Nonlinear Analysis: Modelling and Control, 2013, Vol. 18, No. 4,

Transcript

1 Nonlinear Analysis: Modelling and Conrol, 23, Vol. 8, No. 4, Exisence and uniqueness of soluions for a singular sysem of higher-order nonlinear fracional differenial equaions wih inegral boundary condiions Lin Wang, Xingqiu Zhang, Xinyi Lu School of Mahemaics, Liaocheng Universiy Liaocheng, Shandong 25259, China fengzhimengwl@63.com; zhxq9758@63.com; linda98989@26.com Received: 7 November 22 / Revised: 22 May 23 / Published online: 25 Sepember 23 Absrac. In his paper, we sudy he exisence and uniqueness of soluions for a singular sysem of nonlinear fracional differenial equaions wih inegral boundary condiions. We obain exisence and uniqueness resuls of soluions by using he properies of he Green s funcion, a nonlinear alernaive of Leray Schauder-ype, Guo Krasnoselskii s fixed poin heorem in a cone and he Banach fixed poin heorem. Some examples are included o show he applicabiliy of our resuls. Keywords: fracional differenial equaion, singular sysem, fracional Green s funcion, fixed poin heorem. Inroducion Fracional differenial equaions have been of grea ineres recenly. I is caused boh by he inensive developmen of he heory of fracional calculus iself and by he applicaions of such consrucions in various fields of sciences and engineering such as conrol, porous media, elecromagneic, and oher fields. There are many papers deal wih he exisence and mulipliciy of soluion of nonlinear fracional differenial equaions see [ 8] and he references herein). The paper [] considered he exisence of posiive soluions of singular coupled sysem D s u f, v), < <, D p v g, u), < <, where < s, p <, and f, g :, ] [, ) [, ) are wo given coninuous funcions, lim f, ), lim g, ) and D s, D p are he sandard fracional Riemann Liouville s derivaives. The exisence resuls of posiive soluion The projec is suppored financially by he Naional Naural Science Foundaion of China 9779), he Foundaion for Ousanding Middle-Aged and Young Scieniss of Shandong Province BS2SF4) and a Projec of Shandong Province Higher Educaional Science and Technology Program No. JLA53). Corresponding auhor. c Vilnius Universiy, 23

2 494 L. Wang e al. are obained by a nonlinear alernaive of Leray Schauder-ype and Guo Krasnoselskii s fixed poin heorem in a cone. The paper [3] considered he exisence of posiive soluions of singular coupled sysem D α u) f, v) ), < <, D β v) g, u) ), < <, u) u ) u) v) v ) v), where 2 < α, β 3 and f, g :, ] [, ) [, ) are wo given coninuous funcions, and lim f, ), lim g, ) and D, α D β are he sandard fracional Riemann Liouville s derivaives. The wo sufficien condiions for he exisence of soluions are obained by a nonlinear alernaive of Leray Schauder-ype and Guo Krasnoselskii s fixed poin heorem in a cone. Goodrich [5] considered he exisence of a posiive soluion o sysems of differenial equaions of fracional order subjec eiher o he boundary condiions D v y ) λ a )f y ), y 2 ) ), D v2 y 2) λ 2 a 2 )g y ), y 2 ) ), or y i) y i) ) yi) 2 ), i n 2, [ D k y ) ] [ D k y 2 ) ], α n 2, ) yi) 2 ), i n 2, [ D k y ) ] φ [ y), D k y 2 ) ] φ 2y), α n 2, where v, v 2 n, n] for n > 3 and n N, he coninuous funcionals φ, φ 2 : C[, ]) R represen nonlocal boundary condiions. The papers [9 2] considered he exisence of posiive soluions of so-called k, n k) or p, n p) conjugae boundary value problems BVP). For example, in [9], he auhors discussed he following p, n p) conjugae singular boundary value problem: ) n P ) y n) φ)f, y), < <, y i) ), i p, y j) ), j n p, where n 2, and he nonlineariy f may be singular a y. The exisence resuls of posiive soluion abou singular higher order boundary value problems are esablished. Inspired by he work of he above papers and many known resuls, in his paper, we sudy he exisence of posiive soluions of BVP ). The exisence and uniqueness

3 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 495 resuls of soluions are obained by a nonlinear alernaive of Leray Schauder-ype, Guo Krasnoselskii s fixed poin heorem in a cone and he Banach fixed poin heorem. We consider he singular sysem of nonlinear fracional differenial equaions wih inegral boundary condiions D α u) f, v) ), < <, D β v) g, u) ), < <, u) u ) u n 2) ), v) v ) v n 2) ), u i) ) λ v j) ) b c us) ds, vs) ds, where n < α, β n, n 3, <, c, i, j N, i, j n 2 and i, j are fixed consans, λ α /α >, 2 bc β /β >, {, i, α )α 2) α i), i, {, j, 2 β )β 2) β j), j. f, g :, ] [, ) [, ) are wo given coninuous funcions and singular a ha is, lim f, ), lim g, ) ), and D, α D β are he sandard fracional Riemann Liouville s derivaives. The paper is organized as follows. Firsly, we presen some necessary definiion and preliminaries, and derive he corresponding Green s funcion known as fracional Green s funcion and argue is properies. Secondly, he exisence resuls of posiive soluions are obained by a nonlinear alernaive of Leray Schauder-ype, Guo Krasnoselskii s fixed poin heorem in a cone and he Banach fixed poin heorem. Finally, we consruc some examples o demonsrae he applicaion of our main resuls. 2 Background maerials and Green s funcion For he convenience of he reader, we presen here he necessary definiions, lemmas and heorems from fracional calculus heory o faciliae analysis of BVP ). These definiions, lemmas and heorems can be found in he recen lieraure, see [ 8]. Definiion. The Riemann Liouville fracional inegral of order α > of a funcion y :, ) R is given by I α y) s) α ys) ds provided he righ-hand side is poinwise defined on, ). ) Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

4 496 L. Wang e al. Definiion 2. The Riemann Liouville fracional derivaive of order α > of a coninuous funcion y :, ) R is given by D α y) ) n d Γn α) d ys) ds, s) α n where n [α], [α] denoes he ineger par of he number α, provided ha he righhand side is poinwise defined on, ). From he definiion of he Riemann Liouville derivaive, we can obain he saemen. Lemma. See [7].) Le α >. If we assume u C, ) L, ), hen he fracional differenial equaion D α u) has u) C α C 2 α 2 C N α N, C i R, i, 2,..., N, as unique soluions, where N is he smalles ineger greaer han or equal o α. Lemma 2. See [7].) Assume ha u C, ) L, ) wih a fracional derivaive of order α > ha belongs o C, ) L, ). Then I α D α u) u) C α C 2 α 2 C N α N for some C i R, i, 2,..., N, where N is he smalles ineger greaer han or equal o α. Remark. See [6].) The following properies are useful for our discussion: I α I β f) Iαβ f), D α I α f) f), α, β >. In he following, we presen Green s funcion of he fracional differenial equaion boundary value problem. Lemma 3. Given y C[, ], he problem Du) α y), u) u ) u n 2) ), u i) ) λ us) ds, 2) where α > 2, n < α n, < <,, ], i N, i n 2 and i is a fixed consan, λ/α) α >, {, i, α )α 2) α i), i,

5 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 497 i N and i is a fixed consan, is equivalen o G, s) u) G, s)ys) ds, α s) α i λ α s)α α λ α α ) s) α λ α α ) α s) α i λ α α ) s) α λ α α ) α s) α i λ α s)α α λ α α ), s, s,, s,, s, α s) α i, s, s. λ α α ) Here, G, s) is called he Green s funcion of BVP 2). Obviously, G, s) is coninuous on [, ] [, ]. Proof. We may apply Lemma 2 o reduce 2) o an equivalen inegral equaion u) I α y) C α C 2 α 2 C n α n for some C, C 2,..., C n R. Consequenly, he general soluion of 2) is u) s) α ys) ds C α C 2 α 2 C n α n. By u) u ) u n 2) ), one ges ha C 2 C 3 C n. Then we have u) u i) ) s) α ys) ds C α, s) α i ys) ds C α i. On he oher hand, u i) ) λ us) ds combining wih u i) ) s) α i ys) ds C, 3) Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

6 498 L. Wang e al. yields C us) ds x s x s) α ys) ds dx C x s) α ys) dx ds C s) α α ys) ds C α α, s α ds s α ds s) α i λ s) α λ ys) ds α α ) λ ys) ds. α α )α Therefore, he unique soluion of he problem 2) is u) For, one has u) s) α ys) ds λ s) α α λ ys) ds. α α )α s) α ys) ds s) α i α λ ys) ds α α ) ) λ s) α α λ ys) ds α α )α ) s) α i α λ ys) ds α α ) α s) α i λ α s)α α λ α α ) s) α λ ys) ds α α ) α s) α i λ α s)α α λ ys) ds α α ) α s) α i λ ys) ds α α ) G, s)ys) ds.

7 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 499 For, one has u) ) s) α ys) ds ) s) α i α λ ys) ds α α ) λ s) α α λ ys) ds α α )α α s) α i λ α s)α α λ α α ) s) α λ ys) ds α α ) α ) α s) α i λ α α ) s) α λ ys) ds α α ) α s) α i λ ys) ds α α ) G, s)ys) ds. The proof is complee. Lemma 4. The funcion G, s) defined by 3) saisfies: a) G, s) > for all, s, ); a2) λ α /α)g, s) λ α /α) ) i s s) α i α for all, s [, ]; a3) λ α /α)g, s) n λ α /α λ α /α)s s) α i for all, s [, ]. Proof. For s, s, α G, s) α s) α i λ α s)α α α s) α α s) α λ α α s )α α α s) α Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

8 5 L. Wang e al. α s) α λ α α s) α α α s) α [ λ ] α α s) s) α α α s) α λα ] )[ α s) α α s) α λ α α s s) α α λ ) [ α α s) α 2 α 2 s) s) α 2 s) ] λ α α s s) α α α s) α 2 α 2[ s) s) ] λ α α s s) α α α s) α 2 α 2 s ) λ α α ) i s s) α i α λ α α ) i s s) α i α. α G, s) α s) α i λ α s)α α α s) α λ ) [ α α s) α i α s) α ] λ α α s) α i α λ α s)α α λ ) [ α α s) α i α s) α s) α i] λ α α [ α s) α i s ) α ] s) α i λα [ ) α s) α i α s ) α ] λ α α s) α i [ α s ) α ] λα [ ) α s) α i α s ) n ] ) n ] λ α α s) α i [ α λα α ) s) α i α s

9 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 5 [ s )][ s ) s ) 2 s ) n ] λ α α s) α i [ α s )] [ s ) s ) 2 s ) n ] n α s s) α i α 2 n λ α α s s) α i α n λα α λα ) α s s) α i. For s, λα α ) G, s) α s) α i α s) α λ ) [ α α s) α i α s) α ] λ α α s) α i α λ α α ) [ s) α α s) α ] λ α α s) α i α λ α α ) [ s) α 2 α 2 s) s) α 2 s) ] λ α α s) α i α α s) α 2 α 2[ s) s) ] λ α α s) α i α λα α ) s) α 2 α 2 s ) λ α α s) α i α λ α α ) i s s) α i α. λα α ) G, s) α s) α i α s) α λ α α ) [ s) α i α s) α ] λ α α s) α i α Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

10 52 L. Wang e al. λ α α ) [ s) α i α s) α s) α i] λ α α s) α i α λα [ ) α s) α i α λα [ ) α s) α i α λα α ) s) α i α s ) α ] λ α α s) α i α s ) n ] λ α α s) α i α [ s )][ s ) s ) 2 s ) n ] λ α α s) α i α n α s s) α i α 2 n λ α α s s) α i α n λα α λα ) α s s) α i. For s, λα α ) G, s) α s) α i λ α s)α α α s) α λ α α s ) α α α s) α λ α α s) α α [ λ ] α α s) s) α α α s) α α λ α α s s) α α λ α α ) i s s) α i α. λα α ) G, s) α s) α i λ α s)α α

11 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 53 α s) α i α λ α α s) α i α λ α α s ) α α α s) α i α λ α α [ α s) α i s ) α ] s) α i α s) α i α λα [ α s) α i α α s) α i α λα [ α s) α i α λα α ) s) α i α λα α s) α i α [ [ s n α ) s ) 2 s ) n ] s s) α i α 2 n λ α α s s) α i α n λα α λα ) α s s) α i. For s, s, λα α ) G, s) α s) α i α s) α i α λ α α s) α i α λ α α ) i s s) α i α. λα α ) G, s) s ) α ] s ) n ] s )] α s) α i α s) α i α λ α α s) α i α α s s) α i α 2 λ α α s s) α i α n λα α λα ) α s s) α i. From above, a) a3) are complee. The proof is complee. Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

12 54 L. Wang e al. Similarly, he general soluion of D β v) y), v) v ) v n 2) ), v j) ) b c vs) ds, where < <, n < β n, < c, j N, j n 2 and j is a fixed consan, 2 bc β /β >, {, j, 2 β )β 2) β j), j, j N and j is a fixed consan, is v) G 2, s)ys) ds, where G 2, s) can be obained from G, s) by replacing α, λ,, i wih β, b, c, j, correspondingly, and saisfy properies a) a3) wih α, λ,, i replaced by β, b, c, j in case of G 2, s), correspondingly. Lemma 5. See [3].) Le E be a Banach space and P E be a cone. Assume Ω and Ω 2 be wo bounded open subses in E such ha θ Ω and Ω Ω 2. Le operaor A : Ω 2 \Ω ) P P be compleely coninuous. Suppose ha one of he wo condiions: ) Au u for all u P Ω, Au u for all u P Ω 2 and 2) Au u for all u P Ω, Au u for all u P Ω 2 is saisfied. Then A has a fixed poin in Ω 2 \ Ω ) P. Lemma 6. See [4].) Le E be a Banach space and Ω E be closed and convex. Assume U is a relaively open subse of Ω wih θ U, and le operaor A : U Ω be a coninuous compac map. Then eiher ) A has a fixed poin in U or 2) here exiss u U and ϕ, ) wih u ϕau. 3 Main resuls and proof Le E C[, ] be he Banach space wih he maximum norm u max [,] u). Thus E E, ) is a Banach space wih he norm defined by u, v) max{ u, v } for all u, v) E E. We define he cone P E E by P { u, v) E E u), v), }.

13 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 55 Lemma 7. Le n < α, β n. Le F :, ] [, ) [, ) be coninuous and saisfy lim F, ). Assume ha here exiss < σ < such ha σ F ) is coninuous on [, ]. Then u) G, s)f s) ds is coninuous on [, ]. Proof. From he coninuiy of σ F ) and u) G, s) σ σ F s) ds, we know ha u). If u) u ) when for any [, ], hen he proof is complee. In he following we separae he process ino hree cases. Case,, ]). Owing o he coninuiy of σ F ), here exiss an M > such ha σ F ) M for all [, ], hen u) u) s) α s σ s σ F s) ds s) α i α λ s σ s σ F s) ds α α λ α s)α α λ s σ s σ F s) ds α α s) α s σ s σ F s) ds s) α i α λ s σ s σ F s) ds α α λ α s)α α λ s σ s σ F s) ds α α s) α s σ s σ F s) ds λ α s) α i α λ s σ s σ F s) ds α α M Mα σ s) α s σ ds M λ α λ α α s) α i α s σ ds B σ, α) λ α )Mα λ B σ, α i) α α ) λ α )MΓ σ) α σ λ α ) ). α α )Γ α i σ) Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

14 56 L. Wang e al. B ) menioned in he above funcions represens he Bea funcion. Case 2, ),, ]). u) u ) s) α s σ s σ F s) ds λ α α λ α α λ α α λ α α λ λ s) α i α s σ s σ F s) ds α s)α α s σ s σ F s) ds s) α i α s σ s σ F s) ds α s)α α s σ s σ F s) ds s) α s) α s σ s σ F s) ds s) α i α α λ α α ) λ α s)α α α λ α α ) ) ) s σ s σ F s) ds s σ s σ F s) ds s) α s) α s σ s σ F s) ds λ α λ α α M s) α i α α s) α s σ s σ F s) ds s) α s) α s σ ds ) s) α s σ s σ F s) ds s) α s σ s σ F s) ds s σ s σ F s) ds

15 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 57 M λ α )α α ) λ α α s) α i s σ ds M s) α s σ ds Mα σ α σ ) B σ, α) λ α )Mα α ) λ B σ, α i) α α ) λ α )MΓ σ) α σ λ α α α σ α α ) ). )Γ α i σ) Case 3, ], [, )). Similarly o he proof of Case 2, so we omi i. The proof is complee. From Lemma 3 we can wrie he sysem of BVPs ) as an equivalen sysem of inegral equaions u) v) G, s)f s, vs) ) ds,, G 2, s)g s, us) ) ds,, which can be proved in he same way as Lemma 3.3 in [6]. For convenience, he proof is omied. We define A : E E E E o be an operaor, i.e., Au, v)) G, s)s σ s σ f s, vs) ) ds, : A v), A 2 u) ). G 2, s)s σ2 s σ2 g s, us) ) ds Lemma 8. Le n < α, β n. Le f, g :, ] [, ) [, ) be coninuous and saisfy lim f, ), lim g, ). Assume ha here exiss < σ, σ 2 < such ha σ f, y), σ2 g, y) are coninuous on [, ] [, ). Then he operaor A : P P is compleely coninuous. Proof. For any u, v) P, we have ha u, v P { y E y), }. ) Since A v) G, s)s σ s σ f s, vs) ) ds, Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

16 58 L. Wang e al. we ge ha A : P P by Lemma 7 and he nonnegaiviy of f. Se v P and v c. If v P and v v <, hen v < c : c. By he coninuiy of σ f, y), we ge ha σ f, y) is uniformly coninuous on [, ] [, c], namely, for all ε >, exiss δ > δ < ), when y y 2 < δ, we have σ f, y ) σ f, y 2 ) < ε, for all [, ], y, y 2 [, c]. Obviously, if v v < δ, hen v ), v) [, c] and v) v ) < δ for all [, ]. Hence, we have σ f, v) ) σ f, v ) ) < ε 4) for all [, ], v P, v v < δ. I follows from 4), we can ge A v A v max A v) A v ) [,] max < ε [,] G, s)s σ s σ f s, vs) ) s σ f s, v s) ) ds G, s)s σ ds ε ε n λ α α λ α α ) λ α α ) n λ α α λ α α ) λ s s) α i s σ ds α α ) s) α i s σ ds ε n λ α α λ α α ) λ B2 σ, α i) α α ) ε n λ α α λ α α ) Γ2 σ )Γα i) λ α α ) Γ2 α i σ ). Owing o he arbirariness of v, we know ha A : P P is coninuous. Similarly, we can ge ha A 2 : P 2 P 2 is coninuous. So, we proved A : P P is coninuous. Le M P be bounded. Tha is o say here exiss a consan l > such ha u, v) l for all u, v) M. Since σ f, y), σ2 g, y) are coninuous on [, ] [, ), le L max [,],u,v) M { σ f, v)), σ2 g, u))}. Then, for each u, v) M, we have A v) G, s)s σ s σ f s, vs) ) ds L n λ α α λ α α ) λ s s) α i s σ ds α α ) L n λ α α λ α α ) Γ2 σ )Γα i) λ α α ) Γ2 α i σ ).

17 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 59 Hence, we have A v max A v) L n λ α α λ α α ) Γ2 σ )Γα i) [,] λ α α ) Γ2 α i σ ). Similarly, we have A 2 u max A 2u) L n 2 b β cβ b β cβ ) Γ2 σ 2 )Γβ j) [,] 2 b β cβ )Γβ) Γ2 β j σ 2 ). Thus, { Au, v) max A v, A 2 u } [,] { n λ α max α λ α α ) λ α α ) n 2 b β cβ b β cβ ) 2 b β cβ )Γβ) Γ2 σ )Γα i) Γ2 α i σ ), Γ2 σ 2 )Γβ j) Γ2 β j σ 2 ) } L. Therefore, AM) is bounded. Nex, we prove ha A is equiconinuous. Le, for all ε >, { ε λ α δ min α )Γ α i σ ) 2 n λ α )LΓ σ, ε 2 b } β cβ )Γ β j σ 2 ) ) 2 n 2 b β )LΓ σ. 2) Then, for any u, v) M,, 2 [, ] wih < 2 and < 2 < δ, we have A v 2 ) A v ) G 2, s)f s, vs) ) ds G, s)f s, vs) ) ds 2 2 s) α s σ s σ f s, vs) ) ds s) α i α 2 λ s σ s σ f s, vs) ) ds α α ) λ α s)α α 2 λ α α ) s σ s σ f s, vs) ) ds s) α s σ s σ f s, vs) ) ds Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

18 5 L. Wang e al. 2 s) α i α λ s σ s σ f s, vs) ) ds α α ) λ α s)α α λ α α ) s σ s σ f s, vs) ) ds 2 s) α s) α s σ s σ f s, vs) ) ds s) α i α 2 α λ α α ) λ α s)α α 2 α λ α α ) ) ) 2 s) α s σ s σ f s, vs) ) ds s σ s σ f s, vs) ) ds s σ s σ f s, vs) ) ds 2 s) α s) α s σ s σ f s, vs) ) ds 2 L λ α ) s)α i α 2 α λ α α ) 2 s) α s σ s σ f s, vs) ) ds 2 s) α s) α s σ ds L λ α )α 2 α ) λ α α 2 L 2 s) α s σ ds Lα σ 2 α σ ) ) s) α i s σ ds s σ s σ f s, vs) ) ds B σ, α) λ α )Lα 2 α λ α α ) ) λ α )LΓ σ ) α σ λ α α 2 α σ 2 α α ). )Γ α i σ ) B σ, α i)

19 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 5 Similarly, A2 u 2 ) A 2 u ) 2 b β )LΓ σ 2) β σ 2 2 b β cβ 2 β σ2 β 2 β ). )Γ β j σ 2 ) Hence, we figure on α σ 2 α σ, α 2 α, β σ2 2 β σ2, β 2 β in he following hree cases: Case. If < δ, 2 < 2δ, hen Case 2. If < 2 δ, hen Case 3. If δ < 2, hen α σ 2 α σ α σ ) α 2 α α ) α σ 2 α σ α σ 2 2δ) α σ < 2 n δ, α 2 α α 2 2δ) α < 2 n δ. 2 2 α σ 2 α σ α σ 2 δ α σ < 2 n δ, α 2 α 2 α δ α < 2 n δ. x α σ dx α σ ) 2 ) α σ 2 α σ )δ < 2 n δ, x α 2 dx α ) 2 ) α 2 2 α )δ < 2 n δ. Hence, A v 2 ) A v ) < ε 2 ε 2 ε. Similarly, A2 u 2 ) A 2 u ) < ε. Therefore, AM) is equiconinuous, and by Arzelà Ascoli s heorem, we obain ha AM) is a relaively compac se, hen we prove operaor A : P P is compleely coninuous. Now we give he following hree resuls of his paper. Theorem. Le n < α, β n. Le f, g :, ] [, ) [, ) be coninuous and saisfy lim f, ), lim g, ). Assume ha here exiss < σ, σ 2 < such ha σ f, y), σ2 g, y) are coninuous on [, ] [, ) and Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

20 52 L. Wang e al. here exis, ) and wo posiive consans ρ, ξ subjecing o ρ > max{ξα )/ m α ), ξβ )/m 2 β )}, where Furher suppose: m m 2 λ α α ) i λ α α b β cβ c) j 2 b β cβ i) for all, y) [, ] [, ξ], ii) for all, y) [, ] [, ρ], s σ s) α i ds, s σ2 s) β j ds. σ f, y) ξ m α, σ2 g, y) ξγβ) m 2 β ; σ f, y) ργ2 α i σ ) λ α α ) n λ α α λ α α )Γ2 σ ), σ2 g, y) ργ2 β j σ 2) 2 b β cβ ) n 2 b β cβ b β cβ )Γ2 σ 2 ). Then BVP ) has a leas one posiive soluion. Theorem 2. Le n < α, β n. Le f, g :, ] [, ) [, ) be coninuous and saisfy lim f, ), lim g, ). Assume ha here exiss < σ, σ 2 < such ha σ f, y), σ2 g, y) are coninuous on [, ] [, ). Suppose hey saisfy he following condiions: iii) here exis wo coninuous and nondecreasing funcions ϕ, ψ : [, ), ) such ha σ f, y) ϕy), σ2 g, y) ψy), y) [, ] [, ); iv) here exiss an r >, yielding { r max{ϕr), ψr)} > max n λ α α λ α α ) Γ2 σ ) λ α α Γ2 α i σ ), n 2 b β cβ b } β cβ ) Γ2 σ 2 ) 2 b. β cβ Γ2 β j σ 2 ) Then he BVP ) has a posiive soluion.

21 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 53 Theorem 3. Le n < α, β n. Le f, g :, ] [, ) [, ) be coninuous and saisfy lim f, ), lim g, ). Assume ha here exiss < σ, σ 2 < such ha σ f, y), σ2 g, y) are coninuous on [, ] [, ). Suppose hey saisfy he following condiion: v) here exiss wo posiive consans L, L 2 such ha, for all, ) [, ] [, ), σ f, y) σ f, x) L y x, σ2 g, y) σ2 g, x) L 2 y x, where n λ α α λ α α ) λ α α Γ2 σ )L Γ2 α i σ ) <, n 2 b β cβ b β cβ ) 2 b β cβ Γ2 σ 2 )L 2 Γ2 β j σ 2 ) <. Then he BVP ) has a unique soluion. In addiion, we can ge he unique soluion by consrucing ieraive sequence and error esimae of he n imes ieraed. Proof of Theorem. From he condiions we obain ρ > max{ξα )/m α ), ξβ )/m 2 β )} > ξ. We divide he demonsraion ino wo seps. Sep. Le Ω {u, v) P u < ξ, v < ξ} such ha u), v) ξ for any u, v) P Ω and for all [, ]. By condiion i) and Lemma 4, we ge A v ) G, s)s σ s σ f s, vs) ) ds ) G, s)s σ s σ f s, vs) ds ξ λ α α ) i m α λ α α ) α s s) α i s σ ds ξ. Hence, Similarly, Therefore, A v max A v) ξ v P Ω. [,] A 2 u max A2 u) ξ u P Ω. [,] Au, v) ξ u, v). Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

22 54 L. Wang e al. Sep 2. Le Ω 2 {u, v) P u < ρ, v < ρ}. For any u, v) P Ω 2, [, ], we have ha u), v) ρ. By condiion ii) and Lemma 4, we ge A v) G, s)s σ s σ f s, vs) ) ds ργ2α i σ ) λ α α ) n λ α α λ α α )Γ2 σ ) ρ. Then we obain Similarly, A v ρ u, v) P Ω 2. n λ α α λ α α ) λ s s) α i s σ ds α α ) A 2 u max A 2 u) ρ u, v) P Ω 2. [,] Therefore, Au, v) ρ u, v). Besides, by Lemma 8, operaor A : P P is compleely coninuous. Then wih Lemma 5, our proof is complee. Proof of Theorem 2. Le U {u, v) P u < r, v < r}, so ha U P. By Lemma 8, we ge o know ha operaor A : U P is compleely coninuous. And if here exiss u, v) U and λ, ), we have u, v) λau, v), hen by iii) for [, ], we obain u) λa v) λ Hence, G, s)f s, vs) ) ds < G, s)s σ ϕ vs) ) ds ϕ v ) G, s)s σ ds ϕ v )n λ α α λ α α ) λ α α ) G, s)s σ s σ f s, vs) ) ds s) α i s σ ds ϕ v )n λ α α λ α α ) λ B2 σ, α i) α α ) ϕ u, v) )n λ α α λ α α ) λ α α Γ2 σ ) Γ2 α i σ ). u < ϕ u, v) )n λ α α λ α α ) λ α α Γ2 σ ) Γ2 α i σ ),

23 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 55 i.e., Similarly, u ϕ u, v) ) < n λ α α λ α α ) Γ2 σ ) λ α α Γ2 α i σ ). v ψ u, v) ) < n 2 b β cβ b β cβ ) Γ2 σ 2 ) 2 b β cβ Γ2 β j σ 2 ). Consequenly, { u, v) max{ϕ u, v) ), ψ u, v) )} < max n λ α α λ α α ) Γ2 σ ) λ α α Γ2 α i σ ), n 2 b β cβ b } β cβ ) Γ2 σ 2 ) 2 b. β cβ Γ2 β j σ 2 ) Again by iv) we know u, v) r which conradics ha u, v) U. Then based on Lemma 6, here is a fixed poin u, v) U. Therefore he BVP ) has a posiive soluion. Proof of Theorem 3. We shall use Banach fixed poin heorem. From v), for any v, v 2 P, [, ], we can ge ha A v 2 ) A v ) G, s)s σ s σ f s, v 2 s) ) s σ f s, v s) ) ds L v 2 ) v ) n λ α α λ α α ) λ α α ) s) α i s σ ds L v2 ) v ) n λ α α λ α α ) λ B2 σ, α i) α α ) L v 2 ) v ) n λ α α λ α α ) λ α α Γ2 σ ) Γ2 α i σ ). Similarly, for any u, u 2 P, [, ], A2 u 2 ) A 2 u ) n 2 b β cβ b β cβ ) Γ2 σ 2 ) 2 b β cβ Γ2 β j σ 2 ) L 2u 2 ) u ). Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

24 56 L. Wang e al. So, we have max Au2, v 2 ) Au, v ) where [,] Au2, v 2 ) Au, v ) A v 2, A 2 u 2 ) A v, A 2 u ) A v 2 A v, A 2 u 2 A 2 u ) L v2 v, u 2 u ), { n L λ α max α λ α α ) Γ2 σ )L λ α α Γ2 α i σ ), n 2 b β cβ b β cβ ) 2 b β cβ Γ2 σ 2 )L 2 Γ2 β j σ 2 ) } <. By he conracion mapping principle, he BVP ) has a unique soluion. Example. For any n < α, β n, ake /4, ξ >, and ρ > max{4 α ξα )/m, 4 β ξβ )/m 2 }, ρ >. Choose σ σ 2 /2. Consider he boundary value problem o he singular sysem of fracional equaions D α u) c v, < <, D β v) c 2 u, < <, u) u ) u n 2) ), u) /2 us) ds, 2 v) v ) v n 2) ), v) /2 vs) ds, 2 5) where c, c 2 are consans saisfying µn/α) α 4 α ξ c ρ[γ2 α σ ) α µ α /α) )Γ2 σ )] m [α µn/α)α µ α /α) ]Γ2 σ, ) µn/β) β 4 β ξγβ) c 2 ρ[γ2 β σ 2) β µ β /β) )Γ2 σ 2)] m 2 [β µn/β)β µ β /β) ]Γ2 σ. 2) Denoe f, y) c y)/, g, y) c 2 y)/. Then f, g are coninuous on, ] [, ) and lim f, ), lim g, ). All condiions of Theorem 3 hold. Therefore, BVP 5) has a leas one posiive soluion.

25 Exisence and uniqueness of soluions for singular sysem wih inegral boundary condiions 57 Example 2. Consider he boundary value problem o he following singular sysem of fracional equaions: D 9/2 u) 2 )2 ln2 v)), < <, D 3/3 v) 2 )2 ln2 u)), < <, u) u ) u ) u 3) ), u) /2 us) ds, 2 v) v ) v ) v 3) ), v) /2 vs) ds. 2 6) Denoe f, g are coninuous on, ] [, ) and lim f, ), lim g, ). Choose σ σ 2 /2 and ϕy) ψy) ln2 y), hen we have /2) 2 ln2 v))/ ln2 v)) for all, y) [, ] [, ). ϕ, ψ : [, ), ) are coninuous, nondecreasing funcions, so, condiion iii) of Theorem 2 holds. Nex, se r. Then condiion iv) of Theorem 2 holds. Therefore, BVP 6) has a leas one posiive soluion. Example 3. Consider he boundary value problem o he following singular sysem of fracional equaions: D 3/2 u) )3 v) ) 2, < <, D 2/3 v) 3 )2 u) ) 5 3, < <, u) u ) u 5) ), u ) 2 v) v ) v 5) ), v ) 3 4 /3 2/3 us) ds, vs) ds, 7) where f, v)) /) 3 v) )/2, g, u)) /3) 2 u) )/ 5 3, and f, g are coninuous on, ] [, ) and lim f, ), lim g, ). Choose σ /2, σ 2 /3. Clearly, σ f, v 2 )) σ f, v )) 33/2) v 2 v, σ2 g, u 2 )) σ2 f, u )) 6/35) u 2 u for all [, ]. By a simple calculaion, we know L <. All condiions of Theorem 3 hold. Therefore, BVP 7) has a unique soluion. Nonlinear Anal. Model. Conrol, 23, Vol. 8, No. 4,

26 58 L. Wang e al. References. C. Bai, J. Fang, The exisence of aposiive soluion for a singular coupled sysem of nonlinear fracional differenial equaions, Appl. Mah. Compu., 5:6 62, Z. Bai, H. Lü, Posiive soluions for boundary value problem of nonlinear fracional differenial equaion, J. Mah. Anal. Appl., 3:495 55, W. Feng, S. Sun, Z. Han, Y. Zhao, Exisence of soluions for a singular sysem of nonlinear fracional differenial equaions, Compu. Mah. Appl., 62:37 378, C.S. Goodrich, Exisence of a posiive soluion o a class of fracional differenial equaions, Appl. Mah. Le., 23:5 55, C.S. Goodrich, Exisence of a posiive soluion o sysems of differenial equaions of fracional order, Compu. Mah. Appl., 62:25 268, X. Su, Boundary value problem for a coupled sysem of nonlinear fracional differenial equaions, Appl. Mah. Le., 22:64 69, X. Xu, D. Jiang, C. Yuan, Muliple posiive soluions o singular posione and semiposione Dirichle-ype boundary value problems of nonlinear fracional differenial equaions, Nonlinear Anal. Theory Mehods Appl., 74: , J. Xu, Z. Wei, W. Dong, Uniqueness of posiive soluions for a class of fracional boundary value problems, Appl. Mah. Le., 25:59 593, R.P. Agarwal, D. O Regan, Posiive soluions for p, n p) conjugae boundary value problems, J. Differ. Equaions, 5: , L. Kong, J. Wang, The Green s funcion for k, n k) conjugae boundary value problems and is applicaions, J. Mah. Anal. Appl., 255:44 422, 2.. R. Ma, Posiive soluions for semiposione k, n k) conjugae boundary value problems, J. Mah. Anal. Appl., 252:22 229, L. Zu, D. Jiang, Y. Gai, D. O Regan, H. Gao, Weak singulariies and exisence of soluions o k, n k) conjugae boundary value problems, Nonlinear Anal., Real World Appl., : , M.A. Krasnoselskii, Posiive Soluion of Operaor Equaion, Noordhoff, Groningen, G. Isac, Leray Schauder Type Alernaives Complemenariy Problem and Variaional Inequaliies, Springer, Berlin, 26.