Studies on Sturm-Liouville boundary value problems for multi-term fractional differential equations

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1 CJMS. 4()(25), 7-24 Caspian Journal of Mathematical Sciences (CJMS) University of Mazandaran, Iran ISSN: Studies on Sturm-Liouville boundary value problems for multi-term fractional differential equations Xiaohui Yang, Shengpign Chen 2, Yuji Liu 3 and Xingyuan Liu 4 Department of Mathematics, Guangdong Police College, China 2, 3 Guangdong University of Business Studies, China 4 Department of Mathematics, Shaoyang University, China Abstract. The Sturm-Liouville boundary value problem of the multi-order fractional differential equation D α p(t)d β u(t) q(t)f(t, u(t)) =, t (, ), a lim t t β u(t) b lim t t α p(t)d β u(t) =, c lim t u(t) d lim t p(t)d β u(t) = is studied. Results on the existence of solutions are established. The analysis relies on a weighted function space and a fixed point theorem. An example is given to illustrate the efficiency of the main theorems. Keywords: multi-order fractional differential equation, Sturm- Liouville boundary value problems, fixed-point theorem. 2 Mathematics subject classification: xxxx, xxxx; Secondary xxxx.. Introduction In recent years, many authors have studied the existence of solutions of boundary value problems for fractional differential equations with Riemann-Liouville fractional derivative or Caputo s fractional derivative, refer to Refs -6. Corresponding author: xiaohui yang@sohu.com Received: 8 February 23 Revised: 9 December 23 Accepted: 9 February 25 7

2 8 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu In 4,5, the authors considered the existence and multiplicity of positive solutions of the following boundary value problem of nonlinear fractional differential equation D α u(t) = f(t, u(t), D γ u(t)), < t < u() u () =, u() u () =, (.) where < α 2 is a real number, γ (, α, and D is the Caputo s fractional derivative of order, and f :,, ) R, ) is continuous. By means of a fixed-point theorem on cones, existence and multiplicity results of positive solutions of (.) were obtained. In 6, the authors studied the existence of solutions of the following more generalized boundary value problem of fractional differential equation D α u(t) = f(t, u(t)), t, T, u() u () = T g(s, y)ds, u(t ) u (T ) = (.2) T h(s, y)ds, where T >, < α 2 is a real number, and D α is the Caputo s fractional derivative, and f, g, h :, T R R is continuous. In 7, authors studied the solvability of the following two-point boundary value problem for fractional p Laplace differential equation D β φ p (D α u(t)) = f(t, u(t), D α u(t)), t,, u() =, c D α u() = c D α u(), (.3) where D denotes the Caputo fractional derivatives of order, < α, β, < α β 2, φ p (s) = s p 2 s is a p-laplacian operator, f :, R 2 R is continuous. By using the coincidence degree theory, the existence of solutions for above fractional boundary value problem was obtained. It is well known that properties of fractional differential equations with Riemann-Liouville fractional derivatives are different from those of fractional differential equations with Caputo s fractional derivatives 4. To compare with 4-6, it is meaningful to define and study Sturm- Liouville boundary value problems for fractional differential equations with Riemmann-Liouville fractional derivatives.

3 Sturm-Liouville boundary value problems for MFDEs 9 In this paper, we discuss the existence of solutions of the following boundary value problem (BVP for short) for the nonlinear fractional differential equation with multi-term Riemmann-Liouville fractional derivatives D α p(t)d β u(t) q(t)f(t, u(t)) =, t (, ), a lim t β u(t) b lim t α p(t)d β u(t) =, t t (.4) c lim u(t) d lim p(t)d β u(t) = t t where a, b, c, d, D is the Riemann-Liouville fractional derivative of order, < β, α with α β >, f :, R R satisfies the assumption (H) (see Section 2), p : (, ) (, ) satisfies the assumption (H2) (see Section 2), q : (, ), ) satisfies the assumption (H3) (see Section 2). We obtain the Green s function of BVP(.4) and establish the existence results of solutions of BVP(.4). An example is given to illustrate the efficiency of the main theorem. We clarify the structure of sequential fractional differential equations. It is the lack of commutativity of the fractional derivatives that represents an interesting complication that does not arise in the integerorder setting. In the problem (.4), we have a composition of two fractional derivatives, which gives rise to a sequential problem. Problem (.4) is a natural generalized form of ordinary differential equation p(t)φ(x (t)) f(t, x(t), x (t)) =. In problem (.4), we allow γ (β, α β). The remainder of this paper is as follows: in section 2, we present preliminary results. In section 3, the main theorems and their proof are given. In section 4, an example is given to illustrate the main results. 2. Preliminary results For the convenience of the readers, we firstly present the necessary definitions from the fractional calculus theory that can be found in the literatures 4,7. Denote the Gamma and Beta functions by Γ(α ) = s α e s ds, B(α 2, β 2 ) = ( x) α 2 x β 2 dx, α >, α 2 >, β 2 >. Definition 2.4. The Riemann-Liouville fractional integral of order α > of a function g : (, ) R is given by I α g(t) = Γ(α) provided that the right-hand side exists. (t s) α g(s)ds,

4 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu Definition The Riemann-Liouville fractional derivative of order α > of a continuous function g : (, ) R is given by d n D α g(t) = Γ(n α) dt n g(s) ds, (t s) α n where n < α n, provided that the right-hand side is point-wise defined on (, ). Lemma 2.4. Let n < α n, u C (, ) L (, ). Then I α D α u(t) = u(t) C t α C 2 t α 2 C n t α n, where C i R, i =, 2,... n, and for α, µ >, it holds that I α t µ = Γ(µ ) Γ(µ α ) tµα, D α t µ = Γ(µ ) Γ(µ α ) tµ α. Suppose that (H) f :, R R satisfies that (i) u f(t, t β u) is continuous for all t,, (ii) t f(t, t β u) is measurable on, for all (u, v) R 2 and (iii) for each r > there exists M r such that f ( t, t β u) Mr for all t,, u, v r. (H2) p : (, ) (, ) is continuous and there exists a number M M > such that p(t) for all t (, ). t β ( t) αβ (H3) q : (, ), ) is continuous and there exist k > and l > α with k l > such that q(t) t k ( t) l for all t (, ) (q may be singular at t = and t = ). (H4) m is a positive integer, there exist nonnegative functions ψ, φ L (, ), µ > such that ( ) q(t)f t, t β u ψ ψ(t) u µ, t (, ), u R. For our construction, we let x C(,, X = x : (, R the following limit exists lim t t β x(t) For x X, let x = sup t β x(t). t (,) Then X is a Banach space. Denote σ(s, t) = s. (t w) β (w s) α p(w) dw, = (bc ad)γ(β) acσ(, ).

5 Sturm-Liouville boundary value problems for MFDEs Lemma 2.2. Suppose that, x X and (H)-(H3) hold. Then u is a solution of D α p(t)d β u(t) q(t)f(t, x(t)) =, < t <, a lim t t β u(t) b lim t t α p(t)d β u(t) =, c lim t u(t) d lim t p(t)d β u(t) =, (2.) if and only if where G(t, s) = δ u(t) = G(t, s)q(s)f(t, x(s))ds, (2.2) δσ(s, t) adγ(β)σ(, t)( s) α acσ(, t)σ(s, ) bdγ(β) 2 t β ( s) α bcγ(β)t β σ(s, ), adγ(β)σ(, t)( s) α acσ(, t)σ(s, ) bdγ(β) 2 t β ( s) α bcγ(β)t β σ(s, ), (2.3) Proof. Since x X, we get r = x <. Then (H) implies that there exists a nonnegative number M r such that f(t, x(t)) = f ( t, t β t β x(t)) Mr for all t (, ). s t, t s. If u is a solution of BVP(2.), then we get p(t)d β u(t) = Γ(α) for some c R. Then D β u(t) = p(t) u(t) = t (t s) β Γ(β) s p(s) (t s) α q(s)f(s, x(s))ds c t α, t (, ) (t s) α t Γ(α) q(s)f(s, x(s))ds c α p(t), (s w α Γ(α) q(w)f(w, x(w))dw c s α p(s) ds c 2 t β, t α p(t)d β u(t) = t α (t s) α γ(α) q(s)f(s, x(s))ds c. Since t α (t s)α q(s)f(s, x(s))ds t α (t s)α s k ( s)) l M r ds t α (t s)αl s k M r ds = t kl M r ( w)αl w k dw, t,

6 2 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu and t β t (t s) β Γ(β) s p(s) (s w) α Γ(α) s q(w)f(w, x(w))du c α p(s) ds t β (t s)β M s s β ( s) αβ (s w)α w k ( w) l M r dwds c t β (t s)β M s α s β ( s) αβ ds M r M t β (t s)β s β (t s) α β s (s w)αl w k dwds c M t β (t s)β s αβ 2 (t s) α β ds M r M t β (t s) α s αβkl dsb(α l, k ) c M t β (t s) α s αβ 2 ds = M r M t kl B( α, α β k l)b(α l, k ) c M tb( α, α β ), t. From the boundary conditions in (2.), we get ac 2 bc =, c c Γ(β) s ( s)β p(s) (s w)α q(w)f(w, x(w))dwds sα ( s)β p(s) ds c 2 d Γ(α) ( s)α f(s, x(s))ds c =.

7 Sturm-Liouville boundary value problems for MFDEs 3 It follows that c = ad Γ(α) ( s)α q(s)f(s,x(s))ds bcad ac Γ(β) sα ( s)β p(s) ds ac ( s)β s p(s) (s w)α q(w)f(w,x(w))dwds bcad ac, sα Γ(β) ( s)β p(s) ds c 2 = bd Γ(α) ( s)α q(s)f(s,x(s),d γ x(s))ds bcad ac Γ(β) sα ( s)β p(s) ds bc ( s)β s p(s) (s w)α q(w)f(w,x(w))duds bcad ac. sα Γ(β) ( s)β p(s) ds Therefore, u(t) = t sα (t s)β p(s) ds s (t s)β p(s) (s w)α q(w)f(w, x(w))dwds ac ( s)β s p(s) (s w)α q(w)f(w,x(w))dwds (bcad)γ(β)ac sα ( s)β p(s) ds tβ Γ(α) bdγ(β) ( s)α q(s)f(s,x(s),d γ x(s))ds (bcad)γ(β)ac sα ( s)β p(s) ds bc ( s)β s p(s) (s w)α q(w)f(w,x(w))duds (bcad)γ(β)ac sα ( s)β p(s) ds adγ(β) ( s)α q(s)f(s,x(s))ds (bcad)γ(β)ac sα ( s)β p(s) ds. Since s (t s)β p(s) (s w)α q(w)f(w, x(w))dwds = u (t s)β (s w) α p(s) dsq(w)f(w, x(w))dw = s (t w) β (w s) α p(w) dwq(s)f(s, x(s))ds = σ(s, t)q(s)f(s, x(s))ds,

8 4 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu we get u(t) = σ(s, t)q(s)f(s, x(s))ds σ(,t) adγ(β) ( s)α q(s)f(s,x(s))dsac σ(s,)q(s)f(s,x(s))ds δ tβ bdγ(β) ( s)α q(s)f(s,x(s))dsbc σ(s,)q(s)f(s,x(s))ds Γ(α) δ = δ δσ(s, t) adγ(β)σ(, t)( s) α acσ(, t)σ(s, ) bdγ(β) 2 t β ( s) α bcγ(β)t β σ(s, ) q(s)f(s, x(s))ds δ t adγ(β)σ(, t)( s) α acσ(, t)σ(s, ) bdγ(β) 2 t β ( s) α bcγ(β)t β σ(s, ) q(s)f(s, x(s))ds = G(t, s)q(s)f(s, x(s))ds. Then u satisfies (2.2). Here G is defined by (2.3). It is easy to prove that u X. Reciprocally, let u satisfy (2.2). It is easy to show that u X, and a lim t t β u(t) b lim t t α p(t)d β u(t) =, c lim t u(t)d lim t p(t)d β u(t) =. Furthermore, we have D α p(t)d β u(t)q(t)f(t, x(t)) =. Then u X is a solution of BVP(2.). The proof is complete. Lemma 2.3. Suppose that >, and (H2) holds. Then t β G(s, t) L( s) α, s, t (, ), (2.4) where L = M B( α, α β ) adγ(β)m B( α, α β ) acm 2 B( α, α β )2 bdγ(β) 2 bcγ(β)m B( α, α β ).

9 Sturm-Liouville boundary value problems for MFDEs 5 Proof. From (2.3), one sees for < s t < that t β σ(s, t) = t β (t u) β (u s) α du s p(u) s = t β (t s u) β v α dv p(s v) = t β (t s) αβ ( w) β w α p(s w(t s)) dw t β (t s) αβ M ( w) β w α s w(t s) β dw s w(t s) αβ t β (t s) αβ M ( w) β w α wt β dw ( w)(t s) αβ = M ( w) α w αβ 2 dw = M B( α, α β ). Hence, for < s t <, we get t β G(s, t) t β σ(s, t) adγ(β)t β σ(, t) ( s) α act β σ(, t) σ(s, ) bdγ(β) 2 ( s) α bcγ(β)σ(s, ) M B( α, α β ) adγ(β)m B( α, α β )( s) α acm 2 B( α, α β ) 2 bdγ(β) 2 ( s) α bcγ(β)m B( α, α β ) M B( α, α β ) adγ(β)m B( α, α β ) acm 2 B( α, α β ) 2 bdγ(β) 2 bcγ(β)m B( α, α β ) ( s) α = L( s) α. For < t s <, we can prove similarly that The proof of (2.4) is completed. t β G(s, t) L( s) α. Now, we define the operator T on X, by (T x)(t) = G(t, s)q(s)f(s, x(s))ds. Lemma 2.4. Suppose that >, (H)-(H3) hold. Then T : X X is completely continuous.

10 6 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu Proof. To complete the proof, we must prove that T : X X is well defined, T is continuous and T maps any bounded subsets of X to relative compact sets of X 5. We divide the proof into four steps. Step. We prove that T : X X is well defined. For x X, we have sup t β x(t) = r <. t (,) Hence from (H) there exists a nonnegative function φ r L (, ) such that ( q(s)f(s, x(s)) = q(t)f t, t β t x(t)) β φr (t), t (, ). From (T x)(t) = t sα (t s)β p(s) ds s (t s)β p(s) (s u)α q(u)f(u, x(u))duds ac ( s)β s p(s) (s u)α q(u)f(u,x(u))duds (bcad)γ(β)ac sα ( s)β p(s) ds tβ Γ(α) bdγ(β) ( s)α q(s)f(s,x(s))ds (bcad)γ(β)ac sα ( s)β p(s) ds adγ(β) ( s)α q(s)f(s,x(s))ds (bcad)γ(β)ac sα ( s)β p(s) ds tβ bc ( s)β s p(s) (s u)α q(u)f(u,x(u))duds Γ(α) (bcad)γ(β)ac, sα ( s)β p(s) ds and (H2), we find that (T u) is continuous on (, and there exists the limit lim t t β (T x)(t). Hence T x X. Then T : X X is well defined. Step 2. We prove that T is continuous. Let {y n } n= be a sequence such that y n y in X. Then r = sup y n <. n=,,2, So there exists a nonnegative α well function φ r such that q(t)f(t, y n (t)) φ r (t)

11 Sturm-Liouville boundary value problems for MFDEs 7 holds for t (, ), n =,, 2,. Then for t (, ), we have from Lemma 2.3 that t β (T y n )(t) (T y )(t) = t β G(s, t)q(s)f(s, y n (s))ds t β G(s, t)q(s)f(s, y (s))ds t β G(s, t) q(s)f(s, y n (s), D β y n (t)) q(s)f(s, y (s)) ds L ( s)α q(s)f(s, y n (s)) f(s, y (s)) ds 2L ( s)α φ r (s)ds. Since f is continuous, by the Lebesgue dominated convergence theorem, we get T y n T y as n. Then T is continuous. Step 3. Let M = {y X : y r}. We prove that T M is bounded. It suffices to show that there exists a positive number L > such that for each x M = {y X : y r}, we have T y L. By the assumption, there exists a nonnegative α well function φ r L (, ) such that f(t, y(t)) φ r (t), t (, ). By the definition of T, for y M, we get t β (T y)(t) = L t β G(t, s)q(s)f(s, y(s))ds L( s) α q(s)f(s, y(s)) ds ( s) α φ r (s)ds. It follows that there exists L > such that T y L for each y {y X : y r}. Then T maps bounded sets into bounded sets in X. Step 4. Let M = {y X : y r}. Prove that t β T M is equicontinuous on,. Let t, t 2 (, with t < t 2 and x M = {y X : y r}. By the assumption, there exists a α well function φ r L (, ) such that q(t)f(t, x(t)) φ r (t), t (, ).

12 8 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu Note (T x)(t) = σ(s, t)q(s)f(s, x(s))ds σ(,t) adγ(β) ( s)α q(s)f(s,x(s))dsac σ(s,)q(s)f(s,x(s))ds δ tβ bdγ(β) ( s)α q(s)f(s,x(s))dsbc σ(s,)q(s)f(s,x(s))ds Γ(α) δ. For s (, t, one sees from (7) that t β 2 σ(s, t 2 ) t β σ(s, t ) = = = t β 2 2 s 2 s t β 2 s t β = (t 2 u) β (u s) α p(u) du t β s t β 2 (t 2 u) β t β 2 t (t u) β (u s) α p(u) t β 2 (t 2 s v) β t β 2 s t s (t s v) β v α p(vs) (t u) β (u s) α p(u) du (t u) β (u s) α p(u) du (t s v) β v α p(sv) dv dv du t β 2 (t 2 s) αβ ( w) β t β (t s w(t 2 s)) β (t 2 s) α w α p(sw(t 2 s)) t β (t s) M M t β (t s) M t2 s αβ t s ( w) β w α p(w(t s)s) dw t β 2 (t 2 s) αβ ( w) β t β (t s w(t 2 s)) β (t 2 s) α w α dw t β 2 (t 2 s) αβ t β t β 2 t2 s αβ t s ( t s t 2 s w w ( w) β w α dw t β (t s) αβ ) β ( w)β w α dw dw M 2 s t s ( w) β w α dw

13 Sturm-Liouville boundary value problems for MFDEs 9 uniformly in (, t as t 2 t. We consider three cases: Case. for s (, t, from (2.3) that t β G(s, t ) t β 2 G(s, t 2 ) = δ δt β 2 σ(s, t 2 ) t β σ(s, t ) adγ(β)( s) α t β σ(, t ) t β 2 σ(, t 2 ) acσ(s, )t β σ(, t ) t β 2 σ(, t 2 ) Case 2. For s t, t 2, we have t β G(t, s) t β 2 G(t 2, s) = δ δt β 2 σ(s, t 2 ) adγ(β)( s) α t β σ(, t ) t β 2 σ(, t 2 ) acσ(s, )t β σ(, t ) t β 2 σ(, t 2 ) Case 3. For s t 2,, we have t β G(t, s) t β 2 G(t 2, s) = δ acσ(s, )t β σ(, t ) t β 2 σ(, t 2 ) Then from t β G(t, s) L( s) α, we get t β (T y)(t ) t β 2 (T y)(t 2 ) = t β G(t, s) t β 2 G(t 2, s)q(s)f(s, y(s))ds 2 t t β G(t, s) t β 2 G(t 2, s)q(s)f(s, y(s))ds t 2 t β G(t, s) t β 2 G(t 2, s)q(s)f(s, y(s))ds t β G(t, s) t β 2 G(t 2, s) q(s) f(s, y(s)) ds 2 t t β G(t, s) t β 2 G(t 2, s) q(s) f(s, y(s)) ds t 2 t β G(t, s) t β 2 G(t 2, s) q(s) f(s, y(s)) ds adγ(β)( s) α t β σ(, t ) t β 2 σ(, t 2 )

14 2 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu t β G(t, s) t β 2 G(t 2, s) φ r (s)ds 2 t t β G(t, s) t β 2 G(t 2, s) φ r (s)ds t 2 t β G(t, s) t β 2 G(t 2, s) φ r (s)ds t β G(t, s) t β 2 G(t 2, s) φ r (s)ds 2L 2 t ( s) α φ r (s)ds t 2 t β G(t, s) t β 2 G(t 2, s) φ r (s)ds t δ δt β 2 σ(s, t 2 ) t β σ(s, t ) adγ(β)( s) α t β σ(, t ) t β 2 σ(, t 2 ) acσ(s, )t β σ(, t ) t β 2 σ(, t 2 ) φ r (s)ds 2L 2 t ( s) α φ r (s)ds δ t 2 adγ(β)( s) α t β σ(, t ) t β 2 σ(, t 2 ) acσ(s, )t β σ(, t ) t β 2 σ(, t 2 ) φ r (s)ds t δ δ t β 2 σ(s, t 2 ) t β σ(s, t ) adγ(β) t β σ(, t ) t β 2 σ(, t 2 ) acm t β σ(, t ) t β 2 σ(, t 2 ) ( s) α φ r (s)ds 2L 2 t ( s) α φ r (s)ds δ t 2 adγ(β) t β σ(, t ) t β 2 σ(, t 2 ) acm t β σ(, t ) t β 2 σ(, t 2 ) ( s) α φ r (s)ds δ δ t β 2 σ(s, t 2 ) t β σ(s, t ) t,t 2 s, adγ(β) t β σ(, t ) t β 2 σ(, t 2 ) acm t β σ(, t ) t β 2 σ(, t 2 ) ( s)α φ r (s)ds 2L 2 t ( s) α φ r (s)ds δ adγ(β) t β σ(, t ) t β 2 σ(, t 2 ) acm t β σ(, t ) t β 2 σ(, t 2 ) ( s)α φ r (s)ds

15 Sturm-Liouville boundary value problems for MFDEs 2 Therefore, t β T M is equicontinuous on,. From Steps 3 and 4, the Arzela-Ascoli theorem implies that T (M) is relatively compact for each bounded set M X. Thus, the operator T : X X is completely continuous. 3. Main theorems Now, we prove the main result. Denote Ψ(t) = and L be defined in Section 2. G(s, t)ψ (s)ds Theorem 3.. Suppose that (H)-(H4) hold. Then BVP(.4) has at least one solution if sup y, ) y (y Ψ ) µ > L ( s) α ψ(s)ds. (3.) Proof. From (3.), we can choose r > such that Denote For x M r, we have r (r Ψ ) µ > L M r = {x X : x Ψ r}. ( s) α ψ(s)ds. (3.2) x x Ψ Ψ r Ψ. Furthermore, q(t)f(t, x(t)) ψ (t) ψ(t) t β x(t) µ.

16 22 Xiaohui Yang, Shengpign Chen, Yuji Liu and Xingyuan Liu So (3.2) implies that t β (T x)(t) Ψ(t) = t β G(t, s)q(s)f(s, x(s)) ψ (s)ds L L L L r. t β G(t, s) q(s)f(s, x(s)) ψ (s) ds t ( s) α q(s)f(s, x(s)) ψ (s) ds ( s) α q(s)f(s, x(s)) ψ (s) ds ( s) α ψ (s) s β x(s) µ ds ( s) α ψ(s)ds x µ ( s) α ψ(s)dsr Ψ µ It follows that T x M r. So T (M r ) M r and Schauder fixed point theorem implies that T has a fixed point x M r. This x is a solution of BVP (.4). The proof is complete. Remark 3.. Since sup y, ) y (y Ψ ) µ =, µ =,, µ (, ), (µ ) µ µ µ Ψ µ, µ >, we have the following corollary: Corollary 3.. Suppose that (H)-(H3) hold. Then BVP(.3) has at least one solution if one of the followings conditions holds: (i) µ (, ). (ii) µ = and L ( s)α ψ(s)ds <. (iii) µ > and (µ )µ > L µ µ Ψ µ ( s)α ψ(s)ds. 4. An example In this section, we give an example to illustrate the main theorem.

17 Sturm-Liouville boundary value problems for MFDEs 23 Example 4.. Letλ >. Consider the following BVP D 2 t 2 D 2 u(t) ( t) 4 t 2 νt 3 4 u(t) µ =, t (,, < α < 2, lim t t 2 u(t) lim t D 2 u(t) =, lim t u(t) lim t D 2 u(t) =, (4.) where ν > is a constant. Corresponding to BVP(4.), we find that β = 3 4, α = 2, a = b = c = d =, p(t) = t 2 and f(t, x) = ( t) 4 t 2 νt 3 4 x µ. One can show that f satisfies (H) and p satisfies (H2) with M =, q satisfies (H3). It is easy to see that ) f (t, t 2 x = ( t) 4 t 3 2 νt 4 2 µ x µ. Choose ψ (t) = ( t) 4 t 2 and ψ(t) = νt µ. Then both ψ and ψ are nonnegative measurable functions. So (H4) holds with ψ, ψ and µ defined above. By direct computation, we find and σ(, ) = ( u) 3 4 u 2 du = 4 3, u 2 = (bc ad)γ(β) acσ(, ) = 2Γ(3/4) 2, L = Γ(/2)Γ(3//4) M B(/2, /4) Γ(3/4)B(/2, /4) B(/2, /4) 2 Γ(3/4) 2 Γ(3/4)B(/2, /4), Ψ = sup t 2 Ψ(t) = sup t 2 t (,) t (,) G(s, t)ψ (s)ds L ( s) 2 ψ (s)ds. By Corollary 3., we get BVP(4.) has at least one solution if one of the followings holds: (i) µ (, ). (ii) µ = and νl ( s) 2 s 4 ds = νlb(/2, 5/4) <. (iii) µ > and (µ ) µ µ µ > νl ( s) 2 s 4 ds L = νl µ B(/2, 5/4)B(3/4, /2) µ. µ ( s) 2 ψ (s)ds

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