On fuzzy fractional order derivatives and Darboux problem for implicit differential equations (Q2818493)

The authors investigate the Caputo's fuzzy fractional differential equations NEWLINE\[NEWLINE(^cD^q_0u(x,y))=f(x,y,u, (^cD^q_0u(x,y)),\,\, (x,y)\in [0,a]\times [0,b], NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,0)=\phi (x),\, x\in [0,a],\, u(0,y)=\phi(y),\, y\in [0,b],\, \phi(0)=\psi(0),NEWLINE\]NEWLINE where \(a, b > 0\), \(^cD^q_0\) is the Caputo's fractional derivative of order \(q = (q_1, q_2) \in (0, 1] \times (0, 1]\), \(f : [0,a]\times [0,b] \times E^n \times E^n \to E^n\) is a given continuous function, \(\phi : [0, a] \to E^n\), \(\psi : [0, b] \to E^n\) are given absolutely continuous functions with \(\phi(0)=\psi(0).\)NEWLINENEWLINEThe authors study the existence of solutions for fuzzy partial hyperbolic differential equations involving Caputo derivatives and introduced two results for this problem; the first one is based on the Banach contraction principle and the second one on a fixed point theorem for absolute retract spaces. So the present paper initiates the concept of fuzzy solutions for implicit fractional differential equations.