**8. Conclusions**

In this paper, we considered a nonlocal boundary value problem *Tω* for a weak nonlinear partial differential equation of mixed type with fractional Hilfer operator *D<sup>α</sup>*,*γ* in a positive rectangular domain Ω 1 = {0 < *t* < *b*, 0 < *x*, *y* < *l*} and with spectral parameter *ω* in a negative rectangular domain Ω 2 = {−*a* < *t* < 0, 0 < *x*, *y* < *l*}.

The set of positive solutions of trigonometric Equation (33) with respect to spectral parameter *ω* was called a set of irregular values of the spectral parameter *ω*. The set of the remaining values of the spectral parameter ℵ = (0; ∞) \ was called a set of regular values of the spectral parameter *ω*.

For all regular values of the spectral parameter *ω* the quantity Δ*<sup>n</sup>*, *m* (*ω*) was nonzero. So, for large *n*, *m* the values of Δ*<sup>n</sup>*, *m* (*ω*) could not become quite small and there the problem of "small denominators" did not arise. Therefore, for regular values of the spectral parameter *ω* the quantity Δ*<sup>n</sup>*, *m* (*ω*) was separated from zero and we considered the questions of one value solvability of the considering boundary value problems (1)–(5).

We studied the boundary value problem *Tω* with following assumptions:

$$\begin{aligned} \varphi \left( \mathbf{x}, y \right) & \in \mathbb{C}^3[0; l]^2, \; \varphi\_{xxxx} \left( \mathbf{x}, y \right) \in L\_2[0; l]^2, \; \varphi\_{yyyy} \left( \mathbf{x}, y \right) \in L\_2[0; l]^2; \\\\ f\_i \left( \mathbf{x}, y, u \right) & \in \mathbb{C}^{3, 3, 0}\_{\mathbf{x}, y, u} \left( \left[ 0; l \right]^2 \times \mathbb{R} \right), f\_{i \times x \times x} \left( \mathbf{x}, y, u \right) \in L\_2 \left( \left[ 0; l \right]^2 \times \mathbb{R} \right), \\\\ f\_{i \, y \, y \, y} \left( \mathbf{x}, y, u \right) & \in L\_2 \left( \left[ 0; l \right]^2 \times \mathbb{R} \right); \end{aligned}$$

*χ*1 1 = max *i*=1, 3 max *n*, *m*∈N max *<sup>t</sup>*∈[0; *b*] *t* <sup>1</sup>−*γη inm* (*t*, *ω*) < ∞; *χ*2 1 = max *i*=1, 3 max *n*, *m*∈N max *<sup>t</sup>*∈[−*a*; 0] | *ξinm* (*t*, *ω*)| < ∞; *χ*30 = 99 *ϕ xxxxyyyy*(*<sup>x</sup>*, *y*) 99*L* 2 [0; *l*] 2 < ∞; *χ*3*i* = 99 *fi xxxxyyyy*(*<sup>x</sup>*, *y*, *γ*) 99*L* 2 [0; *l*] 2 < ∞; *f i xxxxyyyy*(*<sup>x</sup>*, *y*, *<sup>γ</sup>*1) − *f i xxxxyyyy*(*<sup>x</sup>*, *y*, *<sup>γ</sup>*2) ≤ *Ki* (*<sup>x</sup>*, *y*)| *γ*1 − *γ*2 |], *K*0*i* = *Ki* (*<sup>x</sup>*, *y*) *L* 2 [0; *l*] 2 < ∞; | Θ*i* (*ξ*, *x*, *y*, *u* 1) − Θ*i* (*ξ*, *x*, *y*, *u* 2)| ≤ Θ1 *i* (*<sup>x</sup>*, *y*)| *u* 1 − *u* 2 | , Θ2 *i* = Θ1 *i* (*<sup>x</sup>*, *y*) *L* 2 [0; *l*] 2 < <sup>∞</sup>, *i* = 1, 2; *ρ* = *γ*2 (*<sup>γ</sup>*1 + *<sup>γ</sup>*3) *γ*4 < 1, *γ*4 = max {*b K*0 1 Θ2 1; *a K*0 2Θ2 2} . If these conditions were fulfilled, then the boundary value problem *Tω* was uniquely

solvable for regular values of the spectral parameter *ω* ∈ ℵ with these solutions represented in the form of the Fourier series (53) and (54) in the domains Ω 1 and Ω 2, respectively. There the series (53), (54) and (57)–(62) were convergen<sup>t</sup> absolutely and uniformly in the corresponding domains Ω 1 or Ω 2.

For irregular values of the spectral parameter *ω* ∈ and for some *k*, *s* = *k* 1 , ..., *k s* the problem *Tω* had an infinite number of solutions in the form of series (64) and (65), if there the condition (63) was fulfilled.

**Author Contributions:** Conceptualization, T.K.Y. and B.J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflicts of interest.
