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 [1–3]. Some recent results on fractional-order boundary value problem can be found in a series of papers [4–12] and the references cited therein. Sequential fractional differential equations have also received considerable attention, for instance see [13–17].
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  and the references cited therein. For some recent works on fractional-order anti-periodic boundary value problems, we refer the reader to [19–23]. However, the study of sequential fractional differential equations equipped with anti-periodic boundary conditions is yet to be initiated.
In this paper, we study new boundary value problems of Liouville-Caputo type sequential fractional differential equation: (1)
subject to anti-periodic type (non-separated) boundary conditions of the form: (2)
and anti-periodic type (non-separated) nonlocal integral boundary conditions: (3)
where c Dα denotes the Liouville-Caputo fractional derivative of order α, k ∈ ℝ+, 0 < η < ξ < T, α1, α2, ρ1, ρ2, β1, β2, λ1, λ2, μ1μ2 ∈ ℝ with α1 + ρ1 ≠ 0, α2 + ρ2e-kT ≠ 0 and f : [0, T] × ℝ → ℝ is a given continuous function. Instead of writing the so-called “Caputo” derivative, we will call it “Liouville-Caputo” derivative as it was introduced by Liouville many decades ago.
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).
2 Preliminaries and auxiliary results
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 [0,∞), where is the gamma function, which is defined by .
The Riemann-Liouville fractional derivative of order q > 0, n — 1 < q < n, n ∈ N, is defined as
where the function f(t) has absolutely continuous derivative up to order (n — 1).
The Liouville-Caputo derivative of order q for a function f : [0,∞) → ℝ can be written as
Let h ∈ C([0, T], ℝ). Then the problem consisting of the equation (4)
and the boundary conditions (2) is equivalent to the integral equation (5)
where the operator is (14)
Here denotes the Banach space of all continuous functions from [0, T] → ℝ endowed with the norm defined by .
Let X be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then has a fixed point in X.
In the first step, we show that the operator defined by (14) is completely continuous. Observe that continuity of follows from the continuity of f. For a positive constant r, let be a bounded ball in . Then for t ∈ [0, T], we have
which consequently implies that
where Q is defined by (15).
Next we show that the operator maps bounded sets into equicontinuous sets of . Let τ1, τ2 ∈ [0, T] with τ1 < τ2 and u ∈ Br. Then we have
As τ1 — τ2 → 0, the right-hand side of the above inequality tends to zero independently of u ∈ Br. Therefore, by the Arzelá-Ascoli theorem, the operator is completely continuous.
Finally, we consider the set and show that V is bounded. For u ∈ V and t ∈ [0, T], we get
Our next existence result is based on Leray-Schauder’s nonlinear alternative.
(Nonlinear alternative for single valued maps [2–5]). Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and 0 ∈ U. Suppose that is a continuous, compact (that is, is a relatively compact subset of C ) map. Then either
has a fixed point in , or
there is a x ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with .
there exists a continuous nondecreasing function and a function p ∈ C([0, T], ℝ+) such that
there exists a constant N > 0 such that (16)
where Q is given by (15).
The next existence result is based on Leray-Schauder’s degree theory .
Let f : [0, T] × ℝ → ℝ be a continuous function. Suppose that (E3) there exist constants 0 ≤ ω < Q-1, and M1 > 0 such that
where Q is given by (15).
Assume that f : [0, T] × ℝ → ℝ is a continuous function satisfying the Lipschitz condition: (E4) there exists a positive number ℓ such that .
where we have used the assumption (E4). Then, for u ∈ Br, we obtain
which implies that . Thus . Next we show that the operator is a contraction. Using the assumption (E4) and (15), we get
In view of the assumption: Q < 1/ℓ, it follows that the operator is a contraction. Thus, by Banach’s contraction mapping principle, we deduce that the operator has a fixed point, which in turn implies that there exists a unique solution for the problem (1)-(2) on [0, T].
(Krasnoselskii’s fixed point theorem ). Let Y be a closed bounded, convex and nonempty subset of a Banach space X. Let B1, B2 be the operators such that (i) B1y1 + B2y2 ∈ Y whenever y1,y2 ∈ Y; (ii) B1 is compact and continuous and (iii) B2 is a contraction mapping. Then there exists z ∈ Y such that z = B1z + B2z.
Let f : [0,T] × ℝ → ℝ be a continuous function satisfying the condition (E4) and that ∀(t,x) ∈ [0, T] × ℝ with g ∈ C([0, T], ℝ+) and . In addition, it is assumed that , where(19)
Consider the following anti-periodic fractional boundary value problem:(20)
Here T = 2, k = 1.5, α1 = 1.25, ρ1 = 4, β1 = —1, α2 = 0.5, ρ2 = —2, β2 = 2.5. With the given values, we find that Q ≈ 7.742915 (Q is given by (15)).
we have . Selecting and , we find that the assumption (E2) holds true for N > 4.064141. As all the conditions of Theorem 3.4 are satisfied, there exists at least one solution of the problem (20) with f(t, u) given by (22) on [0, 2].
(c) Let us take(23)
(d) Let us choose(24)
Clearly as and . 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 with and Q1 ≈ 6.732035 (Q1 is given by (19)). Obviously . 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 [0, 2].
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 α1 = α2 = ρ1 = ρ2 = 1, β1 = β2 = 0, we obtain the existence results for sequential fractional differential equation (1) with anti-periodic boundary condition: u(0) + u(T) = 0, u′(0) + u′(T) = 0. Our results correspond to the ones obtα1ned in  for u(0) = a, u′(0) = u′(1) by taking α1 = 1 = α2, ρ1 = 0, ρ2 = —1, β1 = a, β2 = 0.
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 associated with the problem (1)-(3) as (25)
where B1(t) and B2(t) are given by (12).
Let f : [0,T] × ℝ → ℝ be a continuous functions satisfying the condition (E4) and that , with .In addition, it is assumed that , where(27)
Let f : [0,T] × ℝ → ℝ be a jointly continuous function. Assume that (E5) there exists a continuous nondecreasing function ψ : [0,∞) → (0,∞) and a function φ ∈ C([0, T], ℝ+) such that
(E6) there exists a constant N1 > 0 such that(28)
where is given by (26),(29)
Let f : [0,T] × ℝ → ℝ be a continuous function. Suppose that (E7) there exist constants , and M2 > 0 such that
where is given by (26).
Let f : [0,T] × ℝ → ℝ be a continuous function. Assume that there exists a positive constant L2 suchthat for t ∈ [0,T], u ∈ ℝ. Then the boundary value problem (1)-(3) has at least one solution on [0, T].
Consider the following anti-periodic fractional boundary value problem:(30)
where , T = 2, η = 1, ξ = 1.25, k = 1.5, α1 = 1.25, ρ1 = 4, λ1 = —1, λ2 = —1, λ2 = 0.5, ρ2 = —2, μ1 = 2, μ2 = 2.5.
With the given values, we find that ( is given by (26)) and as . Clearly, .Thus all the conditions of Theorem 4.1 are satisfied. Hence we deduce by the conclusion of Theorem 4.1 that there exists a unique solution for the problem (30).
For the applicability of Theorem 4.2, we find that with and ( is given by (27)). Obviously .Thus all the conditions of Theorem 4.2 are satisfied. Hence the conclusion of Theorem 4.2 applies to the problem (30).
To illustrate Theorem 4.3, we take , , with and ( and are given by (26) and (29) respectively). Thus all the conditions of Theorem 4.3 are satisfied. Hence the conclusion of Theorem 4.3 implies that the problem (30) has at least one solution on [0, T].
Several special cases of the existence results for the problem (1)-(3) follow by fixing the values of the parameters involved in the problem. For example, by taking α1= 1 = α2, ρ1 = 0 = ρ2, λ1 = 1 = μ1, λ2 = 0 = μ2, the results of this section correspond to the conditions: , . In case we take α1 = 0 = α2, ρ1 = 1 = ρ2, λ1 = 1 = μ1, λ2 = 0 = μ2, we obtain the results for terminal-point conditions: , . Letting α1 = 1 = α2, ρ1 = 1 = ρ2, λ1 = 1/eta, μ1 = 1/((T — ξ), λ2 = 0 = μ2, we get the results for the average valued (integral) conditions: , . By taking α = 2, our results correspond to the equation: (D2 + kD)u(t) = f (t, u(t)), 0 < t < T, T > 0, which are also new.
The research of J.J. Nieto was partially supported by the Ministerio de Economia y Competi-tividad of Spain under grant and MTM2013-43014-P, and Xunta de Galicia under grant GRC 2015/004.
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Published Online: 2016-10-16
Published in Print: 2016-01-01