Study of implicit delay fractional differential equations under anti-periodic boundary conditions

Abstract

This research work is related to studying a class of special type delay implicit fractional order differential equations under anti-periodic boundary conditions. With the help of classical fixed point theory due to Schauder and Banach, we derive some results about the existence of at least one solution. Further, we also study some results including Hyers-Ulam, generalized Hyers-Ulam, Hyers-Ulam Rassias, and generalized Hyers-Ulam-Rassias stability. We provide some test problems for illustrating our analysis.

Qualitative analysis of nonlinear coupled pantograph differential equations of fractional order with integral boundary conditions

“…Theorem 4 Let f 1 , f 2 , f 3 be continuous functions, subsequently (r, s) ∈ U is a solution of (1) if and only if (r, s) is the solution of the integral equations given in (15).…”

“…Basically, PDEs form a subclass of differential equations called delay, which gives change in terms of dependent variable at previous time [14]. Not long ago, in this area, some beneficial work has been carried out (see [15][16][17]).…”

Existence and uniqueness results for Φ-Caputo implicit fractional pantograph differential equation with generalized anti-periodic boundary condition

“…i. Observe that, if α 1 = α 2 = 1 and (t) = t, problems (1.4) and coincide with Caputo fractional differential equation considered in [8] and [1,3,14] respectively. ii.…”

“…representing the right and left limits of v(t) at t = t j . Recently, Ali et al [8] examined the existence, uniqueness and stability of the boundary value problem:…”

Qualitative Study of Nonlinear Coupled Pantograph Differential Equations of Fractional Order

“…For these studies, they have used classical fixed point theory, tools of nonlinear analysis and optimization theory (we refer Refs. [13][14][15][16][17][18]. The mentioned area has many applications in science, like probability theory, physics, control theory, light absorption model and cell division, we refer few papers such as Refs.…”