Boundary value problems for hybrid differential equations with fractional order

Abstract

This note is motivated by some papers treating the fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0 < α < 1. An existence theorem for this equation is proved under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle is proved, which will be useful for further study of qualitative behavior of solutions.

A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions

“…Thus, it is very important we increase our abilities in modern modeling by working on complicated fractional integro-differential equations and inclusions. As is well known, there have been studied different types of hybrid equations by many researchers (see, for example, [1][2][3][4] and [5]). One of the significant strategies is reviewing of hybrid models of different phenomena.…”

Dhage iterative principle for quadratic perturbation of fractional boundary value problems with finite delay

“…By using this approach, along with the existence, we attempt to prove the uniqueness and approximate results under weaker partial compactness and partial Lipschitz type conditions by assuming that either the lower or upper solution exists in it. For details about the technique of upper and lower solution, we cite some good references for the readers, see previous works …”

“…For details about the technique of upper and lower solution, we cite some good references for the readers, see previous works. [25][26][27] As far as authors know, this is the first work addressing the nonlinear hybrid multi-point BVP involving the fractional derivatives of unknown functions. In which, by employing some standard classical FPTs, we shall develop the result of existence of solutions for considered problem (2.1).…”

Existence Results to a Class of Hybrid Fractional Differential Equations

“…They develop the sufficient condition forexistence and uniqueness of solution for the aforesaid class of hybrid fractional differential equation. A group researcher, generalized the above results to the following hybrid fractional differential equations with boundary conditions involving Caputo's derivative [42].…”

“…The most significant feather, that attracted the consideration of researchers is to investigates the conditions under the system of HFDEs is positive solutions. For the aforesaid purpose, Dhage, Lakshmikantham and Krasnoselskii are extensively studied the system of HFDEs [38][39][40]. …”