On a problem with a shift for a parabolic-hyperbolic equation with... 75 Uzbek Mathematical Journal 2023, Volume 67, Issue 2, pp.75-87 DOI: 10.29229/uzmj.2023-2-9 ON A PROBLEM WITH A SHIFT FOR A PARABOLIC-HYPERBOLIC EQUATION WITH THE GERASIMOV-CAPUTO OPERATOR IN A DOMAIN WITH DEVIATION OUT OF THE CHARACTERISTIC Islomov B.I., Akhmadov I.A. Abstract. In this paper, we study a problem with a shift for a parabolic-hyperbolic equation with the Gerasimov-Caputo operator in a domain with deviation out of the characteristic. The correctness of the problem in the sense of a classic and strong solution is proved. Keywords: fractional order equation, domain with deviation from characteristic, Wright and Green function, classical solution, strong solution. MSC (2010): 35A02, 35M10, 35S05. 1. Introduction For the first time in 1962, the boundary value problem for the Lavrentiev–Bitsadze equation with a boundary condition connecting the values of the desired function on two independent characteristics in the hyperbolic part of the domain was formulated and investigated in [20]. In works [12],[13] A.M. Nakhushev studied non-local boundary value problems (problems with a shift) for a degenerate equation of hyperbolic and mixed types. Currently, not a little of works are devoted to the study of boundary value problems with a shift for various types and orders of equations. We note the papers [2], [10], [14], [19], [16]. Further in the work of V.A. Eleev [3] considered the classical solvability, which is an analog of the Tricomi problem for mixed parabolichyperbolic equations with a noncharacteristic line of type change. In the work of N.Yu. Kapustin [8] considered the strong solvability of the Tricomi problem for a system of parabolic-hyperbolic equations. In the works of M.A. Sadybekov [17] and A.S. Berdyshev [1] proved the strong solvability and Volterra property (the absence of eigenvalues) of a boundary value problem in a domain with departure from the characteristic for a second-order mixed-type equation. Note that boundary value problems for equations of parabolic and parabolic-hyperbolic types of fractional order have been little studied. Note the works [5], [6], [7], [9], [15]. In this paper, we study a boundary value problem with a shift for a mixed parabolic- hyperbolic equation with a fractional order operator in the sense of Gerasimov-Caputo in a domain with deviation out of the characteristic. The correctness of the formulated problem is proved in the sense of a classical and strong solution. 2. Formulation of the problem Let Ω ⊂ R 2 be finite domain, bounded at y> 0 by segments AA 0 ,A 0 B 0 ,B 0 B and at y< 0 smooth curve AC : y = -γ (x), 0 ≤ x ≤ 1, γ (0) = γ (1) = 0, located inside the characteristic triangle 0 ≤ x + y ≤ x - y ≤ 1, where A = (0, 0), A 0 = (0, 1), B 0 = (1, 1), B = (1, 0), AB = J = {(x, y): 0 <x< 1,y =0} . Consider the equation Lz (x, y)= g(x; y), (2.1)