Chen et al. Boundary Value Problems (2016) 2016:51 DOI 10.1186/s13661-016-0566-y
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Some existence results on boundary value problems for fractional p-Laplacian equation at resonance Taiyong Chen* , Wenbin Liu and Huixing Zhang *
Correspondence:
[email protected] Department of Mathematics, China University of Mining and Technology, Xuzhou, 221116, P.R. China
Abstract Two boundary value problems of the fractional p-Laplacian equation at resonance are considered in this paper. By using the continuation theorem due to Ge, we obtain some existence results for such boundary value problems. MSC: 34A08; 34B15 Keywords: fractional differential equation; p-Laplacian operator; boundary value problem; continuation theorem; resonance
1 Introduction Consider the following fractional p-Laplacian equation: β D+ φp Dα+ x(t) = f t, x(t), Dα+ x(t) ,
t ∈ [, ],
(.)
with the boundary value conditions either x() = x(),
Dα+ x() = ,
(.)
x() = x(),
Dα+ x() = ,
(.)
or
where < α, β ≤ , φp (s) = |s|p– s (p > ), Dα+ is a Caputo fractional derivative, and f : [, ] × R → R is a continuous function. In the last two decades, the theory of fractional calculus has gained popularity due to its wide applications in various fields of engineering and the sciences [–]. Moreover, the p-Laplacian equations often exist in non-Newtonian fluid theory, nonlinear elastic mechanics, and so on. Recently, many important results on the p-Laplacian equations or the fractional differential equations have been given. We refer the reader to [–]. However, as far as we know, there is little work about boundary value problems (BVPs for short) for the fractional differential equations with p-Laplacian operator at resonance. © 2016 Chen et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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Note that BVP (.)-(.) (or BVP (.)-(.)) happens to be at resonance because its associated homogeneous BVP ⎧ ⎨Dβ+ φ (Dα+ x(t)) = , t ∈ [, ], p ⎩x() = x(), Dα+ x() = (or x() = x(), Dα+ x() = ),
has a solution x(t) = c, ∀c ∈ R. The rest of this paper is organized as follows. Section contains some definitions, lemmas and notations. In Section , some related lemmas are stated and proved which are useful in the proof of our main results. In Section and Section , in view of the continuation theorem due to Ge, we establish two theorems about the existence of solutions for BVP (.)-(.) (Theorem .) and BVP (.)-(.) (Theorem .).
2 Preliminaries We give here some definitions and lemmas about the fractional calculus. Definition . [] The Riemann-Liouville fractional integral operator of order α > of a function x : (, +∞) → R is given by Iα+ x(t) =
(α)
t
(t – s)α– x(s) ds,
provided that the right side integral is pointwise defined on (, +∞). Definition . [] The Caputo fractional derivative of order α > of a continuous function x : (, +∞) → R is given by dn x(t) dt n t = (t – s)n–α– x(n) (s) ds, (n – α)
Dα+ x(t) = In–α +
where n is the smallest integer greater than or equal to α, provided that the right side integral is pointwise defined on (, +∞). Lemma . [] Let α > . Assume that x, Dα+ x ∈ L([, ], R). Then the following equality holds: Iα+ Dα+ x(t) = x(t) + c + c t + · · · + cn– t n– , where ci ∈ R, i = , , . . . , n – , and n is the smallest integer greater than or equal to α. Lemma . [] For any u, v ≥ , φp (u + v) ≤ φp (u) + φp (v), if p < ; φp (u + v) ≤ p– φp (u) + φp (v) , if p ≥ .
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Next we introduce an extension of Mawhin’s continuation theorem [, ] which allows us to deal with the more general abstract operator equations, such as BVPs of p-Laplacian equations. Let X and Z be Banach spaces with norms · X and · Z , respectively. Definition . [] A continuous operator M : dom M ∩ X → Z is said to be a quasi-linear operator if () Im M = M(dom M ∩ X) is a closed subset of Z, () Ker M = {x ∈ dom M ∩ X|Mx = } is linearly homeomorphic to Rn with n < ∞. Definition . [] Let Z be a subspace of Z. An operator Q : Z → Z is said to be a semi-projector provided that () Q z = Qz, ∀z ∈ Z, () Q(λz) = λQz, ∀z ∈ Z, λ ∈ R. Set X = Ker M and let X be the complement space of X in X, then X = X ⊕X . Suppose Z is a subspace of Z and Z is the complement space of Z in Z such that Z = Z ⊕ Z . Let P : X → X be a projector and Q : Z → Z a semi-projector, and ⊂ X an open bounded set with the origin θ ∈ . Definition . [] A continuous operator Nλ : → Z, λ ∈ [, ] is said to be M-compact in if there is a vector subspace Z of Z with dim Z = dim X , and an operator R : × [, ] → X being continuous and compact such that (I – Q)Nλ () ⊂ Im M ⊂ (I – Q)Z,
(.)
λ ∈ (, )
(.)
QNλ x = θ ,
⇔
QNx = θ ,
R(·, ) is the zero operator and R(·, λ)|λ = (I – P)|λ , M P + R(·, λ) = (I – Q)Nλ , where λ ∈ [, ], N = N , and
λ
(.) (.)
= {x ∈ |Mx = Nλ x}.
Lemma . [] Suppose M : dom M ∩ X → Z is a quasi-linear operator and Nλ : → Z, λ ∈ [, ] is M-compact in . In addition, if (C ) Mx = Nλ x for every (x, λ) ∈ [(dom M \ Ker M) ∩ ∂] × (, );
for every x ∈ Ker M ∩ ∂; (C ) QNx = (C ) deg{JQN, ∩ Ker M, } = , where N = N and J : Z → X is a homeomorphism with J(θ ) = θ , then the abstract equation Mx = Nx has at least one solution in dom M ∩ . We set Z = C([, ], R) with the norm z = maxt∈[,] |z(t)|, and X = {x ∈ Z|Dα+ x ∈ Z, x() = x(), Dα+ x() = }, X = {x ∈ Z|Dα+ x ∈ Z, x() = x(), Dα+ x() = } with the norm xX = max{x , Dα+ x }. By using linear functional analysis theory, we can prove X, X are Banach spaces.
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3 Related lemmas We will give some lemmas that are useful in the proof of our main results. Define the operator M : dom M ∩ X → Z by β Mx = D+ φp Dα+ x ,
(.) β
where dom M = {x ∈ X|D+ φp (Dα+ x) ∈ Z}. For λ ∈ [, ], we define Nλ : X → Z by Nλ x(t) = λf t, x(t), Dα+ x(t) ,
∀t ∈ [, ].
(.)
Then BVP (.)-(.) is equivalent to the equation Mx = Nx,
x ∈ dom M,
where N = N . Lemma . The operator M, defined by (.), is a quasi-linear operator. Proof The proof will be given in the following two steps. Step . Ker M is linearly homeomorphic to R. β From Lemma ., the homogeneous equation D+ φp (Dα+ x(t)) = has the following solutions: x(t) = d +
φq (d ) α t , (α + )
d , d ∈ R.
Thus, by the boundary value condition Dα+ x() = , one has
Ker M = x ∈ X|x(t) = d, ∀t ∈ [, ], d ∈ R . Obviously, Ker M R. Step . Im M is a closed subset of Z. β Take x ∈ dom M and consider the equation D+ φp (Dα+ x(t)) = z(t). Then we have z ∈ Z and β φp Dα+ x(t) = d + I+ z(t),
d ∈ R.
By the condition Dα+ x() = , one has d = . Thus we get β x(t) = d + Iα+ φq I+ z (t),
d ∈ R,
where φq is understood as the operator φq : Z → Z defined by φq (x)(t) = φq (x(t)). Hence, from the condition x() = x(), we obtain β Iα+ φq I+ z () = .
(.)
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β
Suppose z ∈ Z and satisfies (.). Let x(t) = Iα+ φq (I+ z)(t), then we have x ∈ dom M and β β Mx(t) = D+ φp Dα+ Iα+ φq I+ z (t) = z(t). Hence we obtain s
(s – τ )β– z(τ ) dτ ds = . Im M = z ∈ Z ( – s)α– φq
Obviously, Im M ⊂ Z is closed. Therefore, by Definition ., M is a quasi-linear operator.
Let X = Ker M and define the continuous operators P : X → X, Q : Z → Z by Px(t) = x(), ∀t ∈ [, ], s α– β– ( – s) φq (s – τ ) z(τ ) dτ ds , Qz(t) = φp ρ
∀t ∈ [, ],
α– β(q–) where ρ = β q– s ds > . It is easy to see that P is a projector and Q z = Qz, ( – s) Q(λz) = λQz, ∀z ∈ Z, λ ∈ R, that is, Q is a semi-projector. Moreover, X = Im P and Im M = Ker Q. Lemma . Let ⊂ X be an open bounded set. Then the operator Nλ , defined by (.), is M-compact in . Proof Choose X = Ker P, Z = Im Q and define the operator R : × [, ] → X by β R(x, λ)(t) = Iα+ φq I+ (I – Q)Nλ x (t) t α– = (t – s) φq (α) (β) s · (s – τ )β– λf τ , x(τ ), Dα+ x(τ ) – QNλ x(τ ) dτ ds.
Obviously, dim Z = dim X = . The remainder of the proof will be given in the following two steps. Step . R : × [, ] → X is continuous and compact. By the definition of R, we obtain β Dα+ Rx(t) = φq I+ (I – Q)Nλ x (t). Clearly, the operators R, Dα+ R are compositions of the continuous operators. So R, Dα+ R are continuous in Z. Hence R is a continuous operator, and R(), Dα+ R() are bounded β in Z. Furthermore, there exists a constant T > such that |I+ (I – Q)Nλ x(t)| ≤ T, ∀x ∈ , t ∈ [, ]. Thus, based on the Arzelà-Ascoli theorem, we need only to show R() ⊂ X is equicontinuous.
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For ≤ t < t ≤ , x ∈ , we have Rx(t ) – Rx(t ) β t = (t – s)α– φq I+ (I – Q)Nλ x(s) ds (α) t β – (t – s)α– φq I+ (I – Q)Nλ x(s) ds
q–
T ≤ (α) =
t
(t – s)
α–
– (t – s)
α–
t
ds +
(t – s)
α–
ds
t
T q– α α t – t + (t – t )α . (α + )
As t α is uniformly continuous in [, ], we obtain R() ⊂ Z is equicontinuous. A similar β proof can show that I+ (I – Q)Nλ () ⊂ Z is equicontinuous. This, together with the uniformly continuity of φq (s) on [–T, T], shows that Dα+ R() ⊂ Z is equicontinuous. Thus we find R is compact. Step . Equations (.)-(.) are satisfied. For x ∈ , it is easy to show that Q(I – Q)Nλ x = QNλ x – Q Nλ x = . So (I – Q)Nλ x ∈ Ker Q = Im M. Moreover, for z ∈ Im M ⊂ Z, one has Qz = . Thus z = z – Qz = (I – Q)z ∈ (I – Q)Z. Hence (.) holds. Since QNλ x = λQNx, (.) holds too. For x ∈ λ , we have Mx = Nλ x ∈ Im M = Ker Q. So QNλ x = . From the condition β β Dα+ x() = , one has I+ D+ φp (Dα+ x) = φp (Dα+ x). Thus we obtain β R(x, λ)(t) = Iα+ φq I+ Nλ x (t) β β = Iα+ φq I+ D+ φp Dα+ x (t) = x(t) – x() = (I – P)x(t). Furthermore, when λ = , we have Nλ x(t) ≡ , which yields R(x, )(t) ≡ , ∀x ∈ . Hence (.) holds. For x ∈ , one has β M Px + R(x, λ) (t) = D+ φp Dα+ Px + R(x, λ) (t) β β = D+ φp Dα+ Iα+ φq I+ (I – Q)Nλ x (t) = (I – Q)Nλ x(t), which implies that (.) holds. Therefore, by Definition ., Nλ is M-compact in .
4 Solutions of BVP (1.1)-(1.2) We will give a theorem on the existence of solutions for BVP (.)-(.). Theorem . Let f : [, ] × R → R be continuous. Assume that:
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(H ) there exist nonnegative functions a, b, c ∈ Z such that f (t, x, y) ≤ a(t) + b(t)|x|p– + c(t)|y|p– ,
∀t ∈ [, ], (x, y) ∈ R ;
(H ) there exists a constant A > such that, for ∀x ∈ dom M \ Ker M satisfying |x(t)| > A for ∀t ∈ [, ], we have
( – s)
α–
(s – τ )
φq
s
β–
f
τ , x(τ ), Dα+ x(τ )
dτ ds = ;
(H ) there exists a constant B > such that, for ∀r ∈ R with |r| > B, we have either
s
( – s)α– φq
φq (r)
(s – τ )β– f (τ , r, ) dτ ds >
(.)
(s – τ )β– f (τ , r, ) dτ ds < .
(.)
or
s
( – s)α– φq
φq (r)
Then BVP (.)-(.) has at least one solution, provided that p– b γ := + c < , (β + ) ((α + ))p– p– b + c < , γ := (β + ) ((α + ))p–
if p < ; (.) if p ≥ .
Proof The proof will be given in the following four steps. Step . = {x ∈ dom M \ Ker M|Mx = Nλ x, λ ∈ (, )} is bounded. For x ∈ , one has Nx ∈ Im M = Ker Q. Thus we have
( – s)
α–
(s – τ )
φq
s β–
f
τ , x(τ ), Dα+ x(τ )
dτ ds = .
From (H ), there exists a constant ξ ∈ [, ] such that |x(ξ )| ≤ A. By Lemma ., one has x(t) = x(ξ ) – Iα+ Dα+ x(ξ ) + Iα+ Dα+ x(t), which together with t α– α (t – s) D+ x(s) ds (α) Dα+ x · t α ≤ α (α) α D + x , ∀t ∈ [, ], ≤ (α + )
α α I + D + x(t) =
(.)
and |x(ξ )| ≤ A yields x ≤ A +
α D + x . (α + )
(.)
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Then, from (H ), we have t β– α (t – s) f s, x(s), D x(s) ds + (β) t p– (t – s)β– a(s) + b(s)x(s) ≤ (β) p– ds + c(s)Dα+ x(s)
β I + Nx(t) =
p– p– a + b x + c Dα+ x · t β (β) β p– a + c Dα+ x ≤ (β + ) α p– D + x , ∀t ∈ [, ]. + b A + (α + )
≤
(.)
By Mx = Nλ x, Dα+ x() = , and Lemma ., one has β φp Dα+ x(t) = λI+ Nx(t), which, together with |φp (Dα+ x(t))| = |Dα+ x(t)|p– and (.), implies α p– D + x ≤
p– a + c Dα+ x (β + ) α p– . D +x + b A + (α + )
(.)
If p < , from (.) and Lemma ., we have α p– D + x ≤
a + Ap– b (β + ) p– α p– b D + x + + c . ((α + ))p–
Then, based on (.), one has q– p– α D + x ≤ a + A b := K . ( – γ )(β + )
(.)
Thus, from (.), we have x ≤ A +
K . (α + )
(.)
Similarly, if p ≥ , we obtain q– p– p– α D + x ≤ a + A b := K , ( – γ )(β + ) x ≤ A +
K . (α + )
(.) (.)
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Therefore, combining (.), (.) with (.), (.), we have
xX = max x , Dα+ x
K K ≤ max K , K , A + ,A + := K. (α + ) (α + ) That is, is bounded. Step . = {x ∈ Ker M|QNx = } is bounded. For x ∈ , one has x(t) = d, ∀d ∈ R. Then we have
s
( – s)α– φq
(s – τ )β– f (τ , d, ) dτ ds = ,
which together with (H ) implies |d| ≤ B. Thus we obtain xX ≤ max{B, } = B. Hence is bounded. Step . If (.) holds, then
= x ∈ Ker M|λIx + ( – λ)JQNx = , λ ∈ [, ] is bounded, where J : Im Q → Ker M is a homeomorphism such that J(d) = d, ∀d ∈ R. If (.) holds, then
= x ∈ Ker M|–λIx + ( – λ)JQNx = , λ ∈ [, ] is bounded. For x ∈ , we have x(t) = d, ∀d ∈ R, and λd = –( – λ)φp
s ( – s)α– φq (s – τ )β– f (τ , d, ) dτ ds . ρ
If λ = , then d = . If λ ∈ [, ), we can show |d| ≤ B. Otherwise, if |d| > B, in view of (.), one has ≤ λd = –( – λ)φp · φq
s
φq (d) ρ
( – s)α–
(s – τ )β– f (τ , d, ) dτ ds < ,
which is a contradiction. Hence is bounded. Similar to the above argument, we can show is also bounded. Step . All conditions of Lemma . are satisfied. Define
= x ∈ X|xX < max{K, B} + .
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Clearly, ( ∪ ∪ ) ⊂ (or ( ∪ ∪ ) ⊂ ). From Lemma . and Lemma ., M is a quasi-linear operator and Nλ is M-compact in . Moreover, by the above arguments, we see that the following two conditions are satisfied: (C ) Mx = Nλ x for every (x, λ) ∈ [(dom M \ Ker M) ∩ ∂] × (, );
for every x ∈ Ker M ∩ ∂. (C ) QNx = Now we verify the condition (C ) of Lemma .. Let us define the homotopy H(x, λ) = ±λIx + ( – λ)JQNx. According to the above argument, we know H(x, λ) = ,
∀x ∈ ∂ ∩ Ker M.
Thus we have
deg{JQN, ∩ Ker M, θ } = deg H(·, ), ∩ Ker M, θ
= deg H(·, ), ∩ Ker M, θ = deg{±I, ∩ Ker M, θ } = . So the condition (C ) of Lemma . is satisfied. Therefore, the operator equation Mx = Nx has at least one solution in dom M ∩ . That is, BVP (.)-(.) has at least one solution in X.
5 Solutions of BVP (1.1)-(1.3) We will give a theorem on the existence of solutions for BVP (.)-(.). Define the operator M : dom M ∩ X → Z by β M x = D+ φp Dα+ x ,
(.) β
where dom M = {x ∈ X |D+ φp (Dα+ x) ∈ Z}. Then BVP (.)-(.) is equivalent to the operator equation x ∈ dom M ,
M x = Nx,
where N = N and Nλ : X → Z, λ ∈ [, ] is defined by (.). By similar arguments to Section , we obtain
Ker M = x ∈ X |x(t) = d, ∀t ∈ [, ], d ∈ R ,
Im M = z ∈ Z ( – s)α– φq – ( – τ )β– z(τ ) dτ +
s
(s – τ )β– z(τ ) dτ ds = .
Lemma . The operator M , defined by (.), is a quasi-linear operator.
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Let X = Ker M , define the projector P : X → X and the semi-projector Q : Z → Z by P x(t) = x(), ∀t ∈ [, ], α– ( – s) φq – ( – τ )β– z(τ ) dτ Q z(t) = φp ρ s + (s – τ )β– z(τ ) dτ ds , ∀t ∈ [, ],
(–s)α– φq (–+sβ ) ds < . Furthermore, let ⊂ X be an open bounded β q– choose X = Ker P , Z = Im Q and define the operator R : × [, ] → X by
where ρ = set,
β R (x, λ)(t) = Iα+ φq I+ (I – Q)Nλ x + d˜ (I – Q)Nλ x (t) t (t – s)α– φq = (α) (β) s · (s – τ )β– λf τ , x(τ ), Dα+ x(τ ) – QNλ x(τ ) dτ
–
(β)
( – τ )β– (I – Q)Nλ x(τ ) dτ ds,
where d˜ : Z → R is defined by ˜ = –I β+ z() d(z) ( – s)β– z(s) ds. =– (β) Lemma . The operator Nλ : X → Z, λ ∈ [, ], defined by (.), is M-compact in . Our second result, based on Lemma . and Lemma ., is stated as follows. Theorem . Let f : [, ] × R → R be continuous. Assume that: (H ) there exists a constant A > such that, for ∀x ∈ dom M \ Ker M satisfying |x(t)| > A for ∀t ∈ [, ], we have
( – s)α– φq – ( – τ )β– f τ , x(τ ), Dα+ x(τ ) dτ
s
+
(s – τ )β– f τ , x(τ ), Dα+ x(τ ) dτ ds = ;
(H ) there exists a constant B > such that, for ∀r ∈ R with |r | > B , we have either
φq (r )
( – s)
φq – ( – τ )β– f (τ , r , ) dτ
s
(s – τ )
+
α–
β–
f (τ , r , ) dτ ds >
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or
φq (r )
( – s)α– φq – ( – τ )β– f (τ , r , ) dτ
s
+
(s – τ )β– f (τ , r , ) dτ ds < ,
and (H ) is true. Then BVP (.)-(.) has at least one solution, provided that p– b < , + c (β + ) ((α + ))p– p– b + c < , δ := (β + ) ((α + ))p– δ :=
if p < ; (.) if p ≥ .
Proof Let
= x ∈ dom M \ Ker M |M x = Nλ x, λ ∈ (, ) . Now we prove is bounded. For x ∈ , one has Nx ∈ Im M = Ker Q . Thus we have
( – s)α– φq – ( – τ )β– f τ , x(τ ), Dα+ x(τ ) dτ
s
(s – τ )
+
β–
α f τ , x(τ ), D+ x(τ ) dτ ds = .
From (H ), there exists a constant η ∈ [, ] such that |x(η)| ≤ A . Hence, by (.), one has x ≤ A +
α D + x . (α + )
(.)
Since M x = Nλ x, Dα+ x() = , one has β β φp Dα+ x(t) = –λI+ Nx() + λI+ Nx(t), which together with (.) and (.) implies α p– D + x ≤
p– a + c Dα+ x (β + ) α p– . D +x + b A + (α + )
If p < , from (.) and Lemma ., we have α p– D + x ≤
p– a + A b (β + ) p– α p– b D . + + c x + ((α + ))p–
(.)
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Then, in view of (.), one has q– p– α D + x ≤ (a + A b ) := T . ( – δ )(β + )
(.)
Similarly, if p ≥ , we obtain q– p– p– α D + x ≤ (a + A b ) := T . ( – δ )(β + )
(.)
Therefore, from (.), (.), and (.), we have
xX ≤ max T , T , A +
T T , A + . (α + ) (α + )
That is, is bounded. The remainder of proof are similar to the proof of Theorem ., so we omit the details. Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors contributed equally in this article. They read and approved the final manuscript. Acknowledgements The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the presentation and quality of the manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2015XKMS072). Received: 9 November 2015 Accepted: 17 February 2016 References 1. Agarwal, RP, Belmekki, M, Benchohra, M: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Differ. Equ. 47, Article ID 981728 (2009) 2. Agarwal, RP, Benchohra, M, Hamani, S: Boundary value problems for fractional differential equations. Georgian Math. J. 16, 401-411 (2009) 3. Agarwal, RP, Benchohra, M, Hamani, S: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973-1033 (2010) 4. Delbosco, D, Rodino, L: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609-625 (1996) 5. He, JH: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57-58 (1998) 6. He, JH: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86-90 (1999) 7. Jaradat, OK, Al-Omari, A, Momani, S: Existence of mild solutions for fractional semilinear initial value problems. Nonlinear Anal. 69, 3153-3159 (2008) 8. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) 9. Agarwal, RP, O’Regan, D, Stanek, S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57-68 (2010) 10. Babakhani, A, Gejji, VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 278, 434-442 (2003) 11. Bai, Z: On solutions of some fractional m-point boundary value problems at resonance. Electron. J. Qual. Theory Differ. Equ. 2010, Article ID 37 (2010) 12. Bai, Z, Lü, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) 13. Benchohra, M, Hamani, S, Ntouyas, SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391-2396 (2009) 14. Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012) 15. Chen, T, Liu, W, Hu, Z: New results on the existence of periodic solutions for a higher-order Liénard type p-Laplacian differential equation. Math. Methods Appl. Sci. 34, 2189-2196 (2011) 16. Chen, T, Liu, W, Hu, Z: A boundary value problem for fractional differential equation with p-Laplacian operator at resonance. Nonlinear Anal. 75, 3210-3217 (2012)
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