Existence of solutions for differential inclusions with partial derivatives of fractional order
The paper is dedicated to the investigation of the Darboux problem for differential inclusions with partial derivatives of fractional order of the form \[ D^r_0 u(x,y)\in F \bigl( x,y,u(x,y) \bigr), \] where \(D^r_0 f(x,Y)\) is defined as follows: Consider the singular integral \[ I^\alpha_0 f(x,y)= {1\over \Gamma (\alpha)} \int^x_0 {f(s,y) \over (x-s)^{1- \alpha}} d s,\;x>0, \] and denote \(D^r_0 f(x,y)= D^2_{xy} f_{1-r} (x,y)\), with \(f_{1-r} (x,y)= I_0^{1-r} (x,y)\). Here, \(1-r\) stands for the couple \((1-\alpha, 1-\beta)\), while \(D^2_{xy}\) is the mixed derivative obtained by taking \(D^\alpha_0 f(x,y)= D_xf_{1- \alpha} (x,y)\). The auxiliary data are of the form \(u_{1-r} (x,0)= \overline u_0(x)\) and \(u_{1-r} (0,y)= \overline u_1 (y)\). Three existence theorems are obtained, under various assumptions (measurability and growth conditions). Estimates are given for the solutions, and some interesting integral inequalities are solved as a byproduct of the investigation.
scientific article; zbMATH DE number 1158844
scientific article