Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 432941, 8 pages
http://dx.doi.org/10.1155/2013/432941
Research Article
A Note on Fractional Equations of Volterra Type with
Nonlocal Boundary Condition
Zhenhai Liu
1,2
and Rui Wang
2
1
Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, China
2
College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
Correspondence should be addressed to Zhenhai Liu; zhhliu@hotmail.com
Received 6 March 2013; Accepted 24 June 2013
Academic Editor: Zhanbing Bai
Copyright © 2013 Z. Liu and R. Wang. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We deal with nonlocal boundary value problems of fractional equations of Volterra type involving Riemann-Liouville derivative.
Firstly, by defning a weighted norm and using the Banach fxed point theorem, we show the existence and uniqueness of solutions.
Ten, we obtain the existence of extremal solutions by use of the monotone iterative technique. Finally, an example illustrates the
results.
1. Introduction
Fractional diferential equations arise in many engineering
and scientifc disciplines as the mathematical modeling of
systems and processes in the felds of physics, chemistry,
aerodynamics, and so forth. Tere has been a signifcant
theoretical development in fractional diferential equations
in recent years (see [1–18]). Monotone iterative technique is
a useful tool for analyzing fractional diferential equations.
In [3], Jankowski considered the existence of the solutions
of the following problem:
()= (, (),∫
0
(,)()), 0<<1,
∈(0,],
(0)=,
(1)
where ∈ ([0,] ×
2
,), (0) =
1−
()|
=0
by using
the Banach fxed point theorem and monotone iterative
technique.
Motivated by [3], in this paper we investigate the follow-
ing nonlocal boundary value problem:
()= (, (),∫
0
(,)())
≡(), 0<<1,∈(0,],
(0)=(),
(2)
where ∈([0,]×
2
,), :
1−
([0, ]) → is a conti-
nuous functional, = [0, ], (0) =
1−
()|
=0
, and (,) ∈
(Δ,); here Δ={(,)∈×:0≤≤≤}.
Firstly, the nonlocal condition can be more useful
than the standard initial condition to describe many phys-
ical and chemical phenomena. In contrast to the case
for initial value problems, not much attention has been
paid to the nonlocal fractional boundary value problems.
Some recent results on the existence and uniqueness of
nonlocal fractional boundary value problems can be found
in [1, 2, 12, 14, 18]. However, discussion on nonlocal boundary
value problems of fractional equations of Volterra type
involving Riemann-Liouville derivative is rare. Secondly,
in [3], in order to discuss the existence and uniqueness
of problem (1), Jankowski divided ∈ (0,1) into two
situations to discuss; one is 0 < ≤ 1/2 with an add-
itional condition and the other is 1/2 < < 1. In this
paper, we unify the two situations without using the
additional condition. Tirdly, for the study of diferential
equation, monotone iterative technique is a useful tool
(see [9, 10, 16, 17]). We know that it is important to build
a comparison result when we use the monotone iterative
technique. We transform the diferential equation into inte-
gral equation and use the integral equation to build the
comparison result which is diferent from [3]. It makes the
calculation easier and is suitable for the more complicated
forms of equations.