A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection–diffusion equations

Existence and Nonexistence of Positive Solutions for High-Order Fractional Differential Equation Boundary Value Problems at Resonance

“…en, the fixed point x of A is a solution of RBVP (1). It is clear that L: P ⟶ Q is continuous and the first eigenvalue of L is λ 1 � σ.…”

“…which implies that FBVP (1) happens to be at resonance. By virtue of the widespread applications, various differential equations have been studied by many researchers (see [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein). Fractional-order models can describe many processes more accurately than integer-order models, and a great deal of papers focusing on FBVPs appeared in recent years (see [14][15][16][17][18][19][20][21][22][23]).…”

“…Motivated by the above work, we focus on deducing some necessary conditions and sufficient conditions of the existence of positive solutions to the RBVP (1). is article has some new features compared with some existing papers.…”

“…Firstly, the main tools used in this article are the spectral theory and fixed point index theory which is different from [35,37]. Secondly, some necessary conditions to RBVP (1) are established by virtue of spectral theory. Finally, we deduce some sufficient conditions concerning the behavior of the nonlinearity f at 0 and at ∞.…”

Monotone Iterative Method for Fractional Differential Equations with Integral Boundary Conditions

“…where D α 0+ is the standard Riemann-Liouville fractional derivative and 1 < α < 2 and A(t) is a positive measure function, and it satisfies 1 0 G(t)dA(t) < G (1). Fractional differential equations have been widely used in physics, chemistry, aerodynamics, electrodynamics of complex media, and rheology of polymers [1][2][3][4][5][6][7]. As a result, various nonlinear functional analysis methods have been used to study the existence of solutions for differential equations .…”