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Positive Solutions for a System of Nonlinear Semipositone Boundary Value Problems with Riemann-Liouville Fractional Derivatives.

1. Introduction

In this paper, we investigate the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives

[mathematical expression not reproducible], (1)

where [alpha] [member of] (2, 3] is a real number and [D.sup.[alpha].sub.0+] is the standard Riemann-Liouville fractional derivative of order [alpha]. The nonlinear terms [f.sub.i] [member of] C([0, 1] x [R.sup.4.sub.+], R) ([R.sub.+] = [0, +[infinity]), R = (-[infinity], +[infinity])) are bounded below; that is, [f.sub.i] (i = 1, 2) satisfy the following.

(H1) there exists a real number M [greater than or equal to] 0, such that [f.sub.i](t, [x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]) + M [greater than or equal to] 0, [for all]t [member of] [0, 1], [x.sub.j] [member of] [R.sub.+], i = 1, 2, j = 1, 2, 3, 4.

Existence and multiplicity of solutions for fractional differential equations are widely studied in the literature; see [1-14] and the references therein. For example, in [1], the authors used the Guo-Krasnosel'skii fixed point theorem to investigate the existence of positive solutions for the singular fractional differential system

[mathematical expression not reproducible], (2)

where [f.sub.i] (i = 1, 2) satisfy

[f.sub.i] (t, u, v)/u + v = 0, or [infinity],

as u + v [right arrow] +[infinity], uniformly for t [member of] [0, 1] or a subinterval.

Condition (3) is used to study various types of fractional systems (see [1-12] and the references therein).

In this paper we use the fixed point index to study the existence of positive solutions for the system of nonlinear semipositone fractional boundary value problem (1). Under some appropriate conditions for [f.sub.i] (i = 1, 2), we use the fixed point index to obtain our results. Moreover, nonnegative concave and convex functions are used to depict the coupling behavior of our nonlinearities (see [13-15]), which depend on the unknown functions u, v and their derivatives u', v'.

2. Preliminary

Definition 1 (see [16, 17]). The Riemann-Liouville fractional derivative of order [alpha] > 0 of a continuous function f : (0, +[infinity]) [right arrow] R is given by

[D.sup.[alpha].sub.0+] f(t) = 1/[GAMMA] (n - [alpha]) [(d/dt).sup.n] [[integral].sup.t.sub.0] [(t - s).sup.n-[alpha]-1] f(s) ds, (4)

where n = [[alpha]] + 1 with [[alpha]] denoting the integer part of a number [alpha], provided that the right hand side is pointwise defined on (0, +[infinity]).

We first study the Green functions of problem (1). Let

[mathematical expression not reproducible]. (5)

Then we have

[mathematical expression not reproducible]. (6)

Lemma 2. Let [f.sub.i] (i = 1, 2) be as in (1). Then (1) is equivalent to

[mathematical expression not reproducible], (7)

which takes the form

[mathematical expression not reproducible]. (8)

Let [D.sup.[alpha].sub.0+]u = -x, -[D.sup.[alpha]0+]v = -y. Then an argument similar to that in [18, Lemma 2.7] and [19, Lemma 3] establishes the result (we omit the standard details).

Lemma 3 ([19, Lemma 4]). The functions [G.sub.i](t, s) [member of] C([0, 1] x [0, 1], [R.sub.+]) (i = 1, 2). Moreover, the following inequalities are satisfied:

[mathematical expression not reproducible]. (9)

Lemma 4 ([19, Lemma 5]). Let [phi](t) = t[(1-t).sup.[alpha]-2] for all t [member of] [0, 1]. Let

[mathematical expression not reproducible]. (11)

Then

[mathematical expression not reproducible]. (12)

Lemma 5. (i) If ([x.sub.*](t), [y.sub.*](t)) is a positive solution of (7), then ([x.sub.*](t) + w(t), [y.sub.*](t) + w(t)) is a positive solution of the following differential equation:

[mathematical expression not reproducible], (13)

where

[mathematical expression not reproducible], (14)

and [mathematical expression not reproducible] are continuous, and

[mathematical expression not reproducible]. (15)

(ii) If (x(t), y(t)) is a solution of (13) and x(t) [greater than or equal to] w(t), y(t) [greater than or equal to] w(t), t [member of] [0, 1], then ([x.sub.*](t), [y.sub.*](t)) = (x(t)-w(t), y(t)-w(t)) is a positive solution of (7).

Proof. If ([x.sub.*](t), [y.sub.*](t)) is a positive solution of (7) then (note w(t) = M [[integral].sup.1.sub.0] [G.sub.1](t, s) ds) we obtain [x.sub.*](0) + w(0) = [x'.sub.*](0) + w'(0) = [x'.sub.*](1) + w'(1) = 0 and

[mathematical expression not reproducible], (16)

Similarly, we have

[mathematical expression not reproducible]; (17)

that is, ([x.sub.*](t) + w(t), [y.sub.*](t) + w(t)) satisfies (13). Therefore, (i) holds. Similarly, it is easy to prove (ii). This completes the proof.

From Lemma 5, to obtain a positive solution of (7), we only need to find solutions x(t), y(t) of (13) satisfying (t) [greater than or equal to] w(t), y(t) [greater than or equal to] w(t), t [member of] [0, 1], If x(t), y(t) are solutions of (13), then x(t), y(f)satisfy

[mathematical expression not reproducible], (18)

Let E := C[0, 1], [parallel]x[parallel] = [max.sub.t[member of][0,1]] [absolute value of x(t)], P := {x [member of] E : x(t) [greater than or equal to] 0, t [member of] [0, 1]}. Then (E, [parallel]*[parallel]) is a real Banach space, and P is a cone on E. We denote [B.sub.[rho]] := {x [member of] E : [parallel]x[parallel] < [rho]} for [rho] > 0. Now, note that u, v solve (1) if and only if x := -[D.sup.[alpha].sub.0+]u, y := -[D.sup.[alpha].sub.0+]v are fixed points of operator

[mathematical expression not reproducible]. (19)

Therefore, if (x, y) is a positive fixed for A with x(t) [greater than or equal to] w(t), y(t) [greater than or equal to] w(t) for t [member of] [0, 1], then ([x.sub.*], [y.sub.*]) = (x-w, y-w) is a positive solution for (1). Moreover, from the continuity of [G.sub.i] and [F.sub.i] (i = 1, 2), we know that [A.sub.i] : P x P [right arrow] P, A : P x P [right arrow] P x P are continuous and completely continuous operators.

Lemma 6. Let [P.sub.0] := {x [member of] P : x(t) [greater than or equal to] [t.sup.[alpha]-1] [parallel]x[parallel], t [member of] [0, 1]}. Then [P.sub.0] is a cone in [member of] and A(P x P) [subset] [P.sup.2.sub.0].

Proof. From (9) for t [member of] [0, 1] we have

[mathematical expression not reproducible]. (20)

Also from (9) and the above inequality, for every (x, y) [member of] P x P, we obtain

[mathematical expression not reproducible] (21)

for all t [member of] [0, 1]. Similarly, [A.sub.2](x, y)(t) [greater than or equal to] [t.sup.[alpha]-1] [parallel][A.sub.2](x, y)[parallel]. Therefore A(P x P) [subset] [P.sup.2.sub.0]. This completes the proof.

To obtain a positive solution of (1), we seek a positive fixed point ([x.sup.*], [y.sup.*]) of A with [x.sup.*] [greater than or equal to] w, [y.sup.*] [greater than or equal to] w (note mean that [x.sup.*](t) = [A.sub.1]([x.sup.*], [y.sup.*])(t), [y.sup.*](t) = [A.sub.2]([x.sup.*], y[.sup.*])(t) for t [member of] [0, 1]). From Lemma 6, we have [x.sup.*], [y.sup.*] [member of] [P.sub.0]. For [x.sup.*] [member of] [P.sub.0] we have

[mathematical expression not reproducible]. (22)

As a result, [x.sup.*](t) [greater than or equal to] w(t) for t [member of] [0, 1] if [parallel][x.sup.*][parallel] [greater than or equal to] M/([alpha]-1)[GAMMA]([alpha]) := [k.sub.5]. Similarly, if [parallel][y.sup.*][parallel] [greater than or equal to] [k.sub.5], we have [y.sup.*](t) [greater than or equal to] w(t),for t [member of] [0, 1].

Lemma 7 (see [20]). Let [OMEGA] [subset] [member of] be a bounded open set and A : [bar.[OMEGA]][intersection]P [right arrow] P a continuous or completely continuous operator. If there exists [u.sub.0] [member of] P \ {0} such that u-Au [not equal to] [mu][u.sub.0] for all [mu] [greater than or equal to] 0 and u [member of] [partial derivative][OMEGA] [intersection] P, then i(A, [OMEGA] n P, P) = 0, where i denotes the fixed point index on P.

Lemma 8 (see [20]). Let [OMEGA] [subset] E be a bounded open set with 0 [member of] [OMEGA]. Suppose A : [bar.[OMEGA]] [intersection] P [right arrow] P is a continuous or completely continuous operator. If u [not equal to] [mu]Au for all u [member of] [partial derivative][OMEGA] [intersection] P and 0 [less than or equal to] [mu] [less than or equal to] 1, then i(A, [OMEGA] [intersection] P, P) = 1.

3. Main Results

Let K := [alpha]/[GAMMA]([alpha]) [greater than or equal to] [max.sub.t,s[member of][0,1]] ([G.sub.1](t, s) + [G.sub.2](t, s)), In the sequel, we use [c.sub.1], [c.sub.2], ... and [d.sub.1], [d.sub.2], ... to stand for different positive constants. Now, we list our assumptions on [F.sub.i] (i = 1, 2).

(H2) There exist h, g [member of] C([R.sub.+], [R.sub.+]) such that

(i) h is concave and strictly increasing on [R.sub.+] (and [lim.sub.x[right arrow]+[infinity]] h(x) = +[infinity]);

(ii) there exist [c.sub.1] > 0, [d.sub.1] > 1/[k.sup.2.sub.1][([k.sub.1] + [k.sub.3]).sup.2], for all (t, [x.sub.1], [x.sub.2], [y.sub.1], [y.sub.2]) [member of] [0, 1] x [R.sup.4.sub.+] such that

[mathematical expression not reproducible]; (23)

(iii) h([K.sup.2]g(x)) [greater than or equal to] [K.sup.2]x - [c.sub.1] for x [member of] [R.sub.+].

(H3) For all (t, [x.sub.1], [x.sub.2], [y.sub.1], [y.sub.2]) [member of] [0, 1] x [[0, [k.sub.5]].sup.4], there is a constant [M.sub.1] [member of] (0, [k.sub.5][k.sup.-1.sub.2]) such that

[F.sub.i] (t, [x.sub.1], [x.sub.2], [y.sub.1], [y.sub.2]) [less than or equal to] [M.sub.1], i = 1, 2. (24)

(H4) There exist [beta], [gamma] [member of] C([R.sub.+], [R.sub.+]) such that

(i) [beta] is convex and strictly increasing on [R.sub.+] (and [lim.sub.x[right arrow]+[infinity]] [beta](x) = +[infinity]);

(ii) for all (t, [x.sub.1], [x.sub.2], [y.sub.1], [y.sub.2]) [member of] [0, 1] x [R.sup.4.sub.+],

[mathematical expression not reproducible]; (25)

(iii) there exist [d.sub.2] > 0 such that [beta]([K.sup.2][gamma](x)) [less than or equal to] [K.sup.2]x + [d.sub.2], for x [member of] [R.sub.+].

(H5) There exist Q : [0, 1][right arrow] R, [theta] [member of] (0, 1], [t.sub.0] [member of] [[theta], 1], for all (t, [x.sub.1], [x.sub.2], [y.sub.1], [y.sub.2]) [member of] [[theta], 1] x [[0, [k.sub.5]].sup.4], such that

[f.sub.i] (t, [x.sub.1], [x.sub.2], [y.sub.1], [y.sub.2]) + M [greater than or equal to] Q(t), i = 1, 2, (26)

where

[[integral].sup.1.sub.[theta]] [G.sub.1] ([t.sub.0], s) Q(s) ds > M/([alpha] - 1) [GAMMA]([alpha]). (27)

Theorem 9. Suppose that (H1)-(H3) hold. Then (1) has at least one positive solution.

Proof. We first prove that there exists R > [k.sub.5] such that

(x, y) [member of] A (x, y) + [lambda] ([phi], [phi]), [for all] (x, y) [member of] [partial derivative][B.sub.R] [intersection] (P x P), [lambda] [greater than or equal to] 0, (28)

where [phi] [member of] [P.sub.0] is a given function. Suppose there exist [mathematical expression not reproducible]. From (i), (ii) of (H2) we have

[mathematical expression not reproducible]. (29)

From (ii) of (H2) we have

[mathematical expression not reproducible]. (30)

From (30) and (i) of (H2) we obtain

[mathematical expression not reproducible]. (31)

This together with (iii) of (H2) yields

[mathematical expression not reproducible]. (32)

Then (32) is substituted into (29) and we obtain

[mathematical expression not reproducible]. (33)

Multiplying by [phi](t) for (33) and integrating over [0, 1], we have

[mathematical expression not reproducible], (34)

using the fact that

[mathematical expression not reproducible], (35)

which can be derived from (12) in Lemma 4. From (34) we obtain

[mathematical expression not reproducible]. (36)

Note that x [member of] [P.sub.0] (note that x = [A.sub.1](x, y) + [lambda][phi] and [A.sub.1](x, y) [member of] [P.sub.0] from Lemma 6 and [phi] [member of] [P.sub.0]) and we have

[mathematical expression not reproducible]. (37)

From (29) we have

[mathematical expression not reproducible]. (38)

Multiplying by [phi](t) and integrating over [0, 1] we obtain

[mathematical expression not reproducible]. (39)

Consequently, we have

[mathematical expression not reproducible]. (40)

Note that we may assume y(t) [not equivalent to] 0 or t [member of] [0, 1]. Then [parallel]y[parallel] > 0 and h(K[parallel]y[parallel]) > 0. For y [member of] [P.sub.0], we have

[mathematical expression not reproducible]. (41)

Hence, h(K[parallel]y[parallel]) [less than or equal to] [N.sub.1]/[k.sub.1][GAMMA]([alpha]). Note [lim.sub.z[right arrow]+[infinity]] h(z) = +[right arrow], and thus there exists [N.sub.2] [greater than or equal to] 0 such that [parallel]Ky[parallel] [less than or equal to] [N.sub.2]. Therefore if [mathematical expression not reproducible]. Thus if we take R > max{[k.sub.5], [c.sub.6][k.sub.2]/([d.sub.1][k.sup.3.sub.1] [([k.sub.1] + [k.sub.3]).sup.2] - [k.sub.1]), [N.sub.2]/K] then (28) is true. Lemma 7 implies

i(A, [B.sub.R] [intersection] (P x P), P x P) = 0. (42)

Let [mathematical expression not reproducible]. From (H3) we have

[mathematical expression not reproducible], (43)

so [parallel][A.sub.1](x, y)[parallel] < [parallel]x[parallel]. Similarly [parallel][A.sub.2](x, y)[parallel] < [parallel]y[parallel]. Hence [mathematical expression not reproducible]. Thus

[mathematical expression not reproducible]. (44)

It follows from Lemma 8 that

[mathematical expression not reproducible]. (45)

From (42) and (45) we have

[mathematical expression not reproducible]. (46)

Therefore the operator A has at least one fixed point in [mathematical expression not reproducible] and so (1) has at least a positive solution. This completes the proof.

Theorem 10. Suppose that (H1), (H4), and (H5) hold. Then (1) has at least one positive solution.

Proof. We first show that there exists R > [k.sub.5] such that

[mathematical expression not reproducible]. (47)

Suppose there exist [mathematical expression not reproducible]. From (i), (ii) of (H4), we have

[mathematical expression not reproducible]. (48)

From (ii) of (H4), we get

[mathematical expression not reproducible]. (49)

From (49) and (i), (iii) of (H4) we have

[mathematical expression not reproducible]. (50)

Then substitute (50) into (48) and we obtain

[mathematical expression not reproducible]. (51)

Multiplying by [phi](t) for (51) and integrating over [0, 1], from (35), we obtain

[mathematical expression not reproducible]. (52)

Consequently, we have

[mathematical expression not reproducible]. (53)

Note that x [member of] [P.sub.0] (note x = [lambda][A.sub.1](x, y) and [A.sub.1](x, y) [member of] [P.sub.0]) and we have

[mathematical expression not reproducible]. (54)

From (50) and Lemma 3 we have

[mathematical expression not reproducible], (55)

so [beta](Ky(t)) is bounded. Note [lim.sub.z[right arrow]+[infinity]] [beta](z) = +[infinity], and thus there exists [N.sub.3] [greater than or equal to] 0 such that [parallel]Ky[parallel] [less than or equal to] [N.sub.3]. Therefore if [mathematical expression not reproducible]. Thus if we take [mathematical expression not reproducible] then (47) is true. Lemma 8 implies

i (A, [B.sub.R] [intersection] (P x P), P x P) = 1. (56)

Let [mathematical expression not reproducible]. It follows from (H5) that

[mathematical expression not reproducible], (57)

and, hence, [parallel][A.sub.1](x,y)[parallel] [greater than or equal to] [A.sub.1](x, y)([t.sub.0]) > [parallel]x[parallel]. Similarly [mathematical expression not reproducible]. Therefore [mathematical expression not reproducible]. Thus

[mathematical expression not reproducible]. (58)

It follows from Lemma 7 that

[mathematical expression not reproducible]. (59)

From (56) and (59), we have

[mathematical expression not reproducible]. (60)

Therefore the operator A has at least one fixed point on [mathematical expression not reproducible] and so (1) has at least one positive solution, which completes the proof.

https://doi.org/10.1155/2018/7351653

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (Grant no. 11601048), Natural Science Foundation of Chongqing (Grant no. cstc2016jcyjA0181), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant no. KJ1703050), and Natural Science Foundation of Chongqing Normal University (Grant nos. 16XYY24, 15XLB011).

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Xiaowei Qiu, (1) Jiafa Xu (iD), (1) Donal O'Regan, (2) and Yujun Cui (iD) (3)

(1) School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

(2) School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

(3) State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Jiafa Xu; xujiafa292@sina.com

Received 18 January 2018; Accepted 28 March 2018; Published 6 May 2018

Academic Editor: Gennaro Infante
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