Existence Results of Initial Value Problems for Hybrid Fractional Sum-Difference Equations

Chasreechai, Saowaluck; Sitthiwirattham, Thanin

doi:10.1155/2018/5268528

Cited by 16 publications

(7 citation statements)

References 62 publications

(25 reference statements)

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“…The definition of Hilfer fractional sum and differences are proposed in [34]. As for hybrid equations with discrete fractional operators, the authors in [35] considered the hybrid fractional sumdifference equations and investigated the existence of solutions. The qualitative properties of the discrete fractional hybrid equations are yet to be explored.…”

Section: Introductionmentioning

confidence: 99%

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

et al. 2022

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Pantograph, a device in which an electric current is collected from overhead contact wires, is introduced to increase the speed of trains or trams. The work aims to study the stability properties of the nonlinear fractional order generalized pantograph equation with discrete time, using the Hilfer operator. Hybrid fixed point theorem is considered to study the existence of solutions, and the uniqueness of the solution is proved using Banach contraction theorem. Stability results in the sense of Ulam and Hyers, and its generalized form of stability for the considered initial value problem are established and we depict numerical simulations to demonstrate the impact of the fractional order on stability.

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Section: Introductionmentioning

confidence: 99%

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

et al. 2022

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“…Non-linear differential equations with quadratic perturbations also known as hybrid type differential equations are of great interest to the mathematicians and engineers due to their ability to describe different dynamic models as special cases. The two types of fractional order perturbation equations are discussed in [38] and some recent contributions on hybrid type fractional order equations include [39][40][41][42][43][44][45][46][47][48][49]. The first work on Hyers-Ulam stability of fractional difference equation with boundary condition was carried out by Fulai Chen and Yong Zhou in 2013 [50].…”

Section: Introductionmentioning

confidence: 99%

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

et al. 2022

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The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vital in establishing the existence results by using Krasnoselkii’s fixed point theorem. Banach contraction mapping principle and Krasnoselkii’s fixed point theorem are employed to establish the uniqueness and existence results for solution of fractional order discrete equation. A problem on heat transfer with fins is provided as an application to considered hybrid type fractional order difference equation and the stability results are demonstrated with simulations.

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“…For instance, Kumam et al [16] highlighted the study of EU of solutions for a coupled system of high order hybrid functional DEs using a classical fixed point approach by Krasnosel'skii. Chasreechai et al [17] explored a hybrid fractional sum-difference initial and BVP in a Banach algebra and presented an instructive example.…”

Section: Introductionmentioning

confidence: 99%

Existence of positive solution and Hyers–Ulam stability for a nonlinear singular-delay-fractional differential equation

et al. 2019

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In this article, we consider a study of a general class of nonlinear singular fractional DEs with p-Laplacian for the existence and uniqueness (EU) of a positive solution and the Hyers-Ulam (HU) stability. To proceed, we use classical fixed point theorem and properties of a p-Laplacian operator. The fractional DE is converted into an integral alternative form with the help of the Green's function. The Green's function is analyzed as regards its nature and then, with the help of a fixed point approach, the existence of a positive solution and uniqueness are studied. After the EU of a positive solution, the HU-stability and an application are considered. The suggested singular fractional DE with φ p is more general than the one considered in (Khan et al. in Eur.

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Existence Results of Initial Value Problems for Hybrid Fractional Sum-Difference Equations

Cited by 16 publications

References 62 publications

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

A Study of Generalized Hybrid Discrete Pantograph Equation via Hilfer Fractional Operator

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

Existence of positive solution and Hyers–Ulam stability for a nonlinear singular-delay-fractional differential equation

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