{sign}(\beta^2 - a^2) \cdot a \sin 2nx}{\sqrt{a^2 + \beta^2} \cdot \sqrt{1 + (2n)^{2s}}}, n \in \mathbb{Z}$$

forms the complete orthonormal system in Hilbert space *W<sup>s</sup>* <sup>2</sup>(0, *π*),, and vector system

$$y\_n(\mathbf{x}) = \sqrt{\frac{2}{\pi}} \cdot \frac{\beta \cos \lambda\_n \mathbf{x} + \varepsilon\_n \cdot \text{sign}(\beta^2 - a^2) \cdot a \sin \lambda\_n \mathbf{x}}{\sqrt{a^2 + \beta^2} \cdot \sqrt{1 + (2n)^{2s}}}, n \in Z$$

by virtue of Lemma 3 satisfying the theorem conditions by R. Paley and N. Wiener (see p. 224, [39]). This theorem implies that system of vectors {*yn*(*x*)}*n*∈*<sup>Z</sup>* forms the Riesz basis in space *<sup>W</sup><sup>s</sup>* <sup>2</sup>(0, *π*).

**Lemma 5.** *Operator*

$$Ly = -y''$$

*with domain*

$$D(L) = \{ y(\mathbf{x}) : y(\mathbf{x}) \in \mathbb{C}^2(0, \pi) \cap \mathbb{C}^1[0, \pi], y'' \in L\_2(0, \pi), \\ \}$$

$$\alpha y(0) + \beta y(\pi) = 0, \beta y'(0) + \alpha y'(0) = 0 \}$$

*is a symmetric operator in class L*2(0, *π*).

**Proof.** Indeed, since functions *f* and *g* belong to domain *D*(*L*), we have *L f* ∈ *L*2(0, *π*), *Lg* = *Lg* ∈ *L*2(0, *π*), and the second Green formula

$$\int\_{G} (\mathcal{L}u \cdot v - u \cdot \mathcal{L}v) dx = -\int\_{\partial G} \left( \frac{\partial u}{\partial n} \cdot v - u \cdot \frac{\partial v}{\partial n} \right) ds$$

at *u* = *f* and *v* = *g* takes the form

$$\int\_0^\pi (Lf \cdot \overline{g} - f \cdot \overline{Lg}) dx = -\left(f'(x)\overline{g(x)} - f(x)\overline{g'(x)}\right)\Big|\_0^\pi.$$

Further, functions *f* and *g* satisfy the boundary conditions:

$$\text{a}\,\text{a}f(0) + \beta f(\pi) = 0,\\ \,\,\beta f'(0) + \text{a}f'(\pi) = 0,\\ \,\,\overline{\text{ag}(0)} + \beta \overline{\text{g}(\pi)} = 0,\\ \,\,\beta \overline{\text{g}'(0)} + \text{a}\overline{\text{g}'(\pi)} = 0.$$

By assumption, *α* = 0, *β* = 0. Therefore,

$$f(0) \cdot \overline{\mathcal{g}(\pi)} - f(\pi) \cdot \overline{\mathcal{g}(0)} = 0$$

and

$$f'(0) \cdot \overline{g'(\pi)} - f'(\pi) \cdot \overline{g'(0)} = 0,$$

i.e., *f*(0) · *g*(*π*) = *f*(*π*) · *g*(0) and *f* (0) · *g*(*π*) = *f* (*π*) · *g*(0). For here, we obtain

$$\frac{f(\pi)}{f(0)} = \frac{g(\pi)}{\overline{g(0)}} = k\_0 = -\frac{\alpha}{\beta}$$

and

$$\frac{f'(\pi)}{f'(0)} = \frac{\overline{g'(\pi)}}{\overline{g'(0)}} = k\_1 = -\frac{\beta}{\alpha},\ k\_0 \cdot k\_1 = 1.$$

$$\text{So, } f(\pi) = k\_0 f(0), \overline{g(\pi)} = k\_0 \overline{g(0)} \text{ è } f'(\pi) = k\_1 f'(0), \overline{g'(\pi)} = k\_1 \overline{g'(0)}. \text{ Thus,}$$

$$\int\_0^{\pi} \left( Lf \cdot \overline{g} - f \cdot \overline{Lg} \right) dx = - \left( f'(x) \cdot \overline{g(x)} - f(x) \cdot \overline{g'(x)} \right) \Big|\_0^{\pi} =$$

$$= - \left( f'(\pi) \cdot \overline{\overline{g(\pi)}} - f(\pi) \cdot \overline{\overline{g'(\pi)}} \right) + \left( f'(0) \cdot \overline{\overline{g(0)}} - f(0) \cdot \overline{\overline{g'(0)}} \right) = 0$$

$$= - \left( f'(0) \cdot \overline{\overline{g(0)}} - f(0) \cdot \overline{\overline{g'(0)}} \right) + \left( f'(0) \cdot \overline{\overline{g(0)}} - f(0) \cdot \overline{\overline{g'(0)}} \right) = 0.$$

$$\text{Therefore, } (Lf, g) = (f, Lg), \forall f, g \in D(L). \quad \blacksquare$$

**Theorem 1.** *Let α* = 0, *β* = 0, |*α*| = |*β*| *be real number, and*

$$\rho = \sqrt{\theta^2 + 2\left(\frac{\theta}{\sqrt{2}} + (\varrho + 1)^s - 1\right)^2} \cdot \sigma(s) < 1$$

*where <sup>σ</sup>*(0) = <sup>1</sup> √2 , *<sup>σ</sup>*(*s*) = <sup>1</sup> *at <sup>s</sup>* <sup>&</sup>gt; 0, *<sup>θ</sup>* <sup>=</sup> <sup>√</sup> 2 · max *x*∈[0,*π*] *e <sup>i</sup>ϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup> , *<sup>λ</sup><sup>n</sup>* <sup>=</sup> <sup>2</sup>*<sup>n</sup>* <sup>+</sup> *<sup>ε</sup><sup>n</sup>* · *<sup>ϕ</sup>*, *<sup>ϕ</sup>* <sup>=</sup> <sup>1</sup> *<sup>π</sup>* arccos <sup>−</sup>2*αβ <sup>α</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> , *ε<sup>n</sup>* = *ε*−*<sup>n</sup>* = ±1 *at n* ∈ *Z*. *Then the system of eigenfunctions*

$$\overline{y}\_n(\mathbf{x}) = \sqrt{\frac{2}{\pi}} \cdot \frac{\not{\otimes} \cos \lambda\_n \mathbf{x} + \varepsilon\_n \cdot \operatorname{sign}(\beta^2 - \alpha^2) \cdot \alpha \sin \lambda\_n \mathbf{x}}{\sqrt{\alpha^2 + \beta^2} \cdot \sqrt{1 + \|\lambda\_n\|^{2s}}}, \ n \in \mathbb{Z}\_+$$

*of spectral Problems (9) and (10) form the complete orthonormal system in Sobolev classes W<sup>s</sup>* <sup>2</sup>(0, *π*).

**Proof.** Symmetry of operator *L* implies that eigenfunctions {*yn*(*x*)}*n*∈*<sup>Z</sup>* of operator *L*, corresponding to the different eigenvalues, are orthogonal in classes *L*2(0, *π*).

System of functions {*D<sup>α</sup>yn*(*x*)}*n*∈*<sup>Z</sup>* is also the system of eigenfunctions of a similar operator corresponding to different eigenvalues, which implies that functions of system {*D<sup>α</sup>yn*(*x*)}*n*∈*<sup>Z</sup>* are orthogonal in classes *L*2(0, *π*).

As a result, we see that system of eigenfunctions {*yn*(*x*)}*n*∈*<sup>Z</sup>* of operator *L*, corresponding to different eigenvalues, are orthogonal in the Sobolev classes *W<sup>s</sup>* <sup>2</sup>(0, *π*). It is known that, if a sequence of vectors {*ψn*(*x*)}*n*∈*<sup>Z</sup>* forms the Riesz basis in a Hilbert space *H*, then system of vectors

$$\left\{\widehat{\psi}\_n(\mathfrak{x})\right\}\_{n \in Z} \left(\widehat{\psi}\_n(\mathfrak{x}) = \frac{\psi\_n(\mathfrak{x})}{||\psi\_n(\mathfrak{x})||}, n \in Z\right)$$

also forms the Riesz basis in *H* (see p. 374, [42]).

By virtue of Lemma 4, system of eigenvectors {*yn*(*x*)}*n*∈*<sup>Z</sup>* forms the Riesz basis in space *<sup>W</sup><sup>s</sup>* <sup>2</sup>(0, *π*). The orthogonality of this system implies that {*yn*(*x*)}*n*∈*<sup>Z</sup>* is a complete orthonormal system in the Sobolev classes *W<sup>s</sup>* <sup>2</sup>(0, *π*).

Theorem 1 and the Sobolev embedding theorem imply the following corollaries.

**Corollary 1.** *Let α* = 0, *β* = 0, |*α*| = |*β*| *be real numbers, and*

$$
\rho = \sqrt{\theta^2 + 2\left(\frac{\theta}{\sqrt{2}} + \rho\right)^2} < 1
$$

*where <sup>θ</sup>* <sup>=</sup> <sup>√</sup><sup>2</sup> · max *x*∈[0,*π*] *eiϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup> , *<sup>λ</sup><sup>n</sup>* <sup>=</sup> <sup>2</sup>*<sup>n</sup>* <sup>+</sup> *<sup>ε</sup><sup>n</sup>* · *<sup>ϕ</sup>*, *<sup>ϕ</sup>* <sup>=</sup> <sup>1</sup> *<sup>π</sup>* arccos <sup>−</sup>2*αβ <sup>α</sup>*2+*β*<sup>2</sup> , *<sup>ε</sup><sup>n</sup>* = *<sup>ε</sup>*−*<sup>n</sup>* = ±1 *at <sup>n</sup>* ∈ *<sup>Z</sup>*. *Then, the Fourier series for function f*(*x*) <sup>∈</sup> *<sup>W</sup>*<sup>1</sup> <sup>2</sup> (0, *π*) ∩ *C*[0, *π*] *in orthonormal eigenfunctions*

$$\overline{y}\_n(\mathbf{x}) = \sqrt{\frac{2}{\pi}} \cdot \frac{\beta \cos \lambda\_n \mathbf{x} + \varepsilon\_n \cdot \text{sign}(\beta^2 - a^2) \cdot a \sin \lambda\_n \mathbf{x}}{\sqrt{a^2 + \beta^2} \cdot \sqrt{1 + |\ \lambda\_n|^2}}, \ n \in \mathbb{Z}$$

*of spectral Problems (9) and (10) uniformly converges on segment* [0, *π*] *to function f*(*x*).

**Corollary 2.** *Let α* = 0, *β* = 0, |*α*| = |*β*| *be real numbers, and*

$$
\rho = \sqrt{\theta^2 + 2\left(\frac{\theta}{\sqrt{2}} + (\rho + 1)^s - 1\right)^2} < 1
$$

*where <sup>s</sup>* <sup>&</sup>gt; *<sup>k</sup>*, *<sup>θ</sup>* <sup>=</sup> <sup>√</sup><sup>2</sup> · max *x*∈[0,*π*] *eiϕ<sup>x</sup>* <sup>−</sup> <sup>1</sup> , *<sup>λ</sup><sup>n</sup>* <sup>=</sup> <sup>2</sup>*<sup>n</sup>* <sup>+</sup> *<sup>ε</sup><sup>n</sup>* · *<sup>ϕ</sup>*, *<sup>ϕ</sup>* <sup>=</sup> <sup>1</sup> *<sup>π</sup>* arccos <sup>−</sup>2*αβ <sup>α</sup>*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*<sup>2</sup> , *<sup>ε</sup><sup>n</sup>* <sup>=</sup> *<sup>ε</sup>*−*<sup>n</sup>* <sup>=</sup> <sup>±</sup><sup>1</sup> *at <sup>n</sup>* <sup>∈</sup> *Z. Then the Fourier series for function f*(*x*) <sup>∈</sup> *<sup>W</sup><sup>s</sup>* <sup>2</sup>(0, *<sup>π</sup>*) <sup>∩</sup> *<sup>C</sup>k*[0, *<sup>π</sup>*] *in orthonormal eigenfunctions*

$$\overline{y}\_n(\mathbf{x}) = \sqrt{\frac{2}{\pi}} \cdot \frac{\beta \cos \lambda\_n \mathbf{x} + \varepsilon\_n \cdot \operatorname{sign}(\beta^2 - a^2) \cdot a \sin \lambda\_n \mathbf{x}}{\sqrt{a^2 + \beta^2} \cdot \sqrt{1 + |\lambda\_n|^2}}, \ n \in Z\_n$$

*of spectral Problems (9) and (10) converges in the norm of space Ck*[0, *π*] *to function f*(*x*)*.*

The scalar product in space *Ws*1,*s*<sup>2</sup> <sup>2</sup> ((0, *π*) × (0, *π*)) is introduced in the following way:

$$\begin{split} (f(\mathbf{x},\mathbf{y}),\mathbf{g}(\mathbf{x},\mathbf{y}))\_{W^{s\_1,\mathbf{y}}\_2((0,\pi)\times(0,\pi))} &= (f(\mathbf{x},\mathbf{y}),\mathbf{g}(\mathbf{x},\mathbf{y}))\_{L\_2((0,\pi)\times(0,\pi))} + \\ + (D^{s\_1}\_{\mathbf{x}}f(\mathbf{x},\mathbf{y}),D^{s\_1}\_{\mathbf{x}}\mathbf{g}(\mathbf{x},\mathbf{y}))\_{L\_2((0,\pi)\times(0,\pi))} &+ (D^{s\_2}\_{\mathbf{y}}f(\mathbf{x},\mathbf{y}),D^{s\_2}\_{\mathbf{y}}\mathbf{g}(\mathbf{x},\mathbf{y}))\_{L\_2((0,\pi)\times(0,\pi))} + \\ &+ (D^{s\_1,s\_2}\_{\mathbf{x},\mathbf{y}}f(\mathbf{x},\mathbf{y}),D^{s\_1,s\_2}\_{\mathbf{x},\mathbf{y}}\mathbf{g}(\mathbf{x},\mathbf{y}))\_{L\_2((0,\pi)\times(0,\pi))}. \end{split}$$

Respectively, the norm in this space is introduced as follows:

$$\begin{aligned} \left\| f(\mathbf{x}, \mathbf{y}) \right\|\_{W\_2^{s\_1, \mathbf{y}}((0, \pi) \times (0, \pi))}^2 &= \\ = \left\| f(\mathbf{x}, \mathbf{y}) \right\|\_{L\_2((0, \pi) \times (0, \pi))}^2 + \left\| D\_{\mathbf{x}}^{s\_1} f(\mathbf{x}, \mathbf{y}) \right\|\_{L\_2((0, \pi) \times (0, \pi))}^2 + \\ + \left\| D\_{\mathbf{y}}^{s\_2} f(\mathbf{x}, \mathbf{y}) \right\|\_{L\_2((0, \pi) \times (0, \pi))}^2 + \left\| D\_{\mathbf{x}, \mathbf{y}}^{s\_1, s\_2} f(\mathbf{x}, \mathbf{y}) \right\|\_{L\_2((0, \pi) \times (0, \pi))}^2 \end{aligned}$$

**Lemma 6.** *If* {*ψ*(1) *<sup>m</sup>* (*x*)} *and* {*ψ*(2) *<sup>n</sup>* (*y*)} *are complete orthonormal systems in <sup>W</sup>s*<sup>1</sup> <sup>2</sup> (0, *<sup>π</sup>*) *and <sup>W</sup>s*<sup>2</sup> <sup>2</sup> (0, *π*),*, respectively, then the system of all products*

$$f\_{nm}(\mathfrak{x}, \mathfrak{y}) = \psi\_{m}^{(1)}(\mathfrak{x}) \cdot \psi\_{n}^{(2)}(\mathfrak{y}),$$

*is a complete orthonormal system in Ws*1,*s*<sup>2</sup> <sup>2</sup> ((0, *π*) × (0, *π*))*, where s*1,*s*<sup>2</sup> = 1, 2, 3, ... *and x*, *y* ∈ (0, *π*)

**Proof.** By virtue of the Fubini theorem,

$$\left\| f\_{\mathrm{mn}}(\mathbf{x}, y) \right\|\_{\mathsf{W}^{s\_1, r\_2}\_2((0, \pi) \times (0, \pi))}^2 = \left\| \psi\_{\mathrm{m}}^{(1)}(\mathbf{x}) \right\|\_{L\_2(0, \pi)}^2 \cdot \left\| \psi\_{\mathrm{m}}^{(2)}(y) \right\|\_{L\_2(0, \pi)}^2 + $$

+ 4 4 4 *Ds*<sup>1</sup> *<sup>x</sup> <sup>ψ</sup>*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) · 4 4 <sup>4</sup>*ψ*(2) *<sup>n</sup>* (*y*) 4 4 4 2 *L*2(0,*π*) + 4 4 <sup>4</sup>*ψ*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) · 4 4 4 *Ds*<sup>2</sup> *<sup>y</sup> <sup>ψ</sup>*(2) *<sup>n</sup>* (*y*) 4 4 4 2 *L*2(0,*π*) + + 4 4 4 *Ds*<sup>1</sup> *<sup>x</sup> <sup>ψ</sup>*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) · 4 4 4 *Ds*<sup>2</sup> *<sup>y</sup> <sup>ψ</sup>*(2) *<sup>n</sup>* (*y*) 4 4 4 2 *L*2(0,*π*) = = 4 4 <sup>4</sup>*ψ*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) + 4 4 4 *Ds*<sup>1</sup> *<sup>x</sup> <sup>ψ</sup>*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) · 4 4 <sup>4</sup>*ψ*(2) *<sup>n</sup>* (*y*) 4 4 4 2 *L*2(0,*π*) + + 4 4 <sup>4</sup>*ψ*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) + 4 4 4 *Ds*<sup>1</sup> *<sup>x</sup> <sup>ψ</sup>*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) · 4 4 4 *Ds*<sup>2</sup> *<sup>y</sup> <sup>ψ</sup>*(2) *<sup>n</sup>* (*y*) 4 4 4 2 *L*2(0,*π*) = = 4 4 <sup>4</sup>*ψ*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) + 4 4 4 *Ds*<sup>1</sup> *<sup>x</sup> <sup>ψ</sup>*(1) *<sup>m</sup>* (*x*) 4 4 4 2 *L*2(0,*π*) · · 4 4 <sup>4</sup>*ψ*(2) *<sup>n</sup>* (*y*) 4 4 4 2 *L*2(0,*π*) + 4 4 4 *Ds*<sup>2</sup> *<sup>y</sup> <sup>ψ</sup>*(2) *<sup>n</sup>* (*y*) 4 4 4 2 *L*2(0,*π*) = 1.

If *m* = *m*<sup>1</sup> or *n* = *n*1, by the same theorem

$$\begin{split} & (f\_{\text{fnn}}(x,y), f\_{\text{fn}}v\_{1}(x,y))\_{\text{l}^{1}\_{2}\text{in}^{2}\_{1}((0,\pi)\times(0,\pi))} = \\ & = (f\_{\text{fnn}}(x,y), f\_{\text{fn}}v\_{1}(x,y))\_{\text{l}^{1}\_{2}\text{in}^{2}\_{1}((0,\pi)\times(0,\pi))} + \\ & + (D^{x\_{1}}\_{x}f\_{\text{fnn}}(x,y), D^{x\_{1}}\_{x}f\_{\text{fm}}(x,y))\_{\text{l}^{1}\_{2}((0,\pi)\times(0,\pi))} + \\ & + (D^{y\_{2}}\_{y}f\_{\text{fnn}}(x,y), D^{y\_{2}}\_{y}f\_{\text{fm}}(x,y))\_{\text{l}^{1}\_{2}((0,\pi)\times(0,\pi))} + \\ & + (D^{z\_{1}}\_{x}f\_{\text{fnn}}(x,y), D^{x\_{1}}\_{x}f\_{\text{fm}}(x,y))\_{\text{l}^{1}\_{2}((0,\pi)\times(0,\pi))} = \\ & = (\Psi^{1}\_{m}(x),\Psi^{1}\_{m}(x))\_{\text{l}^{1}\_{2}(0,\pi)} \cdot (\Psi^{2}\_{m}(x),\Psi^{2}\_{m}(y))\_{\text{l}^{1}\_{2}((0,\pi)\times(0,\pi))} \\ & + (D^{z\_{1}}\_{x}\Psi^{1}\_{m}(x),D^{z\_{1}}\_{x}\Psi^{1}\_{m}(x))\_{\text{l}^{2}\_{2}(0,\pi)} \cdot (\Psi^{2}\_{y}(y),\Psi^{2}\_{y}\Psi^{1}\_{m}(y))\_{\text{l}^{2}\_{2}((0,\pi)\times(0,\pi)$$

since scalar product (*fmn*(*x*, *<sup>y</sup>*), *fm*1*n*<sup>1</sup> (*x*, *<sup>y</sup>*))*Ws*1,*s*<sup>2</sup> <sup>2</sup> ((0,*π*)×(0,*π*)) of two variables exist on <sup>Π</sup> = (0, *<sup>π</sup>*) <sup>×</sup> (0, *π*). Let us prove the completeness of system { *fmn*(*x*, *y*)}. Assume that there exists a function *f*(*x*, *y*) in *Ws*1,*s*<sup>2</sup> <sup>2</sup> ((0, *π*) × (0, *π*)) that is orthogonal to all functions *fmn*(*x*, *y*). Set

$$F\_{\mathfrak{m}}(y) = (f(x,y), \psi\_{\mathfrak{m}}^{(1)}(\mathfrak{x}))\_{\mathcal{W}\_2^{s\_1}(0,\mathfrak{m})}.$$

It is easy to see, that function *Fm*(*y*) belongs to class *Ws*<sup>2</sup> <sup>2</sup> (0, *π*). That's why for any *n*,m again applying the Fubini theorem, we obtain

$$\left( \left( F\_m(y), \psi\_n^{(2)}(y) \right)\_{W\_2^{s\_2}(0,\pi)} = \left( f(x,y), f\_{mn}(x,y) \right)\_{W\_2^{s\_1,s\_2}((0,\pi)\times(0,\pi))} = 0.$$

By completeness of system *<sup>ψ</sup>*(2) *<sup>n</sup>* (*y*), for almost all *<sup>y</sup>*

$$F\_{\mathfrak{m}}(y) = 0.$$

But then, for almost every *y*,, equalities

$$\left(f(x,y),\psi\_m^{(1)}(x)\right)\_{\mathcal{W}\_2^{(1)}(0,\pi)} = 0$$

hold for all *<sup>m</sup>*. Completeness of system *<sup>ψ</sup>*(1) *<sup>m</sup>* (*x*) implies that, for almost all *<sup>y</sup>*, the set of those *<sup>x</sup>*, for which

$$f(x,y) \neq 0,$$

has the measure zero. By virtue of the Fubini theorem, this means that, on Π = (0, *π*) × (0, *π*), function *f*(*x*, *y*) is zero almost everywhere.

The scalar product in space *Ws*1,*s*2,...,*sN* <sup>2</sup> (Π) is introduced in the following way:

$$\begin{split} (f(\mathbf{x}), \mathbf{g}(\mathbf{x}))\_{W^{1:\mathcal{V}2\rightarrow\mathcal{V}N}\_{2}(\Pi)} &= (f(\mathbf{x}), \mathbf{g}(\mathbf{x}))\_{L\_{2}(\Pi)} + \\ &+ \sum\_{j\_{1}=1}^{N} (D^{s\_{j\_{1}}}\_{x\_{j\_{1}}} f(\mathbf{x}), D^{s\_{j\_{1}}}\_{x\_{j\_{1}}} g(\mathbf{x}))\_{L\_{2}(\Pi)} + \\ &+ \sum\_{1 \le j\_{1} < j\_{2} \le N} (D^{s\_{j\_{1}}}\_{x\_{j\_{1}}} D^{s\_{j\_{2}}}\_{x\_{j\_{2}}} f(\mathbf{x}), D^{s\_{j\_{1}}}\_{x\_{j\_{1}}} D^{s\_{j\_{2}}}\_{x\_{j\_{2}}} g(\mathbf{x}))\_{L\_{2}(\Pi)} + \cdots + \\ &+ \sum\_{1 \le j\_{1} < j\_{2} < \dots < j\_{N} \le N} (D^{s\_{j\_{1}}}\_{x\_{j\_{1}}} D^{s\_{j\_{2}}}\_{x\_{j\_{2}}} \dots D^{s\_{j\_{N}}}\_{x\_{j\_{N}}} f(\mathbf{x}), D^{s\_{j\_{1}}}\_{x\_{j\_{1}}} D^{s\_{j\_{2}}}\_{x\_{j\_{2}}} \dots D^{s\_{j\_{N}}}\_{x\_{j\_{N}}} g(\mathbf{x}))\_{L\_{2}(\Pi)}. \end{split}$$

Respectively, the norm in this space is introduced as follows:

$$\begin{split} \left\| f(\mathbf{x}) \right\|\_{W^{s\_1 s\_2 \dots s\_N}\_{2}(\Pi)}^2 &= \left\| f(\mathbf{x}) \right\|\_{L\_2(\Pi)}^2 + \sum\_{j\_1=1}^N \left\| D^{s\_{j\_1}}\_{\mathbf{x}\_{j\_1}^\*} f(\mathbf{x}) \right\|\_{L\_2(\Pi)}^2 + \dots \\ &+ \sum\_{1 \le j\_1 < j\_2 \le N} \left\| D^{s\_{j\_1}}\_{\mathbf{x}\_{j\_1}^\*} D^{s\_{j\_2}}\_{\mathbf{x}\_{j\_2}^\*} f(\mathbf{x}) \right\|\_{L\_2(\Pi)}^2 + \\ &+ \dots + \sum\_{1 \le j\_1 < j\_2 < \dots < j\_N \le N} \left\| D^{s\_{j\_1}}\_{\mathbf{x}\_{j\_1}^\*} D^{s\_{j\_2}}\_{\mathbf{x}\_{j\_2}^\*} \dots D^{s\_{j\_N}}\_{\mathbf{x}\_{j\_N}^\*} f(\mathbf{x}) \right\|\_{L\_2(\Pi)}^2. \end{split}$$

Using the method of mathematical induction and Lemma 6, we obtain the following:

**Lemma 7.** *If* {*ψ*(1) *<sup>m</sup>*<sup>1</sup> (*x*1)}, ... , {*ψ*(*N*) *mN* (*xN*)} *are complete orthonormal systems in spaces <sup>W</sup>s*<sup>1</sup> <sup>2</sup> (0, *π*)*, ..., WsN* <sup>2</sup> (0, *π*), *respectively, then system of all products*

$$f\_m(\mathbf{x}) = f\_{m\_1 \dots m\_N}(\mathbf{x}\_1, \dots, \mathbf{x}\_N) = \psi\_{m\_1}^{(1)}(\mathbf{x}\_1) \cdot \dots \cdot \psi\_{m\_N}^{(N)}(\mathbf{x}\_N)$$

*is a complete orthonormal system in Ws*1,*s*2,...,*sN* <sup>2</sup> (Π)*.*

Let us apply Lemma 7 to our orthonormal systems. In space *Ws*1,*s*2,...,*sN* <sup>2</sup> (Π) of functions of *N* variables *f*(*x*) = *f*(*x*1,..., *xN*) all products

$$
\omega\_{m\_1\dots m\_N}(\mathbf{x}\_1,\dots,\mathbf{x}\_N) = \overline{y}\_{m\_1}^{(1)}(\mathbf{x}\_1)\cdot\dots \cdot \overline{y}\_{m\_N}^{(N)}(\mathbf{x}\_N).
$$

form the complete orthonormal system. Here,

$$\mathfrak{F}\_{m\_{\vec{j}}}^{(j)}(\mathbf{x}\_{\vec{j}}) = \sqrt{\frac{2}{\pi}} \cdot \frac{\beta\_{\vec{j}} \cos \lambda\_{m\_{\vec{j}}} \mathbf{x}\_{\vec{j}} + \varepsilon\_{m\_{\vec{j}}} \cdot \operatorname{sign}(\beta\_{\vec{j}}^2 - \alpha\_{\vec{j}}^2) \cdot a\_{\vec{j}} \sin \lambda\_{m\_{\vec{j}}} \mathbf{x}\_{\vec{j}}}{\sqrt{a\_{\vec{j}}^2 + \beta\_{\vec{j}}^2} \cdot \sqrt{1 + |\lambda\_{m\_{\vec{j}}}|^{2s\_{\vec{j}}}}}, \ m\_{\vec{j}} \in Z\_{\vec{j}}$$

at 1 ≤ *j* ≤ *p*,

$$\overline{y}\_{m\_j}^{(j)}(\mathbf{x}\_j) = \frac{1}{\sqrt{\pi}} \frac{1}{\sqrt{1 + |\, 2m\_j \, |}^{2s\_j}} \exp(i2m\_j \mathbf{x}\_j), \ m\_j \in Z\_j$$

at *p* + 1 ≤ *j* ≤ *q*,

$$\overline{y}\_{m\_j}^{(j)}(\mathbf{x}\_j) = \sqrt{\frac{2}{\pi}} \frac{1}{\sqrt{1 + |\ m\_j|^{2s\_j}}} \sin(m\_j \mathbf{x}\_j), \ m\_j \in \mathcal{N}$$

at *q* + 1 ≤ *j* ≤ *N*.

Thus, the following statement is valid:

**Theorem 2.** *Let α<sup>j</sup>* = 0, *β<sup>j</sup>* = 0, |*αj*| = |*βj*| *be real numbers at every* 1 ≤ *j* ≤ *p*, *and*

$$\rho = \max\_{1 \le j \le p} \sqrt{\theta\_j^2 + 2\left(\frac{\theta\_j}{\sqrt{2}} + (\rho\_j + 1)^{s\_j} - 1\right)^2} \cdot \sigma(s\_j) < 1$$

*where <sup>σ</sup>*(0) = <sup>1</sup> √2 , *<sup>σ</sup>*(*sj*) = 1, *at sj* <sup>&</sup>gt; 0, *<sup>θ</sup><sup>j</sup>* <sup>=</sup> <sup>√</sup> 2 · max *x*∈[0,*π*] *e <sup>i</sup>ϕjx* <sup>−</sup> <sup>1</sup> , *λmj* = 2*mj* + *εmj* · *ϕj*, *ϕ<sup>j</sup>* = 1 *<sup>π</sup>* arccos <sup>−</sup>2*αjβ<sup>j</sup> α*2 *<sup>j</sup>* + *<sup>β</sup>*<sup>2</sup> *j* , *εmj* = *ε*−*mj* = ±1 *at mj* ∈ *Z*. *Then, system of eigenfunctions* {*vm*1...*mN* (*x*1,..., *xN*)}(*m*1,...,*mp*)∈*Zp*,(*mp*+1,...,*mq*)∈*Zq*−*p*,(*mq*+1,...,*mN*)∈*NN*−*<sup>q</sup>* <sup>=</sup>

$$= \left\{ \prod\_{j=1}^{p} \sqrt{\frac{2}{\pi}} \frac{\beta\_j \cos \lambda\_{m\_j} \mathbf{x}\_j + \varepsilon\_{m\_j} \text{sign}(\beta\_j^2 - \mathbf{a}\_j^2) \cdot \mathbf{a}\_j \sin \lambda\_{m\_j} \mathbf{x}\_j}{\sqrt{\mathbf{a}\_j^2 + \beta\_j^2} \cdot \sqrt{1 + |\ \lambda\_{m\_j}|^{2s\_j}}} \right\}\_{(m\_1, \dots, m\_p) \in \mathbb{Z}^p} \times \mathbf{1}$$
 
$$\times \left\{ \prod\_{j=p+1}^q \frac{1}{\sqrt{\pi}} \frac{1}{\sqrt{1 + |\ \mathbf{2} m\_j|^{2s\_j}}} \exp(i2m\_j \mathbf{x}\_j) \right\}\_{(m\_{p+1}, \dots, m\_q) \in \mathbb{Z}^{q-p}}$$
 
$$\times \left\{ \prod\_{j=q+1}^N \sqrt{\frac{2}{\pi}} \frac{1}{\sqrt{1 + |\ \mathbf{m}\_j|^{2s\_j}}} \sin(m\_j \mathbf{x}\_j) \right\}\_{(m\_{q+1}, \dots, m\_N) \in \mathbb{N}^{N-q}}$$

*of spectral Problems (7) and (8) forms the complete orthonormal system in Sobolev classes Ws*1,*s*2,...,*sN* <sup>2</sup> (Π).

**Corollary 3.** *Let α<sup>j</sup>* = 0, *β<sup>j</sup>* = 0, |*αj*| = |*βj*| *be real numbers at every* 1 ≤ *j* ≤ *p*, *and*

$$\rho = \max\_{1 \le j \le p} \sqrt{\theta\_j^2 + 2\left(\frac{\theta\_j}{\sqrt{2}} + (\rho\_{\hat{j}} + 1)^{s\_j} - 1\right)^2} \cdot \sigma(s\_{\hat{j}}) < 1$$

*where <sup>σ</sup>*(0) = <sup>1</sup> √2 , *<sup>σ</sup>*(*sj*) = <sup>1</sup> *at sj* <sup>&</sup>gt; 0, *<sup>θ</sup><sup>j</sup>* <sup>=</sup> <sup>√</sup><sup>2</sup> · max *x*∈[0,*π*] *e <sup>i</sup>ϕjx* <sup>−</sup> <sup>1</sup> , *λmj* = 2*mj* + *εmj* · *ϕj*, *ϕ<sup>j</sup>* = 1 *<sup>π</sup>* arccos <sup>−</sup>2*αjβ<sup>j</sup> α*2 *<sup>j</sup>* + *<sup>β</sup>*<sup>2</sup> *j* , *εmj* = *ε*−*mj* = ±1 *at mj* ∈ *Z, sj* > *k* + *N* <sup>2</sup> , *<sup>k</sup>* <sup>≥</sup> 0, *<sup>k</sup>* <sup>∈</sup> *<sup>Z</sup>*. *Then, the Fourier series for function f*(*x*) <sup>∈</sup> *<sup>W</sup>s*1,*s*2,...,*sN* <sup>2</sup> (Π) <sup>∩</sup> *<sup>C</sup>k*(Π) *in orthonormal eigenfunctions*

$$\begin{split} \left\{ \varepsilon\_{m\_{1},\ldots,m\_{N}}(\mathbf{x}\_{1},\ldots,\mathbf{x}\_{N}) \right\}\_{(m\_{1},\ldots,m\_{p})\in\mathbb{Z}^{p},(m\_{p+1},\ldots,m\_{q})\in\mathbb{Z}^{q-p},(m\_{q+1},\ldots,m\_{N})\in\mathbb{N}^{N-q}} &= \\ \left\{ \prod\_{j=1}^{p} \sqrt{\frac{2}{\pi}} \frac{\beta\_{j}\cos\lambda\_{m\_{j}}\mathbf{x}\_{j} + \varepsilon\_{m\_{j}}\operatorname{sign}(\beta\_{j}^{2}-\mathbf{a}\_{j}^{2}) \cdot \mathbf{a}\_{j}\sin\lambda\_{m\_{j}}\mathbf{x}\_{j}}{\sqrt{\mathbf{a}\_{j}^{2}+\beta\_{j}^{2}}\cdot\sqrt{1+\left|\ \lambda\_{m\_{j}}\right|^{2s\_{j}}}} \right\}\_{(m\_{1},\ldots,m\_{p})\in\mathbb{Z}^{p}} &\times \\ \times \left\{ \prod\_{j=p+1}^{q} \frac{1}{\sqrt{\pi}} \frac{1}{\sqrt{1+\left|\ 2m\_{j}\right|^{2s\_{j}}}} \exp(i2m\_{j}\mathbf{x}\_{j}) \right\}\_{(m\_{p+1},\ldots,m\_{q})\in\mathbb{Z}^{p}} &\times \\ &\times \left\{ \prod\_{j=q+1}^{N} \sqrt{\frac{2}{\pi}} \frac{1}{\sqrt{1+\left|\ {m\_{j}}\right|^{2s\_{j}}}} \sin(m\_{j}\mathbf{x}\_{j}) \right\}\_{(m\_{q+1},\ldots,m\_{N})\in\mathbb{N}^{N^{1-q}}} \end{split}$$

*of spectral Problems (7) and (8) converges in the norm of space Ck*(Π) *to function f*(*x*).

The proof of Corollary 3 is carried out using Theorem 2 and the Sobolev embedding theorem. The following are true:

#### **4. Main Results**

In this section, we give the most general case of the works done in [17].

**Theorem 3.** *Let α<sup>j</sup>* = 0, *β<sup>j</sup>* = 0, |*αj*| = |*βj*| *be real numbers at every* 1 ≤ *j* ≤ *p*, *and*

$$\rho = \max\_{1 \le j \le p} \sqrt{\theta\_j^2 + 2\left(\frac{\theta\_j}{\sqrt{2}} + (\varphi\_j + 1)^{s\_j} - 1\right)^2} \cdot \sigma(s\_j) < 1$$

*where <sup>σ</sup>*(0) = <sup>1</sup> √2 , *<sup>σ</sup>*(*sj*) = <sup>1</sup> *at sj* <sup>&</sup>gt; 0, *<sup>θ</sup><sup>j</sup>* <sup>=</sup> <sup>√</sup> 2 · max *x*∈[0,*π*] *e <sup>i</sup>ϕjx* <sup>−</sup> <sup>1</sup> , *λmj* = 2*mj* + *εmj* · *ϕj*, *<sup>ϕ</sup><sup>j</sup>* <sup>=</sup> <sup>1</sup> *<sup>π</sup>* arccos <sup>−</sup>2*αjβ<sup>j</sup> α*2 *<sup>j</sup>* + *<sup>β</sup>*<sup>2</sup> *j* , *<sup>ε</sup>mj* <sup>=</sup> *<sup>ε</sup>*−*mj* <sup>=</sup> <sup>±</sup><sup>1</sup> *at mj* <sup>∈</sup> *Z, sj* <sup>&</sup>gt; *<sup>k</sup>* <sup>+</sup> *<sup>N</sup>* <sup>2</sup> , *k* ≥ 0, *k* ∈ *Z and ϕj*(*x*) ∈ *<sup>W</sup>s*1+*j*<sup>−</sup> *<sup>N</sup>* <sup>2</sup> ,*s*2+*j*<sup>−</sup> *<sup>N</sup>* <sup>2</sup> ,...,*sN*+*j*<sup>−</sup> *<sup>N</sup>* 2

<sup>2</sup> (Π), *<sup>f</sup>*(*x*, *<sup>t</sup>*) <sup>∈</sup> *<sup>W</sup>s*1,*s*2,...,*sN*,*sN*+<sup>1</sup> <sup>2</sup> (Π × (0, +∞)). *Then, the solution of problems (4)–(6) exists, it is unique, and is represented in the form of series*

$$\begin{split} u(\mathbf{x},t) &= \sum\_{m\_1 = -\infty}^{\infty} \cdots \cdot \sum\_{m\_q = -\infty}^{\infty} \sum\_{m\_{q+1} = 1}^{\infty} \cdots \cdot \sum\_{m\_N = 1}^{\infty} \sum\_{j=1}^{n} \varphi\_{j,(m\_1 \dots m\_N)} t^{n-j} E\_{\mathbf{z}, \mathbf{z} \cdots j+1} (-\mu\_{m\_1 \dots m\_N} \cdot t^n) + \\ &+ \int\_0^t (t - \tau)^{\mathbf{z} - 1} \cdot E\_{\mathbf{z}, \mathbf{z}} [-\mu\_{m\_1 \dots m\_N} (t - \tau)^n] f\_{m\_1 \dots m\_N} (\tau) d\tau \cdot \upsilon\_{m\_1 \dots m\_N} (\mathbf{x}\_1, \dots, \mathbf{x}\_N) \end{split} \tag{11}$$

*where coefficients are determined in the following way :*

$$E\_{\mathfrak{a},\mathfrak{a}-j+1}(-\mu\_{m\_1...m\_N}\cdot t^{\mathfrak{a}}) = \sum\_{i=0}^{\infty} \frac{(-\mu\_{m\_1...m\_N}\cdot t^{\mathfrak{a}})^i}{\Gamma(\mathfrak{a}i+\mathfrak{a}-j+1)},$$

$$E\_{a,\mathbf{z}}\left(-\mu\_{m\_1\ldots m\_N}\cdot(t-\tau)^a\right) = \sum\_{i=1}^{\infty} \frac{(-\mu\_{m\_1\ldots m\_N})^{i-1}\cdot(t-\tau)^{a(i-1)}}{\Gamma(a\cdot i)},$$

$$f(\mathbf{x},t) = \sum\_{m\_1=-\infty}^{\infty}\cdots\cdot\sum\_{m\_q=-\infty}^{\infty}\sum\_{m\_{q+1}=1}^{\infty}\cdots\cdot\sum\_{m\_N=1}^{\infty}f\_{m\_1\ldots m\_N}(t)\cdot\upsilon\_{m\_1\ldots\ldots m\_N}(\mathbf{x}\_1,\ldots,\mathbf{x}\_N),$$

$$\varphi\_j(\mathbf{x}) = \sum\_{m\_1=-\infty}^{\infty}\cdots\cdot\sum\_{m\_q=-\infty}^{\infty}\sum\_{m\_{q+1}=1}^{\infty}\cdots\cdot\sum\_{m\_N=1}^{\infty}\varphi\_{j,(m\_1\ldots m\_N)}\cdot\upsilon\_{m\_1\ldots\ldots m\_N}(\mathbf{x}\_1,\ldots,\mathbf{x}\_N),$$

$$j = 1,2,\ldots,n\_r\text{ }\mu\_{m\_1\ldots m\_N} = \lambda\_{m\_1}^2 + \cdots + \lambda\_{m\_N}^2.$$

**Proof.** Since system of eigenfunctions

$$\left\{\varpi\_{\mathfrak{M}\_1\ldots\mathfrak{M}\_N}(\mathfrak{x}\_1,\ldots,\mathfrak{x}\_N)\right\}\_{(\mathfrak{m}\_1,\ldots,\mathfrak{m}\_p)\in\mathbb{Z}^{p},(\mathfrak{m}\_{p+1},\ldots,\mathfrak{m}\_q)\in\mathbb{Z}^{q-p},(\mathfrak{m}\_{q+1},\ldots,\mathfrak{m}\_N)\in\mathbb{N}^{N-q}}\in\mathbb{N}^{N-q}$$

of spectral Problems (7) and (8) forms the complete orthonormal system in Sobolev classes *Ws*1,*s*2,...,*sN* <sup>2</sup> (Π), any function from class *<sup>W</sup>s*1,*s*2,...,*sN* <sup>2</sup> (Π) can be represented as a convergent Fourier series in this system. For any *t* > 0, expand solution *u*(*x*, *t*) of Problems (4)–(6) into the Fourier series in eigenfunctions

$$\{\varpi\_{\mathfrak{M}\_1\dots m\_N}(\mathfrak{x}\_1\dots \mathfrak{x}\_{\mathcal{N}})\}\_{(\mathfrak{m}\_1,\dots,\mathfrak{m}\_p)\in\mathbb{Z}} \in \mathbb{Z}\\
\mathbb{P}\_{\succ}(\mathfrak{m}\_{p+1},\dots,\mathfrak{m}\_q)\in\mathbb{Z}\\
\mathbb{A}^{q-p}, (\mathfrak{m}\_{q+1},\dots,\mathfrak{m}\_N)\in\mathbb{N}^{N-q}$$

of spectral Problems (4) and (5):

$$\mu(\mathbf{x},t) \quad = \sum\_{m\_1=-\infty}^{\infty} \cdots \sum\_{m\_q=-\infty}^{\infty} \sum\_{m\_{q+1}=1}^{\infty} \cdots \sum\_{m\_N=1}^{\infty} T\_{m\_1 \dots m\_N}(t) \cdot \upsilon\_{m\_1 \dots m\_N}(\mathbf{x}),\tag{12}$$

$$T\_{m\_1\dots m\_N}(t) = (\mu(\mathfrak{x}, t), \upsilon\_{m\_1\dots m\_N}(\mathfrak{x})) .$$

By virtue of Problems (4) and (5), unknown functions *Tm*1...*mN* (*t*) must satisfy equation

$$D\_{0t}^{\mathfrak{a}}T\_{m1\ldots m\chi}(t) + \mu\_{m1\ldots m\chi}T\_{m1\ldots m\chi}(t) = f\_{m1\ldots m\chi}(t), \ l - 1 < \mathfrak{a} \le l, \ l \in \mathcal{N} \tag{13}$$

with initial conditions

$$\lim\_{t \to 0} D\_{0t}^{a-k} T\_{m\_1 \dots m\_N}(t) = \varphi\_{k, m\_1 \dots m\_N}, \quad k = 1, 2, \dots, l, \quad \mu\_{m\_1 \dots m\_N} = \lambda\_{m\_1}^2 + \dots + \lambda\_{m\_N}^2. \tag{14}$$

The solution of Cauchy Problems (13) and (14) has the form

$$\begin{aligned} \left(T\_{m1\dots m\_N}(t)\right) &= \sum\_{j=1}^n \varphi\_{j,(m\_1\dots m\_N)} t^{n-j} E\_{a,a-j+1}(-\mu\_{m\_1\dots m\_N} \cdot t^a) + \\\\ &+ \int\_0^t (t-\tau)^{a-1} \cdot E\_{a,a}[-\mu\_{m\_1\dots m\_N}(t-\tau)^a] f\_{m\_1\dots m\_N}(\tau) d\tau \end{aligned} \tag{15}$$

where coefficients are determined as follows:

$$E\_{\mathfrak{a},\mathfrak{a}-j+1}(-\mu\_{m\_1...m\_N}\cdot\mathfrak{t}^{\mathfrak{a}}) = \sum\_{i=0}^{\infty} \frac{(-\mu\_{m\_1...m\_N}\cdot\mathfrak{t}^{\mathfrak{a}})^i}{\Gamma(\mathfrak{a}i+\mathfrak{a}-j+1)}.$$

$$E\_{a,a}\left(-\mu\_{m\_1\ldots m\_N}\cdot (t-\tau)^a\right) = \sum\_{i=1}^{\infty} \frac{(-\mu\_{m\_1\ldots m\_N})^{i-1} \cdot (t-\tau)^{a(i-1)}}{\Gamma(a\cdot i)},$$

$$f(\mathbf{x},t) = \sum\_{m\_1=-\infty}^{\infty} \cdot \dots \cdot \sum\_{m\_q=-\infty}^{\infty} \sum\_{m\_{q+1}=1}^{\infty} \cdot \dots \cdot \sum\_{m\_N=1}^{\infty} f\_{m\_1\ldots m\_N}(t) \cdot \upsilon\_{m\_1\ldots m\_N}(\mathbf{x}\_1,\ldots,\mathbf{x}\_N),$$

$$\sigma\_j(\mathbf{x}) = \sum\_{m\_1=-\infty}^{\infty} \cdot \dots \cdot \sum\_{m\_q=-\infty}^{\infty} \sum\_{m\_{q+1}=1}^{\infty} \cdot \dots \cdot \sum\_{m\_N=1}^{\infty} \theta\_{j,(m\_1\ldots m\_N)} \cdot \upsilon\_{m\_1\ldots m\_N}(\mathbf{x}\_1,\ldots,\mathbf{x}\_N), j = 1,2,\ldots,n.$$

After substituting Problem (15) into Problem (12), we obtain the unique solution of Problems (4)–(6) in the form of Series (8).

Let *ν* > 1. Consider mixed Problems (4)–(6). If we look for a solution *u*(*x*, *t*) to Problems (4)–(6) in the form of Fourier series expansion

$$u(\mathbf{x},t) \quad = \sum\_{m\_1=-\infty}^{\infty} \cdots \sum\_{m\_q=-\infty}^{\infty} \sum\_{m\_{q+1}=1}^{\infty} \cdots \sum\_{m\_N=1}^{\infty} T\_{m\_1 \dots m\_N}(t) \cdot \upsilon\_{m\_1 \dots m\_N}(\mathbf{x}),$$

where are *Tm*1...*mN* (*t*)=(*u*(*x*, *t*), *vm*1...*mN* (*x*)) are the coefficients of the series, {*vm*1...*mN* } is the system of eigenfunctions of spectral Problems (7) and (8).

Differential operator (−Δ)*ν*, generated by a differential expression *<sup>l</sup>* (*ν*)(*v*(*x*)) = (−Δ)*νv*(*x*) with domain definition

$$D\left(( - \Delta)^{\upsilon} \right) = \{ v(\mathbf{x}) : v(\mathbf{x}) \in \mathbb{C}^{2\upsilon}(\Pi) \cap \mathbb{C}^{2\upsilon - 1}(\Pi), l^{(\upsilon)}(v(\mathbf{x})) \in L\_2(\Pi) \}$$

satisfies Condition (8).

Similarly, as Lemma 5, it can be shown that operator (−Δ)*ν*, is a symmetric and positive operator in space *L*2(Π). The eigenvalues of Problems (7) and (8) *μm*1...*mN* ≥ 0, and each *μμ<sup>m</sup>*1...*mN* = *λ*2 *<sup>m</sup>*<sup>1</sup> <sup>+</sup> ··· *<sup>λ</sup>*<sup>2</sup> *m*<sup>1</sup> *ν* corresponds to an eigenvalue of Problems (9) and (10), and the eigenfunctions {*vm*1...*mN* (*x*)} of Problems (7) and (8) and eigenfunctions {*ym*1...*mN* (*x*)} of Problems (9) and (10) coincide, i.e.,

$$
\varphi\_{\mathcal{M}\_1\ldots\mathcal{M}\_N}(\mathbf{x}) \equiv \mathcal{Y}^{\mathcal{M}\_1\ldots\mathcal{M}\_N}(\mathbf{x})\ldots
$$

Therefore, the following theorem is valid:

**Theorem 4.** *Let α<sup>j</sup>* = 0, *β<sup>j</sup>* = 0, |*αj*| = |*βj*| *be real numbers at every* 1 ≤ *j* ≤ *p*, *and*

$$\rho = \max\_{1 \le j \le p} \sqrt{\theta\_j^2 + 2\left(\frac{\theta\_j}{\sqrt{2}} + (\varphi\_j + 1)^{s\_j} - 1\right)^2} \cdot \sigma(s\_j) < 1$$

*where <sup>σ</sup>*(0) = <sup>1</sup> √2 , *<sup>σ</sup>*(*sj*) = <sup>1</sup> *at sj* <sup>&</sup>gt; 0, *<sup>θ</sup><sup>j</sup>* <sup>=</sup> <sup>√</sup> 2 · max *x*∈[0,*π*] *e <sup>i</sup>ϕjx* <sup>−</sup> <sup>1</sup> , *λmj* = 2*mj* + *εmj* · *ϕj*, *<sup>ϕ</sup><sup>j</sup>* <sup>=</sup> <sup>1</sup> *<sup>π</sup>* arccos <sup>−</sup>2*αjβ<sup>j</sup> α*2 *<sup>j</sup>* + *<sup>β</sup>*<sup>2</sup> *j* , *<sup>ε</sup>mj* <sup>=</sup> *<sup>ε</sup>*−*mj* <sup>=</sup> <sup>±</sup><sup>1</sup> *at mj* <sup>∈</sup> *Z, sj* <sup>&</sup>gt; (*<sup>k</sup>* <sup>+</sup> *<sup>N</sup>* <sup>2</sup> )*ν*, *k* ≥ 0, *k* ∈ *Z and*

*<sup>ϕ</sup>j*(*x*) <sup>∈</sup> *<sup>W</sup>*(*s*1+*j*<sup>−</sup> *<sup>N</sup>* <sup>2</sup> )*ν*,(*s*2+*j*<sup>−</sup> *<sup>N</sup>* <sup>2</sup> )*ν*,...,(*sN*+*j*<sup>−</sup> *<sup>N</sup>* <sup>2</sup> )*ν* <sup>2</sup> (Π), *<sup>f</sup>*(*x*, *<sup>t</sup>*) <sup>∈</sup> *<sup>W</sup>s*1,*s*2,...,*sN*,*sN*+<sup>1</sup> <sup>2</sup> (Π × (0, +∞)). *Then the solution of Problems (4)–(6) exists, it is unique, and is represented in the form of series*

$$\begin{split} u(\mathbf{x},t) &= \sum\_{m\_1=-\infty}^{\infty} \cdots \cdot \sum\_{m\_q=-\infty}^{\infty} \sum\_{m\_{q+1}=1}^{\infty} \cdots \cdot \sum\_{m\_N=1}^{\infty} \sum\_{j=1}^{n} \boldsymbol{\varrho}\_{j,(m\_1\dots m\_N)} \mathbf{t}^{n-j} \boldsymbol{E}\_{\mathbf{z}, \mathbf{z}-j+1} (-\boldsymbol{\mu}\_{m\_1\dots m\_N} \cdot \mathbf{t}^n) + \cdots \\ &+ \int\_0^t (t-\tau)^{\mathbf{z}-1} \cdot \boldsymbol{E}\_{\mathbf{z}, \mathbf{z}} [-\boldsymbol{\mu}\_{m\_1\dots m\_N} (t-\tau)^a] f\_{m\_1\dots m\_N}(\tau) d\tau \cdot \boldsymbol{\upsilon}\_{m\_1\dots m\_N} (\mathbf{x}\_1,\dots,\mathbf{x}\_N) \end{split}$$

*where coefficients are determined in the following way:*

$$E\_{a,a-j+1}(-\mu\_{m\_1\ldots m\_N}\cdot t^a) = \sum\_{i=0}^{\infty} \frac{(-\mu\_{m\_1\ldots m\_N}\cdot t^a)^i}{\Gamma(ai+a-j+1)},$$

$$E\_{a,a}\left(-\mu\_{m\_1\ldots m\_N}\cdot (t-\tau)^a\right) = \sum\_{i=1}^{\infty} \frac{(-\mu\_{m\_1\ldots m\_N})^{i-1}\cdot (t-\tau)^{a(i-1)}}{\Gamma(a\cdot i)},$$

$$f(\mathbf{x},t) = \sum\_{m\_1=-\infty}^{\infty} \dots \sum\_{m\_\ell=-\infty}^{\infty} \sum\_{m\_\ell+1=1}^{\infty} \dots \sum\_{m\_N=1}^{\infty} f\_{m\_1\ldots m\_N}(t) \cdot \upsilon\_{m\_1\ldots m\_N}(\mathbf{x}\_1,\ldots,\mathbf{x}\_N),$$

$$\eta\_j(\mathbf{x}) = \sum\_{m\_1=-\infty}^{\infty} \dots \sum\_{m\_\ell=-\infty}^{\infty} \sum\_{m\_\ell+1=1}^{\infty} \dots \sum\_{m\_N=1}^{\infty} \eta\_{j,(m\_1\ldots m\_N)} \cdot \upsilon\_{m\_1\ldots m\_N}(\mathbf{x}\_1,\ldots,\mathbf{x}\_N),\ j = 1,2,\ldots,n\_r$$

$$\mu\_{m\_1\ldots m\_N} = \left(\lambda\_{m\_1}^2 + \dots + \lambda\_{m\_N}^2\right)^\nu.$$
