A nonlinear quantum boundary value problem (q-FBVP) formulated in the sense of quantum Caputo derivative, with fractional q-integro-difference conditions along with its fractional quantum-difference inclusion q-BVP are investigated in this research. To prove the solutions’ existence for these quantum systems, we rely on the notions such as the condensing functions and approximate endpoint criterion (AEPC). Two numerical examples are provided to apply and validate our main results in this research work.

A new theory of analytic functions has been recently introduced in the sense of conformable fractional derivative. In addition, the concept of fractional contour integral has also been developed. In this paper, we propose and prove some new results on complex fractional integration. First, we establish necessary and sufficient conditions for a continuous function to have antiderivative in the conformable sense. Finally, some of the well-known Cauchy′s integral theorems will also be the subject of the extension that we do in this paper.

We investigate the existence of solutions for a system of m-singular sum fractional q-differential equations in this work under some integral boundary conditions in the sense of Caputo fractional q-derivatives. By means of a fixed point Arzelá–Ascoli theorem, the existence of positive solutions is obtained. By providing examples involving graphs, tables, and algorithms, our fundamental result about the endpoint is illustrated with some given computational results. In general, symmetry and q-difference equations have a common correlation between each other. In Lie algebra, q-deformations can be constructed with the help of the symmetry concept.

The conformable derivative and its properties have been recently introduced. In this research work, we propose and prove some new results on the conformable calculus. By using the definitions and results on conformable derivatives of higher order, we generalize the theorems of the mean value which follow the same argument as in the classical calculus. The value of conformable Taylor remainder is obtained through the generalized conformable theorem of the mean value. Finally, we introduce the conformable version of two interesting results of classical multivariable calculus via the conformable formula of finite increments.

In the present research study, for a given multiterm boundary value problem (BVP) involving the nonlinear fractional differential equation (NnLFDEq) of variable order, the uniqueness-existence properties are analyzed. To arrive at such an aim, we first investigate some specifications of this kind of variable order operator and then derive required criteria confirming the existence of solution. All results in this study are established with the help of two fixed-point theorems and examined by a practical example.