Positive Solutions to Periodic Boundary Value Problems of Nonlinear Fractional Differential Equations at Resonance

By Leggett-Williams norm-type theorem for coincidences due to O’Regan and Zima, we discuss the existence of positive solutions tofractionalorderwithperiodicboundaryconditionsatresonance. Atlast,anexampleispresentedtodemonstratethemainresults.


Introduction
Fractional differential equations are generalizations of ordinary differential equations to an arbitrary order. It has played a significant role in many fields, such as viscoelasticity, engineering, physics, and economics; see [1][2][3][4][5]. During the last ten years, there are a large number of papers dealing with the existence of solutions boundary value problem for fractional differential equations; see [6][7][8][9][10].
Recently, there is an increasing tendency on discussion for the existence of positive solutions to boundary value problems of fractional differential equations which enriched many previous results.
In [9], Yang and Wang considered the existence of solutions for the following two-point boundary value problems for fractional differential equations: where 1 < < 2, 0 + denoting the Caputo fractional derivative. By using the Leggett-Williams norm-type theorem for coincidences due to O'Regan and Zima, the authors obtained the existence of positive solutions to the above problem.
Periodic boundary value problems have profound practical background and wide range of applications, such as mechanics, biology, and engineering; see [11][12][13][14]. Recently, periodic boundary value conditions of fractional order have been studied by some authors, such as [15][16][17].
In [16], Chen et al. studied the following periodic boundary value problem for fractional -Laplacian equation: 2 International Journal of Differential Equations where 0 < , ≤ 1, 0 + is a Caputo fractional derivative, and In [17], Hu et al. considered the existence of solutions for the following periodic boundary value problem for fractional differential equation: where 1 < < 2, 0 + denotes the Caputo fractional derivative, and : [0, 1]×R 2 → R is continuous. By using the coincidence degree theory, the authors obtained the existence of solutions.
From the above works, we notice that the study of positive solutions to periodic boundary value problems of fractional order at resonance is poor. Now, the question is as follows: though the existence of solutions to (4) is obtained, how can we get the positive solutions of it? The aim of this paper is to fill the gap in the relevant literature. Our main tool is the recent Leggett-Williams norm-type theorem for coincidences due to O'Regan and Zima [18].
The rest of this paper is organized as follows. Section 2, we give some necessary notations, definitions, and lemmas. In Section 3, we obtain the existence of positive solutions of (4) by Theorem 9. Finally, an example is given to illustrate our results in Section 4.

Preliminaries
First of all, we present the necessary definitions and lemmas from fractional calculus theory. For more details, see [1].
In the following, let us recall some definitions on Fredholm operators and cones in Banach space (see [19]).
Let , be real Banach spaces. Consider a linear mapping : dom ⊂ → and a nonlinear operator : → . Assume that (A1) is a Fredholm operator of index zero; that is, Im is closed and dim ker = codim Im < ∞.
This assumption implies that there exist continuous projections : → and : → such that Im = ker and ker = Im . Moreover, since dim Im = codim Im , there exists an isomorphism : Im → ker . Denote by the restriction of to ker ∩ dom . Clearly, is an isomorphism from ker ∩ dom to Im ; we denote its inverse by : Im → ker ∩ dom . It is known that the coincidence equation = is equivalent to Let be a cone in such that (i) ∈ for all ∈ and ≥ 0, It is well known that induces a partial order in by The following property is valid for every cone in a Banach space .
Lemma 5 (see [18]). Let be a cone in . Then for every ∈ \ {0} there exists a positive number ( ) such that Let : → be a retraction, that is, a continuous mapping such that ( ) = for all ∈ . Set We use the following result due to O'Regan and Zima.
Then the equation = has a solution in the set ∩ (Ω 2 \ Ω 1 ).

Main Results
In this section, we give the existence theorems for problem (4). We write Banach space = = [0, 1] with the norm Define the operator : dom → by where Define the operator Then problem (4) can be written by Let be a constant, which is in (0, 1) and satisfies Lemma 7. The mapping : dom ⊂ is a Fredholm operator of index zero. Furthermore, the operator : Im → dom ∩ ker can be written as where Proof. By Lemma 3, 0 + ( ) = 0 has solution According to the boundary value conditions of (4), we have Let ∈ Im , so there exists a function ( ) ∈ dom which satisfies ( ) = ( ). By Lemma 3, we have By (0) = (1), we can obtain ∫  (28) So we have 2 = . In the same way, 2 = . We notice that Im = ker and ker = Im . It follows from Ind = dim ker − codim Im = 0 that is a Fredholm mapping of index zero. Next, we will prove that the operator is the inverse of | dom ∩ker .

Example
To illustrate how our main result can be used in practice, we present here an example.
Let us consider the following fractional differential equation at resonance: