# Existence of Weak Solutions for Fractional Integrodifferential Equations with Multipoint Boundary Conditions.

1. Introduction

In recent years, fractional differential equations in Banach spaces have been studied and a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology. As long as the Banach space is reflexive, the weak compactness offers no problem since every bounded subset is relatively weakly compact and therefore the weak continuity suffices to prove nice existence results for differential and integral equations [1, 2]. De Blasi [3] introduced the concept of measure of weak noncompactness and proved the analogue of Sadovskiis fixed point theorem for the weak topology (see also [4]). As stressed in [5], in many applications, it is always not possible to show the weak continuity of the involved mappings, while the sequential weak continuity offers no problem. This is mainly due to the fact that Lebesgues dominated convergence theorem is valid for sequences but not for nets. Recall that a mapping between two Banach spaces is sequentially weakly continuous if it maps weakly convergent sequences into weakly convergent sequences.

The theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored. There are many papers dealing with multipoint boundary value problems both on resonance case and on nonresonance case; for more details see [6-11]. However, as far as we know, few results can be found in the literature concerning multipoint boundary value problems for fractional differential equations in Banach spaces and weak topologies. Zhou et al. [12] discuss the existence of solutions for nonlinear multipoint boundary value problem of integrodifferential equations of fractional order as follows:

[mathematical expression not reproducible], (1)

with respect to strong topology, where [sup.c][D.sup.[alpha].sub.0+] denotes the fractional Caputo derivative and the operators given by

(Hx) (t) = [[integral].sup.t.sub.0] g (t, s) x (s) ds,

(Kx) (t) = [[integral].sup.t.sub.0] g (t, s) x (s) ds. (2)

Moreover, theory for boundary value problem of integrodifferential equations of fractional order in Banach spaces endowed with its weak topology has been few studied until now. In [13], we discussed the existence theorem of weak solutions nonlinear fractional integrodifferential equations in nonreflexive Banach spaces E:

[mathematical expression not reproducible], (3)

and obtain a new result by using the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals, where [sup.c][D.sup.[alpha].sub.0+] denotes the fractional Caputo derivative and the operators given by

(Tx) (s) = [[integral].sup.s.sub.0] [k.sub.1] (s, [tau]) g ([tau], x) ([tau])) d[tau]

(Sx) (s) = [[integral].sup.1.sub.0] [k.sub.2] (s, [tau]) h ([tau], x) ([tau])) d[tau]. (4)

Our analysis relies on the Krasnoselskii fixed point theorem combined with the technique of measure of weak noncompactness.

Motivated by the above works, in this paper, we use the techniques of measure of weak noncompactness combine with the fixed point theorem to discuss the existence theorem of weak solutions for a class of nonlinear fractional integrodifferential equations of the form

[mathematical expression not reproducible], (5)

where T and S are two operators defined by

(Tu) (t) = [[integral].sup.t.sub.0] [k.sub.1] (t, s) g (s, u (s)) ds,

(Su) (t) = [[integral].sup.a.sub.0] [k.sub.2] (t, s) h (s, u (s)) ds. (6)

E is a nonreflexive Banach space, [sup.c][D.sup.[alpha].sub.0+] denotes the fractional Caputo derivative, [mathematical expression not reproducible] are given functions satisfying some assumptions that will be specified later, the integral is understood to be the Henstock-Kurzweil-Pettis, and solutions to (5) will be sought in E = C(I, [E.sub.[omega]]).

The problems of our research are different between this paper and paper [13]. In paper [13], we studied two point boundary value problem by using the corresponding Green's function and fixed point theorems; moreover, we get some good results. In this paper, we use the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals to discuss the existence theorem of weak solutions for a class of the multipoint boundary value problem of fractional integrodifferential equations equipped with the weak topology. Our results generalized some classical results and improve the assumptions conditions, so our results improve the results in [13].

The paper is organized as follows: In Section 2 we recall some basic known results. In Section 3 we discuss the existence theorem of weak solutions for problem (5).

2. Preliminaries

Throughout this paper, we introduce notations, definitions, and preliminary results which will be used.

Let I = [0,1] be the real interval, let E be a real Banach space with norm [parallel] x [parallel], its dual space [E.sup.*] also B([E.sup.*]) denotes the closed unit ball in [E.sup.*], and [E.sub.w] = (E,w) = (E, [sigma](E, [E.sup.*])) denotes the space E with its weak topology. Denote by C(I, [E.sub.w]) = (C(I, E), [omega]) the space of all continuous functions from I to E endowed with the weak topology and the usual supremumnorm [parallel]x[parallel] = [sup.sub.t[member of]I] [absolute value of x(i)].

Let [[OMEGA].sub.E] be the collection of all nonempty bounded subsets of E, and let [W.sub.E] be the subset of [[OMEGA].sub.E] consisting of all weakly compact subsets of E. Let [B.sub.r] denote the closed ball in E centered at 0 with radius r > 0. The De Blasi [14] measure of weak noncompactness is the map [beta] : [[OMEGA].sub.E] [right arrow] [0, [infinity]) defined by

[beta](A) = inf {r > 0 : there exists a set W

[member of] [W.sub.E] such that A [subset not equal to] W + [B.sub.r]} (7)

for all A [member of] [[OMEGA].sub.E]. The fundamental tool in this paper is the measure of weak noncompactness; for some properties of [beta](A) and more details see [3].

Now, for the convenience of the reader, we recall some useful definitions of integrals.

Definition 1 (see [15]). A function u : I [right arrow] E is said to be Henstock-Kurzweil integrable on I if there exists an J [member of] E such that, for every [epsilon] >0, there exists [delta]([xi]) : I [right arrow] [R.sup.+] such that, for every 5-fine partition D = [{([I.sub.i], [[xi].sub.i])}.sup.n.sub.i=1], we have

[parallel] [n.summation over (i=1)] u ([[xi].sub.i]) [mu] ([I.sub.i]) - J[parallel] < [epsilon], (8)

and we denote the Henstock-Kurzweil integral J by (HK) [[integral].sup.b.sub.a] u(s)ds.

Definition 2 (see [15]). A function f : I [right arrow] E is said to be Henstock-Kurzweil-Pettis integrable or simply HKP-integrable on I, if there exists a function g : I [right arrow] E with the following properties:

(i) [for all][x.sup.*] [member of] [E.sup.*], [x.sup.*] f is Henstock-Kurzweil integrable on I;

(ii) [for all]t [member of] I, [for all][x.sup.*] [member of] [E.sup.*], [x.sup.*]g(t) = (HK) [[integral].sup.t.sub.0] [x.sup.*] f(s)ds.

This function g will be called a primitive of f and be denote by g(t) = [[integral].sup.t.sub.0] f(t)dt the Henstock-Kurzweil-Pettis integral of f on the interval I.

Definition 3 (see [16]). A family M of functions f : S [right arrow] E is called HK-equi-integrable if each f [member of] M is HK-integrable and for every [epsilon] > 0 there exists a gauge [delta] on S such that, for every 5-fine HK-partition n of S, we have

[parallel] [k.summation over (I,s)[member of][pi]] f(s) [[lambda].sub.m](I) - (HK) [[integral].sub.S] [parallel] [less than or equal to] [epsilon], (9)

for all f [member of] M.

Theorem 4 (see [16]). Let ([f.sub.n]) be a pointwise bounded sequence of HKP integrable functions fn : S [right arrow] E and let f : S [right arrow] E be a function. Assume that,

(i) for every [x.sup.*] [member of] [E.sup.*], [x.sup.*] ([f.sub.n](t)) [right arrow] [x.sup.*] (f(t)) a.e. on S,

(ii) for every sequence ([x.sup.*.sub.k]) c B([E.sup.*]), the sequence [([x.sup.*.sub.k]([f.sub.n])).sub.k,n] is HK-equi-integrable, then f is HKP-integrable and for every I [member of] J, and we have

[mathematical expression not reproducible] (10)

in the weak topology [sigma](E, [E.sup.*]), where F is the HKP-primitive of f and S is a fixed compact nondegenerate interval in [R.sup.n]. Denote by J the family of all closed nondegenerate subintervals of S.

Lemma 5 (see [17]). If B [subset] C(I, E) is equicontinuous, [u.sub.0] [member of] C(I, E), then [bar.co]{B, [u.sub.0]} is also equicontinuous in C(I, E).

Lemma 6 (see [17, 18]). Let E be a Banach space, and let B c C(I,E) be bounded and equicontinuous. Then [beta](B(t)) is continuous on I, and [beta](B) = [max.sub.teI][beta](B(t)).

Lemma 7 (see [14, 19]). Let E be a Banach space and let B [subset] C(I, E) be bounded and equicontinuous. Then the map t [right arrow] [beta](B(t)) is continuous on I and

[mathematical expression not reproducible]. (11)

Lemma 8 (see [17]). Let B c C(I,E) be bounded and equicontinuous. Then [beta](B(t)) is continuous on I and

[beta]([[integral].sub.I] B(s)ds) [less than or equal to] [[integral].sub.I] [beta](B(s))ds. (12)

We give the fixed point theorem, which play a key role in the proof of our main results.

Lemma 9 (see [20]). Let E be a Banach space and [beta] a regular and set additive measure of weak noncompactness on E. Let C be a nonempty closed convex subset of E, [x.sub.0] [member of] C, and [n.sub.0] a positive integer. Suppose F : C [right arrow] C is [beta]-convex power condensing about [x.sub.0] and [n.sub.0]. If F is weakly sequentially continuous and F(C) is bounded, then F has a fixed point in C.

The following we recall the definition of the Caputo derivative of fractional order.

Definition 10. Let x : I [right arrow] E be a function. The fractional HKP-integral of the function x of order [alpha] [member of] [R.sub.+] is defined by

[I.sup.[alpha].sub.0+] x(t) := [[integral].sup.t.sub.0] [(t - s).sup.[alpha]-1]/[GAMMA]([alpha]) x (s)ds. (13)

In the above definition the sign "[integral]" denotes the HKP-integral integral.

Definition 11. The Riemann-Liouville derivative of order a with the lower limit zero for a function f : [0, [infinity]) [right arrow] R can be written as

[mathematical expression not reproducible]. (14)

Definition 12. The Caputo fractional derivative of order a for a function f : [0, [infinity]) [right arrow] E can be written as

[mathematical expression not reproducible], (15)

where n = [[alpha]] + 1 and [[alpha]] denotes the integer part of [alpha].

3. Main Results

In this section, we present the existence of solutions to problem (5) in the space C(I, [E.sub.[omega]]).

Definition 13. A function x [member of] C(I, [E.sub.[omega]]) is said to be a solution of problem (5) if x satisfies the equation [sup.c][D.sup.[alpha].sub.0+]x(t) = f(t, x(t), (Tx)(t), (Sx)(t)) on I and satisfies the conditions [a.sub.1]x(0) - [b.sub.1]x'(0) = [d.sub.1]x([[xi].sub.1]), [a.sub.2]x(0) - [b.sub.2]x'(1) = [d.sub.2]x([[xi].sub.2]).

Lemma 14 (see [21]). Let [alpha] > 0. If one assumes u [member of] C(0,1) [intersection] L(0,1), then the differential equation

[sup.c][D.sup.[alpha].sub.0+] (t) = 0 (16)

has solution u(t) = [c.sub.0] + [c.sub.1]t + [c.sub.2][t.sup.2] + ... + [c.sub.n][t.sup.n-1], [c.sub.i] [member of] R, i = 0,1, ..., n, n = [[alpha]] + 1.

From the lemma above, we deduce the following statement.

Lemma 15 (see [21]). Assume that u [member of] C(0,1) [intersection] L(0,1) with a fractional derivative of order [alpha] >0 that belongs to C(0,1) n L(0,1). Then

[sup.c][D.sup.[alpha].sub.0+] ([sup.c][D.sup.[alpha].sub.0+] (t)) = u(t) + [C.sub.0] + [C.sub.1]t + [C.sub.2][t.sup.2] + ... + [c.sub.n][t.sup.n-1] (17)

for some [c.sub.i] [member of] R, i = 0, 1, ..., n, n = [[alpha]] + 1.

The following we give the corresponding Greens function for problem (5).

Lemma 16. Let [DELTA] = 0, [rho] [member of] C(I, [E.sub.w]) and [alpha] [member of] (1, 2], then the is given by unique solution of

[mathematical expression not reproducible], (18)

x(t) = [[integral].sup.1.sub.0] G(t, s) [rho] (s)ds, (19)

where the Green function G is given by

[mathematical expression not reproducible]. (20)

Proof. Based on the idea of paper [7], assuming that x(i) satisfies (18), by Lemma 15, we formally put

[mathematical expression not reproducible] (21)

for some constants [c.sub.1], [c.sub.2] [member of] R.

On the other hand, by the relations [mathematical expression not reproducible], we get

x'(t) = 1/[GAMMA]([alpha] - 1) [integral].sup.t.sub.0] [(t - s).sup.[alpha]-2] [rho](s) ds - [c.sub.2]. (22)

By the boundary conditions of (18), we have

[mathematical expression not reproducible], (23)

By the proof of paper [12], we get

[mathematical expression not reproducible]. (24)

where [DELTA] = [([b.sub.1] + [d.sub.1] [[xi].sub.1])([a.sub.2] - [d.sub.2]) + ([a.sub.2] + [b.sub.2] - [d.sub.2] [[xi].sub.2])([a.sub.1] - [d.sub.1])] = 0. Substituting the values of [c.sub.1] and [c.sub.2] in (21), we get

[mathematical expression not reproducible]. (25)

This completes the proof.

Let [D.sub.r] = {z [member of] C(I, [E.sub.w], [parallel]z[parallel] [less than or equal to] r}, BV(I, R) denote the space of real bounded variation functions with its classical norm [[parallel] x [parallel].sub.BV].

Problem (5) will be studied under the following assumptions:

(1) For each weakly continuous function x : I [right arrow] E, the functions [k.sub.1](t, x)g(x, x(x)), [k.sub.2](t, x)h(x, x(x)), f(*, x(x), T(x)(x), S(x)(x)) are HKP-integrable, f : I x [E.sup.3] [right arrow] E, g, h : I x E [right arrow] E are weakly-weakly continuous function, and [[integral].sup.t.sub.0] g(s, x(s))ds, [[integral].sup.t.sub.0] h(s, x(s))ds are bounded.

(2)

(i) For any r > 0, there exist a HK-integrable function m : I [right arrow] [R.sup.+] and nondecreasing continuous functions [mathematical expression not reproducible] such that

[mathematical expression not reproducible], (26)

for all s [member of] I, (x, y, z) [member of] [D.sub.r] x [D.sub.r] x [D.sub.r] with

[[integral].sup.1.sub.0] M(s)ds < [[integral].sup.[infinity].sub.0] dr/[[summation].sup.3.sub.i=1] [[psi].sub.i](s). (27)

(ii) For each bounded set X, Y, Z [subset] [D.sub.r], and each for each closed interval J [subset] I, t [member of] I, there exists positive constant l [greater than or equal to] 0 such that

[mathematical expression not reproducible]. (28)

(3) For each t [member of] I, G(t, .), [k.sub.i](t, x) [member of] BV(I, R), i = 1, 2 are continuous; i.e., the maps t [??] G(t,.) and t [??] [k.sub.i](t, x) are [[parallel]x[parallel].sub.BV]-continuous.

(4) The family {[x.sup.*] f(x, x(x), T(x)(x), S(x)(x)) : [x.sup.*] [member of] [E.sup.*], [parallel] [x.sup.*] [parallel] [less than or equal to] 1} is uniformly HK-integrable over I for every x [member of] [D.sub.r].

Remark 17. From assumption (3) and the expression of function G(t, s), it is obvious that it is bounded and let [G.sup.*] = [sup.sub.t[member of]I] [parallel]G(t, x)[parallel].sub.BV].

Now, we present the existence theorem for problem (5).

Theorem 18. Assume that conditions (5)-(20). Then problem (5) has a solution x [member of] C(I, [E.sub.w]).

Proof. Let [mathematical expression not reproducible]; we have

[mathematical expression not reproducible] (29)

and also

sup [[absolute value of [x.sup.*]Tx] : x [member of] [E.sup.*], [parallel] [x.sup.*] [parallel] [less than or equal to] 1} < [r.sub.0]. (30)

So [mathematical expression not reproducible]. Similarly, we prove [mathematical expression not reproducible]. Defining the set

[mathematical expression not reproducible], (31)

it is clear that the convex closed and equicontinuous subset [mathematical expression not reproducible], where

[mathematical expression not reproducible]. (32)

Clearly,

b' (t) = M(t)([3.summation over (i=1)] [[psi].sub.i] (b(i))), and

b(0) = 0 (33)

for all t [member of] l. Also notice that Q is a closed, convex, bounded, and equicontinuous subset of C(I, [E.sub.W]). We define the operator F : C(I, [E.sub.W]) [right arrow] C(I, [E.sub.W]) by

Fx(t) = [[integral].sup.1.sub.0] G(t, s) f (s, x(s),(Tx)(s), (Sx)(s))ds,

t [member of] I, (34)

where G(x, x) is Green's function defined by (20). Clearly the fixed points of the operator F are solutions of problem (5). Since for t [member of] I the function s [??] G(t, s) is of bounded variation, then by the proof of Theorem 3.1 in [13] and assumption (4), the function G(t, x)/(x, x(x), T(x)(x), S(x)(x)) is HKP-integrable on I and thus the operator F makes sense.

We will show that F satisfies the assumptions of Lemma 8; the proof will be given in three steps.

Step 1. We shall show that the operator F maps into itself. To see this, let x [member of] Q, t [member of] I. Without loss of generality, assume that Fx(t) [not equal to] 0. By Hahn-Banach theorem, there exists [x.sup.*] [member of] [E.sup.*] with [parallel] [x.sup.*][parallel] = 1 and [parallel]Fx(t)[parallel] = [absolute value of [x.sup.*](Fx(t))] Thus

[mathematical expression not reproducible]. (35)

Then [parallel]Fx[parallel] = [sup.sub.t[member of]I] [absolute value of Fx(i)] [less than or equal to] [r.sub.0]. Hence F: Q [greater than or equal to] Q.

Let 0 < [t.sub.1] < [t.sub.2] [less than or equal to] 1, without loss of generality; assume that Fx([t.sub.2]) - Fx([t.sub.1]) = 0. By Hahn-Banach theorem, there exists [x.sup.*] [member of] [E.sup.*] with [parallel] [x.sup.*] [parallel] = 1 and

[mathematical expression not reproducible], (36)

and this estimation shows that F maps Q into itself.

Step 2. We will show that the operator F is weakly sequentially continuous. In order to be simple, we denote Tx(t) = [phi](x)(t) = [[integral].sup.1.sub.0] [k.sub.1](t, s)g(s, x(s))ds, Sx(t) = [phi](x)(t) = [[integral].sup.1.sub.0] [k.sub.2](t, s)h(s, x(s))ds. To see this, by Lemma 9 of [22], a sequence [x.sub.n](x) weakly convergent to x(x) [member of] Q if and only if [x.sub.n](x) tends weakly to x(t) for each t [member of] l. From Dinculeanu ([23, p. 380]) [(C(I,E)).sup.*] = M[(I,E.sup.*]), M(I, [E.sup.*]) is the set of all bounded regular vector measures from I to E* which are of bounded variation). Let [x.sup.*] [member of] [E.sup.*], t [member of] I. Put [P.sub.t] = [x.sup.*][[delta].sub.t], where [[delta].sub.t] is the Dirac measure concentrated at the point t. Then [P.sub.t] [member of] M(I, [E.sup.*]). Since [x.sub.n] converges weakly to x [member of] Q, then we have

[mathematical expression not reproducible] (37)

which means that

[mathematical expression not reproducible]. (38)

Thus, for each t [member of] l, [x.sub.n](t) converges weakly to x(t) [member of] E. Since g(s, x),h(s, x) are weakly-weakly sequentially continuous, then g(s, [x.sub.n](s)) and h(s, [x.sub.n](s)) converge weakly to g(s, x(s)) and h(s, x(s)),respectively. Hence, andbyTheorem4andassumptions (1), we have

[mathematical expression not reproducible]. (39)

This relation is equivalent to

[mathematical expression not reproducible]. (40)

Similarly, we have

[mathematical expression not reproducible]. (41)

This relation is equivalent to

[mathematical expression not reproducible]. (42)

Therefore, the operators T, S are weakly sequentially continuous in Q.

Moreover, because f is weakly-weakly sequentially continuous, we have that f(s, [x.sub.n](s), (T[x.sub.n])(s), (S[x.sub.n])(s)) converges weakly to f(s, x(s), (Tx)(s), (Sx)(s)) in E. By assumption (4), for every weakly convergent [mathematical expression not reproducible], the set

[mathematical expression not reproducible] (43)

is HK-equi-integrable. Since for t [member of] l the function s [??] G(t, s) is of bounded variation, and by the proof of Theorem 3.1 in [13], the function G(t, x)/(x, [x.sub.n](x), (T[x.sub.n])(x), (S[x.sub.n])(x)) is HKP-integrable on I for every n [greater than or equal to] 1, and by Theorem 4, we have that [[integral].sup.1.sub.0] G(t, s)f(s, [x.sub.n](s), (T[x.sub.n])(s), (S[x.sub.n])(s))ds converges weakly to [[integral].sup.1.sub.0] G(t, s)f(s, x(s), (Tx)(s), (Sx)(s))ds in E which means that

[mathematical expression not reproducible], (44)

for all m [member of] M(I, [E.sup.*]). This relation is equivalent to

[mathematical expression not reproducible], (45)

Therefore F is weakly-weakly sequentially continuous.

Step 3. We show that there is an integer [n.sub.0] such that the operator F is [beta]-power-convex condensing about 0 and [n.sub.0]. To see this, notice that, for each bounded set H [subset not equal to] Q and for each t [member of] I,

[mathematical expression not reproducible]. (46)

Let [tau] = [G.sup.*] x max{1, [k.sup.*.sub.1], [k.sup.*.sub.2]} x l > 0. Lemma 7 implies (since H is equicontinuous) that

[beta]([F.sup.(1,0)] (H)(t)) [less than or equal to] t[tau][beta](H). (47)

Since [F.sup.(1,0)](H) is equicontinuous, it follows from Lemma 5 that [F.sup.(2,0)] (H) is equicontinuous. Using (47), we get

[mathematical expression not reproducible], (48)

where V = [bar.co]([F.sup.(1,0)] (H) [union] {0}); it is clear that V is equicontinuous set. By Lemma 8, we get

[beta](V(s)) = [beta] ([F.sup.(1,0)] (H) (s)) [less than or equal to] s[tau][beta] (H), (49)

and therefore,

[[integral].sup.1.sub.0] [beta](V(s))ds [less than or equal to] S[tau] [t.sup.2]/2 [beta](H). (50)

Thus,

[beta]([F.sup.(2,0)] (H)(t)) [less than or equal to] [([tau]t)s.up.2] [beta](H). (51)

By induction, we get

[beta]([F.sup.(n,0)] (H)(t)) [less than or equal to] [([tau]t).sup.n]/n! [beta](H). (52)

And by Lemma 7, we have

[beta]([F.sup.(n,0)] (H)(t)) [less than or equal to] [([tau]T).sup.n]/n! [beta](H). (53)

Since [lim.sub.n [right arrow] [infinity]] ([([tau]t).sup.n]/n!) = 0, then there exist an [n.sub.0] with [mathematical expression not reproducible], and we have

[mathematical expression not reproducible]. (54)

Consequently, F is [beta]-power-convex condensing about 0 and [n.sub.0], by Lemma 8, then problem (5) has a solution x [member of] C(I, [E.sub.w]).

4. Conclusions

In this paper, we use the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals to discuss the existence theorem of weak solutions for a class of the multipoint boundary value problem of fractional integrodifferential equations equipped with the weak topology. Our results generalized some classical results.

https://doi.org/10.1155/2018/1203031

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11061031).

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Haide Gou and Baolin Li (iD)

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China Correspondence should be addressed to Baolin Li; ghdzxh@163.com

Received 5 September 2017; Revised 20 November 2017; Accepted 22 July 2018; Published 2 September 2018