Existence criterion for the solutions of fractional order p-laplacian boundary value problems

Transcription

1 Jafari et al. Boundary Value Problems 25 25:64 DOI.86/s R E S E A R C H Open Access Existence criterion for the solutions of fractional order p-laplacian boundary value problems Hossein Jafari,2*,DumitruBaleanu 3,4, Hasib Khan 5,6, Rahmat Ali Khan 5 and Aziz Khan 5 * Correspondence: jafarh@unisa.ac.za Department of Mathematical Sciences, University of South Africa, UNISA, P.O. Box 392, Pretoria, 3, South Africa 2 Department of Mathematical Sciences, University of Mazandaran, Babolsar, , Iran Full list of author information is available at the end of the article Abstract The existence criterion has been extensively studied for different classes in fractional differential equations FDEs through different mathematical methods. The class of fractional order boundary value problems FOBVPs with p-laplacian operator is one of the most popular class of the FDEs which have been recently considered by many scientists as regards the existence and uniqueness. In this scientific work our focus is on the existence and uniqueness of the FOBVP with p-laplacian operator of the form: D γ φ p D θ zt atfzt =, 3 < θ, γ 4, t [,, z = z, ηd α zt t= = z, ξz z =, < α <,φ p D θ zt t= ==φ p D θ zt t=,φ p D θ zt t= = φ 2 pd θ zt t=,φ p D θ zt t= =,where<ξ, η <andd θ, D γ, D α are Caputo s fractional derivatives of orders θ, γ, α, respectively. For this purpose, we apply Schauder s fixed point theorem and the results are checked by illustrative examples. Keywords: FOBVP with p-laplacian operator; fixed point theorems; existence and uniqueness Introduction Fractional calculus has widely been studied by scientists from the era of Leibniz to the present and has drawn the attention of mathematicians, engineers, and physicists in many scientific disciplines based on mathematical modeling, and it was found that the fractional order models are more precise in comparison with integer order models and, therefore, we can see many useful fractional order models in fluid flow, viscoelasticity, signal processing, and many other fields. For instance, see [ 6. In fractional calculus, the existence of a positive solution for FOBVP with p-laplacian operator has extensively attracted the attention of the scientific community. This side of the fractional calculus has a wide range of applications in day life problems and these were investigated for the existence of solutions by different mathematical tools. For instance, Lv [7, has studied existence results for m-point FOBVP with p-laplacian operator with the help of a monotone iterative technique and produced interesting results which were examined by two examples. Prasad and Krushna [8 studied FOBVPs with p-laplacian operator with the help of the Krasnosel skii and five functional fixed point theorems and checked their results by examples. Yuan and Yang [9 studied the existence of a positive solution for a q-fobvp with a p-laplacian operator using the upper and lower solution method through Schauder s fixed point theorem and the results were examined by examples. 25 Jafari et al. 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 Jafari etal. Boundary Value Problems 25 25:64 Page 2 of Some of the interesting results in the existence of positive solution for FOBVP with p-laplacian operator which raised our attention toward this project are, for instance, that Zhang et al. [ has contributed for positive solutions of an FOBVP with p-laplacian z = k τ, z, z, τ [, T, zt=z, z T=z, with the help of degree theory, and they also used the upper and lower solution method for this work. Xu and Xu in [ investigatedthep-laplacian equation for sign changing equations of the form z τ = f τ, z, z, τ,, z==z, 2 using the method of upper and lower solution method with the help of Leray-Schauder degree theory. Wang in [2 considered three solutions of z τ mτf τ, z, z =, <τ <, z =, lim z =. 3 τ In this paper we consider the FOBVP with p-laplacian of the form D γ φ p D θ zt atf zt =, 3<θ, γ 4, t [,, z = z, ηd α zt t= = z, ξz z =, < α <, φ p D θ zt t= == φ p D θ zt t=, 4 D θ zt t= = D θ zt t=, 2 D θ zt t= =, where D θ, D γ,andd α are Caputo s fractional derivatives of fractional order θ, γ, α where θ, γ 3, 4, and α,. φ p s= s p 2 s, p >,φ p = φ q, p =. We recall some basic q definitions and results. For α >,choosen =[αinthecaseα is not an integer and n = α in the case α is an integer. We recall the following definitions of a fractional order integral and a fractional order derivative in Caputo s sense, and some basic results of fractional calculus [2, 3. Definition [3 Forafunctionk :, R and γ >, fractional integral of order γ is defined by I γ kτ= Ɣγ τ τ s γ ks ds, 5 with the condition that the integral converges.

3 Jafari etal. Boundary Value Problems 25 25:64 Page 3 of Definition 2 [3 For γ > the left Caputo fractional derivative of order γ is defined by D γ h zt = Ɣn γ t where n is such that n <γ < n. a t x n γ f n x dx, 6 Lemma 3 [2 For μ, β >,the following relation holds: D μ t ν = Ɣν Ɣν μ tν μ, ν > nand D μ t l =,for l =,,2,...,n. Lemma 4 For Ht C,, the solution of the homogeneous FDE D ω Ht=is Ht=k k 2 t k 3 t 2 k n t n, k i R, i =,2,3,...,n. 7 Definition 5 [4 AconeP is solid in a real Banach space X if its interior is non-empty. Definition 6 [4 LetP be a solid cone in a real Banach space X, T : P P be an operator and < α <.ThenT is called θ-concave operator if Tku k α Tu for any < k < and u P. Lemma 7 [4 Assume that P is a normal solid cone in a real Banach space X, <α <, and T : P P is a α-concave increasing operator. Then T has only one fixed point in P. 2 Main results Lemma 8 For zt C[,, the FOBVP D θ zt=ht, 3 < θ 4, 8 z = z, ηd α z = z, ξz z =, <α <, 9 has a solution of the form zt= Gt, sht ds, where Gt, s= tη η Ɣ2 α t2 ξ 2 ξ tη η Ɣ2 α t2 ξ 2 ξ [ sθ α Ɣθ α ξ s θ ξɣ3 α Ɣθ 2 s θ Ɣθ 2 Ɣθ t sθ, <s t <, [ sθ α Ɣθ α ξ s θ ξɣ3 α Ɣθ 2 s θ, <t s <. Ɣθ 2 Proof Applying the operator I α on the differential equation in 4 and using Lemma 4,we obtain zt=i θ htc c 2 t c 3 t 2 c 4 t 3. 2

4 Jafari etal. Boundary Value Problems 25 25:64 Page 4 of Using boundary conditions 9 for the values of c, c 2, c 3, c 4, z = = z yield to c = =c 4,andbyξz z =, we have c 3 = ξ 2 ξ Iθ 2 h. Using ηd α z = z, we η obtain c 2 = η [I θ α ξ h ξɣ3 α Iθ 2 h. Substituting the values of c, c 2, c 3, c 4,in Ɣ2 α 2, we get zt=i θ ht [ tη η I θ α ξ h ξɣ3 α Iθ 2 h Ɣ2 α t 2 ξ 2 ξ Iθ 2 h. 3 The integral form is given as t t s θ hs ds zt= Ɣθ [ tη η Ɣ2 α ξ ξɣ3 α t 2 ξ s θ hs ds 2 ξ Ɣθ 2 s θ α hs ds Ɣθ α s θ hs ds Ɣθ 2 = Gt, shs ds. 4 Lemma 9 Let 3<θ, γ 4. For zt C[,, the FOBVP with p-laplacian D γ φ p D θ zt atf zt =, 3<θ, γ 4, t [,, z = z, ηd α zt t= = z, ξz z =, < α <, φ p D θ zt t= == φ p D θ zt t=, 5 D θ zt t= = D θ zt t=, 2 D θ zt t= =, has a solution of the form zt= Gt, sφ q Hs, xaxf zx dx ds, 6 where t xγ t2 Ɣγ Ɣγ 2 Ht, x= xγ, <x t <, t 2 Ɣγ 2 xγ, <t x <, 7 and Gt, s is given by. Applying the integral I γ on the differential equation in 5 and using Lemma 9, we obtain φ p D θ zt = I γ atf zt c c 2 t c 3 t 2 c 4 t 3. 8

5 Jafari etal. Boundary Value Problems 25 25:64 Page 5 of The boundary conditions φ p D θ z = φ p D θ z =φ p D α z =leadtoc = c 2 = c 4 =.From8andc = c 2 = c 4 =,wededuce D θ zt = I γ 2 atf zt 2c 3, 9 and the boundary condition φ p D θ z = 2 φ pd θ z yield c 3 = Ɣγ 2 xγ axf zx dx.consequently,8 takes the form φ p D θ zt = Ɣγ = t 2 Ɣγ 2 t x γ axf zx dx x γ axf zx dx Ht, xaxf zx dx. 2 The boundary value problem 5 reduces to the following problem: D α zt=φ q Ht, xaxf zx dx, z = z, ηd α z = y, ξz z =, < α <, 2 which, in view of Lemma 8, yields the required result, zt= Gt, sφ q Hs, xaxf zx dx ds. 22 Lemma Let 3<γ 4. The Green s function Ht, x is a continuous function and satisfies A Ht, x, Ht, x H, x, for t, x,, B Ht, x t γ H, x for t, x,. The continuity of Ht, x is ensured by its definition. For A, considering the case when <x t, we have t xγ t 2 Ht, x = xγ Ɣγ Ɣγ 2 = tγ x γ t t 2 xγ Ɣγ Ɣγ 2 tγ x γ Ɣγ t γ xγ Ɣγ 2 tγ x γ [ Ɣγ Ɣγ 2 >. 23 Ɣγ Ɣγ 2 In the case < t x, the Ht, s>isobvious.nowforht, x H, x, for t, x,, we have t xγ 2 t 2 Ɣγ Ɣγ 2 Ht, x= xγ, <x t, t 2t Ɣγ 2 24 xγ, <t x.

6 Jafari etal. Boundary Value Problems 25 25:64 Page 6 of For x, t,, such that < x t, from 24, we deduce t Ht, x = t xγ 2 Ɣγ 2t xγ Ɣγ 2 = tγ 2 x 2 γ t 2t xγ Ɣγ Ɣγ 2 tγ 2 x γ 2 Ɣγ 2 2tγ xγ Ɣγ 2 tγ 2 x γ 2 [ 2Ɣγ Ɣγ 2, 25 Ɣγ Ɣγ 2 from 25, we have Ht, x. In the case < t x, 24 implies that the relation t Ht, x is obvious, which implies that Ht, x is an increasing function w.r.t. t.hence t Ht, x H, x. Now for part B, using 7inthecase<x t, we proceed thus: Ht, x H, x = = t x γ t2 xγ Ɣγ Ɣγ 2 x γ Ɣγ xγ Ɣγ 2 t γ x t γ t2 xγ Ɣγ Ɣγ 2 x γ Ɣγ t γ x γ Ɣγ x γ Ɣγ Ɣγ = t γ [ xγ x γ Ɣγ xγ Ɣγ 2 tγ xγ Ɣγ 2 xγ Ɣγ 2 Ɣγ 2 xγ xγ Ɣγ 2 = t γ. 26 We can also prove the result for the case of < t x, this completes the proof. Assume that the following hold: J < H, xax dx <. J 2 Thereexist <δ <and ρ >such that f x δlφ p x,for x ρ,where<l φ p ϖ δ H, xax dx. J 3 Thereexistsb >,suchthatf x Mφ p x,foru > b, <M <φ p ϖ 2 q H, x ax dx. J 4 f z is non-decreasing with respect to z. J 5 Thereexists β <such that f Kz φ p K β f z, for any k <and <z <. 2. Existence and uniqueness of solutions Theorem Under the assumptions J and J 2, the FOBVP 4 has at least one positive solution. Proof Define K = {z C[, : zt ρ on [, } aclosedconvexset[5. Define an operator T : K C[, by T zt= Gt, sφ q Hs, xaxf zx dx ds. 27

7 Jafari etal. Boundary Value Problems 25 25:64 Page 7 of By Lemma 9, zt is a solution of the FOBVP with p-laplacian operator 4 ifandonlyif zt isafixedpointoft. The compactness of the operator T can easily be shown. Now consider Gt, s ds = t t sθ ds Ɣθ tη [ sθ α ds η Ɣ2 α Ɣθ α ξ sθ ds ξɣ3 αɣθ 2 = t sθ [ t Ɣθ tη s θ α η Ɣθ α Ɣ2 α ξ s θ 2 ξɣ3 α Ɣθ [ t θ Ɣθ t 2 ξ sθ ds 2 ξ Ɣθ 2 t 2 ξ s θ 2 2 ξ Ɣθ tη = η Ɣθ α Ɣ2 α ξ t 2 ξ ξɣ3 α Ɣθ 2 ξ Ɣθ [ Ɣθ η η Ɣθ α Ɣ2 α ξ ξ ξɣ3 α Ɣθ 2 ξ Ɣθ = ϖ, 28 where ϖ = Ɣθ η [ Ɣ2 α J 2 and Lemma,weobtain T zt = η Ɣθ α ξ ξɣ3 α Ɣθ ξ 2 ξ Gt, sφ q Hs, xaxf zx dx ds Gt, sφ q H, xaxlδφ p ρ dx ds Ɣθ. For any y K,using ϖφ q H, xaxlδφ p ρ dx = ϖφ q Lδ H, xax dx ρ ρ, 29 which implies T K K.BySchauder sfixedpointtheoremt has a fixed point in K. Theorem 2 Under the assumptions J, J 3 the FOBVP with p-laplacian operator 4 has at least one positive solution. Proof Let b >asgiveninj 3. Define F = max x b f x, then F f x for x b. In view of J 3, we have ϖ 2 q φ q Mφ q H, xax dx <. 3

8 Jafari etal. Boundary Value Problems 25 25:64 Page 8 of Choose b > b large enough such that ϖ 2 q φ q F φ q Mb φ q H, xax dx < b. 3 Define K 2 = {zt C[, : zt b on [, }, = {t [, : zt b}, 2 = {t [, : b < zt b }.Then 2 = [, and 2 = ϕ. Forz K,J 3 implies f zt Mφ p zt Mφ p b fort 2 and T zt = Gt, sφ q Hs, xaxf zx dx ds ϖφ q H, xaxf S yx dx H, xaxf zx dx S 2 ϖφ q F H, xax dx Mφ p b H, xax dx S S 2 ϖφ q F Mφ p b φ q H, xax dx. 32 From 3 and using the inequality a b r 2 r a r b r for any a, b, r >,with32 we have T zt ϖ 2 q φ q F φ q Mb φ q H, xax dx b, 33 thus T K 2 K 2. Hence by Schauder s fixed point theorem T has a fixed point z K 2, therefore, the FOBVP with p-laplacian operator 4 has at least one positive solution in K 2. Theorem 3 Assume that J, J 4, J 5 hold. Then the FOBVP with p-laplacian operator 4 has a unique positive solution. Proof Consider the set = {zt C[, : zt on[,}, where is a normal solid cone in C[, with = {zt C[, : zt>on[,}.lett : C[, be defined by 27, we prove that T is a θ-concave and increasing operator. For z, z 2 with z z 2 the assumption J 4 implies that the operator T is an increasing operator, i.e., T z t T z 2 t on t [,. With the help of f kz φ p k α f z, we have T kzt Gt, sφ q Ht, xφ p k α axf zx dx ds = k α Gt, sφ q Ht, xaxf zx dx ds = k α T zt, 34 which implies that T is α-concave operator and with the help of Lemma 7, the operator T has a unique fixed point which is the unique positive solution of the FOBVP with p-laplacian operator 4in.

9 Jafari etal. Boundary Value Problems 25 25:64 Page 9 of 3 Examples Example Consider the following boundary value problem: D 3.5 φ p D 3.5 zt tzt=, z==z, 2 D.5 zt t= z =, φ p D 3.5 zt t= == φ p D 3.5 zt t=, 2 z z =, 35 D 3.5 zt t= = 2 D 3.5 zt, D 3.5 zt t= =. Here we have α = β =3.5,α =.5,ξ = γ =, at=t, f zt = zt. By simple computation, 2 we obtain < L.992, choose L =.5,δ =,p = 2, and then the conditions J andj 2 are satisfied. Hence, by Theorem, the FOBVP with p-laplacian operator 35 has at least one positive solution. Example 2 For the following boundary value problem: D 3.5 φ p D 3.5 zt t 3 zt=, z==z,.d.5 zt t= z =,.z z =, φ p D 3.5 zt t= == φ p D 3.5 zt t=, 36 D 3.5 zt t= = 2 D 3.5 zt, D 3.5 zt t= =, we have α = β =3.5,ξ = η =.,at=t, f ut = 3 ut and by simple computation we get M < and thus by choosing M =4.,b =,andp = q =2,weseethat36 satisfy J andj 3. Hence by Theorem 2, the FOBVP with p-laplacian operator 36 has at least one positive solution. Example 3 The uniqueness of the solution for FOBVP with the p-laplacian operator; we have D 3.5 φ p D 3.5 zt tzt=, z==z, 3 D.5 zt t= z =, φ p D 3.5 zt t= == φ p D 3.5 zt t=, 3 z z =, 37 D 3.5 zt t= = 2 D 3.5 zt, D 3.5 zt t= =. We apply Theorem 3.In37, we have θ = γ =3.5,α =.5,ξ = γ =, at=t, f zt = zt 3 it is clear that 37 satisfies conditions J, J 4. Also considering β = 2,J 5issatisfied. Thus by Theorem 3, thefractionaldifferential equation 37 hasa uniquesolution. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All the authors read and approved the final manuscript.

10 Jafari etal.boundary Value Problems 25 25:64 Page of Author details Department of Mathematical Sciences, University of South Africa, UNISA, P.O. Box 392, Pretoria, 3, South Africa. 2 Department of Mathematical Sciences, University of Mazandaran, Babolsar, , Iran. 3 Department of Mathematics Computer Science, Cankaya University, Ankara, 653, Turkey. 4 Institute of Space Sciences, MG-23, Magurele-Bucharest, 769, Romania. 5 Department of Mathematics, University of Malakand, Dir Lower, P.O. Box 8, Chakdara, Khybarpukhtunkhwa, Pakistan. 6 Shaheed Benazir Bhutto University, Dir Upper, P.O. Box 8, Sheringal, Khybarpakhtunkhwa, Pakistan. Acknowledgements We are thankful to the referees and editor for their valuable comments and remarks. Received: 9 November 24 Accepted: 25 August 25 References. Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore 2 2. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York Oldhalm, KB, Spainer, J: The Fractional Calculus. Academic Press, New York Podlubny, I: Fractional Differential Equations. Academic Press, New York Sabatier, J, Agrawal, OP, Machado, JAT: Advances in Fractional Calculus. Springer, Berlin Lv, ZW: Existence results for m-point boundary value problems of nonlinear fractional differential equations with p-laplacian operator. Adv. Differ. Equ. 24, Prasad, KR, Krushna, BMB: Existence of multiple positive solutions for p-laplacian fractional order boundary value problems. Int. J. Anal. Appl. 6, Yuan, Q, Yang, W: Positive solution for q-fractional four-point boundary value problems with p-laplacian operator. J. Inequal. Appl. 24, Zhang, JJ, Liu, WB, Ni, JB, Chen, TY: Multiple periodic solutions of p-laplacian equation with one side Nagumo condition. J. Korean Math. Soc. 456, Xu, X, Xu, B: Sign-changing solutions of p-laplacian equation with a sub-linear nonlinearity at infinity. Electron. J. Differ. Equ. 23, Wang, B: Positive solutions for boundary value problems on a half line. Int. J. Math. Anal. 35, Herzallah, MAE, Baleanu, D: On fractional order hybrid differential equations. Abstr. Appl. Anal. 24, Article ID Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, Orland Han, Z, Lu, H, Sun, S, Yang, D: Positive solution to boundary value problem of p-laplacian fractional differential equations with a parameter in the boundary. Electron. J. Differ. Equ. 22, 23 22