On solvability of the non-local problem for the fractional mixed-type equation with Bessel operator

Fractional differential equations plays a significant role, because of its multiple applications in engineering, chemistry, biology and other parts of science and modeling the real-life problems [1]-[3]. Studying boundary value problems for linear and non-linear fractional differential equations with Riemann-Liouville and Caputo fractional derivative have becoming interesting targets simultaneously [4]-[7]. The quality and the types of articles have been changed when the generalized Riemann-Liouville differential operators (later called Hilfer derivative) used in the scientific field with its interesting application [8], [9]. To tell the truth, the generalization of the Riemann-Lioville fractional differential operators was already announced by M. M. Dzherbashian and A. B. Nersesian [10] in 1968, but because of some reasons it was not familiar with many mathematicians around the globe till the translation of this work published in the journal of Fract. Calc. Appl. An. [11]. We also refer some papers [12]-[13] devoted studying some problems with the Dzherbashian-Nersesian differential operator which has the following form