Nonlinear sequential fractional differential equations in partially ordered spaces

Abstract

In this paper, some new partially ordered Banach spaces are introduced. Based on those new partially ordered Banach spaces and applying some fixed point theorems, we present a new approach to the theory of nonlinear sequential fractional differential equations. An example illustrating our approach is also discussed. where D α is the classical Riemann-Liouville fractional derivative of order α.

Existence of solutions of infinite system of nonlinear sequential fractional differential equations

“…Recently, Fazli and Nieto [1] investigated the existence and uniqueness of the following interesting problem, which is a model of physical phenomena: where 0 < α ≤ 1, 0 < T < ∞. The term D 2α is for the sequence fractional derivative presented by Miller and Ross [21],…”

“…where D α is the classical Riemann-Liouville fractional derivative of order α. Before giving the weighted Cauchy type problem obtained in [1], let us recall some notions introduced in that work. Let…”

“…A function u ∈ C α 1-α [0, T] is called a lower solution of the initial value problem (1), if D 2α u(x) ≤ f (x, u(x), D α u(x)) for every x ∈ (0, T] and…”

“…(H 2 ) f is non-decreasing in all its arguments except for the first argument and f (x, u, v)f (x, ũ, ṽ) ≤ L 1 (u -ũ) + L 2 (v -ṽ) for some L 1 , L 2 > 0 whenever x ∈ (0, T] and u ≥ ũ, v ≥ ṽ. The weighted Cauchy type problem presented in [1] is given by the following result. Theorem 1.1 Assume that (H 1 )-(H 2 ) hold.…”

An investigation of fractional Bagley-Torvik equation

“…Fractional di erential equations appear naturally in a number of elds such as physics, geophysics, polymer rheology, regular variation in thermodynamics, biophysics, blood ow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity, Bode's analysis of feedback ampli ers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, tting of experimental data, nonlinear oscillation of earthquake, the uid-dynamic tra c model, etc. For more details and applications, we refer the reader to the books [1][2][3][4][5][6] and references [7][8][9][10][11][12][13][14].…”