Existence and Uniqueness of Solutions for a Fractional Order Antiperiodic Boundary Value Problem with a p-Laplacian Operator

Theorem 1. Let f : [0,1] × ℝ → ℝ be continuous. Assume that

•   

  (H) there exist nonnegative functions a, b ∈ C[0,1] such that

  $$ (3)

Definition 2. A function ψ : (−∞, +∞)→[0, +∞) is called a C − N function if it is a continuous nondecreasing function. Again if ψ satisfies ψ(r) < r, r > 0, then ψ is called a nonlinear 𝒟-contraction mapping.

Definition 3. A function f is said to be a Crathéodory’s function if the following conditions hold:

• (i)

  for each x ∈ ℝ, the mapping t ↦ f(t, x) is Lebesgue measurable.

• (ii)

  for a.e. t ∈ [0,1], the mapping x ↦ f(t, x) is continuous on ℝ.

Theorem 4. Let f be a Crathéodory’s function. Assume that

•   

  (S) there exist some constant ϵ ∈ (0, β) such that a(t), b(t) ∈ L^1/ϵ([0,1], [0, +∞)) and a C − N function ψ with

  $$ (5)

Corollary 5. Let f be a Crathéodory’s function. Assume that

•   

  (S^*) there exists some constant ϵ ∈ (0, β) such that a(t) ∈ L^1/ϵ([0,1], [0, +∞)) and

  $$ (7)

Corollary 6. Let f be a Crathéodory’s function. Assume that

•   

  (S^**) there exists some constant ϵ ∈ (0, β) such that b(t) ∈ L^1/ϵ([0,1], [0, +∞)) and a C − N function ψ with

  $$ (9)

Remark 7. The main results of this paper improve the work of Chen and Liu from the following three aspects.

• (1)

  In [13], a stronger condition that f : [0,1] × ℝ → ℝ is continuous is required, but in this paper we only require that f satisfies Crathéodory’s condition, which is a weaker condition than those of paper [13].

• (2)

  In Theorem 4, a(t), b(t) ∈ L^1/ϵ([0,1], [0, +∞)) for some ϵ ∈ (0, β), and a, b can be singular at some zero measure set of [0,1]. However, a(t), b(t) are continuous in paper [13], which are not allowed to have singularity in [0,1].

• (3)

  In the main results of this paper, ψ is only required to be a C − N function. Clearly, a C − N function includes ψ(|u|) = |u|^p−1, p > 1 as special case.

In what follows, we also complement a uniqueness result on the ABVP (2), which is based on the Banach contraction mapping principle and a basic property of the p-Laplacian operator: if q > 2, |x | , |y | ≤ M, then

Theorem 8. Let f be a Crathéodory’s function. Assume that 1 < p < 2 and

•   

  $$ there exist some constant ϵ ∈ (0, β) such that a(t), b(t) ∈ L^1/ϵ([0,1], [0, +∞)) and a 𝒟-contraction mapping ψ with

  $$ (12)

Definition 9 (see [16].)The Riemann-Liouville fractional integral operator of order α > 0 of a function x : (a, +∞) → R is given by

Definition 10 (see [16].)The Caputo fractional derivative of order α > 0 of a continuous function x : (a, +∞) → R is given by

Proposition 11 (see [17].)Let α > 0. Assume that $$. Then the following equality holds:

Lemma 12 (see [13].)Given h ∈ C[0,1], the unique solution of

Lemma 13. Let X be a real Banach space and let Ω be a bounded open subset of X, where θ ∈ Ω, $$ is a completely continuous operator. Then, either there exists x ∈ ∂Ω, λ > 1 such that Tx = λx, or there exists a fixed point $$.

Proof of Theorem 4. Consider the operator F : C[0,1] → C[0,1] defined by (21). For the sake of convenience, we subdivide the proof into two steps.

Step 1.  F : C[0,1] → C[0,1] is completely continuous.

Let Ω ⊂ C[0,1] be any bounded set. We will prove that F(Ω) is also bounded. In fact, for any x ∈ Ω, there exists a l > 0 such that ∥x∥ ≤ l, and then by the Hölder inequality

On the other hand, by the continuity of f on x and the Lebesgue dominated convergence theorem, we can get that F is continuous. Moreover, according to the strategy in [13], we know that F(Ω) ⊂ C[0,1] is equicontinuous. Thus Ascoli-Arzela theorem assures that F is completely continuous.

Step 2. F has at least a fixed point.

Now consider B_r = {x ∈ C[0,1] : ∥x∥ ≤ r}. An application of Leray-Schauder nonlinear alternative theorem yields either that the operator Fx = x has a fixed point or there exists x ∈ ∂B_r such that Fx = λx for some λ > 1. We show that the latter assertion does not hold. Assume the contrary, then there exist a x ∈ ∂B_r and some λ > 1 such that Fx = λx. By (22)–(24) and (6), we have

Proof of Theorem 8. According to (12) and a similar strategy as Theorem 4, we know that F is completely continuous operator. Now we will prove that F is a contraction mapping. By (22), we have

Example 1. Consider the following ABVP for the fractional p-Laplacian equation:

Example 2. Consider the following ABVP for the fractional p-Laplacian equation: