On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order

Abstract

In this manuscript we introduce a quadratic integral equation of the Urysohn type of fractional variable order. The existence and uniqueness of solutions of the proposed fractional model are studied by transforming it into an integral equation of fractional constant order. The obtained new results are based on the Schauder’s fixed-point theorem and the Banach contraction principle with the help of piece-wise constant functions. Although the used methods are very powerful, they are not applied to the quadratic integral equation of the Urysohn type of fractional variable order. With this research we extend the applicability of these techniques to the introduced the Urysohn type model of fractional variable order. The applicability of the new results are demonstrated by providing Ulam–Hyers stability criteria and an example. Moreover, the presented results lead to future progress and expansion of the theory of fractional-order models, as well as of the concept of entropy in the framework of fractional calculus. Further, an example is constructed to demonstrate the reasonableness and effectiveness of the observed results.

New Results Achieved for Fractional Differential Equations with Riemann–Liouville Derivatives of Nonlinear Variable Order

“…Remark 7. Stability results for fractional differential systems with derivatives of variable order have been considered in the existent literature [20,35,38,40]. Hence, the proposed new stability criteria are a contribution to the development of the stability theory of such equations.…”

“…The fractional variable order generalization makes the fractional differential systems in terms of variable order derivatives a more flexible apparatus in modeling various processes and natural phenomena. Hence, there has been an increasing research activity in the theory of such equations [19][20][21][22][23][24], including recently studied applications [25][26][27][28][29] which demonstrated the flexibility of this modeling approach. In fact, fractional differential equations in terms of variable order fractional derivatives have proven to be suitable in modeling numerous phenomena such as anomalous diffusion [28,30], tumor modeling [29], petroleum engineering [31], viscoelastic mechanics [32] and many others [17,[25][26][27].…”

Multiterm Impulsive Caputo–Hadamard Type Differential Equations of Fractional Variable Order

“…One of the most extensively studied classes of fractional differential equations is the class of fractional equations with variable fractional order [9,10]. Different researchers studied properties of fractional equations with variable fractional order of Riemann-Liouville type [11][12][13], Caputo type [14][15][16], and Hadamard type [17][18][19]. The enormous interest in these equations is due mainly to the extended possibilities of their applications [20][21][22].…”

“…Inspired by [11,14,17,19,37,[39][40][41], we deal with the following impulsive boundary value problem (BVP)…”