Differential equations with deviating arguments arise in various branches of science, engineering, economics and so on (see [4, 8] and the references therein). Many researchers have studied the existence, uniqueness, continuous dependence, and stability of solutions of nonlinear fractional differential equations (see [1, 2, 3, 5, 6, 7, 10, 11, 12, 15, 16, 17, 19, 20, 25, 30, 33, 34]). The monotone iterative technique [23] combined with the method of upper and lower solutions provides an effective mechanism to prove constructive existence results for nonlinear differential equations. The monotone technique is an interesting and powerful tool to deal with existence results for fractional differential equations. In 2008, the monotone technique for fractional differential equations with initial conditions was first developed by Lakshmikantham and Vatsala [25]. Later, a series of papers appeared in the literature to prove existence and uniqueness of solution of various problems with initial conditions, boundary conditions, integral boundary conditions, nonlinear boundary conditions, and periodic boundary conditions for fractional differential equations, (see, for example [13, 14, 18, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 35, 37, 38, 39, 40, 41, 42, 43] and the references therein). However, work on fractional differential equations with deviating argument is rare. In this paper, we study the following problem for the Riemann-Liouville fractional differential equation with a deviating argument and integral boundary conditions:

where f ∈ C (J × ℝ²,ℝ), θ ∈ C (J, J), θ(t) ≤ t, t ∈ J, λ ≥ 0, 0 < α < 1. The paper is organized as follows. In Section 2, we introduce some useful definitions and basic lemmas. In Section 3, we study the uniqueness of a solution for the problem (1.1) using the Banach fixed point theorem. In Section 4, we develop the monotone method and apply it to obtain existence and uniqueness results for Riemann-Liouville fractional differential equations with deviating arguments and integral boundary conditions.

For the reader’s convenience, we present some necessary definitions and lemmas from the theory of fractional calculus. In addition, we prove some basic results which are useful for further discussion.

Definition 2.1. ([21, 36])

For α > 0, the integral

is called the Riemann-Liouville fractional integral of order α.

Definition 2.2. ([21, 36])

The Riemann-Liouville derivative of order α (n − 1 < α ≤ n) can be written as

Lemma 2.1. ([21])

Let u ∈ Cⁿ[0, T], α ∈ (n − 1, n), n ∈ ℕ. Then for t ∈ J,

Consider the space C_1−α (J,ℝ) = {u ∈ C ((0, T],ℝ) : t^1−αu ∈ C (J,ℝ)}.

Lemma 2.2. ([9])

Let m ∈ C_1−α(J,ℝ) where for some t₁ ∈ (0, T],

Then it follows that

Lemma 2.3

Let f ∈ C (J × ℝ²,ℝ). A function u ∈ C_1−α (J,ℝ) is a solution of the problem (1.1) if and only if u is a solution of the integral equation

Lemma 2.4

Suppose that {u_ε} is a family of continuous functions defined on J, for each ε > 0, which satisfies

where |f (t, u_ε(t), u_ε(θ(t)))| ≤ M for t ∈ J. Then the family {u_ε} is equicontinuous on J.

In this section, we obtain the uniqueness of solution of the problem (1.1) for Riemann-Liouville fractional differential equations with deviating argument and integral boundary conditions.

Theorem 3.1

Assume that

• f ∈ C (J × ℝ²,ℝ), θ(t) ∈ C (J, J), θ ≤ t, t ∈ J,

• there exists nonnegative constants M and N such that function f satisfies∣f(t,u1,u2)-f(t,υ1,υ2)∣ ≤M∣u1-υ1∣+N∣u2-υ2∣,

  for all t ∈ J, u_i, υ_i ∈ ℝ, i = 1, 2. Ifλ<Γ(α+1)-Tα(M+N)TΓ(α+1), then the problem (1.1) has a unique solution.

Corollary 3.1

Let M, N be constants, σ ∈ C_1−α(J,ℝ). The linear problem

has a unique solution.

In this section, we prove the existence and uniqueness of solution for the problem (1.1) by monotone iterative technique combined with the method of upper and lower solutions. Now we define the functional interval as follows:

First, we prove the following comparison result, which plays an important role in our further discussion.

Lemma 4.1

Let θ ∈ C (J, J) where θ(t) ≤ t on J. Suppose that p ∈ C_1−α(J,ℝ) satisfies the inequalities

where M and N are constants. If

then p(t) ≤ 0 for all t ∈ J.

Definition 4.1

A pair of functions [υ₀, w₀] in C_1−α(J,ℝ) are called lower and upper solutions of the problem (1.1) if

and

Theorem 4.1

Assume that

• f ∈ C (J × ℝ²,ℝ), θ ∈ C (J, J), θ(t) ≤ t, t ∈ J,

• functions υ₀and w₀in C_1−α(J,ℝ) are lower and upper solutions of the problem (1.1) such that υ₀(t) ≤ w₀(t) on J,

• there exists nonnegative constants M, N such that function f satisfies the condition(4.5)f(t,u1,u2)-f(t,υ1,υ2)≥-M(u1-υ1)-N(u2-υ2),

  for υ₀(t) ≤ υ₁ ≤ u₁ ≤ w₀(t), υ₀(θ(t)) ≤ υ₂ ≤ u₂ ≤ w₀(θ(t)).

Then there exists monotone sequences {υ_n(t)} and {w_n(t)} in C_1−α(J,ℝ) such that

for all t ∈ J, where υ and w are minimal and maximal solutions of the problem (1.1) respectively and υ(t) ≤ u(t) ≤ w(t) on J.

Theorem 4.2

Assume that

• all the conditions of the Theorem 4.1 hold,

• there exists nonnegative constants M, N such that the function f satisfies the condition(4.10)f(t,u1,u2)-f(t,υ1,υ2)≤M(u1-υ1)+N(u2-υ2),

  for υ₀(t) ≤ υ₁ ≤ u₁ ≤ w₀(t), υ₀(θ(t)) ≤ υ₂ ≤ u₂ ≤ w₀(θ(t)).

Then the problem (1.1) has a unique solution.

The authors are grateful to the respected anonymous referee and the English language expert Prof. J. T. Neugebauer for their valuable comments and suggestions that greatly improved the presentation of the paper.