Let $M_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in $M_{m,n}$ is >br>    $R_{m,n}=Z\in M_{m,n}:I^{\left(m\right)}-ZZ*ispositivedefinite$. Here $I^{\left(m\right)}\in M_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L_{\alpha }^p\left(R_{m,n}\right)$ is defined to be the space $L^p{R}_{m,n};{\left[det(I^{\left(m\right)}-ZZ*)\right]}^{\alpha }d{\mu }_{m,n}\left(Z\right)$, where ${\mu }_{m,n}$ is the Lebesgue measure in $M_{m,n}$, and $H_{\alpha }^p\left(R_{m,n}\right)\subset L_{\alpha }^p\left(R_{m,n}\right)$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if $Re\beta >(\alpha +1)/p-1$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then     $f\left(\right)=T_{m,n}^{\beta }\left(f\right)\left(\right),\in R_{m,n},$where $T_{m,n}^{\beta }$ is the integral operator defined by (0.13)-(0.14). In the present paper, given 1 ≤ p <...

In this paper, the boundedness of the Riesz potential generated by generalized shift operator $I_{B_k}^{\alpha }$ from the spaces $a=(L_{p_m,\nu }\left({ℝ}_n^k\right),a_m)$ to the spaces $a^{\text{'}}=(L_{q_m,\nu }\left({ℝ}_n^k\right),a_m^{\text{'}})$ is examined.

In mathematical models of some physical phenomena a new class of nonlinear Volterra equations appears ([5],[6]). The equations belonging to this class have u = 0 as a solution (trivial solution), but with respect to their physical meaning, nonnegative nontrivial solutions are of prime importance.

2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05The Caputo fractional derivative is one of the most used definitions of a fractional derivative along with the Riemann-Liouville and the Grünwald- Letnikov ones. Whereas the Riemann-Liouville definition of a fractional derivative is usually employed in mathematical texts and not so frequently in applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville fractional derivatives,...