Existence of solutions for fractional integral boundary value problems with p(t)-laplacian operator

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1 Available online at J. Nonlinear Sci. Appl. 9 (26), 5 5 Research Article Existence of solutions for fractional integral boundary value problems with -Laplacian operator Tengfei Shen, Wenbin Liu College of Sciences, China University of Mining and Technology, Xuzhou 226, P. R. China. Communicated by M. De la Sen Abstract This paper aims to investigate the existence of solutions for fractional integral boundary value problems (VPs for short) with -Laplacian operator. y using the fixed point theorem and the coincidence degree theory, two existence results are obtained, which enrich existing literatures. Some examples are supplied to verify our main results. c 26 All rights reserved. Keywords: Fractional differential equation, boundary value problem, -Laplacian operator, fixed point theorem, coincidence degree theory. 2 MSC: 26A33, 34A8, 345, Introduction In the recent years, fractional differential equations have been applied in many research fields (see [4, 2, 23, 24, 3]). For example, Leszczynski and laszczyk [23] discussed the following fractional mathematical model which can be used to describe the height of granular material decreasing over time in a silo: C D α T D α a h (t) βh (t) =, t [, T ], where C D α T is the right Caputo fractional derivative, D α a is the left Riemann-Liouville fractional derivative, α (, ). Furthermore, some valuable results which are related to the stability of fractional functional equations (see [, 7, 28]) and the existence and multiplicity of solutions for fractional boundary value problems (see [ 3, 5, 6, 5, 8]) have been achieved by some scholars. For example, Cabada and Wang [6] considered the existence of positive solutions for the following fractional integral VP by Guo-Krasnoselskii Corresponding author addresses: stfcool@26.com (Tengfei Shen), cumt_equations@26.com (Wenbin Liu) Received 26--4

2 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), fixed point theorem: C D α x(t) f(t, x(t)) =, t (, ), x() = x () =, x() = λ x(s)ds, where 2 < α 3, < λ < 2, C D α is a Caputo fractional derivative, f : [, ] [, ) [, ) is continuous. It is well-known that p-laplacian operator is a nonlinear operator and occurs in the course of considering glacial sliding (see [27]), torsional creep (see [9]), porous medium (see [22]), etc. For solvability of VPs for integer differential equations with p-laplacian operator, we provide readers with some articles (see [, 3, 4]). Recently, some scholars have paid more attention to fractional p-laplacian equations and got some interesting results (see [7, 9, 6, 25]). For example, Chen and Liu [9] investigated the existence of solutions for the anti-periodic VP of fractional differential equation with p-laplacian operator by Schaefer s fixed point theorem: D β ϕ p (D α x(t)) = f(t, x(t)), t [, ], x() = x(), D α x() = D α x(), where < β, α, < α β 2, D α is a Caputo fractional derivative, ϕ p ( ) is a p-laplacian operator, f : [, ] R R is continuous. Mahmudov and Unul [25] studied the existence and uniqueness of solutions for integral VP of fractional differential equation with p-laplacian operator by Green s functions and some fixed point theorems: D β ϕ p (D α x(t)) = f(t, x(t), D x(t)), t [, ], x() µ x() = σ g(s, x(s))ds, x () µ 2 x () = σ 2 h(s, x(s))ds, D α x() =, D α x() = νd α x(η), where < α 2, < β,, < η <, ν, µ i, σ i > (i =, 2), D α is a Caputo fractional derivative, ϕ p ( ) is a p-laplacian operator, f, g, h are continuous. Motivated by the work above, our paper aims to investigate the existence of solutions for the following fractional integral VP with -Laplacian operator under the non-resonance case and resonance case: D β ϕ (D α x(t)) f(t, x(t)) =, t (, ), x() =, D α x() = I α x(η), D α (.) x() =, where D α is a Riemann-Liouville fractional derivative, < α 2, < β, >, < η <, f : [, ] R R is continuous. ϕ ( ) is a -Laplacian operator, >, C [, ]. Noting that the -Laplacian operator is the non-standard growth operator which arises from nonlinear electrorheological fluids (see [29]), image restoration (see [8]), elasticity theory (see [32]), etc. There are many valuable results with respect to this type problems (see [2, 3] and references therein). Compared with constant growth operator, it will bring many difficulties. It can turn into the well-known p-laplacian operator when = p, so our results extend and enrich some existing papers. Moreover, there are almost no papers which considered fractional integral VPs with -Laplacian operator. For the non-resonance case, by constructing the Green function, we show a new existence result for VP (.) by the Schaefer s fixed point theorem. For the resonance case, by investigating the following equivalence problem (see Lemma 2.8) D α x(t) = ϕ (Iβ f(t, x(t)), t (, ), x() =, D α x() = I α x(η), we obtain a new existence result for VP (.) by the coincidence degree theory of Mawhin (see [26]). 2. Preliminaries For basic definitions of Riemann-Liouville fractional integral and fractional derivative, please see [2]. Here, we show some important properties, lemmas and definitions as follows.

3 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), Lemma 2. ([3]). For any (t, x) [, ] R, ϕ (x) = x 2 x, is a homeomorphism from R to R and strictly monotone increasing for any fixed t. Moreover, its inverse operator ϕ ( ) is defined by ϕ 2 (x) = x x, x R \ }, ϕ () =, x =, which is continuous and sends bounded sets to bounded sets. Definition 2.2 ([26]). Let X and Y be real anach spaces and let L : doml X Y be a Fredholm operator with index zero. Let P : X X, Q : Y Y be continuous linear projectors such that ImP = KerL, KerQ = ImL, X = KerL KerP, Y = ImL ImQ. It follows that L doml KerP : doml KerP ImL is invertible. Its inverse is defined by K P. If Ω is an open bounded subset of X, and doml Ω, the map N : X Y will be called L compact on Ω if QN : Ω Y is bounded and K P,Q N := KP (I Q)N : Ω X is compact. Lemma 2.3 ([26]). Let L : doml X Y be a Fredholm operator of index zero and N : X Y be L compact on Ω. Assume that the following conditions are satisfied: (i) Lx λnx for every (x, λ) [(doml \ KerL)] Ω (, ); (ii) Nx ImL for every x KerL Ω; (iii) deg(qn KerL, KerL Ω, ), where Q : Y Y is a projection such that ImL = KerQ. Then the equation Lx = Nx has at least one solution in doml Ω. Lemma 2.4 ([2]). Some properties for the Riemann-Liouville fractional integral and fractional derivative are as follows: (i) If α, λ >, λ α i, i =, 2,..., [α], we have D α t λ = Moreover, D α t α i =, i =, 2,..., [α]. (ii) If α >, λ >, we have I α t λ = Γ(λ ) Γ(λ α ) tλ α. Γ(λ ) Γ(λ α ) tλα. Lemma 2.5. If y(t) C[, ] and < η 2α 2 < Γ(2α ), the unique solution of D β ϕ (D α x(t)) y(t) =, t (, ), x() =, D α x() = I α x(η), D α x() =, can be expressed as the following integral equation where x(t) = G(t, s)ϕ p(s) (Iβ y(s))ds, (2.) t α Γ(2α ) t α (η s) 2α 2 (t s) α (Γ(2α ) η 2α 2 ) Γ(α)(Γ(2α ) η 2α 2 ), s t, s η, G(t, s) = t α Γ(2α ) t α (η s) 2α 2 Γ(α)(Γ(2α ) η 2α 2 ), t s η <, t α Γ(2α ) (t s) α (Γ(2α ) η 2α 2 ) Γ(α)(Γ(2α ) η 2α 2 ), < η s t, t α Γ(2α ), t s, s η. Γ(α)(Γ(2α ) η 2α 2 )

4 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), Proof. ased on the definitions of Riemann-Liouville fractional integral, we have ϕ (D α x(t)) = I β y(t) ct β, c R. Combining with D α x() =, for fixed t =, we have c = and x(t) = Γ(α) (t s) α ϕ p(s) (Iβ y(s))ds c t α c 2 t α 2, where c i R, i =, 2. y x() =, we obtain c 2 =. In view of Lemma 2.4, it follows that and x(t) = Γ(2α ) I α D α x(t) = (t s) 2α 2 ϕ Γ(α) p(s) (Iβ y(s))ds c Γ(2α ) t2α 2, ϕ p(s) (Iβ y(s))ds c Γ(α). ased on the boundary value condition D α x() = I α x(η), it follows c = [ Γ(2α ) Γ(α)(Γ(2α ) η 2α 2 ϕ ) p(s) (Iβ y(s))ds Γ(2α ) Thus, we have x(t) = Γ(α) Γ(2α ) Therefore, (2.) holds. (t s) α ϕ t α [ Γ(2α ) p(s) (Iβ y(s))ds Γ(α)(Γ(2α ) η 2α 2 ) ] (η s) 2α 2 ϕ p(s) (Iβ y(s))ds. Lemma 2.6. If < η 2α 2 < Γ(2α ), G(t, s) satisfies the condition < G(t, s) < Proof. y definition of G(t, s), it is clear that G(t, s) < hand, if s t, s η, then set We can obtain Γ(2α ) Γ(α)(Γ(2α ) η 2α 2, s, t (, ). ) ] (η s) 2α 2 ϕ p(s) (Iβ y(s))ds. ϕ p(s) (Iβ y(s))ds Γ(2α ) for all s, t (, ). On the other Γ(α)(Γ(2α ) η 2α 2 ) g(t, s) = t α Γ(2α ) t α (η s) 2α 2 (t s) α (Γ(2α ) η 2α 2 ). g(t, s) t α Γ(2α ) t α η 2α 2 (t s) α (Γ(2α ) η 2α 2 ) = (Γ(2α ) η 2α 2 )[t α (t s) α ] (Γ(2α ) η 2α 2 )[t α (t ts) α ] = t α (Γ(2α ) η 2α 2 )[ ( s) α ] >. For other situations, it is clear that G(t, s) >, so we omit the proof. Remark 2.7. Noting that if η 2α 2 = Γ(2α ), VP (.) is the resonance case. However, the Mawhin s continuation theorem is not suitable for the nonlinear operator case. Thus, we need the following lemma to turn the nonlinear operator case into the linear operator case.

5 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), Lemma 2.8. VP (.) is equivalent to the following problem D α x(t) = ϕ (Iβ f(t, x(t)), t (, ), x() =, D α x() = I α x(η). Proof. On one hand, based on the boundary condition D α x() =, we conclude (2.2) from (.). On the other hand, if D α x(t) = ϕ (Iβ f(t, x(t)), substituting t = into the above equality, we obtain D α x() =. Moreover, applying the operator ϕ and Dβ to the both side of above equality, we have D β ϕ (D α x(t)) f(t, x(t)) =. Thus, the claim is proved. (2.2) 3. Main result 3.. The non-resonance case Let X = C[, ] with norm x = max t [,] x(t). In the sequel, we assume that < η 2α 2 < Γ(2α ). Furthermore, in order to state Theorem 3., let P L := min t [,], P M := max t [,]. Theorem 3.. Assume that the following condition holds. (H ) There exist constants a, b > such that f(t, x) a b x θ, < θ P L. Then VP (.) has at least one solution, provided that 2 P L Γ(2α ) maxb P L, b P M } P (Γ(β )) M Γ(α)(Γ(2α ) η 2α 2 ) <. (3.) Proof. The operator T : C[, ] C[, ] is defined by T x(t) = G(t, s)ϕ p(s) (Iβ f(s, x(s)))ds. y the continuity of f, it is easy to find that T is continuous. Let Ω be any bounded open subset of C[, ]. Since ϕ ( ) and f are continuous, there exists a constant M > such that ϕ (Iβ f(t, x(t))) M on [, ] Ω. Thus, we can obtain MΓ(2α ) T x = max T x t [,] Γ(α)(Γ(2α ) η 2α 2 ) ds = MΓ(2α ) Γ(α)(Γ(2α ) η 2α 2 ). Thus, T Ω is uniformly bounded. On the other hand, for all t, t 2 [, ], assume that t t 2, for any x Ω, we have T x(t 2 ) T x(t ) = G(t 2, s)ϕ p(s) (Iβ f(s, x(s)))ds G(t, s)ϕ p(s) (Iβ f(s, x(s)))ds = M Γ(α) [(t 2 s) α (t s) α ]ds M Γ(α) (Γ(2α) η 2α )M Γ(α)[Γ(2α) (2α )η 2α 2 ] (tα 2 t α ) 2 t (t 2 s) α ds M Γ(α ) (tα 2 t α (Γ(2α) η 2α )M ) Γ(α)[Γ(2α) (2α )η 2α 2 ] (tα 2 t α ).

6 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), Thus, one has T x(t 2 ) T x(t ) uniformly as t t 2. Therefore, T is equicontinuous on Ω. y the Arzelá-Ascoli theorem, we can obtain T is completely continuous. Define V = x X x = λt x, λ (, )}. According to Schaefer s fixed point theorem, we just need to prove that V is bounded. For x V, we have I β f(t, x(t)) Γ(β) Γ(β) (t s) β f(s, x(s)) ds (t s) β (a b x(s) θ )ds (a b x θ ). Γ(β ) y the basic inequality (x y) p 2 p (x p y p ) for x, y, p > (see [2]), we have x(t) = λ T x(t) Since G(t, s)ϕ p(s) (Iβ f(s, x(s)) )ds Γ(2α ) Γ(α)(Γ(2α ) η 2α 2 ) 2 p(s) [ (Γ(β )) p(s) (a p(s) b θ (, ], by the basic inequality xκ x for x >, κ (, ], we have θ p(s) p(s) x )]ds. x 2 P L Γ(2α ) P (Γ(β )) M Γ(α)(Γ(2α ) η 2α 2 ) [a p(s) b p(s) ( x )]ds. y (3.), there exists a constant M > such that x M. Thus, the operator T has a fixed point, which implies VP (.) has at least one solution. Example 3.2. Consider the following VP: D 3 4 ϕ t 2 2(D 3 2 x(t)) f(t, x(t)) =, t (, ), x() =, D 2 x() = 2 I 2 x( 2 ), D 3 2 x() =, where α = 3 2, β = 3 4, = t2 2, θ = 2, η = 2, = 9 2, f(t, x(t)) = 64 x2, a =, b = Clearly, (H ) holds. Moreover, in view of (3.), we have Thus, it has at least one solution The resonance case Γ( 3 2 )(Γ( 7 4 )) 2 In this part, let X = Y = C[, ] with the norm x = max t [,] x(t). Noting that if η 2α 2 = Γ(2α ), VP (.) turns into resonance case. y Lemma 2.8, the original problem can be turned into the following problem: D α x(t) = ϕ (Iβ f(t, x(t)), t (, ), x() =, D α x() = I α x(η). <.

7 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), where Define the operator L : doml X Y by Lx = D α x, doml = x X D α x(t) Y, x() =, D α x() = I α x(η)}. Let N : X Y be the Nemytski operator Nx(t) = ϕ (Iβ f(t, x(t)), t [, ]. Then VP (.) is equivalent to the operator equation Lx = Nx, x doml. where It is clear that KerL = x X x(t) = ct α, t [, ], c R}, ImL = y Y y(s)ds Γ(2α ) (η s) 2α 2 y(s)ds = }. Define the linear continuous projection operators P : X X and Q : Y Y : P x(t) = Dα x() t α, t [, ], Γ(α) Qy(t) = [ := y(s)ds Γ(2α) Γ(2α) η 2α = It is easy to find that P 2 = P, Q 2 = Q, Noting that Γ(2α ) (η s) 2α 2 y(s)ds], t [, ], Γ(2α) Γ(2α) ηγ(2α ) = 2α 2α η >. X = KerL KerP, and Y = ImL ImQ. dim KerL = dim ImQ = codim ImL =. Thus, L is a Fredholm operator of index zero. Let K P : ImL doml KerP, which can be given by K P y = I α y. K P is the inverse of L doml KerP. Since f and ϕ ( ) are continuous, it is easy to obtain that N is L-compact on Ω whose proof is similar to some parts of Lemma 3.3 in [5], so we omit it here. Theorem 3.3. Assume that (H ) and the following condition hold. (H 2 ) there exists a constant > such that if x(t) > for any t [η, ], either or sgnx(t)}qn(x(t)) <, sgnx(t)}qn(x(t)) >. Then VP (.) has at least one solution, provided that b2 θ ( η α ) θ Γ(β )(Γ(α )η α <. (3.2) ) θ

8 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), Proof. Let Ω = x doml \ KerL Lx = λnx, λ (, )}. Since x Ω, we have Nx ImL = KerQ. Thus, QNx =. y (H 2 ), there exists a constant ξ [η, ] such that x(ξ). y x() =, we have and where Hence, we can obtain c [ x(ξ) ξα Γ(α) ased on Lu = λnu, we can get ξ x x(t) = I α D α x(t) c t α. (ξ s) α D α x(s) ds] η α ( Γ(α ) Dα x ), ηα Γ(α )η α Dα x. (3.3) ηα D α x(t) = λϕ (Iβ f(t, x(t))). Applying the operator ϕ to the both side of above equality, one has From (H ) and λ (, ), we have D α x(t) Γ(β) In view of (3.3), we have D α x(t) ϕ (D α x(t)) = λ (I β f(t, x(t))). (t s) β f(t, x(t)) ds [a b( Γ(β ) η α Γ(α )η α Dα x y the basic inequality (x y) p 2 p (x p y p ), x, y, p >, we have Hence, we can obtain Since D α x(t) Λ Λ 2 D α x θ, Λ = b(2)θ aη (α )(θ ) Γ(β )η (α )(θ ), Λ 2 = D α x 2 (Λ (a b x θ Γ(β ) ). η α )θ ]. b2 θ ( η α ) θ Γ(β )(Γ(α )η α ) θ. θ Λ2 D α x ). θ (, ], by the basic inequality xκ x for x >, κ (, ], we have D α x 2 Λ 2 Λ 2 ( D α x ). In view of (3.2), we can obtain that there exists a constant M 2 > such that D α x M 2, x ηα Γ(α )η α M 2 η α := M 3.

9 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), So, Ω is bounded. Let Ω 2 = x x KerL, Nx ImL}. For x Ω 2, we have x(t) = ct α, c R and Nx ImL. Thus, we can obtain QN(ct α ) =, which together with (H 2 ) implies c. Hence, Ω η α 2 is bounded. Let Ω 3 = x KerL λjx ( λ)qnx =, λ [, ]}, where J : KerL ImQ is defined by J(ct α ) = c, c R, t [, ]. Thus, we have [ λc ( λ) ϕ p(s) (Iβ f(s, cs α ))ds Γ(2α ) If λ =, by the first part of (H 2 ), we have c Otherwise, if c > η α, in view of the first part of (H 2 ), one has [ λsgn(ct α )c ( λ) sgn(ct α ) Γ(2α ) for any t [η, ]. Thus, by choosing t =, we obtain [ λsgn(c)c ( λ) sgn(c) Γ(2α ) which contradicts to (3.4). Thus, Ω 3 is bounded. Let ] (η s) 2α 2 ϕ p(s) (Iβ f(s, cs α ))ds =. (3.4) η α. If λ (, ], we can also obtain c ϕ p(s) (Iβ f(s, cs α ))ds ] > (η s) 2α 2 ϕ p(s) (Iβ f(s, cs α ))ds ϕ p(s) (Iβ f(s, cs α ))ds (η s) 2α 2 ϕ p(s) (Iβ f(s, cs α ))ds Ω 3 = x KerL λjx ( λ)qnx =, λ [, ]}. ] >, η α. Similarly, we can prove that Ω 3 is bounded by the second part of (H 2). Let Ω = x X x < maxm 3, } }. Since L is a Fredholm operator of index zero and N is η α L-compact on Ω, then by the previous proof, we have (i) Lx λnx, for every (x, λ) [(doml \ KerL) Ω] (, ). (ii) Nx / ImL, for every x KerL Ω. Let H(x, λ) = ±λj(x) ( λ)qnx. We can obtain H(x, λ) for x KerL Ω. Therefore, deg(qn KerL, Ω KerL, ) = deg(h(, ), Ω KerL, ) = deg(h(, ), Ω KerL, ) = deg(±j, Ω KerL, ). Hence, the conditions of Lemma 2.3 are satisfied. Therefore, we can obtain that Lx = Nx has at least one solution in doml Ω. Then VP (.) has at least one solution.

10 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), Example 3.4. Consider the following VP: D 2 3 ϕ (t 2 2)(D 3 2 x(t)) x(t) =, t (, ), x() =, D 2 x() = I 2 x(η), D 2 x() =, where = t 2 2, α = 3 2, β = 2 3, θ = 2, f(t, x(t)) = x(t) a = b =. Clearly, (H ) holds. y choosing = 2, η = 2, = 2, it is easy to verify Γ(2α ) = η2α 2. Moreover, if x(t) > 2, we have f(t, x(t)) = x(t) >. Thus, we obtain Nx(t) = ϕ (t 2 2) (I 2 3 f(t, x(t))) > and Hence, we have QNx(t) = [ = [ = 2 Nx(s)ds 2 Γ(2) Nx(s)ds Nx(s)ds ( 2 s)nx(s)ds] ( 2s)Nx(s)ds] [ ( 2s)]Nx(s)ds} >. sgnx(t)}qn(x(t)) >. (3.5) If x(t) < 2, it can be found that (3.5) is also true. Thus, the first part of (H 2 ) holds. In view of (3.2), we have 2 2 5Γ( 5 3 )Γ( 5 2 ) <. Therefore, it has at least one solution. Acknowledgment This research is supported by the National Natural Science Foundation of China (No.27364). References []. Ahmad, R. P. Agarwal, Some new versions of fractional boundary value problems with slit-strips conditions, ound. Value Probl., 24 (24), 2 pages. [2] C. ai, Existence of positive solutions for a functional fractional boundary value problem, Abstr. Appl. Anal., 2 (2), 3 pages. [3] C. ai, Existence of positive solutions for boundary value problems of fractional functional differential equations, Electron. J. Qual. Theory Differ. Equ., 2 (2), 4 pages. [4] J. ai, X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 6 (27), [5] Z. ai, Y. Zhang, Solvability of fractional three-point boundary value problems with nonlinear growth, Appl. Math. Comput., 28 (2), [6] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (22), [7] G. Chai, Positive solutions for boundary value problem of fractional differential equation with p-laplacian operator, ound. Value Probl., 22 (22), 2 pages. [8] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (26), [9] T. Chen, W. Liu, An anti-periodic boundary value problem for the fractional differential equation with a p- Laplacian operator, Appl. Math. Lett., 25 (22),

11 T. Shen, W. Liu, J. Nonlinear Sci. Appl. 9 (26), [] M. De la Sen, About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory Appl., 2 (2), 9 pages. [] Z. Du, X. Lin, C. C. Tisdell, A multiplicity result for p-lapacian boundary value problems via critical points theorem, Appl. Math. Comput., 25 (28), [2] X. Fan, Q. Zhang, D. Zhao, Eigenvalues of p(x)-laplacian Dirichlet problem, J. Math. Anal. Appl., 32 (25), [3] W. Ge, oundary value problems for ordinary nonlinear differential equations, Science Press, eijing, China, (27). [4] W. Ge, J. Ren, An extension of Mawhin s continuation theorem and its application to boundary value problems with a p-laplacian, Nonlinear Anal., 58 (24), [5] Z. Hu, W. Liu, Solvability for fractional order boundary value problems at resonance, ound. Value Probl., 2 (2), pages., 3.2 [6] Z. Hu, W. Liu, J. Liu, Existence of solutions of fractional differential equation with p-laplacian operator at resonance, Abstr. Appl. Anal., 24 (24), 7 pages. [7] J.. Hu, G. P. Lu, S.. Zhang, L. D. Zhao, Lyapunov stability theorem about fractional system without and with delay, Commun. Nonlinear Sci. Numer. Simul., 2 (25), [8] W. Jiang, The existence of solutions to boundary value problems of fractional differential equations at resonance, Nonlinear Anal., 74 (2), [9]. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 4 (99), 22. [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North- Holland Mathematics Studies, Elsevier Science.V., Amsterdam, (26)., 2, 2.4 [2] J. C. Kuang, Applied Inequalities, Shandong Science and Technology Press, Jinan, China, (24). 3. [22] L. S. Leibenson, General problem of the movement of a compressible fluid in a porous medium, (Russian), ull. Acad. Sci. URSS. Sér. Géograph. Géophys., 9 (983), 7. [23] J. S. Leszczynski, T. laszczyk, Modeling the transition between stable and unstable operation while emptying a silo, Granular Matter, 3 (2), [24] R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Comput. Math. Appl., 59 (2), [25] N. I. Mahmudov, S. Unul, Existence of solutions of fractional boundary value problems with p-laplacian operator, ound. Value Probl., 25 (25), 6 pages. [26] J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, Topological methods for ordinary differential equations, Lecture Notes in Math., Springer, erlin, (993)., 2.2, 2.3 [27] M. C. Pélissier, L. Reynaud, Étude d un modèle mathématique d écoulement de glacier, (French), C. R. Acad. Sci. Paris Sér. A, 279 (974), [28] R. Rakkiyappan, J. Cao, G. Velmurugan, Existence and uniform stability analysis of fractional-order complexvalued neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst., 26 (25), [29] M. R užička, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, Springer- Verlag, erlin, (2). [3] E. Szymanek, The application of fractional order differential calculus for the description of temperature profiles in a granular layer, Adv. Theory Appl. Non-integer Order Syst., Springer Inter. Publ., Switzerland, (23). [3] Q. Zhang, Y. Wang, Z. Qiu, Existence of solutions and boundary asymptotic behavior of p(r)-laplacian equation multi-point boundary value problems, Nonlinear Anal., 72 (2), , 2. [32] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 5 (986),