In this article, we develop the distributed order fractional hybrid differential equations (DOFHDEs) with linear perturbations involving the fractional Riemann-Liouville derivative of order $0 < q < 1$ with respect to a nonnegative density function. Furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved under the mixed $varphi$-Lipschit...

Linear differential systems ẋ(t) = Ax(t) (A ∈ Rn×n, x0 = x(0) ∈ Rn, t ≥ 0) whose solutions become and remain nonnegative are studied. It is shown that the eigenvalue of A furthest to the right must be real and must possess nonnegative right and left eigenvectors. Moreover, for some a ≥ 0, A+aI must be eventually nonnegative, that is, its powers must become and remain entrywise nonnegative. Init...

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation ∂tu + (−∆) s u = 0, 0 < s < 1, posed in the whole space R with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative d...

Nonnegative rectangular matrices having nonnegative Unweighted group inverses are characterized. Our techniques suggest an interesting approach to extend the earlier known results on X-monotone square matrices to rectangular ones. We also answer a question of characterizing nonnegative matrices having a nonnegative solution Y where (1) A = AXA, (2) X = XAX, (3) (AX) is O-symmetric, (4) ( XA) is...

Abstract. We consider the aggregation equation ut+∇· [(∇K)∗u)u]=0 with nonnegative initial data in L(R)∩L(R) for n≥2. We assume that K is rotationally invariant, nonnegative, decaying at infinity, with at worst a Lipschitz point at the origin. We prove existence, uniqueness, and continuation of solutions. Finite time blow-up (in the L norm) of solutions is proved when the kernel has precisely a...