Existence of solutions for differential inclusions with partial derivatives of fractional order
type: Item
instance of: scholarly article
title: Existence of solutions for differential inclusions with partial derivatives of fractional order
zbMATH Open document ID: 0905.35102
author: Aleksandr N. Vityuk
published in: Russian Mathematics
publication date: 1999-02-02T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35R70
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 1158844
zbMATH Keywords: differential inclusion
zbMATH Keywords: partial derivatives of fractional order
zbMATH Keywords: existence
zbMATH Keywords: integral inequalities
reviewed by: Constantin Corduneanu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1158844