Existence of solutions for multi-point boundary value problem of fractional q-difference equation

This paper is mainly concerned with the existence of solutions for a multi-point boundary value problem of nonlinear fractional q-difference equations by means of Banach contraction principle and Krasnoselskii's fixed point theorem. Further, an example is presented to illustrate the main results

Existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions. Our analysis relies on two fixed point theorems of a sum operator in partial ordering Banach space. The main results obtained can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it

Three-point boundary value problems with delta Riemann-Liouville fractional derivative on time scales

In this paper, we establish the criteria for the existence and uniqueness of solutions of a three-point boundary value problem for a class of fractional differential equations on time scales. By using some well known fixed point theorems, sufficient conditions for the existence of solutions are established. An illustrative example is also presented

On the Existence and Uniqueness of Solutions for Q-Fractional Boundary Value Problem

Abstract We discuss in this paper the existence and uniqueness of solutions for boundary value problem in a Banach space. Under certain conditions on f , the existence of solutions is obtained by applying Banach fixed point theorem and Schaefer's fixed point theorem

Existence and uniqueness of some Cauchy Type Problems in fractional q-difference calculus

In this paper we derive a sufficient condition for the existence of a unique solution of a Cauchy type q-fractional problem (involving the fractional q-derivative of Riemann-Liouville type) for some nonlinear differential equations. The key technique is to first prove that this Cauchy type q-fractional problem is equivalent to a corresponding Volterra q-integral equation. Moreover, we define the $q$-analogue of the Hilfer fractional derivative or composite fractional derivative operator and prove some similar new equivalence, existence and uniqueness results as above. Finally, some examples are presented to illustrate our main results in cases where we can even give concrete formulas for these unique solutions

On the solutions of some fractional q-differential equations with the Riemann-Liouville fractional q-derivative

This paper is devoted to explicit and numerical solutions to linear fractional q -difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q -derivative in q -calculus. The approaches based on the reduction to Volterra q -integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q -difference equations. For simplicity, we give results involving fractional q -difference equations of real order a > 0 and given real numbers in q -calculus. Numerical treatment of fractional q -difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection

On the solutions of some fractional q-differential equations with the Riemann-Liouville fractional q-derivative

This paper is devoted to explicit and numerical solutions to linear fractional q-difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q-derivative in q-calculus. The approaches based on the reduction to Volterra q-integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q-difference equations. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection

Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

In this article we discuss some of the qualitative properties of fractional difference operators. We especially focus on the connections between the fractional difference operator and the monotonicity and convexity of functions. In the integer-order setting, these connections are elementary and well known. However, in the fractional-order setting the connections are very complicated and muddled. We survey some of the known results and suggest avenues for future research. In addition, we discuss the asymptotic behavior of solutions to fractional difference equations and how the nonlocal structure of the fractional difference can be used to deduce these asymptotic properties

Development and Application of Difference and Fractional Calculus on Discrete Time Scales

The purpose of this dissertation is to develop and apply results of both discrete calculus and discrete fractional calculus to further develop results on various discrete time scales. Two main goals of discrete and fractional discrete calculus are to extend results from traditional calculus and to unify results on the real line with those on a variety of subsets of the real line. Of particular interest is introducing and analyzing results related to a generalized fractional boundary value problem with Lidstone boundary conditions on a standard discrete domain N_a. We also introduce new results regarding exponential order for functions on quantum time scales, along with extending previously discovered results. Finally, we conclude by introducing and analyzing a boundary value problem, again with Lidstone boundary conditions, on a mixed time scale, which may be thought of as a generalization of the other time scales in this work. Advisers: Lynn Erbe and Allan Peterso