In this paper the quantum Hahn difference operator and the quantum Hahn integral operator are defined via the quantum shift operator θnew fractional integral inequalities are established by using the quantum Hahn integral for one and two functions bounded by quantum integrable functions. The Hermite-Hadamard type of ordinary and fractional quantum Hahn integral inequalities as well as the Pólya-Szegö type fractional Hahn integral inequalities and the Grüss-Cebyšev type fractional Hahn integral inequality are also presented. MSC: Primary 39A10; 39A13; secondary 39A70 Keywords: Hahn difference operator; Hahn integral operator; Hahn difference inequalities; Hermite-Hadamard quantum Hahn integral inequality; Pólya-Szegö type fractional Hahn integral inequalities; Grüss-Cebyšev type fractional Hahn integral inequality 1 Introduction and preliminaries Let be f defined on an interval I ⊆ R containing ω 0 := ω 1-q . The Hahn difference operator D q,ω , introduced in [1], is defined as D q,ω f (t) = ⎧ ⎨ ⎩ f (qt+ω)-f (t) t(q-1)+ω , t = ω 0 , f (ω 0 ), t = ω 0 ,

In this paper, we study the existence and uniqueness results for noninstantaneous impulsive fractional quantum Hahn integro-difference boundary value problems with integral boundary conditions, by using Banach contraction mapping principle and Leray–Schauder nonlinear alternative. Examples are included illustrating the obtained results. To the best of our knowledge, no work has reported on the existence of solutions to the Hahn-difference equation with noninstantaneous impulses.

In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ1-Hilfer and ψ2-Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results.

In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type ψ1-Hilfer and ψ2-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As far as we know, this combination of ψ1-Hilfer and ψ2-Caputo fractional derivatives in coupled systems is new in the literature. The uniqueness result is achieved via the Banach contraction mapping principle, while the existence result is established by applying the Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.

In this work, we initiate the study of a new class of impulsive boundary value problems consisting of mixed type fractional quantum and Hadamard derivatives. We will establish existence and uniqueness results by using tools from the functional analysis. We prove the uniqueness result via Banach’s contraction mapping principle, while we will use the Leray-Schauder nonlinear alternative to establish an existence result. We also present examples to illustrate the obtained results.