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For all α 1 , α 2 ∈ (1 , 2) with 1 / α 1 + 1 / α 2 > 5 / 3 , we show that the number of pairs (n 1 , n 2) of positive integers with N = ⌊ n 1 α 1 ⌋ + ⌊ n 2 α 2 ⌋ is equal to Γ (1 + 1 / α 1) Γ (1 + 1 / α 2) Γ (1 / α 1 + 1 / α 2) − 1 N 1 / α 1 + 1 / α 2 − 1 + o (N 1 / α 1 + 1 / α 2 − 1) as N → ∞ , where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when α 1 , α 2 ∈ (1 , 2) lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way. [ABSTRACT FROM AUTHOR]

The aim of this work is to derive a symbol calculus on L 2 (R n) for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second named author. [ABSTRACT FROM AUTHOR]

In this paper, we study results of well-posedness and regularity of higher order in time abstract non-autonomous semilinear Cauchy problems associated with Newton’s binomial theorem and the theory of sectorial operators. Our approach to parabolic problems of arbitrarily order n apparently has never been addressed earlier in the existing literature. Also, we present applications to evolutionary equations involving the fractional Laplacian in bounded smooth domains of R N . [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the existence and uniqueness of pseudo S -asymptotically periodic mild solutions for a class of neutral fractional evolution equations with finite delay. We essentially use the fractional powers of closed linear operators, the semigroup theory and some classical fixed point theorems. Furthermore, we provide an example to illustrate the applications of our abstract results. [ABSTRACT FROM AUTHOR]

In this work, we discuss the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces. We assume that the corresponding linear part is approximately controllable. The results are obtained by using fractional power theory and α-norm, the measure of noncompactness and the Mönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. As a result, we obtain a generalization of the work of Mahmudov [N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 2008, 3, 536–546], without assuming the compactness of the resolvent operator. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results. [ABSTRACT FROM AUTHOR]

In this paper, the sufficient conditions on summability degree p are found under which the Schrödinger operator with a potential, being singular on manifolds, is a positive one in Banach spaces Lp. It is also demonstrated that the domains of various degrees on this operator form an interpolation pair. In addition to this, the sufficient conditions on p are established which ensure that fractional powers σ, 0 < σ < 1, of this operator are bounded from to Lp. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the existence and uniqueness of pseudo S-asymptotically periodic mild solutions for a class of neutral fractional evolution equations with finite delay. We essentially use the fractional powers of closed linear operators, the semigroup theory and some classical fixed point theorems. Furthermore, we provide an example to illustrate the applications of our abstract results.

A generalization of the ultra-hyperbolic time-fractional diffusion-wave equation introduced by the author in Dorrego [The Mittag–Leffler function and its application to the ultra-hyperbolic time-fractional diffusion-wave equation. Integral Transforms Spec. Funct. 2016;27(5):392–404] is presented and studied. The results obtained here extend the previous to the whole space R n . The problem is solved by applying the integral transforms method and the analytical solution is given by the Fox H-function. Some interesting particular cases are analysed. [ABSTRACT FROM AUTHOR]

This work is concerned the study of a class of ill-posed Cauchy problems associated with a densely defined linear operatorΑ in a Banach space Ε. It is proved that if - A is the generator of an analytic semigroup, then there exists a family of regularizing operators by using the quasi-reversibility method, fractional powers and semigroups of linear operators. [ABSTRACT FROM AUTHOR]