In a recent paper Al-Salam(1) has denned fractional q -integral operator by the basic integral (1) Where α ≠ 0, −1, −2, …. Using series definition of integrals, (1·1) is written as valid for all

We establish sufficient conditions for the existence of mild solutions some densely defined semilinear functional differential equations and inclusions involving Riemann-Liouville fractional derivative. Our approach is based on -semigroups theory combined with suitable fixed point theorems.

By giving a counter-example, we point out gap in the paper by Karapinar (Adv. Theory Nonlinear Anal. Its Appl. 2018, 2, 85–87) where given fixed may be not unique and present corrected version. We also state celebrated theorem of Reich–Rus–Ćirić framework complete partial metric spaces, taking interpolation theory into account. Some examples are provided main result papers Reich (Can. Math. Bull. 1971, 14, 121–124; Boll. Unione Mat. Ital. 1972, 4, 26–42 1–11.) is applicable.

Incorporating two delays ( $\tau_{1}$ represents the maturity of predator, $\tau_{2}$ top predator), we establish a novel delayed three-species food-chain model with stage structure in this paper. By analyzing characteristic equations, constructing suitable Lyapunov functional, using Lyapunov–LaSalle's principle, comparison theorem and iterative technique, investigate existence nonnegative equilibria their stability. Some interesting findings show that have great impacts on dynamical...

Abstract Fractional calculus was born in 1695 on September 30 due to a very deep question raised letter of L’Hospital Leibniz. The prophetical answer Leibniz that encapsulated huge inspiration for all generations scientists and is continuing stimulate the minds contemporary researchers. During 325 years existence, fractional has kept attention top level mathematicians, during last period time it become useful tool tackling dynamics complex systems from various branches science engineering....

The objective of this paper is to study oscillation fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new conditions are obtained which insure that all solutions the studied equation oscillatory. Our results complement some known for equations. An illustrative example included.

In this paper we are interested in the fractional-order form of Chua's system. A discretization process will be applied to obtain its discrete version. Fixed points and their asymptotic stability investigated. Chaotic attractor, bifurcation chaos for different values parameter discussed. We show that proposed method is from other methods, such as predictor-corrector Euler sense our an approximation right-hand side system under study.

By means of the familiar incomplete gamma functions γ(s, x) and Γ(s, x), we introduce Pochhammer symbols that lead us to a natural generalization decomposition class hypergeometric other related which are potentially useful in closed-form representations definite semi-infinite integrals various special functions. Applications these (for example) communication theory, probability theory groundwater pumping modelling shown.

In this paper, we study the oscillatory behavior of a class third-order nonlinear delay differential equations $$ (a(t) (b(t) y'(t))')' + q(t) y^\gamma(\tau(t)) = 0. Some new oscillation criteria are presented by transforming equation to first-order delayed and advanced equations. Employing suitable comparison theorems establish results on studied equation. Assumptions in our less restrictive, these improve those recent paper [Appl. Math. Comput., 202 (2008), 102-112] related contributions...