In this article, we study the existence and uniqueness of solutions for the following nonlinear fractional integro-differential equation:

D α u ( t ) = f ( t , u ( t ) , ( ϕ u ) ( t ) , ( ψ u ) ( t ) ) , t ∈ 0 , T , α ∈ 1 , 2 ,

subject to the boundary conditions of fractional order given by

D α - 2 u ( 0 + ) = 0 ,

D α - 1 u ( 0 + ) = ν I α - 1 u ( η ) , 0 < η < T , ν is a constant ,

where D ^α denotes the Riemann-Liouville fractional derivative of order α, f: [0, T] × ℝ × ℝ × ℝ → ℝ is continuous, and

( ϕ x ) ( t ) = ∫ 0 t γ ( t , s ) x ( s ) d s , ( ψ x ) ( t ) = ∫ 0 t δ ( t , s ) x ( s ) d s ,

with γ and δ being continuous functions on [0, T] × [0, T].

Boundary value problems for nonlinear fractional differential equations have recently been investigated by several researchers. As a matter of fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes (see 1 ) and make the fractional-order models more realistic and practical than the classical integer-order models. Fractional differential equations arise in many engineering and scientific disciplines, such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. (see 1 2 ). For some recent development on the topic, (see 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 and references therein).

Let us recall some basic definitions (see 20 21 ).

Definition 2.1 The Riemann-Liouville fractional integral of order α > 0 for a continuous function u: (0, ∞) → ℝ is defined as

I α u ( t ) = 1 Γ ( α ) ∫ 0 t ( t - s ) α - 1 u ( s ) d s ,

provided the integral exists.

Definition 2.2 For a continuous function u: (0, ∞) → ℝ, the Riemann-Liouville derivative of fractional order α > 0, n = [α] + 1 ([α] denotes the integer part of the real number α) is defined as

D α u ( t ) = 1 Γ ( n - α ) 1 d t n ∫ 0 t ( t - s ) n - α - 1 u ( s ) d s = d d t n I n - α u ( t ) ,

provided it exists.

For α < 0, we use the convention that D ^α u = I ^-α u. Also for β ∈ [0, α), it is valid that D ^β I ^a u = I ^α-β u.

Note that for λ >-1, λ ≠ α - 1, α - 2,..., α - n, we have

D α t λ = Γ ( λ + 1 ) Γ ( λ - α + 1 ) t λ - α ,

and

D α t α - i = 0 , i = 1 , 2 , … , n .

In particular, for the constant function u(t) = 1, we obtain

D α 1 = 1 Γ ( 1 - α ) t - α , α ∉ ℕ .

For α ∈ ℕ, we get, of course, D ^α 1 = 0 because of the poles of the gamma function at the points 0, -1, -2,....

For α > 0, the general solution of the homogeneous equation

D α u ( t ) = 0

in C(0, T) ∩ L(0, T) is

u ( t ) = c 0 t α - n + c 1 t α - n - 1 + ⋯ + c n - 2 t α - 2 + c n - 1 t α - 1 ,

where c _i , i = 1, 2,..., n - 1, are arbitrary real constants.

We always have D ^α I ^α u = u, and

I α D α u ( t ) = u ( t ) + c 0 t α - n + c 1 t α - n - 1 + ⋯ + c n - 2 t α - 2 + c n - 1 t α - 1 .

To define the solution for the nonlinear problem (1.1) and (1.2)-(1.3), we consider the following linear equation

D α u ( t ) = σ ( t ) , α ∈ 1 , 2 , t ∈ 0 , T , T > 0 ,

where σ ∈ C[0, T].

We define

A = ν ∫ 0 η s α - 1 ( η - s ) α - 2 Γ ( α - 1 ) d s = ν Γ ( α ) η 2 α - 2 Γ ( 2 α - 1 ) ,

such that A ≠ Γ(α).

The general solution of (2.1) is given by

u ( t ) = c 1 t α - 1 + c 0 t α - 2 + I α σ ( t ) ,

with I ^α the usual Riemann-Liouville fractional integral of order α.

From (2.3), we have

D α - 1 u ( t ) = c 1 Γ ( α ) + I 1 σ ( t ) ,

D α - 2 u ( t ) = c 1 Γ ( α ) t + c 0 Γ ( α - 1 ) + I 2 σ ( t ) .

Using the conditions (1.2) and (1.3) in (2.4) and (2.5), we find that c ₀ = 0 and

c 1 = ν Γ ( α ) - A ∫ 0 η ( η - s ) α - 2 Γ ( α - 1 ) ∫ 0 s ( s - x ) α - 1 Γ ( α ) σ ( x ) d x d s ,

where A is defined by (2.2).

Substituting the values of c ₀ and c ₁ in (2.3), the unique solution of (2.1) subject to the boundary conditions (1.2)-(1.3) is given by

u ( t ) = ∫ 0 t ( t - s ) α - 1 Γ ( α ) σ ( s ) d s + ν t α - 1 Γ ( α ) - A ∫ 0 η ( η - s ) α - 2 Γ ( α - 1 ) ∫ 0 s ( s - x ) α - 1 Γ ( α ) σ ( x ) d x d s = ∫ 0 t ( t - s ) α - 1 Γ ( α ) σ ( s ) d s + ν t α - 1 Γ ( α ) - A I 2 α - 1 σ ( η ) .

Let C = C ( [ 0 , T ] , ℝ ) denotes the Banach space of all continuous functions from [0, T] → ℝ endowed with the norm defined by ║u║ = sup{|u(t)|, t ∈ [0, T]}.

If u is a solution of (1.1) and (1.2)-(1.3), then

u ( t ) = ∫ 0 t ( t - s ) α - 1 Γ ( α ) f s , u s , ϕ u s , ψ u s d s + ν 1 t α - 1 ∫ 0 η ( η - s ) 2 α - 2 Γ ( 2 α - 1 ) f s , u s , ϕ u s , ψ u s d s ,

where

ν 1 = ν Γ ( α ) - A .

Define an operator P : C → C as

P u ( t ) = ∫ 0 t ( t - s ) α - 1 Γ ( α ) f s , u s , ϕ u s , ψ u s d s + ν 1 t α - 1 ∫ 0 η ( η - s ) 2 α - 2 Γ ( 2 α - 1 ) f s , u s , ϕ u s , ψ u s d s , t ∈ 0 , T .

Observe that the problem (1.1) and (1.2)-(1.3) has solutions if and only if the operator equation P u = u has fixed points.

Lemma 3.1 The operator P is compact.

Proof

(i) Let B be a bounded set in C[0, T]. Then, there exists a constant M such that |f(t,u(t), (φu)(t), (ψu)(t))| ≤ M, ∀u ∈ B, t∈[0, T]. Thus

| P u ( t ) | ≤ M ∫ 0 t ( t - s ) α - 1 Γ ( α ) d s + M | ν 1 | t α - 1 ∫ 0 η ( η - s ) 2 α - 2 Γ ( 2 α - 1 ) d s ≤ M T α - 1 T Γ ( α + 1 ) + | ν 1 | η 2 α - 1 Γ ( 2 α ) ,

which implies that

| | P u | | ≤ M T α - 1 T Γ ( α + 1 ) + | ν 1 | η 2 α - 1 Γ ( 2 α ) < ∞ .

Hence, P ( B ) is uniformly bounded.

(ii) For any t ₁, t ₂ ∈ [0, T], u ∈ B, we have

| P u t 1 - P u t 2 | = ∫ 0 t 1 t 1 - s α - 1 Γ ( α ) f s , u s , ϕ u s , ψ u s d s - ∫ 0 t 2 t 2 - s α - 1 Γ ( α ) f s , u s , ϕ u s , ψ u s d s + ν 1 t 1 α - 1 - t 2 α - 1 ∫ 0 η η - s 2 α - s Γ 2 α - 1 f s , u s , ϕ u s , ψ u s d s ≤ M 1 Γ ( α ) ∫ 0 t 1 t 1 - s α - 1 - t 2 - s α - 1 d s - 1 Γ ( α ) ∫ t 1 t 2 t 2 - s α - 1 d s + ν 1 t 1 α - 1 - t 2 α - 1 ∫ 0 η η - s 2 α - s Γ 2 α - 1 d s → 0 as t 1 → t 2 .

Thus, P ( B ) is equicontinuous. Consequently, the operator P is compact. This completes the proof.   □

We need the following known fixed point theorem to prove the existence of solutions for the problem at hand.

Theorem 3.1 ( 22 ) Let E be a Banach space. Assume that T: E → E be a completely continuous operator and the set V = {x ∈ E | x = μTx, 0 < μ < 1} be bounded.

Then, T has a fixed point in E.

Theorem 3.2 Assume that there exists a constant M > 0 such that

| f t , u t , ϕ u t , ψ u t | ≤ M , ∀ t ∈ 0 , T , u ∈ ℝ .

Then, the problem (1.1) and (1.2)-(1.3) has at least one solution on [0,T].

Proof We consider the set

V = u ∈ ℝ | u = μ P u , 0 < μ < 1 ,

and show that the set V is bounded. Let u ∈ V, then u = μ P u , 0 < μ < 1. For any t ∈ [0, T], we have

| u t | ≤ μ ∫ 0 t t - s α - 1 Γ ( α ) | f s , u s , ϕ u s , ψ u s | d s + | ν 1 | t α - 1 ∫ 0 η η - s 2 α - 2 Γ ( 2 α - 1 ) | f s , u s , ϕ u s , ψ u s | d s .

As in part (i) of Lemma 3.1, we have

∥ P u ∥ ≤ M T α - 1 T Γ α + 1 + | ν 1 | η 2 α - 1 Γ ( 2 α ) < ∞ .

This implies that the set V is bounded independently of μ ∈ (0,1). Using Lemma 3.1 and Theorem 3.1, we obtain that the operator P has at least a fixed point, which implies that the problem (1.1) and (1.2)-(1.3) has at least one solution. This completes the proof.

Theorem 3.3 Assume that

(A₁) there exist positive functions L ₁(t), L ₂(t), L ₃(t) such that

| f t , u t , ϕ u t , ψ u t - f t , v t , ϕ v t , ψ v t | ≤ L 1 t | u - v | + L 2 t | ϕ u - ϕ v | + L 3 t | ψ u - ψ v | , ∀ t ∈ 0 , 1 , u , v ∈ ℝ .

(A₂) Λ = (ξ ₁ + |ν ₁|T ^α-1 ξ ₂)(1 + γ ₀ + δ ₀) < 1, where

γ 0 = sup t ∈ 0 , 1 | ∫ 0 t γ t , s d s | , δ 0 = sup t ∈ 0 , 1 | ∫ 0 t δ t , s d s | , ξ 1 = sup t ∈ 0 , T | I q L 1 t | , | I q L 2 t | , | I q L 3 t | , ξ 2 = max | I 2 α - 1 L 1 η | , | I 2 α - 1 L 2 η | , | I 2 α - 1 L 3 η | ,

Then the problem (1.1) and (1.2)-(1.3) has a unique solution on C[0, T].

Proof Let us set sup _{t∈[0, T]} |f(t,0,0,0)| = M, and choose

r ≥ ε M 1 - Λ .

Then we show that P B r ⊂ B r , where B r = x ∈ C : ∥ u ∥ ≤ r . For x ∈ B _r , we have

∥ P u t ∥ = sup t ∈ 0 , T | ∫ 0 t t - s α - 1 Γ α f s , u s , ϕ u s , ψ u s d s + ν 1 t α - 1 ∫ 0 η η - s 2 α - 2 Γ 2 α - 1 f s , u s , ϕ u s , ψ u s d s | ≤ sup t ∈ 0 , T ∫ 0 t t - s α - 1 Γ α | f s , x s , ϕ x s , ψ x s - f s , 0 , 0 , 0 | + | f s , 0 , 0 , 0 | d s + | ν 1 | t α - 1 ∫ 0 η η - s 2 α - s Γ 2 α - 1 | f s , x s , ϕ x s , ψ x s - f s , 0 , 0 , 0 + | f s , 0 , 0 , 0 | d s ≤ sup t ∈ 0 , T ∫ 0 t t - s α - 1 Γ α L 1 s | x s | + L 2 s | ϕ x s | + L 3 s | ψ x s | + M d s + | ν 1 | t α - 1 ∫ 0 η η - s 2 α - 2 Γ 2 α - 1 L 1 s | x s | + L 2 s | ϕ x s | + L 3 s | ψ x s | + M d s ≤ sup t ∈ 0 , T ∫ 0 t t - s α - 1 Γ α L 1 s | x s | + γ 0 L 2 s | x s | + δ 0 L 3 s | x s | + M d s + | ν 1 | t α - 1 ∫ 0 η η - s 2 α - 2 Γ 2 α - 1 L 1 s | x s | + γ 0 L 2 s | x s | + δ 0 L 3 s | x s | + M d s ≤ sup t ∈ 0 , T I α L 1 ( t ) + γ 0 I α L 2 t + δ 0 I α L 3 t r + M t q Γ q + 1 + | ν 1 | t α - 1 I ( 2 α - 1 ) L 1 ( η ) + γ 0 I ( 2 α - 1 ) L 2 ( η ) + δ 0 I ( 2 α - 1 ) L 3 ( η ) r + M η 2 α - 1 Γ ( 2 α ) ≤ ξ 1 + | ν 1 | T α - 1 ξ 2 1 + γ 0 + δ 0 r + M T α - 1 T Γ α + 1 + | ν 1 | η α - 1 Γ ( 2 α ) = Λ r + M ε ≤ r

In view of (A ₁), for every t ∈ [0, T], we have

| P u t - P v t | ≤ sup t ∈ 0 , T ∫ 0 t t - s α - 1 Γ α | f s , u s , ϕ u s , ψ u s - f s , v s , ϕ v s , ψ v s | d s + | ν 1 | t α - 1 ∫ 0 η η - s 2 α - 2 Γ 2 α - 1 | f s , u s , ϕ u s , ψ u s - f s , υ s , ϕ v s , ψ v s | d s ≤ sup t ∈ 0 , T ∫ 0 t t - s α - 1 Γ α L 1 s | u - v | + L 2 s | ϕ v | + L 3 s | ψ u - ψ v | d s + | ν 1 | t α - 1 ∫ 0 η η - s 2 α - 2 Γ 2 α - 1 | L 1 ( s ) | u - v | + L 2 ( s ) | ϕ u - ϕ v | + L 3 ( s ) | ψ u - ψ v | ∥ d s ≤ sup t ∈ 0 , T I α L 1 ( t ) + γ 0 I α L 2 ( t ) + δ 0 I α L 3 ( t ) ∥ u - v ∥ + | ν 1 | T α - 1 I ( 2 α - 1 ) L 1 ( η ) + γ 0 I ( 2 α - 1 ) L 2 ( η ) + δ 0 I ( 2 α - 1 ) L 3 ( η ) ∥ u - v ∥ ≤ ξ 1 + | ν 1 | T α - 1 ξ 2 1 + γ 0 + δ 0 ∥ u - v ∥ = Λ ∥ u - v ∥

By assumption (A ₂), Λ < 1, therefore, the operator P is a contraction. Hence, by Banach fixed point theorem, we deduce that P has a unique fixed point which in fact is a unique solution of problem (1.1) and (1.2)-(1.3). This completes the proof.   □

Theorem 3.4 (Krasnoselskii's fixed point theorem 22 ). Let M be a closed convex and nonempty subset of a Banach space X. Let A, B be the operators such that (i) A x + B y ∈ M whenever x , y ∈ M ; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then, there exists z ∈ M such that z = Az + Bz.

Theorem 3.5 Assume that f: [0, T] × ℝ × ℝ × ℝ → ℝ is a continuous function and the following assumptions hold:

(H ₁ )

| f t , u t , ϕ u t , ψ u t - f t , v t , ϕ v t , ψ v t | ≤ L 1 ( t ) | u - v | + L 2 ( t ) | ϕ u - ϕ v | + L 3 ( t ) | ψ u - ψ v | , ∀ t ∈ 0 , T , u , v ∈ ℝ .

(H ₂ ) |f (t,u)| ≤ μ(t), ∀(t,u)∈[0, T] × ℝ, and μ ∈ C([0, T],ℝ⁺).

If

| ν 1 | T α - 1 η 2 α - 1 Γ 2 α < 1 ,

then the boundary value problem (1.1) and (1.2)-(1.3) has at least one solution on [0, T].

Proof Letting sup_{t∈[0, T]} |μ(t)| = ||μ||, we fix

r ̄ ≥ ∥ μ ∥ T α - 1 T Γ ( α + 1 ) + | ν 1 | η 2 α - 1 Γ ( 2 α ) ,

and consider B r ̄ = u ∈ C : ∥ u ∥ ≤ r ̄ . We define the operators P 1 and P 2 on B r ̄ as

P 1 u ( t ) = ∫ 0 t t - s α - 1 Γ ( α ) f s , u ( s ) , ( ϕ u ) ( s ) , ( ψ u ) ( s ) d s , P 2 u ( t ) = ν 1 t α - 1 ∫ 0 η η - s 2 α - s Γ 2 α - 1 f s , u s , ϕ u ( s ) , ( ψ u ( s ) ) d s .

For u , v ∈ B r ̄ , we find that

∥ P 1 u + P 2 v ∥ ≤ ∥ μ ∥ T α - 1 T Γ ( α + 1 ) + | ν 1 | η 2 α - 1 Γ ( 2 α ) ≤ r ̄ .

Thus, P 1 u + P 2 v ∈ B r ̄ . It follows from the assumption (H ₁) together with (3.1) that P 2 is a contraction mapping. Continuity of f implies that the operator P 1 is continuous.

Also, P 1 is uniformly bounded on B r ̄ as

∥ P 1 u ∥ ≤ ∥ u ∥ T α Γ α + 1 .

Now we prove the compactness of the operator P 1 .

In view of (H ₁), we define sup ( t , x , ϕ x , ψ x ) ∈ 0 , T × B r × B r × B r | f ( t , x , ϕ x , ψ x ) | = f ̄ , and consequently we have

| P 1 u t 1 - P 2 u ( t 2 ) | = 1 Γ ( α ) ∫ 0 t 1 ( t 1 - s ) α - 1 - ( t 2 - s ) α - 1 s , u ( s ) , ( ϕ u ) ( s ) , ( ψ u ) ( s ) d s - 1 Γ ( α ) ∫ t 1 t 2 ( t 2 - s ) α - 1 f s , u ( s ) , ( ϕ u ) ( s ) , ( ψ u ) ( s ) d s ≤ f ̄ Γ ( α + 1 ) | 2 ( t 2 - t 1 ) α + t 1 α - t 2 α | ,

which is independent of u and tends to zero as t ₂ → t ₁. So, P 1 is relatively compact on B r ̄ . Hence, by the Arzelá-Ascoli Theorem, P 1 is compact on B r ̄ . Thus, all the assumptions of Theorem 3.4 are satisfied. So the conclusion of Theorem 3.4 implies that the boundary value problem (1.1) and (1.2)-(1.3) has at least one solution on [0, T]. This completes the proof.   □

The authors declare that they have no competing interests.

Both authors, BA and JJN, contributed to each part of this work equally and read and approved the final version of the manuscript.