Positive solutions for nonlocal boundary value problems of fractional differential equation

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1 Positive solutions for nonlocal boundary value problems of fractional differential equation YITAO YANG Tianjin University of Technology Department of Applied Mathematics No. 39 BinShuiWest Road, Xiqing District, Tianjin CHINA FANWEI MENG Qufu Normal University Department of Applied Mathematics No. 57 West Jingxuan Xilu, Qufu CHINA Abstract: In this paper, we investigate the existence, uniqueness and multiplicity of positive solutions for nonlocal boundary value problems of fractional differential equation. Firstly, we reduce the problem considered to the equivalent integral equation. Secondly, the existence and uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions. Thirdly, by means of some fixed point theorems, some results on the multiplicity of positive solutions are obtained. Key Words: Riemann-Liouville fractional derivative; Nonlocal boundary value problem; Fixed point theorem; successive iteration; Positive solution. Introduction In this paper, we investigate the positive solutions for nonlocal boundary value problems for fractional differential equations of the form D α ut ft, ut, < t <, u u, u uξ, 2 where D α is the Riemann-Liouville fractional derivative of order 2 < α 3 and, ξ, f satisfying H : >, < ξ <, > ; H 2 : f C [, ] [, [,. Fractional-order models are found to be more adequate than integer -order models in some real world problems. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, etc. involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For details and examples, see [-4] and the reference therein. Boundary value problems for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory. A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems. Multi-point nonlinear boundary value problems, which refer to a different family of boundary conditions in the study of disconjugacy theory [5] and take into account the boundary data at intermediate points of the interval under consideration, have been addressed by many authors, for example, see [6-25] and the references therein. The multi-point boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval under consideration dissipate or add energy according to the sensors located at intermediate points. In recent years, differential equations of fractional order have been addressed by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. Fractional differential equations appear naturally in various fields of science and engineering such as physics, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, etc. In consequence, fractional differential equations have been of great interest. For details, see [26-29] and the references therein. Recently, E-ISSN: Issue 2, Volume 2, December 23

2 two-point boundary value problems of fractional differential equation have been extensively studied in the literature. By the use of some fixed point theorems on cones, Zhang [3] obtain the existence of positive solutions for the equation D α ut ft, ut, < t <, with the boundary condition u u u u. Using the adomian decomposition method, Jafari [3] discuss the following fractional boundary value problem D α uxµfx, ux, < x <, < α 2, u, u c. Recently, in [34], using a fixed point theorem on cone, Bai established the eigenvalue intervals of the following problem C D α ut λhtfut, < t <, u u u, where 2 < α 3, C D α is the standard Caputo differentiation. Sun etc [33] studied the existence of solutions for the boundary value problem of fractional hybrid differential equations [ ] D α xt gt, xt, < t <, ft, xt x x, where < α 2, D α is the Riemann-Liouville fractional derivative. Also, the existence and multiplicity of solutions for three-point boundary value problems of fractional differential equations have been studied by some authors. In [32], Bai investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation where D α ut ft, ut, < t <, u, u uη, < α 2, < η α <, < η <, and D α is the standard Riemann-Liouville fractional derivative. Applying the Schauder fixed point theorem, in [35], an existence result is proved for the following system where D α ut ft, vt, D p vt, < t <, D α vt gt, ut, D q ut, < t <, u, v, u γuη, v γvη, < α, < 2, p, q, γ >, < η <, α q, p, γη α <, γη <, and D α is the standard Riemann-Liouville fractional derivative. We find that the above papers considered threepoint fractional boundary value problem for < α 2. To the best knowledge of the authors, no work has been done to get positive solution of the three-point fractional boundary value problem for 2 < α 3. The aim of this paper is to fill the gap in the relevant literatures. Such investigations will provide an important platform for gaining a deeper understanding of our environment. 2 Preliminaries and Lemmas The material in this section is basic in some sense. For the reader s convenience, we present some necessary definitions from fractional calculus theory and preliminary results. Definition [32] The Riemann-Liouville fractional integral operator of order q > for function y L R is defined as I q yt t t s q ysds, q >. Γq Definition 2 [32] The Riemann-Liouville fractional derivative of order q >, n < α < n, n N for function y is defined as D q yt d n t t s n α fsds, Γn q dt where the function ft has absolutely continuous derivatives up to order n. Lemma 3 [32] The equality D q Iq ft ft, q > holds for f L,. E-ISSN: Issue 2, Volume 2, December 23

3 Lemma 4 [32] Let q >, then the fractional differential equation has solutions D q ut ut c t α c 2 t α 2 c 3 t α 3 c n t α n, c i R, i, 2,, n, n [q]. Lemma 5 [32] Let q >, then I q Dq ut ut c t α c 2 t α 2 c 3 t α 3 c n t α n for some c i R, i, 2,, n, n [q]. Lemma 6 Suppose that H, H 2 hold. Then for ht C[, ], the following boundary value problem D α ut ht, 2 < α 3, < t <, 3 u u, u uξ, 4 has a unique solution where ut Gt, s Γα Gt, shsds tα Gξ, shsds, 5 t α s α t s α, s t, t α s α, t s. 6 Proof: We can apply Lemma 5 and Definition to reduce D α ut ht to an equivalent integral equation ut t t s α hsds c t α Γα c 2 t α 2 c 3 t α 3. By the boundary conditions of 4, there are c 2 c 3 and C [ Γα s α hsds ξ ] ξ s α hsds. Therefore, problem 3,4 has a unique solution t ut t s α hsds Γα [ Γα s α t α hsds ξ ] ξ s α t α hsds t t s α hsds Γα s α t α hsds Γα ξ α Γα Γα t Γα Γα ξ s α t α hsds ξ s α t α hsds [ s α t α t s α ]hsds t t α Γα t α s α t α hsds ξ s α ξ α hsds Γα ξ s α hsds t [ s α t α t s α ]hsds Γα s α t α hsds Γα t t α [ ξ Γα [ s α ξ α ξ s α ]hsds ] s α ξ α hsds ξ Gt, shsds tα Gξ, shsds. The proof is completed. Lemma 7 The function Gt, s defined by 6 satisfies Gt, s > for t, s,. Proof: We divide the proof into two cases. Case : when s t, t α s α t ts α > t s α, so, t α s α t s α >. E-ISSN: Issue 2, Volume 2, December 23

4 Case 2: when t s, The proof is completed. t α s α >. Theorem 8 Let P be a cone in a real Banach space E, P c {x P : x < c}, θ is a nonnegative continuous concave functional on P such that θx x, for all x P c, and P θ, b, d {x P : b θx, x d}. Suppose that T : P c P c is completely continuous and there exist positive constants < a < b < d c such that C : {x P θ, b, d : θx > b} and θt x> b for x P θ, b, d; C 2 : T x < a for x P a ; C 3 : θt x > b for x P θ, b, c with T x > d. Then T has at least three fixed points x, x 2 and x 3 with x < a, b < θx 2, a < x 3 with θx 3 < b. Remark 9 If d c, then condition C implies condition C 3. 3 Existence and uniqueness solution of the problem, 2 Let E C[, ] be endowed with the ordering u v if ut vt for all t [, ] and the imum norm u ut. Define the cone P E by t P {u E ut for t [, ]}. Lemma Let T : P E be the operator defined by T ut tα ξ α Gt, sfs, usds Gξ, sfs, usds. 7 Then T : P P is completely continuous. Proof: The operator T : P P is continuous in view of nonnegativeness and continuity of Gt, s and ft, u. Let Ω P be bounded. i.e., there exists a positive constant M > such that u M, for all u Ω. Let L ft, u, then for u Ω, we have t, um T ut tα ξ α L Gt, sfs, usds Gξ, sfs, usds t Gt, sds. Hence, T Ω is bounded. On the other hand, given ε >, setting δ min, Γαεα 2Lξ α ξ α 2 α α, then for each u Ω, t, t 2 [, ], t < t 2 and t 2 t < δ, one has T ut 2 T ut < ε. That is to say, T Ω is equicontinuity. In fact, T ut 2 T ut [Gt 2, s Gt, s]fs, usds tα 2 tα tα 2 t α Gξ, sfs, usds Gξ, sfs, usds [Gt 2, s Gt, s]fs, usds L Gt 2, s Gt, s ds L Gξ, sfs, usds Gξ, sdst α 2 t α L Gt 2, s Gt, s ds L ξ α ξ α Γα t α 2 t α. α In the following, we divide the proof into two cases. Case : δ t < t 2 <, we have t α 2 t α α δ 2 α t 2 t α δ α. Case 2: t < δ, t 2 < 2δ, we have t α 2 t α t α 2 < 2δ α. Consequently, we have t α 2 t α 2 α δ α. On the one hand, Gt, s is uniformly continuous in [, ] [, ] because of the continuity of Gt, s. So, for the given ε >, whenever t 2 t < δ, we have Gt 2, s Gt, s < ε 2L. E-ISSN: Issue 2, Volume 2, December 23

5 < On the other hand, < ε 2. L Γα L Γα ξ α ξ α t α 2 t α α ξ α ξ α 2 α δ α α Therefore, T ut 2 T ut < ε, which implies that {T u : u Ω} is equicontinuous. By means of the Arzela-Ascoli theorem, we have that T : P P is completely continuous. The proof is completed. Lemma Suppose that H, H 2 hold. Then u is a solution of problem, 2 if and only if u is a fixed point of T. Theorem 2 Assume that ft, u satisfies ft, u ft, v < λt u v, t [, ], u, v [,. 8 Then problem, 2 has a unique positive solution if s α s α λsds < Γα. 9 Proof: In the following, we will prove that T n is a contraction operator for n sufficiently large. Indeed, from 6, 7, for u, v P, we have the estimate, < T u T vt Gt, s fs, us fs, vs ds tα Gξ, s fs, us fs, vs ds u v Γα t [t α s α t s α ]λsds t t α s α λsds tα u v Γα ξ [ξ α s α ξ s α ]λsds ξ ξ α s α λsds < u v tα Γα ξα u v t α Γα s α λsds s α λsds. Let L sα λsds, we have Consequently, < < T u T vt < T 2 u T 2 vt L u v tα Γα. Gt, s fs, T us fs, T vs ds Gξ, s fs, T us fs, T vs ds L u v t Γ 2 α [t α s α tα t t s α ]s α λsds t α s α λss α ds Ltα u v ξ Γ 2 α 2 [ξ α s α ξ s α ]s α λsds ξ α s α λss α ds ξ L u v t α Γ 2 α ξ α Lξα u v t α ξ α 2 Γ 2 α LM u v tα Γ 2 α s α s α λsds LMξα u v t α 2 Γ 2 α LM u v t α Γ 2 α 2. s α s α λsds where M sα s α λsds. By induction, we have T n u T n vt LM n u v t α [Γα ] n. Taking into account that sα s α λsds < Γα, we have LM n [Γα ] n < 2 E-ISSN: Issue 2, Volume 2, December 23

6 for sufficiently large n, and therefore, T n u T n v u v, 2 which gives the proof. In the following, we will give the existence of solution of problem, 2. We set λ lim u then < λ. η η Theorem 3 Suppose t [,] ft, u, u λ ξ α Γαα, λ ξ α Γαα. 2 η <, 3 then problem, 2 has one positive solution. Proof: According to Lemma, we just need to verify that the operator T has a fixed point in E. From 3, choose λ > λ, such that η <. 4 In view of the definition of λ, there exists N > such that ft, u < λ u, t [, ], u N. 5 So we have ft, u λ u R, t [, ], u [,, 6 where R ft, u <. t [,], u N From 7, 3-6, we have T ut Gt, sfs, usds t [,] tα t [,] Gξ, sfs, usds Gt, s fs, us ds Gξ, s fs, us ds λ u R Gt, s ds Gξ, s ds λ t α u R Γα α ξα ξ α α λ u R Γα α ξα ξ α α λ ξ α u R Γαα λ ξ α Γαα u R η u R, 7 in which η is given by 2, R is a positive constant. 7 imply that T u η u R, u E, where η <. So, we can choose r > such that T P r P r, where P r {u P u r}. By using Schauder fixed point theorem, we have that the operator T has a fixed point in P r. Therefore, problem, 2 has one positive solution. Remark 4 If when u, ft, u u then condition 3 satisfied. η., t [, ], 8 4 Existence of three positive solutions of the problem, 2 We define the nonnegative continuous concave functional αu min ut, u P, t [ξ,] N Γαα ξα ξ α, N 2 Γαα ξα ξ α ξ α ξ α. Theorem 5 Assume that H, H 2 hold and there exist constants with < a < b < c d such that the following conditions hold A ft, u < N a, for t, u [, ] [, a], A 2 ft, u N 2 b, for t, u [ξ, ] [b, c], A 3 ft, u N c, for t, u [, ] [, c]. E-ISSN: Issue 2, Volume 2, December 23

7 Then problem, 2 has at least three positive solutions u, u 2 and u 3 with u t < a, t [,] b < min u 2t < u 2t c, t [ξ,] t [,] a < u 3t c, t [,] min u 3t < b. t [ξ,] Proof: We first show that C 2 of Theorem 8 holds. If u P c, then u c. By 6, 7 and A 3, we get T ut Gt, sfs, usds t [,] tα t [,] N c t [,] Gξ, sfs, usds Gt, s fs, us ds Gξ, s fs, us ds Gt, s ds Gξ, s ds t α t α N c Γα t [,] α ξ α ξ α α N c Γα α ξ α N c Γαα c, ξ α ξ α α which implies that T u c, u P c. Hence, T : P c P c. In view of Lemma, T : P c P c is completely continuous. In the same way, we can show that if A holds, then T P a P a. Hence condition C 2 of Theorem 8 is satisfied. Next we show that C of Theorem 8 holds. Choose u t bc 2, t [, ]. It is easy to see that u P, u bc 2 c, αu bc 2 > b. That is u {u P α, b, d : αu > b}. Moreover, if u P α, b, d, we have b ut c, for t [ξ, ]. By A 2 and 7, we have αt u min T ut t [ξ,] min Gt, sfs, usds t [ξ,] ξ ξα ξ α ξ α N 2 b ξ α Gξ, sfs, usds Gξ, sfs, usds Gξ, sds ξ α ξ α N 2 b b. αγα Hence condition C of Theorem 8 is satisfied. By Remark 9, condition C 3 of Theorem 8 is satisfied. To sum up,all the hypotheses of Theorem 8 are satisfied. By Theorem 8, problem, 2 has at least three positive solutions u, u 2 and u 3 with u t < t [,] a, b < min u 2t < u 2t c, a < t [ξ,] t [,] u 3t c, min u 3t < b. The proof is t [,] t [ξ,] completed. 5 Existence and uniqueness of successive iteration positive solutions of the problem, 2 Theorem 6 Assume that H, H 2 hold. If there exists a positive number Λ such that the following conditions hold B For each t [, ], ft, is nondecreasing; B 2 ft, Λ N Λ, for all t [, ]; B 3 ft, is not identically zero on any compact subinterval of[, ]. Then problem, 2 has at least one iteration solution u P Λ with u Λ, lim T n ω u or n lim T n υ u, where ω t Λ, υ,t [, ]. n Proof: We now show that T : P Λ P Λ. If u P Λ, then u Λ, we have ut u Λ. t [,] By B, B 2 and 7, we have T ut Gt, sfs, usds t [,] tα t [,] Gξ, sfs, usds Gt, s fs, us ds E-ISSN: Issue 2, Volume 2, December 23

8 N Λ t [,] Gξ, s fs, us ds Gt, s ds Gξ, s ds N Λ t α t α Γα t [,] α ξ α ξ α α N Λ Γα α ξ α N Λ Γαα Λ. ξ α ξ α α Thus we assert that T : P Λ P Λ. Let ω n T n ω, υ n T n υ, then ω n, υ n P Λ, n, 2,. Since T is completely continuous, we assert that {ω n } n, {υ n} n are sequentially compact sets. By B, B 2 and 7, we have ω t T ω t tα Gt, sfs, Λds Gξ, sfs, Λds N Λ Gt, sds t α N Λ Γα N Λ Γα Gξ, sds α ξ α ξ α α α ξ α N Λ Γαα Λ ω t. Similarly, ξ α ξ α α ω 2 t T ω t T ω t ω t, t [, ]. By induction, then ω n t T ω n t T ω n t ω n t, t [, ], n, 2,. Hence, there exists u P Λ, such that u lim n ω n. Applying the continuity of T and ω n T ω n, we get u T u. By B 3, we get Similarly, υ t T υ t Gt, sfs, υ sds tα Gt, sfs, ds tα > υ t. Gξ, sfs, υ sds Gξ, sfs, ds υ 2 t T υ t T υ t υ t, t [, ]. By induction, then υ n t T υ n t T υ n t υ n t, t [, ], n, 2,. Hence, there exists u P Λ, such that u lim υ n. n Applying the continuity of T and υ n T υ n, we get u T u. The proof is completed. 6 Conclusion This paper is motivated from some recent papers treating the boundary value problems for three-point fractional differential equations. We first give some notations, recall some concepts and preparation results. Second, we establish a general framework to find the existence and uniqueness solution of the problem, 2. Third, some sufficient conditions for the existence of three positive solutions of the problem, 2 are established by applying fixed point methods. In the last, the existence and uniqueness of successive iteration positive solutions of the problem, 2 are obtained. To the best of our knowledge, no work has been done to get positive solution of the three-point fractional boundary value problem for 2 < α 3. The aim of this paper is to fill the gap in the relevant literatures. Such investigations will provide an important platform for gaining a deeper understanding of our environment. Acknowledgements: The research was supported by Tianjin University of Technology and in the case of the first author, it was also supported by the National Natural Science Foundation of China No.778 and the Tianyuan Fund of Mathematics in China No The authors thank the referee for his/her careful reading of the paper and useful suggestions. The authors thank the referee for his/her careful reading of the paper and useful suggestions. E-ISSN: Issue 2, Volume 2, December 23

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