Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations

Abstract

We study the existence of positive solutions for the boundary value problem of nonlinear fractional differential equationsD0+αu(t)+λf(u(t))=0,0<t<1,u(0)=u(1)=u'(0)=0, where2<α≤3is a real number,D0+αis the Riemann-Liouville fractional derivative,λis a positive parameter, andf:(0,+∞)→(0,+∞)is continuous. By the properties of the Green function and Guo-Krasnosel'skii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional differential equation boundary value problem are considered, some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established. As an application, some examples are presented to illustrate the main results.

On existence and uniqueness of positive solutions to a class of fractional boundary value problems

“…In a recent paper [18], the authors study the existence of positive solutions of a particular case of Problem (2). More precisely, they study the following fractional autonomous boundary value problem…”

“…Recently, in the paper [18] to appear in this special issue, the authors studied the existence of positive solutions for the following autonomous boundary value problem of fractional order…”

“…Notice that in [18] the question of uniqueness of solutions is not treated. In our study, the main tool is a fixed-point theorem in partially ordered sets, which gives us uniqueness of the solution.…”

Existence and non-existence of positive solutions for integral boundary value problems of high-order fractional differential equations with generalised p-Laplacian operator

“…In recent years, with the wide applications in the various fields in science and engineering such as rheology, dynamical processes in self-similar and porous structures, heat conduction, control theory, electroanalytical chemistry, chemical physics, economics, etc., the study for boundary value problems of fractional differential equations as abstracted from practical problems attracts much attention of many mathematicians (Zhai and Xu, 2014;Li, Sun and Li, 2014;Zhao et al, 2011;Yan et al, 2014Yan et al, , 2013.…”

“…Being different from Zhai and Xu (2014), Li, Sun and Li (2014), Zhao et al (2011), Yan et al (2014 and Yan et al (2013), using monotone iterative technique, we not only study the existence of the minimal and maximal positive solutions for BVP (1) and (2) but also develop two computable explicit monotone iterative sequences for approximating the two positive solutions of BVP (1) and (2). In addition, to start our work, we employ the monotone iterative method (Wang, Liu and Zhang, 2014;Yao et al, 2013;Jiang and Zhong, 2014;Sun and Zhao, 2014), which is indeed an interesting and effective technique for investigating this topic and different from the ones used in relevant papers.…”