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Springerplus. 2016; 5(1): 957.
Published online 2016 July 2. doi:  10.1186/s40064-016-2665-8
PMCID: PMC4930441

On the existence of positive solutions for fractional differential inclusions at resonance


In this paper, we discuss the existence of positive solutions for a boundary value problem of fractional differential inclusions with resonant boundary conditions. By using the Leggett–Williams theorem for coincidences of multi-valued operators due to O’Regan and Zima, results on the existence of positive solutions are established. An example is given to illustrate the efficiency of the main theorems.

Keywords: Fractional differential inclusions, Multi-valued operator, Positive solution, Resonance


In this article, we investigate the existence of positive solutions of fractional differential inclusions with two-point boundary conditions:


where n - 1 < α < nn ≥ 2, D0+α denotes the Caputo fractional derivative, f:[0, 1] × ℝ → ℱ(ℝ), ℱ(ℝ) denotes the family of nonempty compact and convex subsets of .

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary order. The fractional differential equations play an important role in various fields of science and engineering, such as chemistry, biology, control theory, viscoelastic materials, signal processing, finance, life science and so on, see Kilbas et al. (2006), Samko et al. (1993), Podlubny (1999) and Orsingher and Beghin (2004).

During the last 10 years, boundary value problems for fractional differential equations are one of the most active fields in the researches of nonlinear differential equations theories. For further details, see Bai and Lü (2005), Zhang (2006), Caballero et al. (2011), Xu et al. (2009), Lin (2007) and Goodrich (2010). Meanwhile, fractional boundary value problems at resonance have been extensively studied. For some recent works on the topic, see Kosmatov (2008, 2010), Bai (2011), Bai and Zhang (2011) and Yang and Wang (2011) and references therein. It is well known that differential inclusions have proved to be valuable tools in the modeling of many realistic problems, such as economics, optimal control and so on. Recently, fractional differential inclusions have been investigated by several researchers, we refer the reader to Agarwal et al. (2010) and Chen et al. (2013).

As shown in the above mentioned works, we can see two facts. Firstly, although the boundary value problems for fractional differential equations at resonance have been studied by some authors, the existence of positive solutions to fractional differential equations at resonance are seldom considered. Secondly, there are few papers to deal with fractional differential inclusions under resonant conditions. The study of positive solutions for higher-order fractional differential inclusions under resonant conditions has yet to be initiated.

To fill this gap, we discuss the fractional differential inclusions (1) by using the Leggett–Williams theorem for coincidences of multi-valued operators due to O’Regan and Zima (2008).

The rest of this paper is organized as follows. “Preliminaries” section, we give some necessary notations, definitions and lemmas. In “Main results” section, we obtain the existence of positive solutions of (1) by Theorem 1. Finally, an example is given to illustrate our results in “Example” section.


First of all, we present the necessary definitions and lemmas from fractional calculus theory. For more details, see Kilbas et al. (2006), Samko et al. (1993) and Podlubny (1999).

Definition 1

(Kilbas et al.2006) The Riemann–Liouville fractional integral of order α > 0 of a function f:(0, ) → ℝ is given by


provided that the right-hand side is pointwise defined on (0, ).

Definition 2

(Kilbas et al.2006) The Caputo fractional derivative of order α > 0 of a continuous function f:(0, ) → ℝ is given by


where n - 1 < α ≤ n, provided that the right-hand side is pointwise defined on (0, ).

Lemma 1

(Kilbas et al. 2006) The fractional differential equation


has solutiony(t) = c0c1t +  ⋯  + cn-1tn-1ci ∈ ℝ,i = 0, 1, …, n - 1,n = [α] + 1.

Furthermore, fory ∈ ACn[0, 1],




Lemma 2

(Kilbas et al. 2006) The relation


is valid in following case:β > 0,  αβ > 0,  f ∈ L1(ab).

In the following, let us recall some definitions on Fredholm operators and cones in Banach space (see Mawhin 1979).

Let X, Y be real Banach spaces. Consider a linear mapping L:domL ⊂ X → Y and a nonlinear multivalued mapping N:X → 2Y. Assume that

  1. L is a Fredholm operator of index zero, that is, ImL is closed and dim (KerL) = codim(ImL) < ,
  2. N:X → 2Y is an upper semicontinuous mapping with nonempty compact convex values.

The assumption (A1) implies that there exist continuous projections P:X → X and Q:Y → Y such that ImP = KerL and KerQ = ImL. Moreover, since dim (ImQ) = codim (ImL), there exists an isomorphism J:ImQ → KerL. Denote by Lp the restriction of L to KerP ∩ domL. Clearly, Lp is an isomorphism from KerP ∩ domL to ImL, we denote its inverse by Kp:ImL → KerP ∩ domL. It is known that the inclusion Lx ∈ Nx is equivalent to

x ∈ (PJQN)xKP(IQ)Nx.

Let C be a cone in X such that

  1. μx ∈ C for all x ∈ C and μ ≥ 0,
  2. x,  - x ∈ C implies xθ.

It is well known that C induces a partial order in X by

x ⪯ y if and only if yx ∈ C.

The following property is valid for every cone in a Banach space X.

Lemma 3

Let C be a cone in X. Then for everyu ∈ C\{0}there exists a positive numberσ(u)such that

xu‖ ≥ σ(u)‖u‖ for all x ∈ C.

Let γ:X → C be a retraction, that is, a continuous mapping such that γ(x) = x for all x ∈ C. Set

Ψ: = PJQNKp(IQ)N and Ψγ: = Ψ ∘ γ.

We use the following result due to O’Regan and Zima.

Theorem 1

(O’Regan and Zima 2008) Let C be a cone in X and letΩ1,Ω2be open bounded subsets of X withΩ¯1Ω2andC(Ω¯2\Ω1). Assume that (A1), (A2) hold and the following assumptions hold:

  • (A3) QN:X → 2Yis bounded on bounded subsets ofC and Kp(IQ)N:X → 2Xbe compact on every bounded subset of C,
  • (A4) γmaps subsets ofΩ¯2into bounded subsets of C,
  • (A5) Lx ∉ λNxfor allx ∈ C ∩ Ω2 ∩ domLandλ ∈ (0, 1),
  • (A6)  deg {[I - (PJQN)γ]|kerL, kerL ∩ Ω2, 0} ≠ 0,
  • (A7) there existsu0 ∈ C\{0}such thatx‖ ≤ σ(u0)‖yforx ∈ C(u0) ∩ Ω1andy ∈ Ψx, whereC(u0) = {x ∈ C:μu0 ⪯ x for some μ > 0}andσ(u0)such thatxu0‖ ≥ σ(u0)‖xfor everyx ∈ C,
  • (A8) (PJQN)γ(Ω2) ⊂ C,
  • (A9) Ψγ(Ω¯2\Ω1)C,
  • (A10) x ∉ (PJQN)γx for x ∈ Ω2 ∩ KerL.

Then the equationLx ∈ Nxhas at least one solution in the setC(Ω¯2\Ω1).

Main results

In this section, we state our result on the existence of positive solutions for (1).

For simplicity of notation, we set


By the monotonicity of the function, it is easy to verify that G(ts) > 0, ts ∈ [0, 1]. Here, we omit the proof. Moreover, κ is a constant which satisfies


Thus, we get 1 - κG(ts) > 0, ts ∈ [0, 1].

Theorem 2

Assume that:

  1. f:[0, 1] × ℝ → ℱ(ℝ),f(t, u)is continuous for everyu ∈ ℝ,  t ∈ [0, 1],
  2. for eachr > 0, there existsαr ∈ L1[0, 1]such that|f(tu)| ≤ αr(t)for a.e.t ∈ [0, 1]and everyu ∈ [0, r], where|f(tu)| = sup{|w|:w ∈ f(tu)},
  3. there exist positive constantsb1b2b3c1c2Bwith
    such that
    κx ≤ w ≤  - c1xc2 and w ≤  - b1|w| + b2xb3
    for allx ∈ [0, B]andw ∈ f(tx)witht ∈ [0, 1],
  4. there existb ∈ (0, B),t0 ∈ [0, 1],ρ ∈ (0, 1],δ ∈ (0, 1)and the functionq ∈ L1[0, 1],q(t) ≥ 0, t ∈ [0, 1],h ∈ C((0, b], ℝ+)such thatw(tu) ≥ q(t)h(u)for(tu) ∈ [0, 1] × (0, b]andw ∈ f(tu).h(u)uρis non-increasing on (0, b] with

Then the problem (1) has at least one positive solution on [0, 1].


We use the Banach space XYC[0, 1] with the supremum norm x‖ = maxt∈[0,1]|x(t)|.

Define L:domL → X and N:X → 2Y with domL=xX:D0+αx(t)C[0,1],x(i)(0)=0,x(0)=x(1),i=1,2,,n-1 by



Nu(t) = {y ∈ Y:y(t) ∈ f(tu(t)) a.e. on[0, 1]}.

Then the problem (1) can be written by

Lu ∈ Nuu ∈ domL.

By Lemma 1, D0+αu(t)=0 has solution

u(t) = c0c1t +  ⋯  + cn-1tn-1

where ci ∈ ℝ, i = 0, 1, …, n - 1. According to the boundary conditions of (1), we get ci = 0, i = 1, 2, …, n - 1. Thus, we obtain

KerL = {u ∈ domL:u(t) = c ∈ ℝ}.

Let y ∈ ImL, so there exists u ∈ domL which satisfies Luy. By Lemma 1, we have


By the definition of domL, we have ci = 0,  i = 1, 2, …, n - 1. Hence,


Taking into account u(0) = u(1), we obtain


On the other hand, suppose y satisfies the above equation. Let u(t)=I0+αy(t), and we can easily prove u(t) ∈ domL. Thus, we get


Define the linear continuous projector operator P:X → X by


Next, we define the operator Q:Y → Y by


Noting that


then we have P2P. Similarly, we have Q2Q.

Then, one has ImP = KerL and KerQ = ImL. It follows from IndL = dim(kerL) - codim(ImL) = 0 that L is a Fredholm mapping of index zero. Then, (A1) holds.

We consider the mapping KP:ImL → domL ∩ KerP by




Now, we will prove that is KP the inverse of L|domL ∩ KerP. In fact, for x ∈ domL ∩ KerP, we have D0+αx(t)=y(t)ImL and α01(1-s)α-1x(s)ds=0.

By Lemma 1, one has


According to the definition of domL, we get ci = 0, i = 1, 2, …, n - 1. Furthermore, by α01(1-s)α-1x(s)ds=0, we have c0=-Γ(1+α)(I0+2αy)(1).



Obviously, LKPyy. Moreover, for x ∈ domL ∩ KerP, we get α01(1-s)α-1x(s)ds=0 and


Thus, we know that KP = (L|domL∩KerP)-1. Moreover, it is easy to see that

|k(ts)| ≤ 3(1-s)α-1,  ∀ ts ∈ [0, 1].

Consider the cone

C = {x ∈ X:x(t) ≥ 0,  t ∈ [0, 1]}.

It is clear that (H1) and (H2) imply (A2) and (A3).



Clearly, Ω1 and Ω2 are bounded and open sets and


Moreover, C(Ω¯2\Ω1). Let JI and (γx)(t) = |x(t)| for x ∈ X, then γ is a retraction and maps subsets of Ω¯2 into bounded subsets of C, which means that (A4) holds.

Next, we will show (A5) holds. Suppose that there exist u0 ∈ Ω2 ∩ C ∩ domL and λ0 ∈ (0, 1) such that Lu0 ∈ λ0Nu0, then D0+αu0(t)λ0f(t,u0(t)) for all t ∈ [0, 1]. In view of (H3), we get that there exists w ∈ f(tu0(t)) such that




From (4), we obtain


which gives


From (5), we obtain


From (3) and the equation

u0 = (IP)u0Pu0KPL(IP)u0Pu0Pu0KPLu0

we can get


Then, we have


which contradicts (H3). Hence (A5) holds.

To prove (A6), consider xKerLΩ¯2, then x(t) ≡ c on [0, 1]. Let


for c ∈ [ - BB] and λ ∈ [0, 1]. It is easy to show that 0 ∈ H(cλ) implies c ≥ 0. Suppose 0 ∈ H(Bλ) for some λ ∈ (0, 1]. Then,


where w ∈ f(tB),  t ∈ [0, 1]. So (H3) leads to


which is a contradiction. In addition, if λ = 0, then B = 0, which is impossible. Thus, H(xλ) ≠ 0 for x ∈ KerL ∩ Ω2, λ ∈ [0, 1]. As a result,


So (A6) holds.

Next, we prove (A7). Letting u0(t) ≡ 1, so we have u0 ∈ C\{0} and C(u0) = {x ∈ C:x(t) > 0, t ∈ [0, 1]}. We can take σ(u0) = 1. For x ∈ C(u0) ∩ Ω1, we get x(t) > 0, 0 < ‖x‖ ≤ b and x(t) ≥ δx‖,  t ∈ [0, 1].

By (H3) and (H4), for every x ∈ C(u0) ∩ Ω1 and v ∈ Ψx, there exits w ∈ Nx such that


Thus, x‖ ≤ σ(u0)‖Ψx for all x ∈ C(u0) ∩ Ω1, i.e., (A7) holds.

Since for x ∈ Ω2 and w ∈ Nγx, from (H2) we have


Thus, (PJQN)γx ⊂ C for x ∈ Ω2. Then (A8) holds.

Next, we prove (A9). Let xΩ¯2\Ω1

Ψγx(t)=vX:wNγxsuch thatv=α01(1-s)α-1|x|ds+01G(t,s)(1-s)α-1w(s,|x|)ds.

According to (H3) and (2), for xΩ¯2\Ω1 and v ∈ Ψγx, there exits w ∈ Nγx such that


Hence, ΨγΩ¯2\Ω1C; i.e., (A9) holds.

To prove (A10), suppose there exists u0 ∈ Ω2 ∩ KerL, i.e., u0c ∈ ℝ and |c| = B such that c ∈ (PJQN)γu. For w ∈ Nγc, we have


Hence, we get c ∈ (PJQN)γu implies c ≥ 0. Then for cB and w ∈ NγB, we have




On the other hand, from (H3), we have


This contradiction implies (A10) holds.

Hence, applying Theorem 1, BVP (1) has a positive solution u on [0, 1] with b ≤ ‖u‖ ≤ B. This completes the proof.


To illustrate how our main result can be used in practice, we present here an example.

Let us consider the following fractional differential inclusion at resonance


where f(t,u)=w(t,u)+125v:v[0,1], w(t,u)=13001+2t-2t2u2-4u+3u.

Corresponding to BVP (1), we have that α = 1.5 and


It is easy to see that G(ts) ≥ 0 for ts ∈ [0, 1].

Let κ = 0.003 and B = 2. By the monotonicity of the function, for x ∈ [0, 2] and w ∈ f(tx), t ∈ [0, 1], we can prove that




Then, we can choose c1=130, c2=117, b1=83, b2=130, b3 = ¼. By calculation, we have


Take q(t)=12401+2t-t2 and h(x) = x. We see that q ∈ L1[0, 1], q(t) ≥ 0 and h ∈ C((0, b], ℝ+),  where b = 1/2 ∈ (0, B) = (0, 2). Furthermore, for (tu) ∈ [0, 1] × (0, 1/2] and w ∈ f(tu), by a simple computation, we get that

w(tu) ≥ q(t)h(u).

Choose ρ = 1, so we have h(u)uρ1 which is non-increasing on (0, b]. By Choosing t0 = 0, δ = 0.997, with simple calculations, we can get


Therefore, (H1)–(H4) of Theorem 2 are satisfied. Then BVP (6) has a positive solution on [0, 1].


In this paper, we have obtained the existence of positive solutions for a boundary value problem of fractional differential inclusions at resonance. By using the Leggett–Williams theorem for coincidences of multi-valued operators due to O’Regan and Zima, we have found the existence results. Our results are new in the context of fractional differential inclusions and positive solutions. As applications, an example is presented to illustrate the main results. In the future, we will consider the the uniqueness of positive solutions for the fractional differential equations at resonance.


The research was supported by the Science Foundation of Shandong Jiaotong University (Z201429).

Competing interests

The author declares that he has no competing interests.


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