Vallée-Poussin theorem for Hadamard fractional functional differential equations

Abstract

We propose necessary and sufficient conditions for the negativity of the two-point boundary value problem in the form of the Vallée-Poussin theorem about differential inequalities for the Hadamard fractional functional differential problem \[ \begin{cases} \left({}^{\rm H}{D}{^{\alpha}}x\right)(t) + (Tx)(t)=f(t),\\ x(1)=x(e)=0. \end{cases} \] Here, the operator $ T:C\rightarrow L_{\infty } $ can be an operator with deviation (of delayed or advanced type), an integral operator or various linear combinations and superpositions. For example, the operator can be of the forms $ (Tx)(t)=q(t)x(t-\tau (t)) $ , $ (Tx)(t)=\int _{1}^{e} Q(t,s)x(\theta (s)){\rm d}s $ or $ (Tx)(t)=\int _{1}^{e} x(s){\rm d}_{s}Q(t,s) $ . We obtain explicit tests of negativity of Green's function in the form of algebraic inequalities. Our paper is the first one where a general form of the operator is considered with Hadamard fractional derivatives.

Keywords

• hadamard fractional derivative
• green's function
• vallée-poussin theorem
• differential inequality
• existence and uniqueness
• two-point fractional boundary value problem