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Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions

Abstract

This article investigates a boundary value problem of Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Some new existence results are obtained by applying standard fixed point theorems.

2010 Mathematics Subject Classification: 26A33; 34A34; 34B15.

1 Introduction

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 ,
(1.1)

subject to the boundary conditions of fractional order given by

D α - 2 u ( 0 + ) = 0 ,
(1.2)
D α - 1 u ( 0 + ) = ν I α - 1 u ( η ) , 0 < η < T , ν is a constant ,
(1.3)

where Dαdenotes the Riemann-Liouville fractional derivative of order α, f: [0, T] × × × 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 [319] and references therein).

2 Preliminaries

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, ∞) → 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, ∞) → , 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 Dαu = Iu. Also for β [0, α), it is valid that DβIau = Iα-βu.

Note that for λ >-1, λα - 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 .

For α , we get, of course, Dα1 = 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 DαIαu = 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 ,
(2.1)

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 ) ,
(2.2)

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 ) ,
(2.3)

with Iαthe 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 ) ,
(2.4)
D α - 2 u ( t ) = c 1 Γ ( α ) t + c 0 Γ ( α - 1 ) + I 2 σ ( t ) .
(2.5)

Using the conditions (1.2) and (1.3) in (2.4) and (2.5), we find that c0 = 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 c0 and c1 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 ) .
(2.6)

3 Main results

Let C=C ( [ 0 , T ] , ) denotes the Banach space of all continuous functions from [0, T] → 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:CC 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 Pu=u has fixed points.

Lemma 3.1 The operator P is compact.

Proof

  1. (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.

  1. (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: EE 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 .

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 | u = μ P u , 0 < μ < 1 ,

and show that the set V is bounded. Let u V, then u=μPu, 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 L1(t), L2(t), L3(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 .

(A2) Λ = (ξ1 + |ν1|Tα-1ξ2)(1 + γ0 + δ0) < 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 supt[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 (A1), 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 (A2), 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]). LetMbe a closed convex and nonempty subset of a Banach space X. Let A, B be the operators such that (i)Ax+ByMwheneverx,yM; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then, there existszMsuch that z = Az + Bz.

Theorem 3.5 Assume that f: [0, T] × × × 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 .

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

If

| ν 1 | T α - 1 η 2 α - 1 Γ 2 α <1,
(3.1)

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

Proof Letting supt[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 (H1) 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 (H1), 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 t2t1. 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.   □

References

  1. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.

    Google Scholar 

  2. Sabatier J, Agrawal OP, Machado JAT, (eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.

  3. Agarwal RP, Andrade B, Cuevas C: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal Real World Appl 2010, 11: 3532-3554. 10.1016/j.nonrwa.2010.01.002

    Article  MathSciNet  Google Scholar 

  4. Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal 2010, 72: 2859-2862. 10.1016/j.na.2009.11.029

    Article  MathSciNet  Google Scholar 

  5. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound Value Probl 2009, 2009: 11. (Article ID 708576)

    MathSciNet  Google Scholar 

  6. Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput Math Appl 2009, 58: 1838-1843. 10.1016/j.camwa.2009.07.091

    Article  MathSciNet  Google Scholar 

  7. Ahmad B, Nieto JJ: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol Methods Nonlinear Anal 2010, 35: 295-304.

    MathSciNet  Google Scholar 

  8. Ahmad B: Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations. Appl Math Lett 2010, 23: 390-394. 10.1016/j.aml.2009.11.004

    Article  MathSciNet  Google Scholar 

  9. Ahmad B, Alsaedi A: Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Appl 2010, 2010: 17. (Article ID 364560)

    Article  MathSciNet  Google Scholar 

  10. Ahmad B, Nieto JJ: Riemann-Liouville fractional differential equations with fractional boundary conditions. Fixed Point Theory, in press.

  11. Ahmad B, Nieto JJ: Anti-periodic fractional boundary value problems. Comput Math Appl 2011, 62: 1150-1156. 10.1016/j.camwa.2011.02.034

    Article  MathSciNet  Google Scholar 

  12. Arciga-Alejandre MP: Asymptotics for nonlinear evolution equation with module-fractional derivative on a half-line. Bound Value Probl 2011, 2011: 29. (Article ID 946143) 10.1186/1687-2770-2011-29

    Article  Google Scholar 

  13. Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal 2010, 72: 916-924. 10.1016/j.na.2009.07.033

    Article  MathSciNet  Google Scholar 

  14. Belmekki M, Nieto JJ, Rodríguez-López R: Existence of periodic solution for a nonlinear fractional differential equation. Bound Value Probl 2009, 2009: 18. (Article ID 324561)

    Article  Google Scholar 

  15. De la Sen M: About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory. Fixed Point Theory Appl 2011, 2011: 19. (Article ID 867932) 10.1186/1687-1812-2011-19

    Article  Google Scholar 

  16. Nieto JJ: Maximum principles for fractional differential equations derived from Mittag-Leffler functions. Appl Math Lett 2010, 23: 1248-1251. 10.1016/j.aml.2010.06.007

    Article  MathSciNet  Google Scholar 

  17. Wang Y, Wang F, An Y: Existence and multiplicity of positive solutions for a nonlocal differential equation. Bound Value Probl 2011, 2011: 5. 10.1186/1687-2770-2011-5

    Article  MathSciNet  Google Scholar 

  18. Wei Z, Li Q, Che J: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J Math Anal Appl 2010, 367: 260-272. 10.1016/j.jmaa.2010.01.023

    Article  MathSciNet  Google Scholar 

  19. Zhang S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput Math Appl 2010, 59: 1300-1309.

    Article  MathSciNet  Google Scholar 

  20. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. In North-Holland Mathematics Studies. Volume 204. Elsevier, Amsterdam; 2006.

    Google Scholar 

  21. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon; 1993.

    Google Scholar 

  22. Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1980.

    Google Scholar 

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Acknowledgements

This study was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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

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Ahmad, B., Nieto, J.J. Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound Value Probl 2011, 36 (2011). https://doi.org/10.1186/1687-2770-2011-36

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