Existence of solutions of fractional boundary value problems with p-laplacian operator

Transcription

1 Mahmudov and Unul Boundary Value Problems 25 25:99 OI.86/s R E S E A R C H Open Access Existence of solutions of fractional boundary value problems with p-laplacian operator Nazim I Mahmudov and Sinem Unul * * Correspondence: sinem.unul@emu.edu.tr Eastern Mediterranean University, Gazimagusa, TRNC, Mersin, Turkey Abstract In this paper, the existence of the solutions of the fractional differential equation with p-laplacian operator and integral conditions is discussed. By Green s functions and the fixed point theorems, we state and prove the existence and uniqueness results of the problem. Two examples are given to illustrate the results. Keywords: existence and uniqueness; fractional calculus; p-laplacian Introduction ifferential equations are useful in modern physics, engineering, and in various fields of science. In these days, the theory on existence and uniqueness of boundary value problems of linear and/or nonlinear fractional equations has attracted the attention of many authors. There are comprehensive studies in this area. At the same time, it is known that the p- Laplacian operator is also used in analyzing mechanics, physics and dynamic systems, and the related fields of mathematical modeling. However, there are few studies of the existence and uniqueness of boundary conditions of fractional differential equations with the p-laplacianoperator, see [ 27] and the references therein. Zhang et al. [4] studied the eigenvalue problem for a class of singular p-laplacian fractional differential equations involving a Riemann-Stieltjes integral boundary condition: β t φp t x t=λf t, xt, t,, x =, t x =, x = xs das, where β t and t are standard Riemann-Liouville derivatives with < 2, < β, A is a function of the bounded variation, and xs das is the standard Riemann-Stieltjes integral. In their study, the results are based on upper and lower solution methods and the Schauder fixed point theorem. In [5], Su et al. studied the existence criteria of non-negative solutions of nonlinear p-laplacian fractional differential equations with first order derivative, 25 Mahmudov and Unul. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 2 of 6 ϕ p c ut = ϕ p λf t, ut, u t, for t,, k u k u =, m u m u =, x r =, r =2,3,...,[], where ϕ p is p-laplacian operator, i.e. ϕ p s= s p 2 s, p >,andϕ p = ϕ q, p q =,c is the Caputo derivative and we have the function f t, u, u :[,] [,, [, which satisfies the Carathéodory type conditions. Moreover, the nonlinear alternative of Leray-Schauder type and Banach fixed point theorems are used. Han et al. [6] studied nonlinear fractional differential equations with p-laplacian operator and boundary value conditions, ϕp ut atf u=, for<t <, u = uξλ, ϕ p u = ϕ p u = ϕp u, where <, 2 < β 3, and, β are Caputo fractional derivatives, ϕ ps= s p 2 s, p >,andϕ p = ϕ q, p =, and the parameters are <,ξ, λ >.The q continuous functions a :, [, andf :[, [, are given. The Green s function properties and the Schauder fixed point theorem are used. In [2], Liu et al. studied the solvability of the Caputo fractional differential equation with boundary value conditions involving the p-laplacian operator. The existence and uniqueness of the problem is found by the Banach fixed point theorem. The problem is given in the following: ϕp xt = f t, xt, fort,, with boundary value conditions x = r x, x = r x, x j =, where i =2,3,...,[].Here,ϕ p is the p-laplacian operator and is the Caputo fractional derivative, < R, and the nonlinear function f C[, ] R, Risgiven. In [7], Lu et al. studied the existence of nonnegative solutions of a nonlinear fractional boundary value problem with the p-laplacianoperator: β ϕp ut = f t, ut, for<t <, u = u = u =, u = u =, where 2 < 3, < β 2, and, β are the standard Riemann-Liouville fractional derivatives. Green s functions, the Guo-Krasnoselskii theorem, and the Leggett-Williams fixed point theorems are used.

3 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 3 of 6 In [], Wang and Xiang used upper and lower solutions method to find the existence results of at least one non-negative solution of the p-laplacian fractional boundary value problem, which is given in the following: φp ut = f t, ut, for<t <, u =, u =, u = auξ, u = b uη, where <, 2, a, b, < ξ, η <,andalso, are Riemann-Liouville fractional operators. In this paper, we focus on the existence of solutions of the fractional differential equation β φ p ut = f t, ut, ut, with the p-laplacian operator and integral boundary conditions, u μ u = σ g s, us ds, 2 u μ 2 u = σ 2 h s, us ds, u =, u = υ uη, where, β are for the Caputo fractional differential equation with < 2, < β 2, ν, μ i, σ i i =, 2 are non-negative parameters. f, g, h are continuous functions. By the Green s functions and fixed point theorems, we state and prove the existence and uniqueness results of the solutions. Two examples are given to illustrate the results. 2 Preliminaries The basic definitions are given in the following. efinition The Riemann-Liouville fractional integral of order > for a function f :, R is defined as I f t= Ɣ t s f s ds, provided that the right hand side of the integral is pointwise defined on, andɣ is the gamma function. efinition 2 The Caputo derivative of order > for a function f :, R is written as f t= Ɣn t s n f n s ds, where n =[],[] is the integral part of.

4 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 4 of 6 Lemma 3 Let u C, L, with the fractional derivative of order >that belongs to C, L,. Then I ut=utc t c 2 t 2 c n t n, for c i R i =,2,...,n, where n is the smallest integer greater than or equal to. Lemma 4 Let >.Then the differential equation f t=has solutions f t=k k t k 2 t 2 k n t n and I f t=f tk k t k 2 t 2 k n t n, where k i Randi=,2,...,n =[]. The Caputo fractional derivative of order n < < n for t is given by t = { Ɣ Ɣ t, N and n or / N and > n,, {,,...,n }. 3 Also, for brevity, we set ω = σ σ 2 μ μ μ μ 2, ω 2 = σ 2, μ 2 c η= υ p η β υ p ηɣβ, L = ctβ c η. We use the following properties of the p-laplacian operator: φ p u = u p 2 u, p >,and φ p = φ q, p q =. L If <p <2, uv >, u, v r >,then φ p u φ p v p r p 2 u v. L2 If p >2, u, v R then φ p u φ p v p R p 2 u v. We define two Green s functions Gt, s and Ht, s, Gt, s= t τ μ μ 2 tμ 2 μ Ɣ μ μ 2 τ Ɣ μ μ 2 τ 2 μ μ 2 Ɣ, t τ, μ μ 2 tμ 2 μ μ μ 2 τ Ɣ μ μ 2 τ 2 μ μ 2 Ɣ, t τ,

5 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 5 of 6 and Ht, s= [t s] β t sβ, s t, η s, Ɣβ υ p ηɣβ [t s] β t sβ Ɣβ tυp η s β υ p ηɣβ υ p ηɣβ t s β υ p ηɣβ t s β tυp η s β υ p ηɣβ υ p ηɣβ, s t, η s,, t s, η s,, t s, η s. Lemma 5 Let f, g, h C,, and with < 2 we have the following fractional boundary value problem: β φ p ut = f t, 4 { u μ u = σ gs ds, u μ 2 u = σ 2 hs ds, 5 u =, u = ν uη, 6 it has a unique solution which is given by T ut= Gt, sφ q Ht, τf τ dτ ds ω ω 2 t, with ω = σ σ 2 μ μ μ μ 2 and ω 2 = σ 2 μ 2. Proof By applying I β to both sides of 4, we get φ p ut t s β = f s ds b b 2 t, b, b 2 R, Ɣβ ut=φ t s β q f s ds b b 2 t. Ɣβ Using the boundary conditions u = and u = ν uη, we have φ q b = b =, and secondly, s β φ q Ɣβ Moreover, since φ p is one-to-one, f s ds b 2 = νφ q η η s β Ɣβ η = φ q ν q η s β Ɣβ f s ds b 2 η f s ds b 2 η. I β f b 2 = ν p I β f η b 2η = ν p I β f η νp b 2 η,

6 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 6 of 6 I β f νp I β f η= ν p η b 2. Then b 2 = = ν p η Iβ f ν p ν p η Iβ f η s β ν p η Ɣβ Since φ p ut = Iβ f t b b 2 t, f s ds ν p η ν p η η s β f s ds. Ɣβ φ p ut t s β = f s, us t s β ds f s, us ds Ɣβ ν p η Ɣβ tν p η η s β f s, us ds ν p η Ɣβ = Ht, sf s ds, ut=φ q Ht, sf s ds, ut= t τ φ q Ɣ Ht, sf s ds dτ c c 2 t. 7 By the boundary conditions 5, we get τ c μ φ p Hτ, sf s ds dτ c c 2 = σ gs ds, Ɣ τ μ φ p Hτ, sf s ds dτ c 2 μ σ gs ds = c μ, Ɣ μ τ μ c = φ p Hτ, sf s ds dτ c 2 μ Ɣ μ c 2 = σ μ μ 2 μ 2 gs ds, 8 τ 2 Ɣ φ p σ 2 Hτ, sf s ds dτ μ 2 hs ds. Inserting c 2 into 8, we get the values of c, and inserting c and c 2 into 7, we have t τ ut= φ p Ht, sf s, us ds dτ Ɣ μ μ 2 tμ 2 μ τ φ p Hτ, sf s, us ds dτ μ μ 2 Ɣ μ μ 2 τ 2 μ μ 2 Ɣ φ p Hτ, sf s, us ds dτ σ g s, us σ2 μ tσ 2 μ ds h s, us ds. μ μ μ 2

7 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 7 of 6 Lemma 6 The functions Gt, s and Ht, s are continuous on [, ] [, ] and Ht, s satisfies the following properties: Ht, s, for t, s [, ], 2 Ht, s Hs, s, for t, s [, ], 3 the Green s function Ht, s satisfies the following condition: Ht, s Bβ, β ds υ p ηɣβ, where B is the Beta function. Proof The proofs of properties -2 are given in []. Thus we will prove property 3 for any t, s [, ]. The Green s function Ht, s is negative. Therefore, Ht, s ds Hs, s Bβ, β ds υ p ηɣβ. 3 Existence and uniqueness results In this section, we state and prove existence and uniqueness results of the fractional BVP -2 by using the Banach fixed point theorem. Our study concerns the space C [, ], R = { u C [, ], R, u C [, ], R }, which is shown in the form u = u c u c, where c is the sup norm in C[, ], R. The following notations will be used throughout this paper: = 2 = [ μ μ 2 μ 2 μ Ɣ μ μ 2 Ɣ [ μ 2 Ɣ2 μ 2 ], ] [ Ɣ g = σ μ, h = σ 2 μ μ σ 2, h2 = μ μ 2 Ɣ2 μ 2. μ μ 2 μ μ 2 To state and prove our first result, we pose the following conditions: A The function f :[,] R R R is jointly continuous. A2 There exists a function l f L τ [, ], R suchthat f t, u, u 2 f t, v, v 2 lf t u v u 2 v 2, ], for all t, u, u 2, t, v, v 2 [, ] R R.

8 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 8 of 6 A3 The functions g and h are jointly continuous and there exists l g, l h L [, ], R such that gt, u gt, v lg t u v and ht, u ht, v lh t u v, for each t, u, t, v [, ] R. Next, we define an operator, T which is T : C[, ] C[, ] as follows: T xt=φ q Ht, sf s, xs, xs ds. Lemma 7 Assume A-A3 hold and q >2.There exists a constant l T >such that T ut T vt l T u v, for all u, v B r. We have l T =q L q 2 H l f Hs, s q 2 ds q L H l Bβ, β f υ p ηɣβ. Proof If p >2andt > we have the following estimation: Ht, sf s, us, us ds Ht, s f s, us, us ds Ht, s lf s us us f s,, ds l f u M Hs, s ds l f r M Hs, s ds = L H, where M = max s [,] f s,,. Now using the property L2, we get the desired inequality, T ut T vt = φ q Ht, sf s, us, us ds φ q Ht, sf s, vs, ds vs q L q 2 H Ht, s f s, us, us f s, vs, vs ds q L q 2 H l f u v Hs, s ds

9 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 9 of 6 q L q 2 H l f Bβ, β υ p ηɣβ u v = l T u v. Theorem 8 Assume A-A3 hold. If { 2 2 } l T i g l g h i l h <, 9 i= i= then our BVP -2 has a unique solution on [, ]. Proof Let us define the operator T : C [, ], R C [, ], RtotransformourBVP- 2intoafixedpointproblem, T ut = t s Ɣ μ μ T f s, us, us ds s Ɣ μ μ 2 μ μ 2 μ 2t s μ 2 Ɣ σ μ T f s, us, us ds s 2 Ɣ T f s, us, us ds T f s, us, us ds g s, us ds σ 2μ μ t μ 2 μ Taking the th fractional derivative, we get h s, us ds. T ut t s = Ɣ T f s, us, us ds μ 2 t μ 2 Ɣ2 σ 2 t μ 2 Ɣ2 s Ɣ T f s, us, us ds h s, us ds for t [, ]. Since f, g, h are continuous, the expression and arewelldefined. Clearly, the fixed point of the operator T is the solution of the problem -2. To show the existence and uniqueness of the solution, the Banach fixed point theorem is used and then we show T is contraction. We have T ut T vt t s l T u v ds Ɣ μ μ s l T u v ds Ɣ

10 Mahmudovand Unul Boundary Value Problems 25 25:99 Page of 6 μ μ 2 s 2 μ μ 2 Ɣ l T u v ds μ 2 s l T u v ds μ 2 Ɣ σ μ σ μ {l T l g s us vs us vs ds σ 2 μ μ l h s us vs μ 2 μ us vs ds t s {l T ds μ μ 2 μ 2 μ s ds Ɣ μ μ 2 Ɣ μ μ 2 s 2 μ μ 2 Ɣ ds } l h s ds u v l g s ds σ 2 μ μ μ 2 μ Ɣ μ μ 2 μ 2 μ Ɣ μ μ 2 μ μ 2 Ɣ μ μ 2 σ μ l g σ 2 μ μ l h } u v. 2 μ 2 μ By using the Hölder inequality, we get T ut T vt { } lt g l g h l h u v, 3 T ut T vt t s Ɣ l T u v ds μ 2 t s μ 2 Ɣ2 Ɣ l T u v ds { t σ 2 μ 2 Ɣ2 l T Ɣ t s ds l h s us vs us vs ds l T t μ 2 s ds Ɣ Ɣ2 μ 2 σ 2 t } l h s ds u v μ 2 Ɣ2 {l T Ɣ μ 2 Ɣ Ɣ2 μ 2 σ 2 } l h s ds u v μ 2 Ɣ2 { l T μ 2 Ɣ Ɣ2 μ 2 σ 2 μ 2 Ɣ2 l h } u v. 4

11 Mahmudovand Unul Boundary Value Problems 25 25:99 Page of 6 Similarly, T ut T vt { l T 2 h2 l h } u v. 5 With the help of 3-5, we find that Tu Tv { } l T 2 g l g h h2 l h u v { 2 2 = l T i g l g hi l h } u v. i= i= Thus T is a contraction mapping by condition 9. By the Banach fixed point theorem, T has a fixed point which is the solution of the BVP. 4 Existence results Theorem 9 Assume: iv There exist non-decreasing functions ϕ :[, [, [, and ψ i :[, [,, i =,2,and the functions l f L τ [, ], R and l g, l h L [, ], R such that f t, u, v lf tϕ u v, gt, u lg tψ u, ht, u lh tψ 2 u, for all t [, ] and u, v R. v There exists a constant N >such that [ N ϕ u l T 2 i= i ψ u l g g ψ 2 u l h 2 i= h i Thus problem -2 has at least one solution on [, ]. ] >. 6 Proof Let B r = {u C [, ], R: u r}. Step : Let the operator T : C [, ], R C [, ], R be given in which defines B r to be a bounded set. For all u B r,weget T ut ϕr Ɣ l T t s ds μ μ 2 μ 2 μ μ μ 2 ϕr Ɣ l T s ds μ μ 2 ϕr μ μ 2 Ɣ l T s 2 ds σ μ ψ r lg s σ 2 μ μ ds ψ 2 r μ 2 μ lh s ds

12 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 2 of 6 and T ut ϕr Ɣ l T t s ds μ 2 ϕr μ 2 Ɣ2 Ɣ l T s ds σ 2 t μ 2 Ɣ2 ψ 2r lh s ds. t By the Hölder inequality, T ut ϕrl T Ɣ μ μ 2 μ 2 μ ϕrl T μ μ 2 ϕrl T μ μ 2 Ɣ μ μ 2 Ɣ σ ψ r l g μ σ 2 μ μ ψ 2 r l h μ 2 μ ϕrl T g ψ r l g h ψ 2 r l h, T ut ϕrl T Ɣ μ 2 ϕrl T μ 2 Ɣ2 Ɣ σ 2 ψ 2 r l h μ 2 Ɣ2 ϕrl T 2 h2 ψ 2 r l h. Therefore, T u ϕrl T 2 g ψ r l g h h2 ψ 2 r l h. Step 2: The families {T u :u B r } and { T u :u B r} are equicontinuous. For t < t 2,weget T ut2 T ut Similarly, ϕrl T Ɣ [ μ 2 t 2 t ϕrl T μ 2 Ɣ t s t 2 s ds 2 σ 2 μ t 2 t ψ 2 r μ 2 μ T ut 2 T ut ϕrl T Ɣ [ s ds t ] t 2 s ds lh s ds ast2 t. t s t 2 s ds 2 t ] t 2 s ds

13 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 3 of 6 ϕrl T μ 2 t 2 t Ɣ μ 2 Ɣ2 σ 2 t 2 t ψ 2 r μ 2 Ɣ2 s ds lh s ds ast2 t. Thus {T u:u B r } and { T u:u B r} are equicontinuous and relatively compact in C[, ], R by the Arzela-Ascoli theorem. Therefore T B r isarelativelycompactsubset of C [, ], R and the operator T is compact. Step 3: Let u = λt u andu = λ T u for < λ <.Foreacht [, ], define M = { u C [, ], R, u < N }.Thenweget u c = λt u c ϕ u lt g ψ u lg h ψ 2 u lh, u c = λ T u c ϕ u lt 2 h2 ψ 2 u lh. Thus u ϕ u lt That means 2 i ψ u lg g ψ 2 u lh i= 2 hi. i= u ϕ u l T 2 i= i ψ u l g g ψ 2 u l h 2 i= h i. For a non-negative N and u < N, the operator T which is defined in M to be C [, ], R is continuous and compact. Therefore T has a fixed point in M. 5 Examples Example Consider the following boundary value problem of a fractional differential equation: Here 3 ut 2 ut, ut φ p 3 ut=l f ut 3 2 u.u =. us ds, s 2 u.u =. es us 2e s 2 ds. 7, β =.5, μ, μ 2 =., σ, σ 2 =., υ, η =.3, τ =.4, =., and f t, u, v= u u v v,

14 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 4 of 6 gt, u= u s, ht, u= e s u 2 2e s 2. Since.88 < Ɣ.5 <.89, Ɣ2 =, Ɣ2.5 =, we find =.89, 2 =.82, g =.9, h =.99, h2 =.9, l g = l h =, with simple calculations. Therefore { lt 2 2 g l g h h2 l h } <.73l T.4 <. Then we can choose l T <.562. Thus all assumptions of Theorem 8 satisfied. Therefore the problem has a unique solution on [, ]. Example Consider the following boundary value problem of fractional differential equation: 3 2 φ p 3 2 ut= ut 3 9 ut 3 3 u.u =, us ds, 3s 2 u.u =, e s us ds, 3e s u =, u =, 3 3 u, 3, where f is given by We have Here f t, u, v= 3 sin 2 ut 3 9sin 2 ut 2, u 3 9 u 3 3 sin v 9sin v 2. f t, u, v u 3 9 u 3 3 sin v 9sin v 2, u R. 8, β =.5, μ, μ 2 =., σ, σ 2 =., υ, η =.3, τ =.4, =., =.89, 2 =.82, g =.9,

15 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 5 of 6 h =.99, h2 =.9, l g = l h =., and gt, u= us, ht, u= es us, ϕn =ψ 3s 2 3e s 2 N =ψ 2 N =N.If N ϕn ψ N..9 ψ 2 N >, N N.96 N.9 N.9 >, N.9628N >,.4 >, then 6 issatisfied. Then thereexistsat leastonesolution ofthebvp on [, ]. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the manuscript and typed, read, and approved the final manuscript. Acknowledgements The authors would like to thank the anonymous referees for their valuable comments and suggestions. Received: 9 March 25 Accepted: 25 May 25 References. Wang, J, Xiang, H: Upper and lower solutions method for a class of singular fractional boundary value problems with p-laplacian operator. Abstr. Appl. Anal. 2, Liu, X, Jia, M, Xiang, X: On the solvability of a fractional differential equation model involving the p-laplacian operator. Comput. Math. Appl. 64, Ahmad, B: Nonlinear fractional differential equations with anti-periodic type fractional boundary conditions. iffer. Equ. yn. Syst. 24, Zhang, X, Liu, L, Wiwatanapataphee, B, Wu, Y: The eigenvalue for a class of singular p-laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition. Appl. Math. Comput. 235, Su, Y, Li, Q, Liu, X: Existence criteria forpositivesolutions of p-laplacian fractional differential equations with derivative terms. Adv. iffer. Equ. 23, Han, Z, Lu, H, Sun, S, Yang, : Positive solutions to boundary-value problems of p-laplacian fractional differential equations with a parameter in the boundary. Electron. J. iffer. Equ. 22, Han, Z, Lu, H, Sun, S, Liu, J: Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with p-laplacian. Adv. iffer. Equ. 23, doi:.86/ Zhang, J, Liangand, S: Existence and uniqueness of positive solutions for integral boundary problems of nonlinear fractional differential equations with p-laplacian operator. Rocky Mt. J. Math. 443, Khan, RA, Khan, A, Samad, A, Khan, H: On existence of solutions for fractional differential equation with p-laplacian operator. J. Fract. Calc. Appl. 52, Wang, L, Zhou, Z, Zhou, H: Positive solutions for singular p-laplacian fractional differential system with integral boundary conditions. Abstr. Appl. Anal. 24, Wei, L, uan, L, Agarwal, RP: Existence and uniqueness of the solution to integro-differential equation involving the generelized p-laplacian operator with mixed boundary conditions. J. Math. 33, Wei, L, Agarwal, RP, Wong, PJY: Study on integro-differential equation with generalized p-laplacian operator. Bound. Value Probl. 22, Wei, L, Agarwal, RP: iscussion on the existence of solutions to nonlinear boundary value problems with generalized p-laplacian operator. Acta Math. Sci. Ser. A Chin. Ed. 32, Wei, L, Zhou, H, Agarwal, RP: Existence of solutions to nonlinear Neumann boundary value problems with p-laplacian operator and iterative construction. Acta Math. Appl. Sinica Engl. Ser. 27, Wei, L, Agarwal, RP, Wong, PJY: Existence of solutions to nonlinear parabolic boundary value problems with generalized p-laplacian operator. Adv. Math. Sci. Appl. 22, Wei, L, Agarwal, RP: Existence of solutions to nonlinear Neumann boundary value problems with generalized p-laplacian operator. Comput. Math. Appl. 562, Agarwal, RP, O Regan,, Papageorgiun, NS: On the existence of two nontrivial solutions of periodic problems with operators of p-laplacian type. iffer. Equ. 432, Jiang, Q, O Regan,, Agarwal, RP: A generalized upper and lower solution method for singular discrete boundary value problems for the one-dimensional p-laplacian. J. Appl. Anal., Aktuğlu, H, Özarslan, MA: Solvability of differential equations of order 2 < 3 involving the p-laplacian operator with boundary conditions. Adv. iffer. Equ. 23,

16 Mahmudovand Unul Boundary Value Problems 25 25:99 Page 6 of 6 2. Kong, X, Wang,, Li, H: Existence of unique positive solution to a two-point boundary value problem of fractional-order switched system with p-laplacian operator. J. Fract. Calc. Appl. 52, Khan, RA, Khan, A, Samad, A, Khan, H: On existence of the solution for fractional differential equation with p-laplacian operator. J. Fract. Calc. Appl. 52, Hu, Z, Liu, W, Liu, J: Existence of solutions for a coupled system of fractional p-laplacian equations at resonance. Adv. iffer. Equ. 23, Chang, X, Qiao, Y: Existence of periodic solutions for a class of p-laplacian operators. Bound. Value Probl. 23, Chen, T, Liu, W: Solvability of some boundary value problems for fractional p-laplacian equation. Abstr. Appl. Anal. 23, Chen, T, Liu, W, Liu, J: Existence of solutions for some boundary value problems of fractional p-laplacian equation at resonance. Bull. Belg. Math. Soc. Simon Stevin 2, Liu, Y, Qian, T: Existence of solutions of periodic-type boundary value problems for multi-term fractional differential equations. Math. Methods Appl. Sci. 22. doi:.2/mma Tang, X, Yang, C, Liu, Q: Existence of the solutions of two-point boundary value problems for fractional p-laplacian differential equations at resonance. J. Appl. Math. Comput. 4, doi:.7/s