SURVEY AND NEW RESULTS ON BOUNDARY-VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS

Electronic Journal of Differential Equation, Vol. 26 26, No. 296, pp. 77. ISSN: 72-669. URL: http://ejde.math.txtate.edu or http://ejde.math.unt.edu SURVEY AND NEW RESULTS ON BOUNDARY-VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS YUJI LIU Abtract. Firtly we prove exitence and uniquene of olution of Cauchy problem of linear fractional differential equation LFDE with two variable coefficient involving Caputo fractional derivative, Riemann-Liouville derivative, Caputo type Hadamard derivative and Riemann-Liouville type Hadamard fractional derivative with order q [n, n by uing the iterative method. Secondly we obtain exact expreion for piecewie continuou olution of the linear fractional differential equation with a contant coefficient and a variable one. Thee reult provide new method to tranform an impulive fractional differential equation IFDE to a fractional integral equation FIE. Thirdly, we propoe four clae of boundary value problem of ingular fractional differential equation with impule effect. Sufficient condition are given for the exitence of olution of thee problem. We allow the nonlinearity ptft, x in fractional differential equation to be ingular at t,. Finally, we point out ome incorrect formula of olution in cited paper. A new Banach pace and the compact propertie of ubet are proved. By etablihing a new framework to find the olution for impulive fractional boundary value problem, the exitence of olution of three clae boundary value problem of impulive fractional differential equation with multi-term fractional derivative are etablihed. Content. Introduction 2 2. Related definition 9 3. Preliminarie 2 3.. Baic theory for linear fractional differential equation 2 3.2. Exact piecewie continuou olution of LFDE 52 3.3. Preliminarie for BVP.7 7 3.4. Preliminarie for BVP.8 77 3.5. Preliminarie for BVP.9 82 3.6. Preliminarie for BVP. 88 4. Solvability of BVP.7. 92 2 Mathematic Subject Claification. 34A8, 26A33, 39B99, 45G, 34B37, 34B5, 34B6. Key word and phrae. Higher order ingular fractional differential ytem; impulive boundary value problem; Riemann-Liouville fractional derivative; Caputo fractional derivative; Riemann-Liouville type Hadamard fractional derivative; Caputo type Hadamard fractional derivative; fixed point theorem. c 26 Texa State Univerity. Submitted February 24, 25. Publihed November 8, 26.

2 Y. LIU EJDE-26/296 5. Application of main reult 2 5.. Impulive multi-point boundary value problem 8 5.2. Impulive Sturm-Liouville boundary value problem 2 5.3. Impulive anti-periodic boundary value problem 33 6. Comment on ome publihed article 38 6.. Corrected reult from [36 38 6.2. Corrected reult from [26 4 6.3. Corrected reult from [27 43 6.4. Corrected reult from [2, 3 45 6.5. Corrected reult from [5 47 6.6. Corrected reult from [67, 3, 33 5 6.7. Corrected reult from [28, 29, 35 58 7. Application of impulive fractional differential equation 68 Acknowledgment 72 Reference 72 8. Addendum poted February 3, 27 78. Introduction One know that the fractional derivative Riemann-Liouville fractional derivative, Caputo fractional derivative and Hadamard fractional derivative and other type ee [58 are actually nonlocal operator becaue integral are nonlocal operator. Moreover, calculating time fractional derivative of a function at ome time require all the pat hitory and hence fractional derivative can be ued for modeling ytem with memory. Fractional order differential equation are generalization of integer order differential equation. Uing fractional order differential equation can help u to reduce the error ariing from the neglected parameter in modeling real life phenomena. Fractional differential equation have many application ee [88, Chapter, and book [58, 57, 88, 94. In recent year, there have been many reult obtained on the exitence and uniquene of olution of initial value problem or boundary value problem for nonlinear fractional differential equation, ee [25, 27, 74, 8, 85, 86, 93, 8, 25, 38. Dynamic of many evolutionary procee from variou field uch a population dynamic, control theory, phyic, biology, and medicine. undergo abrupt change at certain moment of time like earthquake, harveting, hock, and o forth. Thee perturbation can be well approximated a intantaneou change of tate or impule.thee procee are modeled by impulive differential equation. In 96, Milman and Myhki introduced impulive differential equation in their paper [82. Baed on their work, everal monograph have been publihed by many author like Samoilenko and Peretyuk [95, Lakhmikantham et al. [6, Bainov and Simeonov [2, 2, Bainov and Covachev [9, and Benchohra et al. [28. Fractional differential equation were extended to impulive fractional differential equation, ince Agarwal and Benchohra publihed the firt paper on the topic [4 in 28. Since then many author [6, 39, 42, 55, 72, 68, 66, 84, 93, 7, 8, 24, 73, 7 tudied the exitence or uniquene of olution of impulive initial

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 3 or boundary value problem for fractional differential equation. For example, impulive anti-periodic boundary value problem ee [5, 6, 4, 69,, impulive periodic boundary value problem ee [5, 26, 5, impulive initial value problem ee [3, 38, 83, 98, two-point, three-point or multi-point impulive boundary value problem ee [2, 47, 6, 36, 6, 34, impulive boundary value problem on infinite interval ee [3. Feckan and Zhou [43 pointed out that the formula of olution for impulive fractional differential equation in [3,, 24, 29 i incorrect and gave their correct formula. In [6,, the author etablihed a general framework to find the olution for impulive fractional boundary value problem and obtained ome ufficient condition for the exitence of the olution to a kind of impulive fractional differential equation. In [3, the author illutrated their comprehenion for the counterexample in [43 and criticized the viewpoint in [43, 6,. Next, in [44, Feckan et al. expanded for the counterexample in [43 and provided further explanation in the paper. In a fractional differential equation, there exit two cae concerning the derivative: the fir cae i D α D α, i.e., the fractional derivative ha a ingle tart point t. The other cae i D α D α, i.e., the fractional derivative ha a t i multiple tart point t t i i N[, m. There have been many author concerning the exitence and uniquene of olution of boundary value problem of impulive fractional differential equation with multiple tart point t t i i N[, m. Recently, Wang [ conider the econd cae in which D α ha multiple tart point, i. e., D α D α. They tudied the exitence and uniquene of olution of t i the following initial value problem of the impulive fractional differential equation C D α ut ft, ut, t i t t i, t i, i N[, p, u j u j, j N[, n, u j t i I ji ut i, i N[, p, j N[, n,. where α n, n with n being a poitive integer, C D α repreent the tandard t i Caputo fractional derivative of order α, N[a, b {a, a,..., b} with a, b being integer, t < t < < t p < t p, I ji CR, R i N[, p, j N[, n, f : [, T R R i a continuou function. Henderon and Ouahab [5 tudied the exitence of olution of the following problem and C D α ut ft, ut, t i t t i, t i, i N[, p, u j u j, j N[,, u j t i I ji ut i, i N[, p, j N[,, C D α ut ft, ut, t t t i, t i, i N[, p, i u j u j b, j N[,, u j t i I ji ut i, i N[, p, j N[,,

4 Y. LIU EJDE-26/296 where α, 2, b >, t < t < < t p < t p b, f : [, b R R, I ji : R R are continuou function. Reader hould alo refer [4. Zhao and Gong [32 tudied exitence of poitive olution of the nonlinear impulive fractional differential equation with generalized periodic boundary value condition C D q ut ft, ut, t, T \ {t t,..., t p }, i ut i I i ut i, u t i J i ut i, i N[, p, αu βu, αu βu,.2 where q, 2, C D q repreent the tandard Caputo fractional derivative of t i order q, α > β >, t < t < < t p < t p, N[a, b {a, a,..., b} with a, b being integer,i i, J i C[,, [, i N[, p, f : [, [, [, i a continuou function. Wang, Ahmad and Zhang [2 tudied the exitence and uniquene of olution of the periodic boundary value problem for nonlinear impulive fractional differential equation C D α ut ft, ut, t, T \ {t t,..., t p }, i ut i I i ut i, u t i Ii ut i, i N[, p, u θ ut but, u θ ut,.3 where α, 2, C D α repreent the tandard Caputo fractional derivative of t i order α, θ, 2, N[a, b {a, a,..., b} with a, b being integer, t < t < < t p < t p T, I i, Ii CR, R i N[, p, f : [, T R R i a continuou function. Zou and Feng, Li and Shang [3, 64, 39 tudied the exitence of olution of the nonlinear boundary value problem of fractional impulive differential equation C D α xt wtft, xt, x t, t, \ {t t,..., t p }, i xt i I i xt i, x t i J i xt i, i N[, p, α x β u g x, α 2 x β 2 x g 2 x,.4 where α, 2, C D α repreent the tandard Caputo fractional derivative of t i order α, α, α 2, β, β 2 R with α α 2 α β 2 α 2 β, N[a, b {a, a,..., b} with a, b being integer, t < t < < t p < t p, I i, J i CR, R i N[, p, f : [, T R 2 R i continuou, w : [, [, i a continuou function, g, g 2 : P C, R are two continuou function.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 5 Liu and Li [7 invetigated the exitence and uniquene of olution for the nonlinear impulive fractional differential equation C D α ut ft, ut, u t, u t, t t t i, t i, i N[, p, i T u λ ut ξ q, u, u, u d, T u λ 2 u T ξ 2 q 2, u, u, u d, T u λ 3 u T ξ 3 q 3, u, u, u d, ut i A i ut i, u t i B i ut i, u t i C i ut i,.5 for i N[, p, where α 2, 3, C D α repreent the tandard Caputo fractional t i derivative of order α, N[a, b {a, a,..., b} with a, b being integer, t < t < < t p < t p T, λ i, ξ i R i, 2, 3 are contant, A i, B i, C i CR, R i N[, p, f : [, T R 3 R i continuou. Recently, in [32, to extend the problem for impulive differential equation u t λut ft, ut, u ut, u t i I i uti, in[, p to impulive fractional differential equation, the author tudied the exitence and the multiplicity of olution for the Dirichlet boundary value problem for impulive fractional order differential equation C DT α C D α xt atxt λft, xt, t [, T, t t i, i N[, m, C D α T C D α xt i µi i xt i, i N[, m, x xt,.6 where α /2,, λ, µ > are contant, N[a, b : {a, a,..., b with a b, t < t < < t m < t m T, f : [, T R R i a continuou function, I i : R Ri N[, m are continuou function, C D α or C D α T i the tandard left or right Caputo fractional derivative of order α, a C[, T and there exit contant a, a 2 > uch that a at a 2 for all t [, T, x tti lim t t xt lim i t t xt xt i xt i and xt i, xt i repreent i the right and left limit of xt at t t i repectively, a, b, x a contant with a b. One know that the boundary condition ax bxt x become x xt x a when a b, that i o called nonhomogeneou periodic type boundary condition. For impulive fractional differential equation whoe derivative have ingle tart point t, there ha been few paper publihed. In [9, author preented a new method to converting the impulive fractional differential equation with the Caputo fractional derivative to an equivalent integral equation and etablihed exitence and uniquene reult for ome boundary value problem of impulive fractional differential equation involving the Caputo fractional derivative with ingle tart point. The exitence and uniquene of olution of the following initial or boundary value problem were dicued in [9: C D α xt ft, xt, t, \ {t,..., t p }, xt i I i xt i, x t i J i xt i, i N[, p, x x, x x ;

6 Y. LIU EJDE-26/296 C D α xt ft, xt, t, \ {t,..., t p }, xt i I i xt i, x t i J i xt i, i N[, p, x φx x, x x ; C D β xt ft, xt, t, \ {t,..., t p }, xt i I i xt i, i N[, p, ax bx, C D α xt ft, xt, t, \ {t,..., t p }, and xt i I i xt i, x t i J i xt i, i N[, p, ax bx x, cx dx x ; C D α xt ft, xt, t, \ {t,..., t p }, xt i I i xt i, x t i J i xt i, i N[, p, x axξ x bxη, where α, 2, β,, D i the Caputo fractional derivative with order and ingle tart point t, f : [, R R, I i, J i : R R are continuou function, a, b, c, d, x, x R are contant, φ : P C, R i a functional. We oberved that in the above-mentioned work, the author all require that the fractional derivative are the Caputo type derivative, the nonlinear term f and the impule function are continuou. It i eay to ee that thee condition are very retrictive and difficult to atify in application. To the author knowledge, there ha been no paper publihed dicued the exitence of olution of boundary value problem of impulive fractional differential equation involving other fractional derivative uch a the Riemann-Liouville fractional derivative, Hadamard fractional derivative. In thi paper, we tudy the exitence of olution of four clae of impulive boundary value problem of ingular fractional differential equation. The firt cla i the impulive Dirichlet type integral boundary value problem RL D β xt λxt ptft, xt, lim t2 β xt t φg, xd, x a.e., t t i, t i, i N[, m, ψh, xd, lim t t i 2 β xt It i, xt i, RL D β t t xt i Jt i, xt i, i for i N[, m, where.7.a < β < 2, λ R, RL D β i the Riemann-Liouville fractional derivative of order β,.a2 m i a poitive integer, t < t < t 2 < < t m < t m, N[a, b {a, a, a 2,..., a n} with a, b being integer and a b,.a3 φ, ψ :, R are meaurable function,.a4 p :, R i continuou and there exit number k > and l max{ β, 2 k, uch that pt t k t l for all t,,.a5 f, G, H defined on, R are impulive II-Carathéodory function, I, J : {t i : i N[, m} R R i a dicrete II-Carathéodory function.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 7 The econd cla i the impulive mixed type integral boundary value problem C D β xt λxt ptft, xt, lim xt t φg, xd, x a.e., t t i, t i, i N[, m, ψh, xd, xt i It i, xt i, x t i Jt i, xt i, i N[, m,.8 where.a6 < β < 2, λ R, C D β i the Caputo fractional derivative of order β, m, t i, N[a, b atifie.a2, φ, ψ :, R atify.a3,.a7 p :, R i continuou and there exit number k > β and l max{ β, β k, uch that pt t k t l for all t,,.a8 f, G, H defined on, R are impulive I-Carathéodory function, I, J : {t i : i N[, m} R R are dicrete I-Carathéodory function. We emphaize that much work on fractional boundary value problem involve either Riemann-Liouville or Caputo type fractional differential equation ee [8, 9,, 6. Another kind of fractional derivative that appear ide by ide to Riemann- Liouville and Caputo derivative in the literature i the fractional derivative due to Hadamard introduced in 892 [48, which differ from the preceding one in the ene that the kernel of the integral in the definition of Hadamard derivative contain logarithmic function of arbitrary exponent. Recent tudie can be een in [33, 34, 35. Thirdly we tudy the following impulive periodic type integral boundary value problem of ingular fractional differential ytem RLH D β xt λxt ptft, xt, lim t log t2 β xt xe a.e., t t i, t i, i N[, m, e lim RLH D β t xt RLH D β xe lim log t log t i 2 β xt It i, xt i, t t i φg, xd, e ψh, xd, RLH D β xt i Jt i, xt i,.9 for i N[, m, where.a9 < β < 2, λ R, RLH D β i the Hadamard fractional derivative of order β,.a m i a poitive integer, t < t < t 2 < < t m < t m e, φ, ψ :, e R are meaurable function, p :, e R i continuou and atifie pt log t k log t l with k >, l, 2 k l >, N[a, b {a, a, a 2,..., a n} with a, b being integer and a b,.a f, G, H defined on, er are impulive III-Carathéodory function, I, J : {t i : i N[, m} R R are dicrete III-Carathéodory function.

8 Y. LIU EJDE-26/296 Finally we tudy the following impulive Neumann type integral boundary value problem of ingular fractional differential ytem CH D β xt λxt ptft, xt, a.e., t t i, t i, i N[, m, t d dt xt t t d dt xt te lim t t i e e φg, xd, ψh, xd, xt xt i It i, xt i,. for i N[, m, where lim t d t t dt xt t d dt xt tti Jt i, xt i, i.a2 < β < 2, λ R, CH D β i the Caputo type Hadamard fractional derivative of order β, t d dt xt tx t,.a3 m, t i, N[a, b atify.a, φ, ψ :, e R are meaurable function, p :, e R i continuou and atifie pt log t k log t l with k >, l, β k l >,.A4 f, G, H defined on, e R are impulive I-Carathéodory function, I, J : {t i : i N[, m} R R are dicrete I-Carathéodory function. A function x :, R i called a olution of BVP.7 or of BVP.8 if x ti,t ii,, j N[, m i continuou, the limit below exit lim t t i 2 β xt, i N[, m, or lim t t i t t i xt i N[, m and x atifie.7 or.8. A function x :, e R i called a olution of BVP.9 or of BVP. if x ti,t ii N[, m i continuou, the limit below exit lim log t 2 β xt, i N[, m, or lim t t t i i t t i xt, i N[, m and x atifie.9 or.. To obtain olution of a boundary value problem of fractional differential equation, we firtly define a Banach pace X, then we tranform the boundary value problem into a integral equation and define a nonlinear operator T on X by uing the integral equation obtained, finally, we prove that T ha fixed point in X. The fixed point are jut olution of the boundary value problem. Three difficultie occur in known paper: one i how to tranform the boundary value problem into a integral equation; the other one i how to define and prove a Banach pace and the completely continuou property of the nonlinear operator defined; the third one i to chooe a uitable fixed point theorem and impoe uitable growth condition on function to get the fixed point of the operator. To the bet of the author knowledge, no one ha tudied the exitence of trong weak or weak olution of BVP.7.. Thi paper fill thi gap. Another purpoe of thi paper i to illutrate the imilarity and difference of thee three kind of fractional differential equation. We obtain reult on the exitence of at leat one olution for BVP.7.. For implicity we only conider the left-ided operator here. The right-ided operator can be treated imilarly. For clarity and

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 9 brevity, we retrict our attention to BVP with one impule, the difference between the theory of one or an arbitrary number of impule i quite imilar. The remainder of thi paper i organized a follow: in Section 2, we preent related definition. In Section 3 ome preliminary reult are given one purpoe i to etablih exitence and uniquene of continuou olution of linear fractional differential equation Subection 3., the econd purpoe i to get exact expreion of piecewie continuou olution of the linear fractional differential equation with a contant coefficient and a variable force term Subection 3.2, the third purpoe i to prove preliminary reult for etablihing exitence reult of olution of.7.in Subection 3.3, 3.4, 3.5 and 3.6, repectively, we tranform them into correponding integral equation and define completely continuou nonlinear operator. In Section 4, the main theorem and their proof are given we etablih exitence reult for olution of BVP.7.. In Section 5, we preet application of theorem obtained in Subection 3.2, the olvability of multi-point boundary value problem, Sturm-Liouville boundary value problem and anti-periodic boundary value problem for fractional differential equation with impule effect are dicued, repectively. In Section 6, ome mitake happened in cited paper are howed. Corrected expreion of olution are given. Finally, in Section 7, we urvey ome example and application of fractional differential equation in variou field: population dynamic, control theory, phyic, biology, medicine. 2. Related definition For convenience of the reader, we firtly preent the neceary definition from the fractional calculu theory. Thee definition and reult can be found in [58, 88, 94. Let the Gamma function, Beta function and the claical Mittag-Leffler pecial function be x α e x dx, Bp, q E δ,σ x k x k Γδk σ x p x q dx, repectively for α >, p >, q >, δ >, σ >. We note that E δ,δ x > for all x R and E δ,δ x i trictly increaing in x. Then for x > we have E δ,σ x < E δ,σ Γσ < E δ,σx. Definition 2. [58. Let c R. The Riemann-Liouville fractional integral of order α > of a function g : c, R i I α c gt provided that the right-hand ide exit. c t α gd, Definition 2.2 [58. Let c R. The Riemann-Liouville fractional derivative of order α > of a function g : c, R i d n RL Dc α gt Γn α dt n c g d, t α n where α < n < α, i.e., n α, provided that the right-hand ide exit.

Y. LIU EJDE-26/296 Definition 2.3 [58. Let c R. The Caputo fractional derivative of order α > of a function g : c, R i C D α c gt Γn α c g n d, t α n where α < n < α, i.e., n α, provided that the right-hand ide exit. Definition 2.4 [58. Let c >. The Hadamard fractional integral of order α > of a function g : [c, R i H I α c gt provided that the right-hand ide exit. c log t α g d, Definition 2.5 [58. Let c >. The Hadamard fractional derivative of order α > of a function g : [c, R i RLH D α c gt Γn α t d dt n c log t n α g d, where α < n < α, i.e., n α, provided that the right-hand ide exit. Definition 2.6 [53. Let c >. The Caputo type Hadamard fractional derivative of order α > of a function g : [c, R i CH D α c gt Γn α c log t n α d d n g d, where α < n α, i.e., n α, provided that the right-hand ide exit. Definition 2.7. We call F : m i t i, t i R R an impulive I-Carathéodory function if it atifie i t F t, u i meaurable on t i, t i i N[, m for any u R, ii u F t, u are continuou on R for almot all t t i, t i i N[, m, iii for each r > there exit M r > uch that F t, u M r, t t i, t i, u r, i N[, m. Definition 2.8. We call F : m i t i, t i R R an impulive II-Carathéodory function if it atifie i t F t, t t i β 2 u i meaurable on t i, t i i N[, m for any u R, ii u F t, t t i β 2 u are continuou on R for almot all t t i, t i i N[, m, iii for each r > there exit M r > uch that F t, t t i β 2 u M r, t t i, t i, u r, i N[, m. Definition 2.9. We call F : m i t i, t i R R an impulive III-Carathéodory function if it atifie i t F t, log t t i β 2 u i meaurable on t i, t i i N[, m for any u R, ii u F t, log t t i β 2 u are continuou on R for all t t i, t i i N[, m, iii for each r > there exit M r > uch that F t, log t t i β 2 u M r, t t i, t i, u r, i N[, m.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES Definition 2.. We call I : {t i : i N[, m} R R a dicrete I-Carathéodory function if it atifie i u It i, u i N[, m are continuou on R, ii for each r > there exit M r > uch that It i, u M r, u r for i N[, m. Definition 2.. We call I : {t i : i N[, m} R R a dicrete II-Carathéodory function if it atifie i u It i, t i t i β 2 u i N[, m are continuou on R, ii for each r > there exit M r > uch that It i, t i t β 2 i u M r, u r for i N[, m. Definition 2.2. We call I : {t i : i N[, m}r R a dicrete III-Carathéodory function if it atifie i u It i, log t i log t i β n u i N[, m are continuou on R, ii for each r > there exit M r > uch that It, log ti t i β n u M r, u r for i N[, m. Definition 2.3 [79. Let E and F be Banach pace. A operator T : E F i called a completely continuou operator if T i continuou and map any bounded et into relatively compact et. Suppoe that n α < n. The following Banach pace are ued: Let a < b be contant. Ca, b denote the et of continuou function on a, b with lim t a xt exiting, and the norm x up xt. t a,b Let a < b be contant. C n α a, b the et of continuou function on a, b with lim t a t a n α xt exiting, the norm x n α up t a,b t a n α xt. Let < a < b. LC n α a, b denote the et of all continuou function on a, b with the limit lim t a log t a n α xt exiting, and the norm x up log t t a,b a n α xt. For a poitive integer m let N[, m {,, 2,..., m}, with t < t < < t m < t m. The following Banach pace are alo ued in thi paper: P m C n α, {x :, R : x ti,t i C n αt i, t i : i N[, m} with the norm x x PmC n α { max up t t i,t i } t t i n α xt : i N[, m. P m C, {x :, R : x ti,t i Ct i, t i : i N[, m} with the norm { } x x PmC, max xt : i N[.m. up t t i,t i For a poitive integer m let N[, m {,, 2,..., m}, with t < t < < t m < t m e. We alo ue the Banach pace { LP m C n α, e x :, e R : x Ct ti,t i i, t i, i N[, m,

2 Y. LIU EJDE-26/296 with the norm with the norm x x LPmC n α and lim log t } n α xt exit for i N[, m t t t i i { max up t t i,t i log t } n α xt, i N[, m. t i P m C, e { x :, e R : x ti,t i Ct i, t i, i N[.m } { x x PmC max up t t i,t i 3. Preliminarie } xt, i N[, m. In thi ection, we preent ome preliminary reult that can be ued in next ection for obtain olution of.7.. 3.. Baic theory for linear fractional differential equation. Lakhmikantham et al. [6, 62, 63, 59 invetigated the baic theory of initial value problem for fractional differential equation involving Riemann-Liouville differential operator of order q,. The exitence and uniquene of olution of the following initial value problem of fractional differential equation were dicued under the aumption that f C r [,. We will etablih exitence and uniquene reult for thee problem under more weaker aumption ee 3.A 3.A4 below. Suppoe that n < α < n and η j Rj N[, n, F, A :, R and B, G :, e R are continuou function. We conider the following four clae of initial value problem of non-homogeneou linear fractional differential equation: C D α xt Atxt F t, a.e. t,, RL D α lim xj t η j, j N[, n, t xt Atxt F t, a.e. t,, η n lim tn α xt t Γα n, lim RL D α j t xt η j, j N[, n, RLH D α xt Btxt Gt, lim log t n α xt t a.e. t, e, η n Γα n, lim RLH D α j t xt η j, j N[, n, CH D α xt Btxt Gt, a.e. t, e, lim d t t dt j xt η j, j N[, n. 3. 3.2 3.3 3.4 where t d dt j xt t dt d dt j xt dt for j 2, 3,.... To obtain olution of 3., we need the following aumption: 3.A there exit contant k i > α n, l i with l i > max{ α, α k i }i, 2, M A and M F uch that At M A t k t l and F t M F t k2 t l2 for all t,.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 3 Chooe the Picard function equence a φ i t n j η j t j j! Claim. φ i C[,. φ t Proof. One ee φ C[,. Then n j η j t j, t [,, j! t α [Aφ i F d, t,, i, 2,.... t α [Aφ F d t α [M A φ k l M F k2 l2 d t αl t αl2 M A φ k d M F k2 d w αl M A φ t αkl w k dw w αl2 M F t αk2l2 w k2 dw M A φ t Bα l, k αkl M F t Bα l 2, k αk2l2 2 a t. It follow that φ i continuou on, and lim t φ t exit. So φ C[,. By mathematical induction, we can prove that φ i C[,. Claim 2. {φ i } i convergent uniformly on [,. Proof. For t [, we have So φ t φ t t α [Aφ F d t α t α M A φ k l d M F k2 l2 d t αl t αl2 M A φ k d M F k2 d M A φ t Bα l, k αkl M F t Bα l 2, k αk2l2 2. φ 2 t φ t t α A[φ φ d

4 Y. LIU EJDE-26/296 t α M A k l M A φ Bα l, k αkl M F Bα l 2, k αk2l2 2 d φ MA 2 t αl Bα l, k α2kl d t αl M A M F Bα l 2, k αkk2l2 2 d φ M 2 At 2α2k2l Bα l, k M A M F t 2αkk2ll2 Bα l 2, k 2 Now uppoe that Then we have φ j t φ j t φ j t φ j t j φ M j A tjαjkjl i Bα l, α 2k l Bα l, α k k 2 l 2. Bα l, iα i k il M j A M F t jαj kk2j ll2 Bα l 2, k 2 j i Bα l, iα ik k 2 i l l 2. t α A[φ j φ j d t α j M A φ M j A jαjkjl i M j A M F jαj kk2j ll2 Bα l 2, k 2 j i Bα l, iα i k il Bα l, iα ik k 2 i l l 2 k l d t αl j M A φ M j A jαjkjl i M j A M F jαj kk2j ll2 Bα l 2, k 2 j i Bα l, iα ik k 2 i l l 2 k d Bα l, iα i k il

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 5 φ M j A tjαjkjl j i M j A M F t Bα l 2, k jαjkk2jll2 2 j Bα l, iα ik k 2 i l l 2. i Bα l, iα i k il Uing mathematical induction, for every i, 2,... we obtain φ i t φ i t φ M i A tiαikil i j MAM i F t Bα l 2, k iαikk2ill2 2 i Bα l, jα jk k 2 j l l 2 j φ M i A i j M i AM F Bα l 2, k 2 for t [,. Conider i i u i v i Bα l, jα j k jl i i φ M i A i j i i Bα l, jα j k jl Bα l, jα jk k 2 j l l 2, M i AM F Bα l 2, k 2 i j Bα l, iα i k il, Bα l, jα jk k 2 j l l 2. One ee that for ufficiently large n with δ, 2, u i Bα l, i α i k i l M A u i M A x αl x iαikil dx δ M A x αl x iαikil dx M A x αl dx δ M A x αl dxδ iαikil M A δ αl α l M A δ iαikil M A δ αl. α l α l δ

6 Y. LIU EJDE-26/296 It i eay to ee that for any ɛ > there exit δ, 2 uch that M A αl δ αl < ɛ 2. For thi δ, there exit an integer N > ufficiently large uch that M A α l δ iαikil < ɛ 2 for all i > N. So < ui u i < ɛ 2 ɛ 2 ɛ for all i > N. It follow that lim i u i /u i. Then i u i converge. Similarly we obtain i v i converge. Hence φ t [φ t φ t [φ 2 t φ t [φ i t φ i t..., t [, i uniformly convergent. Then {φ i t} i convergent uniformly on [,. Claim 3. φt lim i φ i t defined on [, i a unique continuou olution of the integral equation xt n j η j j! t α [Ax F d, t [,. 3.5 Proof. From φt lim i φ i t and the uniformly convergence, we ee that φt i continuou on [,. From t α t α [Aφ p F d [Aφ q F d t α M A φ p φ q k l d M A φ p φ q t Bα l, k αkl M A φ p φ q Bα l, k a p, q, we have φt lim φ it i lim i n j n j n j η j t j j! η j t j j! η j t j j! [ n j η j t j j! lim i t α t α [Aφ i F d t α [Aφ i F d [ A lim i φ i F d t α [Aφ F d. Then φ i a continuou olution of 3.5 defined on [,. Suppoe that ψ defined on [, i alo a olution of 3.5. Then ψt n j η j j! t α [Aψ F d, t,.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 7 We need to prove that φt ψt on [,. Then ψt φ t Furthermore, t α Aψ F d φ M A t αkl Bα l, k ψt φ t M F t Bα l 2, k αk2l2 2. t α A[ψ φ d φ MAt 2 Bα l, k 2α2k2l Bα l, α 2k l M A M F t 2αkk2ll2 Bα l 2, k 2 Now uppoe that Then j ψt φ j t φ M j A tjαjkjl Hence, i Bα l, α k k 2 l 2. Bα l, iα i k il M j A M F t jαj kk2j ll2 Bα l 2, k 2 j i Bα l, iα ik k 2 i l l 2. ψt φ j t t α A[ψ φ j d j φ M j Bα l, iα i k il A tjαjkjl i M j A M F t Bα l 2, k jαjkk2jll2 2 j Bα l, iα ik k 2 i l l 2. i ψt φ i t φ M i A tiαikil i j M i AM F t iαikk2ill2 Bα l 2, k 2 Bα l, jα j k jl

8 Y. LIU EJDE-26/296 i j φ M i A Bα l, jα jk k 2 j l l 2 i j M i AM F Bα l 2, k 2 Bα l, jα j k jl i j for i, 2,.... Similarly we have lim φ M i i A i j Bα l, jα jk k 2 j l l 2 Bα l, jα j k jl lim M i Bα l 2, k 2 AM F i i Bα l, jα jk k 2 j l l 2. j, Then lim i φ i t ψt uniformly on [,. Then φt ψt. Then 3.5 ha a unique olution φ. The proof i complete. Theorem 3.. Suppoe that 3.A hold. Then x i a olution of IVP 3. if and only if x i a olution of the integral equation 3.5. Proof. Suppoe that x i a olution of 3.. Then lim t xt η and x r <. From 3.A, we have for t, x t α n Γα n [Ax F d t α n Γα n A d t α n F d Γα n t α n Γα n [M Ar k l M F k2 l2 d t α n Γα n [M Ar k t l M F k2 t l2 d M A r t l t αk n M F t l2 t αk2 n w α n Γα n wk dw w α n Γα n wk2 dw M A r t l t αk n Bα n, k M F t l2 t αk2 n Bα n, k 2. by t w

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 9 So t that lim t t α n Γα n [Ax F d i defined on,. k i > α n implie t α n t α n Axd lim F d. 3.6 Γα n t Γα n Furthermore, for t, t 2, with t < t 2 we have 2 t α [Ax F d t t 2 α Ax F d 2 t α t 2 α Ax F d [ 2 t 2 α M A r k l d t t α t 2 α k l d [ 2 t 2 α M F k2 l2 d t t α t 2 α k2 l2 d [ 2 t 2 αl M A r k d t t α t 2 α k t 2 l d [ 2 t 2 αl2 M F k2 d t t α t 2 α k2 t 2 l2 d [ M A r t αkl 2 t t 2 wαl w k dw t αl k d M F [t αk2l2 2 t t 2 t αl2 k2 d [ M A r t αkl 2 t αkl 2 t2 t t 2 t 2 αl wαl2 w k2 dw t 2 αl2 wαl w k dw t αkl w αl w k dw t 2 α [Ax F d k d k2 d w αl w k dw

2 Y. LIU EJDE-26/296 M F [t αk2l2 2 t αk2l2 [ M A r t αkl 2 t t 2 wαl2 w k2 dw w αl2 t t 2 w k2 dw t αk2l2 2 wαl w k dw t2 t αkl t αkl 2 Bα l, k t αkl 2 M F [t αk2l2 2 t t 2 wαl2 w k2 dw t αk2l2 t αk2l2 2 Bα l 2, k 2 t αk2l2 2 a t t 2. So t w αl2 t t2 t t2 w k2 dw wαl w k dw wαl2 w k2 dw t α [Ax F d i continuou on,, by defining t α [Ax F d lim t t We have I α C D α xt I α [Atxt F t. So t α [Ax F d. t α [Ax F d I α [Atxt F t Iα C D α xt t α w α x n wdw d Γn α interchange the order of integration Γn α Γn α uing Bα, α w t α w n α dx n wdw uing w t w u t u n u α u n α dux n wdw Γ α Γ t u n x n wdw n! [t u n x n w t n! n t u n 2 x n wdw η n n! n 2!... t u n 2 x n wdw

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 2 n η j j! j n x η j d xt j!. j Then x C, i a olution of 3.5. On the other hand, if x i a olution of 3.5. From Cae, 2 and 3, we have x C, and lim t x j t η j j N[, n. So x C[,. Furthermore, from 3.6 we have C D α xt Γn α Γn α t n α x n d t n α n j η j j j! w α nd [Awxw F wdw t t n α w α nd [Awxw F wdw Γn α t n α Γn α Γα n d w α n [Awxw F wdw [ t n α Γα n Γn α d w α n [Awxw F wdw [t n α w α n [Awxw F wdw t Γα n Γn α n α t n α w α n [Awxw F w dw d [ t n α Γα n Γn α w α n [Awxw F w dw d by 3.6 [ Γα n Γn α [Awxw F wdw w t n α w α n d by changing the order of integration [ u n α u α n du[awxw F wdw Γα n Γn α becaue w t w u

22 Y. LIU EJDE-26/296 [ [Awxw F wdw by uing Bn α, α n Γn αγα n Atxt F t in the lat equality. So x C[, i a olution of 3.. The proof i complete. Theorem 3.2. Suppoe that 3.A hold. Then 3. ha a unique olution. If there exit contant k 2 > αn, l 2 with l 2 > max{ α, α k 2 }, M F uch that F t M F t k2 t l2 for all t,, then the following problem ha a unique olution n xt η j E α,j λt α t j j C D α xt λxt F t, a.e., t,, lim t x j t η j, j N[, n 3.7 t α E α,α λt α F d, t,. 3.8 Proof. i From Claim, 2 and 3, Theorem 3. implie that 3. ha a unique olution. ii From the aumption and At λ, it i eay to ee that 3.A hold with k l and k 2, l 2 mentioned. Thu 3.7 ha a unique olution. From the Picard function equence we have φ i t n j n j n j η j t j j! η j t j j! λ λ u α η j t j j! n λ j λ 2 t α λ n j η j t j j! λ2 2 λ 2 t α t α t α φ i d n F udu d η j j! n η j λ j! tαj j u u t α F d j η j j j! λ t α F d t α F d t α j d u α φ i 2 udu u α φ i 2 u du d u α t α F u du d F d w α w j dw t α u α dφ i 2 udu t α u α df udu

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 23 n j η j t j j! λ Γ2α n λη j t αj Γα j j t u 2α F udu n η j t j Γα j j... n j n j t α i η j t j v t α i η j t j v λt α Γ2α λ v t αv Γvα j i v λ2 Γ2α λt α Γα j F d λi Γiα t u 2α φ i 2 udu t α F d λ v t αv F d Γv α λ v t αv Γvα j i t α n η j t j i j v v λ2 Γ2α λ i n Γiα j λ v t αv F d Γv α λ v t αv Γvα j n η j t j E α,j λt α j t u iα φ udu j v η j t αnj t u 2α φ i 2 udu t α i λ v t αv F d Γv α t α E α,α λt α F d, a m. Then xt lim i φ i t i a unique olution of 3.7. So x atifie 3.8. The proof i complete. To obtain olution of 3.2, we need the following aumption: 3.A2 There exit contant k i >, l i with l > max{ α, α k }, l 2 > max{ α, n k 2 }, M A and M F uch that At M A t k t l and F t M F t k2 t l2 for all t,. We chooe Picard function equence a φ i t n v φ t for t,, i, 2,.... Claim. φ i C n α [,. n v η v Γα v tα v η v Γα v tα v, t,, t α [Aφ i F d,

24 Y. LIU EJDE-26/296 Proof. It i eay to ee that φ C n α [,. We have t n α t α [Aφ F d t n α t α k l α n [ n α φ n d M F t n α t α k l t n α t αl φ αk n d M F t n α t αl2 k2 d φ t Bα l, α k αkl n M F t Bα l 2, k nk2l2 2. Then t t α lim t tn α [Aφ n F d i convergent on, and t α [Aφ n F d. We ee that φ C n α [,. By mathematical induction, we can prove that φ n C n α [,. Claim 2. {t t n α φ i t} converge uniformly on [,. Proof. A in Cae, for t [, we have So t n α φ t φ t t α [Aφ F d φ t Bα l, α k αkl n t n α φ 2 t φ t t Bα l 2, k nk2l2 2. t α A[φ φ d t n α t α M A k l α n φ Bα l, α k αkl n Bα l 2, k nk2l2 2 d M A φ t n α t αl 2α n2kl d Bα l, α k n M A M F t n α t αl αkk2l2 d Bα l 2, k 2 M A φ t 2α2k2l Bα l, 2α n 2k l M A M F t αnklk2l2 Bα l, α k k 2 l 2 Bα l, α k n Bα l 2, k 2.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 25 Furthermore, t n α φ 3 t φ 2 t t α A[φ 2 φ d t n α t α M A k l α n M A φ 2α2k2l Bα l, 2α n 2k l Bα l, α k n M A M F Bα l, α k αnklk2l2 k 2 l 2 Bα l 2, k 2 d MA φ 2 t n α t αl 3α n3k2l d Bα l, 2α n 2k l Bα l, α k n MAM 2 F t n α t αl 2α2klk2l2 d Bα l, α k k 2 l 2 Bα l 2, k 2 MA φ 2 t Bα l, 3α n 3k 3α3k3l 2l Bα l, 2α n 2k l Bα l, α k n M 2 AM F t 2αn2k2lk2l2 Bα l, 2α 2k l k 2 l 2 Bα l, α k k 2 l 2 Bα l 2, k 2. Similarly by the mathematical induction, for every i, 2,... we obtain t n α φ i t φ i t M i A φ t iαikil Bα l, α k n i 2 j Bα l, j α j k jl n M m A M F t i αni ki lk2l2 Bα l 2, k 2 i 2 j Bα l, j α j k jl k 2 l 2 M i A φ Bα l, α k n

26 Y. LIU EJDE-26/296 i 2 j Bα l, j α j k jl n M i A M Bα l 2, k 2 F i 2 j Bα l, j α j k jl k 2 l 2, t [,. Similarly we can prove that both i i u i v i i i 2 j i i 2 j are convergent. Hence, M i A φ Bα l, α k n Bα l, j α j k jl n, M i A M Bα l 2, k 2 F Bα l, j α j k jl k 2 l 2 t n α φ tt n α [φ t φ tt n α [φ 2 t φ t t n α [φ i t φ i t..., for t [,, i uniformly convergent. Then {t t n α φ i t} i convergent uniformly on,. Claim 3. φt t α n lim i t n α φ i t defined on, i a unique continuou olution of the integral equation n η t v t α xt Γα v tα v [AxF d, t,. 3.9 v Proof. By lim i t n α φ i t t n α φt and the uniformly convergence, we ee φt i continuou on,. From t n α t α [Aφ p F d t α [Aφ q F d M A φ p φ q t n α t α k l α n d M A φ p φ q t n α t αl αk n d M A φ p φ q t Bα l, α k αkl n M A φ p φ q Bα l, α k n

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 27 we know that φt t α n lim i t n α φ i t t α n lim i uniformly a p, q, [t n α n v t n α t α n η v Γα v tα v v n v η v Γα v tα v η v Γα v tα v [Aφ i F d lim i t α [Aφ i F d t α [Aφ F d. Then φ i a continuou olution of 3.9 defined on,. Suppoe that ψ defined on, i alo a olution of 3.9. Then n η t v t α ψt Γα v tα v [Aψ F d, t [,. v We need to prove that φt ψt on,. Then t n α ψt φ t t n α t α Aψ F d ψ t αkl Bα l, α k n Furthermore, we have t n α ψt φ t t n α t α A[ψ φ d M A φ t 2α2k2l Bα l, 2α n 2k l M A M F t αnklk2l2 Bα l, α k k 2 l 2 Uing mathematical induction, we have t n α ψt φ i t t n α t α A[ψ φ i 2 d MA φ i t Bα l, α k iαikil n i 2 j Bα l, j α j k jl n t Bα l 2, k nk2l2 2. Bα l, α k n Bα l 2, k 2.

28 Y. LIU EJDE-26/296 M m A M F t i αni ki lk2l2 Bα l 2, k 2 i 2 j Bα l, j α j k jl k 2 l 2 M i A φ Bα l, α k n i 2 j Bα l, j α j k jl n M i A M Bα l 2, k 2 F i 2 j Bα l, j α j k jl k 2 l 2, t [,. Hence, t n α ψt φ i t M i A φ Bα l, α k n i 2 j Bα l, j α j k jl n M i A M Bα l 2, k 2 F i 2 j Bα l, j α j k jl k 2 l 2, for i, 2,.... Similarly we have lim i t n α φ i t t n α ψt uniformly on,. Then φt ψt on,. Then 3.9 ha a unique olution φ. The proof i complete. Theorem 3.3. Suppoe that 3.A2 hold. Then x C n α, i a olution of IVP 3.2 if and only if x C n α, i a olution of the integral equation 3.9. Proof. Suppoe that x C n α, i a olution of 3.2. Then t t n α xti continuou on, by defining t n α xt t lim t t n α xt and x r <

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 29. So from w u, we obtain lim lim w n α xwdw lim ξn α xξ w n α w α n w n α xwdw w n α w α n dw by mean value theorem with ξ, ξn α xξ lim u n α u α n du η n Bn α, α n. Γα n From 3.A2, we have imilarly to Cae that t n α t α [Ax F d t n α t α [A α n n α x F d t n α t α [M A r α n k l M F k2 l2 d rm A t αkl Bα l, α k n So t t n α t α lim t tn α M F t Bα l 2, k nk2l2 2. [Ax F d i defined on, and 3. t α [Ax F d. 3. Furthermore, we have imilarly to Theorem 3. that t t α [Ax F d i continuou on,. So t t n α t t α [Ax F d i continuou on [, by defining t n α t α [Ax F d t lim t tn α t α [Ax F d. We have I α RL D α xt I α [Atxt F t. So t α [Ax F d I α [Atxt F t Iα RL D α xt t t α [ Γn α w n α xwdw n d t α w n α n d xwdw Γn α

3 Y. LIU EJDE-26/296 t α d RL D α x t α RL D α x t Γn αγα n d w xwdw n α Γn αγα Γn αγα 2 η tα... t α 2 t α 2 n d w n α η xwdw tα t α 3 n 2d w xwdw n α η 2 Γα tα 2 Γn αγα n t α n n η v Γα v tα v v [ t α n Γn αγα n 2 w n α xwdw d w n α xwdw d n η v Γα v tα v v [t α n w n α t xwdw Γn αγα n 2 α n t α n w n α xw dw d n η v Γα v tα v uing 3. v [ t α n w n α dxwdw Γn αγα n η n Γα n tα n n v [ Γn αγα n n η v Γα v tα v t α n v xt n v η v Γα v tα v. u η v Γα v tα v w α n w n α dwxwdw Then x C n α, i a olution of 3.9.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 3 On the other hand, if x C n α, i a olution of 3.9. Then 3. implie lim t t n α xt. Furthermore, we have η n Γα n RL D α xt n t n α xd Γn α t n α n η v Γn α Γα v α v v u α n [Auxu F udu d n η t v t n α α v d Γn α Γα v v t n α u α n [Auxu F u du d n η v Γn α Γα v tn v w n α w α v dw v n α uα t d[auxu F udu u n η v Γn α Γα v tn v w n α w α v dw v t u n w Atxt F t. n α wα n dw[auxu F udu n So x C n α, i a olution of IVP3.2. The proof i complete. Theorem 3.4. Suppoe that 3.A2 hold. Then 3.2 ha a unique olution. If At λ and there exit contant k 2 >, l 2 with l 2 > max{ α, n k 2 } and M F uch that F t M F t k2 t l2 for all t,, then the problem RL D α xt λxt F t, a.e. t,, lim t n α η n xt t Γα n, lim RL D α j t xt η j, j N[, n ha a unique olution n xt η v t α v E α,α v λt α t α E α,α λt α F d, 3.3 v for t,. 3.2 Proof. i From Claim, 2 and 3, and Theorem 3.3, we ee that 3.2 ha a unique olution. ii From the aumption and At λ, one ee that 3.A2 hold with k l and k 2, l 2 mentioned. Thu 3.2 ha a unique olution. From the Picard function equence we have φ i t

32 Y. LIU EJDE-26/296 n η t v t α t α Γα v tα v λ φ i d F d v n η t v t α n η v Γα v tα v λ v Γα v α v v u α u α λ φ i 2 udu F udu d n v λ 2 λ n v t α F d η v Γα v tα v λ u u n v η v Γα v t α α v d t α u α dφ i 2 udu t α u α t α df udu F d n η v Γ2α v t2α v η v Γα v tα v λ λ 2 v t u 2α φ i 2 udu λ Γ2α t α F d n η v t α v Γα v v... n v v t α η v t α v i j λt α Γ2α t α m j t u 2α F udu Γ2α λt α Γ2α v λ2 F d λ j t αj λ i Γjα α v j λ j t αj F d Γj α n i η v t α v λ j t αj Γjα α v n η v t α v E α,α v λt α v t u 2α φ i 2 udu Γ2α t u iα φ udu Γmα t α i λ j t αj F d Γj α j t α E α,α λt α F d. Then we obtain xt lim i φ i t i a unique olution of 3.2. Then x atifie 3.3. The proof i complete. To obtain olution of 3.3, we need the following aumption:

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 33 3.A3 There exit contant k i >, l i with l > max{ α, α k }, l 2 > max{ α, n k 2 }, M B and M G uch that Bt M B log t k log t l and Gt M G log t k2 log t l2 for all t, e. We chooe Picard function equence a n η v φ t Γα v log tα v, φ i t n v v η v Γα v log tα v t, e, i, 2,.... t, e, log t α [Bφ i G d, Claim. φ i LC n α, e. Proof. We have φ LC n α [, e and log t n α log t α [Bφ G d log t n α log t α [ M B φ log α n log k log l d M G log k2 log l2 log t n α M B φ log t αl log log t n α M G log t αl2 log k2 d αk n d M B φ log t αkl Bα l, α k n M G log t nkl Bα l 2, k 2 a t, we know that t t log t α [Bφ G d i continuou on, e and lim t log t n α φ t exit. Then φ LC n α [, e. By mathematical induction, we can how φ i LC n α [, e. Claim 2. {t log t n α φ i t} i convergent uniformly on [, e. Proof. A above, for t [, e we have log t n α φ t φ t log tn α log tn α log t α [Bφ G d log t α [ φ M B log α nk log l M G log k2 log l2 d M B φ log t αkl Bα l, α k n

34 Y. LIU EJDE-26/296 So Then M G log t Bα l 2, k nk2l2 2. log t n α φ 2 t φ t log tn α log tn α log t α B[φ φ d log t αl M B log kα n M B φ log Bα l, α n k αkl M G log nk2l2 Bα l 2, k 2 d φ M 2 Blog t 2α2k2l Bα l, 2α 2k l n Bα l, α k n M B M G log t Bα l, α k αnklk2l2 k 2 l 2 Bα l 2, k 2. log t n α φ 3 t φ 2 t log tn α log t α B[φ 2 φ d log tn α int t log t αl M B log kα n φ MBlog 2 Bα l, α 2k αn2k2l l Bα l, α k n Bα l, n k k 2 l 2 M B M G log 2nklk2l2 Bα l 2, k 2 d φ MBlog 3 t Bα l, 3α 3k 3α3k3l 2l n Bα l, 2α 2k l n Bα l, α k n M 2 BM G log t 2αn2k2lk2l2 Bα l, 2α 2k l k 2 l 2 Bα l, α k k 2 l 2 Bα l 2, k 2.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 35 Furthermore, log t n α φ 4 t φ 3 t log tn α log tn α log t α B[φ 3 φ 2 d log t αl M B log kα n φ M 3 Blog α2n3k3l Bα l, α n 3k 2l Bα l, α k n Bα l, 2n 2k l k 2 l 2 Bα l, n k k 2 l 2 Bα l, α 2k l M 2 BM G log 3n2k2lk2l2 Bα l 2, k 2 d φ MBlog 4 t Bα l, 4α 4k 4α4k4l 3l n Bα l, 3α 3k 2l n Bα l, 2α 2k l n Bα l, α k n M 3 BM G log t 3αn3k3lk2l2 Bα l, 3α 3k 2l k 2 l 2 Bα l, 2α 2k l k 2 l 2 Bα l, α k k 2 l 2 Bα l 2, k 2. Similarly by mathematical induction, for every i, 2,... we obtain log t n α φ i t φ i t φ M i Blog t iαikil Bα l, α k n i j Bα l, j α j k jl n M i B M Glog t i αni ki lk2l2 Bα l 2, k 2 i j Bα l, jn jk j l k 2 l 2 φ MB i Bα l, α k n i j Bα l, j α j k jl n

36 Y. LIU EJDE-26/296 M i B M Bα l 2, k 2 G i j Similarly we can prove that both i i converge. Hence, Bα l, jn jk j l k 2 l 2, t, e. u i v i i i j i i j φ MB i Bα l, α k n Bα l, j α j k jl n, M m B M G Bα l 2, k 2 Bα l, jn jk j l k 2 l 2 log t n α φ t log t n α [φ t φ t log t n α [φ i t φ i t..., for t, e converge uniformly. Then {t log t n α φ i t} converge uniformly on [, e. Claim 3. φt log t α n lim i log t n α φ i t defined on [, e i a unique continuou olution of the integral equation n η v xt Γα v log tα v log t v α [Bx G d, 3.4 for t, e. Proof. From lim i log t n α φ i t log t n α φt and the uniformly convergence, we ee that φt i continuou on [, e. From log t n α log t α [Aφ p F d log t α [Bφ q G d M B φ p φ q log t n α log t αl log k log M B φ p φ q log t n α log t αl log M B φ p φ q log t nkl Bα l, α k M B φ p φ q Bα l, α k n uniformly a p, q, αk n d α n d

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 37 we know that φt log t α n lim log i tn α φ i t [ n η v lim log tα v i Γα v v v log t α [Bφ i G d n η t v Γα v log tα v lim i v n η v Γα v log tα v v log t α [Bφ i G d log t α [Bφ G d. Then φ i a continuou olution of 3.4 defined on, e. Suppoe that ψ defined on, e i alo a olution of 3.4. Then n η v ψt Γα v log tα v log t α [Bψ G d, for t, e. We need to prove that φt ψt on, e. Then Furthermore, log t n α ψt φ t log t n α log t α Bψ G d M B φ log t Bα l, α k αkl n M G log t Bα l 2, k nk2l2 2. log t n α ψt φ t log t n α log t α B[ψ φ d φ MBlog 2 t Bα l, 2α 2k 2α2k2l l n Bα l, α k n M B M G log t αnklk2l2 Bα l, α k k 2 l 2 By mathematical induction, we obtain log t α ψt φ i t log t n α Bα l 2, k 2. log t α B[ψ φ i d φ M i Blog t iαikil Bα l, α k n

38 Y. LIU EJDE-26/296 i j Bα l, j α j k jl n M i B M Glog t i αni ki lk2l2 Bα l 2, k 2 i j Bα l, jn jk j l k 2 l 2 φ MB i Bα l, α k n i j Bα l, j α j k jl n i M i B M Bα l 2, k 2 G j Bα l, jn jk j l k 2 l 2, for t, e, i, 2,.... Similarly we have lim i log t n α φ i t log t n α ψt uniformly on, e. Then φt ψt on, e. Then 3.4 ha a unique olution φ. The proof i complete. Theorem 3.5. Suppoe that 3.A3 hold. Then x i a olution of IVP 3.3 if and only if x LC n α, e i a olution of the integral equation 3.4. Proof. Suppoe that x i a olution of 3.3. Then t log t n α xti continuou on, e by defining log t n α xt t lim t log t n α xt and x r <. So lim lim log w n α xw dw w log w n α log w α n log w n α xw dw w lim log ξn α xξ log w n α log w α n dw w by the mean value theorem with ξ, lim log ξn α xξ u n α u α n du becaue log w log u η n Bn α, α n. Γα n and for v N[, n we have lim d t d n v log w n α xw dw w From 3.A3, we have log t n α Γn v α v lim t RLH D α v xt Γn αη v. log t α [Bx G d

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 39 log t n α log t α [M B rlog α n log k log l M G log k2 log l2 d log t n α M B r log t αl log log t n α M G log t αl2 log αk n d k2 d M B rlog t αkl Bα l, k α M G log t nkl Bα l 2, k 2. So t log t n α t log t α [Bx G d lim tn α t log i defined on, e and log t α [Bx G d. 3.5 Furthermore, imilarly to Theorem 3. we have t log t α [BxG d i continuou on, e. So t log t n α t log t α [Bx G d i continuou on [, e by defining log t n α log t α [Bx G d t. 3.6 We have H I α RLH D α xt H I α [Btxt Gt. So log t α [Bx G d H I α [Btxt Gt H I α RLH D α xt t log t Γn α α d d n log w n α xw dw d w t log t Γn α α d [ d d n log w n α xw dw w [log t Γn α α d d n log w n α xw dw t w α log t α 2 d d n log w n α xw dw d w log tα lim d Γn α t d n log w n α xw dw w t log t Γα Γn α α 2 d d n log w n α xw dw w η log tα Γα Γn α log w n α xw dw d w... v log t α 2 d d n n η v Γα v log tα v Γα n Γn α log t α n d

4 Y. LIU EJDE-26/296 log w α xw dw d w n v η v Γα v log tα v Γα n 2 Γn α t [ log t α n log w α xw dw w Big d n η v Γα v log tα v v [log t α n log w n α xw dw w t α n n [ v lim t η v Γn αγα n 2 t log t α n log w n α xw dw w Γα v log tα v tα n log α n u Γn αγα n 2 t d log w n α xw dw w log t α n log d n α w xwdw w n η v Γα v log tα v Γn αγα n 2 t v [ η n Bn α, α n log tα n Γα n α n xt n v u α n u n α duxw dw w η v Γα v log tα v. Then x LC n α, e i a olution of 3.4. On the other hand, if x i a olution of 3.4, Cae, 2, 3 and 3.5 imply lim t log t n α η xt n Γα n. Then x LC n α, e. Furthermore, by Definition 2.5 we have RLH D α xt Γn α t d dt n log t n α x d Γn α t d dt n[ log t n n α η v Γα v v log w α [Awxw F w dw d w n η v Γn α Γα v t d dt n log t n α log v log α v α v d

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 4 Γn α t d dt n n Γn α v Γn α t d dt n Γn α t d dt n Γn α t d dt n log t n α log w α [Bwxw Gw dw w η v Γα v t d dt n log t n v [Bwxw Gw dw w w n α w α v dw log t n α log w α [Bwxw Gw dw w u log t n α log w log t w n w n α w α dw Γn t d dt n log t w n [Bwxw Gw dw w Btxt Gt. α d [Bwxw Gwdw w So x LC n α, e i a olution of IVP3.3. The proof i complete. Theorem 3.6. Suppoe that 3.A3 hold. Then 3.4 ha a unique olution. If Bt λ and there exit contant k 2 >, l 2 with l 2 > max{ α, n k 2 } and M G uch that Gt M G t k2 t l2 for all t, e, then following problem RLH D α xt λxt Gt, a.e., t, e, η n lim t log tn α xt Γα n, 3.7 lim RLH D α j t xt η j, j N[, n ha a unique olution n xt η v log t α v E α,α v λlog t α v log t u α E α,α λlog t u α G d, t, e. d d 3.8 Proof. i From Claim, 2 and 3, 3.4 ha a unique olution. ii From the aumption and Bt λ, one ee that 3.A3 hold with k l and k 2, l 2 mentioned in aumption. Thu 3.7 ha a unique olution. From the Picard function equence we obtain φ i t n η v Γα v log tα v λ v n v log t α G d η v Γα v log tα v λ v log t α φ i d n η v Γα v log t α log α v d

42 Y. LIU EJDE-26/296 λ 2 log t α log u α φ i 2 u du u λ log t α log u α Gu du d u n η v n Γα v log η v tα v λ Γα v v v w α w α v dw λ 2 λ log t u α log u n η v n Γα v log tα v λ v λ2 Γ2α λ Γ2α n v v log t u 2α φ i 2 u du u log t u 2α Gu du u η v log t α v Γα v... n v log t u α λ Γ2α log t u α η v log t α v i j log t i u α j n η v log t α v i v j log t i u α j u d log t2α v log t α log u α d Gudu u η v log t2α v Γ2α v λlog tα Γ2α v log t α G d λ2 Γ2α G d λ j log t jα λi Γjα α v Γiα λ j Γj α log t u jα G d λ j log t jα Γjα α v λ j Γj α log t u jα G d n η v log t α v E α,α v λlog t α v log t α G d α d φ i 2u du u log t α G d log t u 2α φ i 2 u du u log t u iα φ u du u log t u α E α,α λlog t u α G d. Then xt lim i φ i t i the unique olution of 3.7. x i jut a in 3.8. The proof i complete. To obtain olution of 3.4, we need the following aumption: 3.A4 there exit contant k i > α n, l i with l i > max{ α, α k i }, M B and M G uch that Bt M B log t k log t l and Gt M G log t k2 log t l2 for all t, e.

EJDE-26/296 SURVEY AND NEW RESULTS ON BVPS FOR IFDES 43 We chooe the Picard function equence a φ i t n j for t, e, i, 2,.... Claim. φ i C, e. φ t n j η j j! log tj Proof. On ee that φ C[, e. From η j j! log tj, t, e, log t α [Bφ G d log t α [M B φ log k log l M G log k2 log l2 d M B φ log t α log k log M G log t α log k2 log log t α [Bφ i G d, l2 d l d M B φ log t αkl Bα l, k M G log t αk2l2 Bα l 2, k 2 a t, we obtain that lim t φ exit and φ i continuou on, e. Then φ C[, e. By mathematical induction, we ee that φ i C[, e. Claim 2. φ i converge uniformly on [, e. Proof. For t [, e we have φ t φ t φ M B log t α [Bφ G d log t α [M B φ log k log l M G log k2 log l2 d log t αl log k d φ M B log t Bα l, k αkl M G log t Bα l 2, k αk2l2 2. So φ 2 t φ t M G log t αl2 log k2 d