Positive solutions for integral boundary value problem of two-term fractional differential equations

Abstract

In this paper, we investigate a class of nonlinear two-term fractional differential equations involving two fractional orders δ ∈ (1, 2] and τ ∈ (0, δ) with integral boundary value conditions. By the Schauder fixed point theorem we obtain the existence of positive solutions based on the method of upper and lower solutions. Then we obtain the uniqueness result by the Banach contraction mapping principle. Examples are given to illustrate our main results.

Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance

“…where C D α 1-and D become a popular research field. At present, many researchers study the existence of solutions of fractional differential equations such as the Riemann-Liouville fractional derivative problem at nonresonance [6][7][8][9][10][11][12][13][14][15][16], the Riemann-Liouville fractional derivative problem at resonance [17][18][19][20][21][22][23], the Caputo fractional boundary value problem [6,24,25], the Hadamard fractional boundary value problem [26][27][28], conformable fractional boundary value problems [29][30][31][32], impulsive problems [33][34][35], boundary value problems [8,[36][37][38][39][40][41][42][43], and variational structure problems [44,45].…”

“…In recent paper [9], the existence and uniqueness results for integral boundary value problem of two-term fractional differential equations…”

Boundary value problems of fractional q-difference equations on the half-line

“…Boundary value problems on a half-line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and models of gas pressure in a semiinfinite porous medium. Though much of the work on fractional calculus deals with finite domain, there is a considerable development on the topic involving unbounded domain [11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Iterative positive solutions to a coupled fractional differential system with the multistrip and multipoint mixed boundary conditions

“…In the past few years the fractional boundary value problems are found to be popular in the research community because of their numerous applications in many disciplinary areas, such as optics, thermal, mechanics, control theory, nuclear physics, economics, signal and image processing, medicine, and so on [1][2][3][4]. To meet the practical application needs, many different theoretical approaches have been taken to study the existence, uniqueness, and multiplicity of solutions to fractional-order boundary value problems, for instance, the method of upper and lower solutions [5][6][7][8][9], the fixed point theory [10][11][12][13], the monotone iterative technique [14][15][16][17][18][19], the coincidence degree theory [20][21][22], etc. In comparison, the monotone iterative technique has more advantages, such as it not only proves the existence of positive solutions but also can obtain approximate solutions that can meet different accuracy requirements.…”