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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 ... Here, the operator T:C→L∞ 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-τ(t)), (Tx)(t)=∫e1Q(t,s)x(θ(s))ds or (Tx)(t)=∫e1x(s)dsQ(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. [ABSTRACT FROM AUTHOR]

In this a Lyapunov‐type inequality is obtained for the fractional differential equation with Caputo derivative CDaγx(t)+q(t)x(t)=0,a

In this paper, we investigate the existence of at least one solution to the following higher order Riemann–Liouville fractional differential equation with Riemann–Stieltjes integral boundary condition at resonance: −(D0+αx)(t)=f(t,x(t),D0+α−1x(t)),n−1<α≤n,t∈[0,1],x(0)=x′(0)=...=x(n−2)(0)=0,x(k)(1)=∫01x(k)(t)dA(t),$$ {\displaystyle \begin{array}{cc}\hfill -\left({D}_{0+}^{\alpha }x\right)(t)=& f\left(t,x(t),{D}_{0+}^{\alpha -1}x(t)\right),n-1<\alpha \le n,t\in \left[0,1\right],\hfill \\ {}\hfill x(0)=& {x}^{\prime }(0)=\dots ={x}^{\left(n-2\right)}(0)=0,{x}^{(k)}(1)=\int_0^1{x}^{(k)}(t) dA(t),\hfill \end{array}} $$by using Mawhin's coincidence degree theory. Here, D0+α$$ {D}_{0+}^{\alpha } $$ is the standard Riemann–Liouville fractional derivative of order α,f:[0,1]×ℝ2→ℝ$$ \alpha, f:\left[0,1\right]\times {\mathbb{R}}^2\to \mathbb{R} $$, and ∫01x(k)(t)dA(t)$$ {\int}_0^1{x}^{(k)}(t) dA(t) $$ is the Riemann–Stieltjes integral of x(k)$$ {x}^{(k)} $$ with respect to A$$ A $$. Our choice of k$$ k $$ in the boundary condition can be any integer between 0 and n−1$$ n-1 $$, which supplements many boundary conditions assumed in the literature. Several examples are given to strengthen our result. [ABSTRACT FROM AUTHOR]