EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS

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1 Electronic Journal of Differential Equations, Vol. 28(28), No. 146, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) EXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS TINGTING QIU, ZHANBING BAI Abstract. In this article, we establish the existence of a positive solution to a singular boundary-value problem of nonlinear fractional differential equation. Our analysis rely on nonlinear alternative of Leray-Schauder type and Krasnoselskii s fixed point theorem in a cone. 1. Introduction Many papers and books on fractional calculus differential equation have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function and to problems of analyticity in the complex domain(see, for example [2, 8]). Moreover, Delbosco and Rodino [3] considered the existence of a solution for the nonlinear fractional differential equation D α u f(t, u), where < α < 1, and f : [, a] R R, < a is a given function,continuous in (, a) R. They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Recently, Zhang [11] considered the existence of positive solution for equation D α u f(t, u), where < α < 1, and f : [, 1] [, ) [, ), is a given continuous function, by using the sub-and super-solution method. In this article, we discuss the existence of a positive solution to boundary-value problems of the nonlinear fractional differential equation D α u(t) f(t, u(t)), < t < 1, u() u (1) u (), (1.1) where 2 < α 3, D α is the Caputo s differentiation, and f : (, 1] [, ) [, ) with lim f(t, ) (that is f is singular at t ). We obtain two results t about this boundary-value problem, by using Krasnoselskii s fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone. For existence theorems for fractional differential equation and applications, we refer the reader to the survey by Kilbas and Trujillo [6]. Concerning the definitions 2 Mathematics Subject Classification. 34B15. Key words and phrases. Boundary value problem; positive solution; singular fractional differential equation; fixed-point theorem. c 28 Texas State University - San Marcos. Submitted April 4, 28. Published October 24, 28. Supported by grants from the National Nature Science Foundation of China, and from the Mathematics Tianyuan Foundation of China. 1

2 2 T. QIU, Z. BAI EJDE-28/146 of fractional integral and derivative and related basic properties, we refer the reader to Samko, Kilbas, and Marichev [5] and Delbosco and Rodino [3]. 2. Preliminaries For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the literature. Definition 2.1. The Riemann-Liouville fractional integral of order α > of a function f : (, ) R is given by I α f(t) 1 (t s) α1 f(s)ds, provided that the right-hand side is pointwise defined on (, ). Definition 2.2. The Caputo fractional derivative of order α > of a continuous function f : (, ) R is given by D α f(t) 1 Γ(n α) f (n) (s) ds, (t s) αn1 where n 1 < α n, provided that the right-hand side is pointwise defined on (, ). Lemma 2.3 ([1]). Let n 1 < α n, u C n [, 1]. Then where C i R, i 1, 2,... n. Lemma 2.4 ([1]). The relation is valid in following case I α D α u(t) u(t) C 1 C 2 t C n t n1, I α ai β aϕ I αβ a ϕ Reβ >, Re(α β) >, ϕ(x) L 1 (a, b). Lemma 2.5. Given f C[, 1], and 2 < α 3, the unique solution of is where G(t, s) D α u(t) f(t), < t < 1, u() u (1) u (). u(t) G(t, s)f(s)ds { (α1)t(1s) α2 (ts) α1, s t 1, t(1s) α2 Γ(α1), t s 1. (2.1) (2.2) Proof. We may apply Lemma2.3 to reduce Eq.(2.1) to an equivalent integral equation u(t) If(t) α C 1 C 2 t C 3 t 2 for some C i R, i 1, 2, 3. By Lemma2.4 we have u (t) D 1 I α f(t) C 2 2C 3 t D 1 I 1 I α1 f(t) C 2 2C 3 t I α1 f(t) C 2 2C 3 t

3 EJDE-28/146 EXISTENCE OF POSITIVE SOLUTIONS 3 u (t) D 1 I α1 f(t) 2C 3 D 1 I 1 I α2 f(t) 2C 3 I α2 f(t) 2C 3. From u() u (1) u (), one has C 1, C 2 1 Therefore, the unique solution of problem (2.1) is u(t) 1 (t s) α1 1 f(s)ds [ ] t(1 s) α2 (t s)α1 f(s)ds G(t, s)f(s)ds (1 s) α2 f(s)ds, C 3. For G(t, s), since 2 < α 3, s t 1 we can obtain t t(1 s) α2 f(s)ds t(1 s) α2 f(s)ds (α 1)t(1 s) α2 t(1 s) α2 t(t s) α2 (t s) α1 obviously,we get G(t, s) >. The proof is complete. Lemma 2.6 ([7]). Let E be a Banach space, P E a cone, and Ω 1, Ω 2 are two bounded open balls of E centered at the origin with Ω 1 Ω 2. Suppose that A : P (Ω 2 \ Ω 1 ) P is a completely continuous operator such that either (i) Ax x, x P Ω 1 and Ax x, x P Ω 2, or (ii) Ax x, x P Ω 1 and Ax x, x P Ω 2 holds. Then A has a fixed point in P (Ω 2 \ Ω 1 ). Lemma 2.7 ([4]). Let E be a Banach space with C E closed and convex. Assume U is a relatively open subset of C with U and A : U C is a continuous compact map. Then either (1) A has a fixed point in U; or (2) there exists u U and λ (, 1) with u λau. 3. Main Results For our construction, we let E C[, 1] and u max u(t) which is a Banach t 1 space. We seek solutions of (1.1) that lie in the cone Define operator T : P P, by P {u E : u(t), t 1}. T u(t) G(t, s)f(s, u(s))ds. Lemma 3.1. Let < σ < 1, 2 < α 3, F : (, 1] R is continuous and lim F (t). Suppose that t tσ F (t) is continuous function on [, 1]. Then the function is continuous on [, 1]. H(t) G(t, s)f (s)ds

4 4 T. QIU, Z. BAI EJDE-28/146 Proof. By the continuity of t σ F (t) and H(t) G(t, s)sσ s σ F (s)ds It is easily to check that H(). The proof is divided into three cases: Case 1: t, t (, 1]. Since t σ F (t) is continuous in [, 1], there exists a constant M >, such that t σ F (t) M, for t [, 1]. Hence H(t) H() M t (α 1)t(1 s) α2 (t s) α1 s σ s σ F (s)ds t(1 s) α2 sσ s σ F (s)ds t(1 s) α2 sσ s σ F (s)ds t(1 s) α2 sσ s σ F (s)ds t(1 s) α2 sσ ds M Mt B(1 σ, α 1) M Γ(1 σ)mt where B denotes the beta function. Case 2: t (, 1), for all t (t, 1] H(t) H(t ) t M(t t ) Γ(1 σ)mtασ Γ(1 α σ) (t s) α1 s σ s σ F (s)ds (t s) α1 s σ s σ F (s)ds (α 1)t(1 s) α2 (t s) α1 s σ s σ F (s)ds M t(1 s) α2 sσ s σ F (s)ds (α 1)t (1 s) α2 (t s) α1 t(1 s) α2 sσ s σ F (s)ds t t (1 s) α2 s σ s σ F (s)ds (t t )(1 s) α2 s σ s σ F (s)ds (t s) α1 (t s) α1 M(t t ) (1 s) α2 M t (t s) α1 B(1 σ, α 1) Mtασ (t s) α1 tασ B(1 σ, α) (as t ) t (1 s) α2 s σ s σ F (s)ds s σ s σ F (s)ds (t s) α1 s σ s σ F (s)ds (t s) α1 s σ s σ F (s)ds s σ s σ F (s)ds (t s) α1 s σ s σ F (s)ds t [ (t s) α1 (t s) α1] B(1 σ, α) Mtασ B(1 σ, α)

5 EJDE-28/146 EXISTENCE OF POSITIVE SOLUTIONS 5 Γ(1 σ)m(t t ) Γ(1 σ)mtασ Γ(1 α σ) Γ(1 σ)mtασ Γ(1 α σ) (as t t ). Case 3: t (, 1], for all t [, t ). The proof is similar to that of Case 2; we omitted it. Lemma 3.2. Let < σ < 1, 2 < α 3, f : (, 1] [, ) [, ) is continuous and lim f(t, ), t tσ f(t, u(t)) is continuous function on [, 1] [, ), then the operator T : P P is completely continuous. Proof. For each u P, let T u(t) G(t, s)f(s, u(s))ds. By Lemma3.1 and the fact that f, G(t, s) are non-negative, we have T : P P. Let u P and u C, if u P and u u < 1, then u < 1 C C. By the continuity of t σ f(t, u(t)), we know that t σ f(t, u(t)) is uniformly continuous on [, 1] [, C]. Thus for all ɛ >, there exists δ > (δ < 1), such that t σ f(t, u 2 )t σ f(t, u 1 ) < ɛ, for all t [, 1], and u 1, u 2 [, C] with u 2 u 1 < δ. Obviously, if u u < δ, then u(t), u (t) [, C] and u(t) u (t) < δ, for all t [, 1]. Hence, t σ f(t, u(t)) t σ f(t, u (t)) < ɛ, for all t [, 1]. (3.1) u P, with u u < δ. It follows from (3.1) that T u T u max T u(t) T u (t) t 1 max < ɛ ɛ t 1 G(t, s)s σ s σ f(s, u(s)) s σ f(s, u (s)) ds G(t, s) (α 1)(1 s) α2 (1 s) α1 ɛ (1 s) α2 ɛ Γ(1 σ)ɛ B(1 σ, α 1). By the arbitrariness of u, T : P P is continuous. Let M P be bounded; i.e., there exists a positive constant b such that u b, for all u p. Since t σ f(t, u) is continuous in [, 1] [, ), let Then 1 T u(t) L max t 1,u M tσ f(t, u) 1, u M. G(t, s)s σ s σ f(s, u(s)) ds L Γ(1α)L Γ(ασ) Γ(1α)L Γ(1ασ) G(1, s) thus T u max T u(t) Γ(1 σ)l t 1. So, T (M) is equicontinuous. For ɛ > set { ɛ ɛ δ min, 2LΓ(1 σ), Γ(1α)L Γ(ασ) ɛ Γ(1α)L2α Γ(1ασ) Γ(1 σ)l ; }.

6 6 T. QIU, Z. BAI EJDE-28/146 For u M, t 1, t 2 [, 1], with t 1 < t 2, for < t 2 t 1 < δ, we have T u(t 2 ) T u(t 1 ) L L 2 1 [ L (t 2 t 1 ) 1 L (t 2 t 1 ) L G(t 2, s)f(s, u(s))ds G(t 1, s)f(s, u(s))ds [ G(t2, s) G(t 1, s) ] s σ s σ f(s, u(s))ds G(t 2, s) G(t 1, s) (α 1)t 2 (1 s) α2 (t 2 s) α1 (α 1)t 1 (1 s) α2 (t 1 s) α1 (1 s) α2 sσ ds (t 1 s) α1 ] 1 L (t 2 t 1 )Γ(1 σ) Case 1: t 1, t 2 < δ. s σ (1 s) α2 ds s σ (t 1 s) α1 ds 2 t 2 t 1 (t 2 s) α1 t 2 (1 s) α2 t 1 (1 s) α2 L 2 s σ (t 2 s) α1 ds LΓ(1 σ) Γ(1 α σ) (tασ 2 t1 ασ ). T u(t2 ) T u(t 1 ) t 2 Γ(1 σ) LΓ(1 σ) L Γ(1 α σ) tασ 2 Case 2: δ t 1 < t 2 < 1. < L δγ(1 σ) LΓ(1 σ) Γ(1 α σ) δ < ɛ. T u(t2 ) T u(t 1 ) δγ(1 σ) LΓ(1 σ) < L Γ(1 α σ) δασ LδΓ(1 σ) LΓ(1 σ)δασ 2LδΓ(1 σ) < < ɛ. Case 3: < t 1 < δ, t 2 < 2δ. T u(t2 ) T u(t 1 ) δγ(1 σ) LΓ(1 σ) < L Γ(1 α σ) 2α δ < ɛ. Therefore, T (M) is equicontinuous. The Arzela-Ascoli theorem implies that T (M) is compact. Thus, the operator T : P P is completely continuous.

7 EJDE-28/146 EXISTENCE OF POSITIVE SOLUTIONS 7 Theorem 3.3. Let < σ < 1, 2 < α 3, f : (, 1] [, ) [, ) is continuous and lim f(t, ), t tσ f(t, y) is continuous function on [, 1] [, ). Assume that there exist two distinct positive constant ρ, µ(ρ > µ) such that (H1) t σ f(t, ω) ρ Γ(ασ) Γ(1σ), for (t, ω) [, 1] [, ρ]; (H2) t σ f(t, ω) µ Γ(ασ) Γ(1σ), for (t, ω) [, 1] [, µ]. Then (1.1) has at least one positive solution. Proof. From Lemma 3.2 we have T : P P is completely continuous. We divide the proof into the following two steps. Step1: Let Ω 1 {u P : u < ασ1 ασ µ}, for u K Ω 1 and all t [, 1], we have u(t) ασ1 ασ µ. It follows from (H2) that Hence, T u(1) G(1, s)f(s, u(s))ds G(1, s)s σ s σ f(s, u(s))ds µ G(1, s) Γ(1 σ) [ (α 1)(1 s) α2 (1 s) α1 µ Γ(1 σ) [ (1 s) α1 (1 s) α1 µ Γ(1 σ) sσ ds [B(1 σ, α 1) B(1 σ, α 1) ] µ Γ(1 σ) α σ 1 α σ µ u. ] ] α σ 1 T u max T u(t) t 1 α σ µ u, for u P Ω 1. Step 2: Let Ω 2 {u P : u < ρ}, for u K Ω 2 and all t [, 1], we have u(t) ρ. By assumption (H1), T u(t) G(t, s)f(s, u(s))ds G(t, s)s σ s σ f(s, u(s))ds [ (α 1)t(1 s) α2 (t s) α1 ρ Γ(1 σ) t(1 s) α2 ] t sσ ds [ t(1 s) α2 (t s) α1 ] ρ Γ(1 σ) sσ ds [ t 1 ] ρ s σ (1 s) α2 ds Γ(1 σ)

8 8 T. QIU, Z. BAI EJDE-28/146 B(1 σ, α 1) ρ Γ(1 σ) Γ(1 σ) ρ Γ(1 σ) ρ. So T u(t) u, for u P Ω 2. Therefore, by (ii) of Lemma 2.6, we complete the proof. Theorem 3.4. Let < σ < 1, 2 < α 3, f : (, 1] [, ) [, ) is continuous and lim f(t, ), t tσ f(t, y) is continuous function on [, 1] [, ). Suppose the following conditions are satisfied: (H3) there exists a continuous, nondecreasing function ϕ : [, ) (, ) with t σ f(t, ω) ϕ(ω), for (t, ω) [, 1] [, ) (H4) there exists r >, with Then (1.1) has one positive solution. r ϕ(r) > Γ(ασ) Γ(1σ) Proof. Let U {u P : u < r}, we have U P. From Lemma 3.2, we know T : U P is completely continuous. If there exists u U, λ (, 1) such that By (H3) and (3.2), for t [, 1] we have u(t) λt u(t) λ ϕ( u ) ϕ( u ) G(t, s)s σ ϕ(u(s))ds u λt u, (3.2) G(t, s)f(s, u(s))ds G(t, s) G(1, s) (α 1)t(1 s) α2 (t s) α1 ϕ( u ) tb(1 σ, α 1) ϕ( u ) Γ(1 σ) ϕ( u ). Consequently, u ϕ( u ) Γ(1σ) Γ(ασ) ; namely, u Γ(1 σ) ϕ( u ). G(t, s)s σ s σ f(s, u(s))ds Combining (H4) and the above inequality, we have u r, which is contradiction with u U. According to Lemma 2.7, T has a fixed point u U, therefore, (1.1) has a positive solution.

9 EJDE-28/146 EXISTENCE OF POSITIVE SOLUTIONS 9 As an example, consider the fractional differential equation D α u(t) (t 1 2 )2 ln(2 u) t σ, < t < 1 u() u (1) u (), where < σ < 1, 2 < α 3. Then (3.3) has a positive solution. (3.3) References [1] Z. B. Bai; Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (25), [2] L. M. C. M. Campos; On the solution of some simple fractional differential equation, Int. J. Math. Sci. 13 (199), [3] D. Delbosco and L. Rodino; Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 24 (1996), [4] A. Granas, R. B. Guenther, J. W. Lee; Some general existence principle in the Caratheodory theory of nonlinear system, J. Math. Pure. Appl. 7 (1991), [5] A. A. Kilbas, O. I. Marichev, and S. G. Samko; Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland, [6] A. A. Kilbas and J. J. Trujillo; Differential equations of fractional order: methods, results and problems-i, Applicable Analysis 78 (21), [7] M. A. Krasnoselskii; Positive Solution of Operator Equation, Noordhoff,Groningen, [8] Y. Ling, S. Ding; A Class of Analytic Function Defined by Fractive Derivative, J. Math. Anal. Appl. 186 (1994), [9] K. S. Miller, B. Ross; An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley, New York, [1] S. G. Samko, A. A. Kilbas, O. I. Marichev; Fractional Integral and Derivative. Theory and Applications, Gordon and Breach, [11] S. Q. Zhang; The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2), Tingting Qiu College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao, 26651, China address: qiutingting19833@163.com Zhanbing Bai College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao, 26651, China address: zhanbingbai@163.com