A coupled system of fractional differential equations on the half-line

Abstract

In this paper, we consider a new fractional differential system on an unbounded domain D α u(t) + ϕ(t, v(t), D γ 1 v(t)) = 0, t ∈ [0, +∞), α ∈ (2, 3], D β v(t) + ψ(t, u(t), D γ 2 u(t)) = 0, t ∈ [0, +∞), β ∈ (2, 3], subject to the conditions I 3-α u(t)| t=0 = 0, D α-2 u(t)| t=0 = h 0 g 1 (s)u(s) ds, D α-1 u(+∞) = Mu(ξ) + a, I 3-β v(t)| t=0 = 0, D β-2 v(t)| t=0 = h 0 g 2 (s)v(s) ds, D β-1 v(+∞) = Nv(η) + b. The nonlinear terms ϕ and ψ are dependent on the fractional derivative of lower order γ i ∈ (0, 1), i = 1, 2, which creates additional complexity to verify the existence of solutions. Moreover, a proper choice of Banach space allows the solutions to be defined on the half-line. From some standard fixed point theorems, sufficient conditions for the existence and uniqueness of solutions to boundary value problems are developed. Finally, the main result is applied to an illustrative example.

Explicit monotone iterative sequences for positive solutions of a fractional differential system with coupled integral boundary conditions on a half-line

“…Proof The proof is similar to that of Lemma 2.4 in [29], so we omit it. Lemma 2.9 (see [44]) Let U ⊂ X be a bounded set.…”

“…For some applications of this method to nonlinear fractional differential equations, see [16][17][18][19][20][21][22][23][24]. We also note that there are some results about monotone iterative solution of a single fractional order equation on a half-line, see [25][26][27][28][29].…”

“…A large number of results about fractional differential equations with integral boundary condition have been obtained, see [9,10,[30][31][32][33][34][35][36][37][38][39][40][41][42][43] and the references cited therein. Meanwhile, we note that the coupled systems of fractionalorder differential equations have also attracted much attention due to their extensive applications, we refer to [3,9,10,14,22,29,[32][33][34][35][36][37][38][39][40][41][42][43].…”

Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control

“…In addition, in order to describe a physical process model with discontinuous jumps or mutations in disease prevention and control, earthquake and shock absorption systems, and other aspects of research, many researchers have investigated the impulsive problems, see [6][7][8][9][10][11][12]. Moreover, in recent years, optimal control problem to all kinds of differential equations has attracted many researchers.…”

Solvability for p‐Laplacian generalized fractional coupled systems with two‐sided memory effects

“…In recent years, fractional differential coupled system has become a hotspot because this kind of equations produce broader application value in practice; see previous studies. [14][15][16][17][18][19][20][21][22] Most works done so far deal with Riemann-Liouville or Caputo fractional derivative, and only a few works report on Riesz-Caputo-type fractional derivative. Gua et al 23 investigated a new class of fractional differential equation with Riesz-Caputo derivative…”