Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions

Abstract We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.


Introduction
Recently, there has been an utterly great interest in developing theoretical analysis for boundary value problems of nonlinear fractional-order differential equations supplemented with a variety of boundary conditions. It has been mainly due to nonlocal nature of fractional-order differential operators which take into account memory and hereditary properties of some important and useful materials and processes. Fractional calculus has played a key role in improving the mathematical modelling of several phenomena occurring in engineering and scientific disciplines, such as blood flow problems, control theory, aerodynamics, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, polymer rheology, regular variation in thermodynamics etc. For more details and explanation, see, for instance [1][2][3]. Some recent results on fractional-order boundary value problem can be found in a series of papers [4][5][6][7][8][9][10][11][12] and the references cited therein. Sequential fractional differential equations have also received considerable attention, for instance see [13][14][15][16][17].
Anti-periodic boundary conditions are found to be quite significant and important in the mathematical modeling of certain physical processes and phenomena, for example trigonometric polynomials in the study of interpolation problems, wavelets, physics etc. For more details, see [18] and the references cited therein. For some recent works on fractional-order anti-periodic boundary value problems, we refer the reader to [19][20][21][22][23]. However, the study of sequential fractional differential equations equipped with anti-periodic boundary conditions is yet to be initiated.
The rest of the paper is organized as follows. In Section 2, we recall some basic concepts of fractional calculus and obtain the integral solution for the linear variants of the given problems. Section 3 contains the existence results for problem (1)-(2) obtained by applying Schaefer's fixed point theorem, Leray-Schauder's nonlinear alternative, Leray-Schauder's degree theory, Banach's contraction mapping principle and Krasnoselskii's fixed point theorem. In Section 4, we provide the outline for the existence results of problem (1)-(3).

Preliminaries and auxiliary results
This section is devoted to some basic definitions of fractional calculus [1,2] and auxiliary lemmas.
Definition 2.1. The fractional integral of order q with the lower limit zero for a function f is defined as provided the right hand-side is point-wise defined on OE0; 1/, where . / is the gamma function, which is defined by Definition 2.2. The Riemann-Liouville fractional derivative of order q > 0; n 1 < q < n; n 2 N , is defined as where the function f .t / has absolutely continuous derivative up to order .n 1/.
Definition 2.3. The Liouville-Caputo derivative of order q for a function f W OE0; 1/ ! R can be written as t k kŠ f .k/ .0/ ! ; t > 0; n 1 < q < n: f .n/ .s/ .t s/ qC1 n ds D I n q f .n/ .t /; t > 0; n 1 < q < n: To define the fixed point problems associated with problems (1)- (2) and (1) . c D˛C k c D˛ 1 /u.t / D h.t //; 1 <˛Ä 2; 0 < t < T; T > 0; (4) and the boundary conditions (2) is equivalent to the integral equation k.˛1 C 1 /.˛2 C 2 e kT / ; Proof. As argued in [13], the general solution of (4) can be written as where A 0 and A 1 are arbitrary constants and Using the boundary conditions (2) in (7) and (8), we get Solving the system (9) and (10) for A 0 and A 1 ; we get Substituting the values of A 0 and A 1 in (7) yields the solution (5). Conversely, by direct computation, it can be established that (5) satisfies the equation (4) and boundary conditions (2). This completes the proof.
Proof. Since the proof is similar to that of Lemma 2.5, we omit it.

Existence results for the problem (1)-(2)
In view of Lemma 2.5, we introduce a fixed point problem associated with the problem (1)-(2) as follows: where the operator H W E ! E is Here E D C.OE0; T ; R/ denotes the Banach space of all continuous functions from OE0; T ! R endowed with the norm defined by kuk D supfju.t /j; t 2 OE0; T g.
Observe that that problem (1)-(2) has solutions if the operator equation .13/ has fixed points. For computational convenience, we set the notation: Now we are in a position to present our main results for the problem (1)-(2). The first one deals with Schaefer's fixed point theorem [24].
Lemma 3.1. Let X be a Banach space. Assume that T W X ! X is a completely continuous operator and the set Y D fu 2 X j u D T u; 0 < < 1g is bounded. Then T has a fixed point in X: Then the boundary value problem (1)-(2) has at least one solution on OE0; T : Proof. In the first step, we show that the operator H defined by (14) is completely continuous. Observe that continuity of H follows from the continuity of f: For a positive constant r, let B r D fu 2 E W kuk Ä rg be a bounded ball in E. Then for t 2 OE0; T ; we have which consequently implies that where Q is defined by (15). Next we show that the operator H maps bounded sets into equicontinuous sets of E: Let 1 ; 2 2 OE0; T with 1 < 2 and u 2 B r : Then we have As 2 1 ! 0, the right-hand side of the above inequality tends to zero independently of u 2 B r . Therefore, by the Arzelá-Ascoli theorem, the operator H W E ! E is completely continuous.
Finally, we consider the set V D fu 2 E W u D Hu; 0 < < 1g and show that V is bounded. For u 2 V and t 2 OE0; T ; we get Therefore, V is bounded. Hence, by Lemma 3.1, the problem (1)-(2) has at least one solution on OE0; T .
Our next existence result is based on Leray-Schauder's nonlinear alternative.
Theorem 3.4. Assume that .E 1 / there exists a continuous nondecreasing function W OE0; 1/ ! .0; 1/ and a function p 2 C.OE0; T ; R C / such that where Q is given by (15). Then the boundary value problem (1)-(2) has at least one solution on OE0; T : Proof. We complete the proof in different steps. We first show that the operator H defined by (14) maps bounded sets (balls) into bounded sets in E. For a positive constant r, let B r D fu 2 E W kuk Ä rg be a bounded ball in E. Then, for t 2 OE0; T ; we have which implies that k.Hu/k Ä k 1 k C .r/kpkQ: In the second step, we establish that the operator H maps bounded sets into equicontinuous sets of E: As in the proof of the previous result, for 1 ; 2 2 OE0; T with 1 < 2 and u 2 B r ; we can have j.Hu/. 2 / .Hu/. 1 /j ! 0 as 2 1 ! 0; independently of u 2 B r . Therefore, it follows by the Arzelá-Ascoli theorem that the operator H W E ! E is completely continuous. Let u be a solution. Then, for t 2 OE0; T ; we have that kuk Ä .kuk/kpkQ C k 1 k: In view of .E 2 /, there exists N such that kuk ¤ N: Let us set U D fu 2 E W kuk < N g: We see that the operator H W U ! E is continuous and completely continuous. From the choice of U, there is no u 2 @U such that u D Â Hu for some Â 2 .0; 1/. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.3), we deduce that H has a fixed point u 2 U which is a solution of the problem (1)- (2). This completes the proof.
The next existence result is based on Leray-Schauder's degree theory [25].
where Q is given by (15). Then the boundary value problem (1)-(2) has at least one solution on OE0; T : Proof. We have to show the existence of at least one solution u 2 E satisfying the fixed point problem where the operator H W E ! E is defined by (14). Introduce a ball B R E as with a constant radius R > 0: Hence, we will show that the operator H W B R ! E satisfies the condition u ¤ Â Hu; 8u 2 @B R ; 8Â 2 OE0; 1: where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, we have h 1 .u/ D u Hu D 0 for at least one u 2 B R : Let us assume that u D Â Hu for some Â 2 OE0; 1 and for all t 2 OE0; T : Then, using the assumption .E 3 /; it is easy to find that which implies that kuk Ä , the inequality .18/ holds. This completes the proof.
Next we show the existence of a unique solution of the problem (1)-(2) by applying Banach's contraction mapping principle.
Theorem 3.6. Assume that f W OE0; T R ! R is a continuous function satisfying the Lipschitz condition: .E 4 / there exists a positive number`such that jf .t; u/ f .t; v/j Ä`ju vj; 8t 2 OE0; T ; u; v 2 R: Then the boundary value problem (1)-(2) has a unique solution on OE0; T if Q < 1=`; where Q is given by (15).
Proof. Consider a set B r D fu 2 E W kuk Ä rg with r QM C k 1 k 1 `Q ; where M D sup t 2OE0;T jf .t; 0/j and Q is given by (15). In the first step, we show that HB r B r ; where the operator H is defined by (14). For any u 2 B r ; t 2 OE0; T ; observe that where we have used the assumption .E 4 /. Then, for u 2 B r ; we obtain which implies that Hu 2 B r : Thus HB r B r : Next we show that the operator H is a contraction. Using the assumption .E 4 / and (15), we get In view of the assumption: Q < 1=`; it follows that the operator H is a contraction. Thus, by Banach's contraction mapping principle, we deduce that the operator H has a fixed point, which in turn implies that there exists a unique solution for the problem (1)-(2) on OE0; T : In the following theorem, we show the existence of solutions for the problem (1)-(2) by applying Krasnoselskii's fixed point theorem.
Lemma 3.7 (Krasnoselskii's fixed point theorem [24]). Let Y be a closed bounded, convex and nonempty subset of a Banach space X: Let B 1 ; B 2 be the operators such that (i) B 1 y 1 C B 2 y 2 2 Y whenever y 1 ; y 2 2 Y ; (ii) B 1 is compact and continuous and (iii) B 2 is a contraction mapping. Then there exists z 2 Y such that z D B 1 z C B 2 z: Theorem 3.8. Let f W OE0; T R ! R be a continuous function satisfying the condition .E 4 / and that jf .t; x/j Ä g.t/; 8.t; x/ 2 OE0; T R with g 2 C.OE0; T ; R C / and sup t2OE0;T jg.t /j D kgk: In addition, it is assumed that Then problem (1)-(2) has at least one solution on OE0; T : Proof. Consider B a D fu 2 E W kuk Ä ag; where a Qkgk C k 1 k with Q given by (15). We define the operators H 1 and H 2 on B a as For u; v 2 B a ; it is easy to verify that kH 1 uCH 2 vk Ä QkgkCk 1 k; where Q is given by (15). Thus, H 1 uCH 2 v 2 B a : Using the assumption .E 4 / and (19), we can get kH 2 u H 2 vk Ä`Q 1 k u v k; which implies that H 2 is a contraction in view of the given condition: Q 1 < 1=`: Notice that continuity of f implies that the operator H 1 is continuous. Also, H 1 is uniformly bounded on B a as kH 1 uk Ä .1 e kT /T˛ 1 kgk k.˛/ : Next, it will be shown that the operator H 1 is compact. Fixing sup .t;u/2OE0;T B a jf .t; u/j D f a and for t 1 ; t 2 2 OE0; T .t 1 < t 2 /; consider independent of u: This implies that H 1 is relatively compact on B a : Hence, by the Arzelá-Ascoli Theorem, the operator H 1 is compact on B a : Thus all the assumptions of Lemma (3.7) are satisfied. In consequence, by the conclusion of Lemma (3.7), the problem (1)-(2) has at least one solution on OE0; T : Example 3.9. Consider the following anti-periodic fractional boundary value problem:
(a) Let Clearly jf .t; u.t //j Ä 3 D L 1 for all t 2 OE0; 2; u 2 R: Thus, by Theorem 3.2, the problem (20) with f .t; u/ given by (21) has at least one solution on OE0; 2: (b) Letting (c) Let us take Clearly`D 1=10 as jf .t; u/ f .t; v/j Ä 1 10 ju vj and`Q 0:774292 < 1: Thus all the conditions of Theorem 3.6 are satisfied. Hence we deduce by the conclusion of Theorem 3.6 that there exists a unique solution for the problem (20) with f .t; u/ given by (24).
For the applicability of Theorem 3.8, we find that jf .t; u/j Ä g.t / D =20 C cos t with kgk D .20 C /=20 and Q 1 6:732035 (Q 1 is given by (19)). Obviously`Q 1 0:673203 < 1: Thus all the conditions of Theorem 3.8 are satisfied. Hence the conclusion of Theorem 3.8 implies that the problem (20) with f .t; u/ given by (24) has at least one solution on OE0; 2: Remark 3.10. By fixing the parameters involved in the boundary conditions (2), we can obtain some new special results for different problems arising from the problem (1)- (2). For instance, for˛1 D˛2 D 1 D 2 D 1;ˇ1 Ď

Existence results for the problem (1)-(3)
In this section, we present some existence results for the problem (1)- (3). We omit the proofs as the method of proof is similar to the one employed in the previous section. First of all, by Lemma 2.6, we define a fixed point operator G W E ! E associated with the problem (1)-(3) as where B 1 .t/ and B 2 .t / are given by (12).
Using the operator (25) and the method of proof for the results obtained in the last section, we can establish the following results for the problem (1)-(3).