Existence and nonexistence of positive solutions for fractional-order two-point boundary value problems
- PDF / 235,733 Bytes
- 11 Pages / 595.28 x 793.7 pts Page_size
- 98 Downloads / 253 Views
RESEARCH
Open Access
Existence and nonexistence of positive solutions for fractional-order two-point boundary value problems Yongping Sun* and Xiaoping Zhang * Correspondence: [email protected] College of Electron and Information, Zhejiang University of Media and Communications, Hangzhou, Zhejiang 310018, China
Abstract The purpose of this paper is to establish some results on the existence and nonexistence of positive solutions for a type of nonlinear fractional-order two-point boundary value problems. The main tool is a fixed point theorem of the cone expansion and compression of functional type due to Avery et al. Some examples are presented to illustrate the availability of the main results. MSC: 34A08; 34B10; 34B15; 34B18 Keywords: positive solution; existence and nonexistence; fractional differential equations; boundary value problems; fixed point theorem
1 Introduction This paper investigates the fractional boundary value problem (FBVP for short): ⎧ ⎨c Dα+ u(t) + f (t, u(t)) = ,
t ∈ (, ), ⎩u() = u () = u () = · · · = u(n–) () = ,
u () = ,
(.)
where c Dα+ is Caputo’s fractional derivative. Throughout this paper, we assume that n ≥ is a fixed integer, α ∈ (n – , n], and f : [, ] × [, ∞) → [, ∞) is continuous. Fractional differential equations can be extensively applied to various disciplines such as physics, mechanics, chemistry, engineering, and many other branches of science. Recently, there have been some papers dealing with the existence and multiplicity of solutions (or positive solutions) of nonlinear fractional differential equations with various boundary conditions (see [–] and the references therein). For example, Agarwal et al. [] and Tian and Liu [] investigated the singular fractional boundary value problem of the form: ⎧ ⎨c Dα+ u(t) + λf (t, u(t)) = ,
t ∈ (, ),
⎩u() = u () = u () = · · · = u(n–) () = ,
u () = ,
where α ∈ (n – , n] and n ≥ is an integer, c Dα+ is Caputo’s fractional derivative, f : (, ) × (, ∞) → [, ∞) is continuous, that is, f (t, u) may be singular at t = , and u = . By constructing a special cone and using an approximation method and fixed point index theory, the authors obtained some results on the existence or nonexistence of one or more positive solutions. ©2014 Sun and Zhang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Sun and Zhang Advances in Difference Equations 2014, 2014:53 http://www.advancesindifferenceequations.com/content/2014/1/53
Page 2 of 11
Fixed point theorems have been applied to various boundary value problems to show the existence and multiplicity of positive solutions in the last two decades. Recently, Avery et al. [] generalized the fixed point theorem of a cone expansion and compression of norm type by replacing the norms with two function
Data Loading...
