Boundary value problems for fractional differential equations with nonlocal boundary conditions

Abstract

In this paper, we establish some sufficient conditions for the existence of solutions to two classes of boundary value problems for fractional differential equations with nonlocal boundary conditions. Our goal is to establish some criteria of existence for the boundary problems with nonlocal boundary condition involving the Caputo fractional derivative, using Banach's fixed point theorem and Schaefer's fixed point theorem. Finally, we present four examples to show the importance of these results.

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

“…In the literature, single-term fractional differential equations of the form D δ x(t) = f (t, x(t)) have been studied by many researchers (see [15][16][17][18][19][20][21][22][23]). In practical problems the equation contains more than one fractional differential term, for example, the classical …”

Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications

“…Observe that, when = 1, we recover the classical solution (15). We analyze growth rates in Lithuania and Qatar, countries with one of the lowest and the highest growth rates, respectively.…”

“…For related results concerning FDEs with different type of fractional derivatives, we refer to other works. [9][10][11][12][13][14][15] The outline of the paper is the following. In Section 2, we present the main definition of this work: the -Caputo fractional derivative, that is, a Caputo-type derivative of a function with respect to another function; in Theorem 1, we prove that this operator is the left inverse of the fractional integral.…”

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.…”