and J. J. Trujillo

doi:10.1155/2011/690653

Research Article

Existence of Solutions to Fractional Mixed Integrodifferential Equations with Nonlocal Initial Condition

A. Anguraj,

P. Karthikeyan,

and J. J. Trujillo

1Department of Mathematics, PSG College of Arts and Science, Coimbatore 641 014, India

2Department of Mathematics, KSR College of Arts and Science, Tiruchengode 637 215, India

3Departamento de An´alisis Matem´atico, Universidad de La Laguna, Tenerife 38271 La Laguna, Spain

Correspondence should be addressed to J. J. Trujillo,[email protected] Received 14 November 2010; Accepted 6 January 2011

Academic Editor: Dumitru Baleanu

Copyrightq2011 A. Anguraj et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the existence and uniqueness theorem for the nonlinear fractional mixed Volter- ra-Fredholm integrodifferential equation with nonlocal initial conditiond^αxt/dt^α ft, xt, t

0kt, s, xsds,1

0ht, s, xsds, x0 1

0gsxsds, wheret ∈ 0,1, 0 < α < 1, andf is a given function. We point out that such a kind of initial conditions or nonlocal restrictions could play an interesting role in the applications of the mentioned model. The results obtainded are applied to an example.

1. Introduction

Recently it have been proved that the differential models involving nonlocal derivatives of fractional order arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in many fields, for instance, physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forthsee1–

6. In fact, such models can be considered as an efficient alternative to the classical nonlinear differential models to simulate many complex processes see7. For instance, fractional differential equations are an excellent tool to describe hereditary properties of viscoelastic materials and, in general, to simulate the dynamics of many processes on anomalous media.

Theory of fractional differential equations has been extensively studied by several authors as Delbosco and Rodino 8, Kilbas et al. 6, Lakshmikantham et al. 9–11, and also see 2,12–16.

Recently Mophou and N’Gu´er´ekata17, studied the Cauchy problem with nonlocal conditions

D^qxt Axt tⁿft, xt, Bxt, t∈0, T, n∈Z ,

x0 gx x0, 1.1

in general Banach space X with 0 < q < 1, and A is the infinitesimal generator of a C0- semigroup of bounded linear operators. By means of the Krasnoselskii’s Theorem, existence of solutions was also obtained.

Subsequently several authors have investigated the problem for different types of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces.

Very recently N’Gu´er´ekata 2, 18discussed the existence of solutions of fractional abstract differential equations with nonlocal initial condition. The nonlocal Cauchy problem is discussed by authors in 15 using the fixed-point concepts. Tidke 19 studied the nonfractional mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions using Leray-Schauder Theorem.

Motivated by the above mentioned works in this manuscript we discuss the existence and the uniqueness of the solution for the following fractional integrodifferential equation with nonlocal integral initial condition in Banach Space:

d^αxt dt^α f

t, xt,

_t

0

kt, s, xsds, ₁

0

ht, s, xsds

,

x0 ₁

0

gsxsds,

1.2

wheret∈J 0,1, 0< α <1,gt∈0,1,g∈L¹0,1,R ,x∈Y CJ, Eis a continuous function onJ with values in the Banach spaceEandxY max_t∈JxtE, andf :J×E× E×E → E,k : D×E → E, andh: D0×E → Eare continuousE-valued functions. Here D {t, s ∈R² : 0≤ s ≤ t ≤ 1}, andD₀ J×J. The operatord^α/dt^αdenotes the Caputo fractional derivative of order α.

For the sake of the shortness let Kxt

_t

0

kt, s, xsds, Hxt ₁

0

ht, s, xsds. A

The paper is organized as follows. In Section 2, some definitions, lemmas and preliminary results are introduced to be used in the sequel.Section 3will involve the assumptions, main results and proofs of existence problem of 1.2, together with a nonlocal initial condition.

Finally an example is presented.

2. Preliminaries

LetEbe a real Banach space andθthe zero element ofE. LetL¹0,1, Ebe the Banach space of measurable functionsx : 0,1 → Ewhich are Lebesgue integrable, equipped with the

normxL¹₁

0xsds. We will use the following notationR 0,∞,andR 0,∞. A functionx∈C0,1, Eis called a solution of1.2if it satisfies1.2.

Definition 2.1. A real functionftis said to be in the spaceCα, α ∈ Rif there exists a real numberp > α, such thatft t^pgt, whereg ∈C0,∞, whileftis said to be in the space C^m_α if and only iff^m∈C_α, m∈N.

Definition 2.2. The fractionalarbitraryorder Riemann-Liouville integralon the right and on the leftof the functionf∈L¹a, b, R of orderα, a, b∈R is defined by

I_a^αft 1 Γα

_t

a

t−s^α−1

Γα fsds, I_b−^α ft 1 Γα

_b

t

s−t^α−1

Γα fsds, 2.1

whereΓis the Gamma function of Euler.

Whena0, we writeI^αft I₀ ^αft f∗φαt, whereφαt t^α−1/Γαfort >0, φαt 0 fort≤0, and∗represents the Convolution of Laplace. Then, it is well known that φ_αt → δtasα → 0, whereδis the Delta function.

Definition 2.3. The Riemann-Liouville fractional integral operator of orderα >0, of a function f∈Cμ, μ≥ −1 is defined as

I^αft φα∗ft, α >0, t >0,

I⁰fx fx. 2.2

Definition 2.4. The Caputo’s derivative of fractional orderα >0 for a suitable functionftis defined by

_c D^αf

t 1 Γn−α

_t

0

fⁿs

t−s^{α−n 1}ds, n−1< α < n, n α 1, 2.3 whereαdenotes the integer part of real number α.

It is obvious that the Caputo’s derivative of a constant is equal to 0.

Lemma 2.5. Letα >0 andn α 1. Then

I^α_c D^αf

t ft−ⁿ⁻¹

k0

f^k0

k! t^k. 2.4

Lemma 2.6. IfQτ, α ΓαI₁₋^αgτ ₁

τgss−τ^α−1dsforτ ∈0,1, and ifg ∈L¹0,1, R satisfies 0≤gs≤1 for 0≤s≤1 andα >0, then

Qτ, α Γα < e,

_t

0t−s^α−1ds

Γα < e. 2.5

Proof. A direct computation shows

Qτ, α Γα

₁

τgss−τ^α−1ds _∞

0 s^α−1e^−sds ₁

τs−τ^α−1ds _∞

0 s^α−1e^−sds

_1−τ

0 s^α−1ds _∞

0 s^α−1e^−sds ≤ e_1−τ

0 s^α−1e^−sds _∞

0 s^α−1e^−sds < e, _t

0t−s^α−1ds Γα

_t

0s^α−1ds _∞

0 s^α−1e^−sds ≤ e_t

0s^α−1e^−sds _∞

0 s^α−1e^−sds < e.

2.6

Theorem 2.7Krasnoselkii. LetXbe a Banach space, letSbe a bounded closed convex subset ofX and letA,Bbe maps ofSintoXsuch thatAx By ∈Sfor every pairx, y ∈S. If A is completely continuous and B is a contraction then the equationAx Bxxhas a solution on S.

3. Main Results

We assume the following.

A1Iff ∈C0,1×E×E×E, Eand a nonnegative, boundedp_f ∈L¹0,1, R , there existM >0,p_ft≤Mfort∈0,1such that

ft, x, Kx, Hx ≤pftx forx∈E. 3.1

A2There exist positive constantsL₁,L₂, andLsuch that f

t, x1, y1, z1

−f

t, x2, y2, z2 ≤L1

x1−x2 y1−y2 z1−z2

3.2 for allx1, y1, z1, x2, y2, z2∈Y,L2max_t∈Jft,0,0,0, andLmax{L1, L2}.

A3There exist positive constantsN₁,N₂, andNsuch that

kt, s, x1−kt, s, x2 ≤N1x1−x2 3.3 for allx1, x2∈Y,N2max_t,s∈Dkt, s,0, andNmax{N1, N2}.

A4There exist positive constantsC₁,C₂, andCsuch that

ht, s, x1−ht, s, x2< C1x1−x2 3.4 for allx1, x2∈Y,C2max_t,s∈D0ht, s,0, andCmax{C1, C2}.

A5p L/Γα 11 C N/α 1is such that 0≤p <1.

Firstly, we obtain the following lemmas to prove the main results on the existence of solutions to1.2.

Lemma 3.1. If (A1) holds withμ ₁

0gsds, then the problem1.2is equivalent to the following equation:

xt 1

1−μ Γα

₁

0

Qτ, αfτ, xτ, Kxτ, Hxτdτ φαt∗ft, xt, Kxt, Hxt.

3.5 Proof. ByLemma 2.5and1.2, we have

xt x0 1 Γα

_t

0

t−s^α−1fs, xs, Kxs, Hxsds. 3.6

Therefore,

x0 ₁

0

gsxsds

₁

0

gsdsx0 1 Γα

₁

0

gs _s

0

s−τ^α−1fτ, xτ, Kxτ, Hxτdτ ds.

3.7

So,

x0 1

1−μ Γα

₁

0

fτ, xτ, Kxτ, Hxτ ₁

τ

s−τ^α−1gsds

dτ

1 1−μ

Γα ₁

0

Qτ, αfτ, xτ, Kxτ, Hxτdτ,

3.8

and then

xt 1

1−μ Γα

₁

0

Qτ, αfτ, xτ, Kxτ, Hxτdτ φ_αt∗ft, xt, Kxt, Hxt.

3.9 Conversely, ifxis a solution of3.5, then for everyt∈0,1, according toDefinition 2.4we have

cD^αxt ^cD^α

1 1−μ

Γα ₁

0

Qτ, αfτ, xτ, Kxτ, Hxτdτ

φαt∗ft, xt, Kxt, Hxt θ ^cD^α

I^αft, xt, Kxt, Hxt ft, xt, Kxt, Hxt.

3.10

It is obvious thatx0 ₁

0gsxsds. This completes the proof.

Lemma 3.2. If (A3) and (A4) are satisfied,K,Hare defined inA, then the conditions Kxt ≤tN1x N2,

Kx1t−Kx₂t ≤N₁tx1−x₂, Hxt ≤C1x C₂, Hx1t−Hx2t ≤C1x1−x2,

3.11

are satisfied for anyt∈J, andx, x₁, x₂∈Y. Proof. ByA3, we have

Kxt ≤ _t

0

kt, s, xsds

_t

0

kt, s, xs−kt, s,0 kt, s,0ds

≤ _t

0

kt, s, xs−kt, s,0ds _t

0

kt, s,0ds

≤N₁tx N₂t≤N1x N₂.

3.12

On the other hand,

Kx1t−Kx2t ≤ _t

0

kt, s, x1s−kt, s, x2sds

≤N1

_t

0

x1s−x2sds

≤N₁tx1−x₂.

3.13

Similarly, for the other conditions, we use assumptionA4, to get Hxt ≤

₁

0

ht, s, xsds≤C1x C₂, Kx1t−Kx2t ≤C1x1−x2.

3.14

Theorem 3.3. If (A1)–(A5) are satisfied, then the fractional integrodifferential equation1.2has a unique solution continuous inJ.

Proof. We use the Banach contraction principle to prove the existence and uniqueness of the solution to1.2. LetBr {x∈Y :x ≤r} ⊆ Y, wherer ≥eM/1−μ L/Γα 11 C N/α 1and define the operatorΨon the Banach spaceY by

Ψxt 1 1−μ

Γα ₁

0

Qτ, αfτ, xτ, Kxτ, Hxτdτ 1

Γα _t

0

t−s^α−1fs, xs, Kxs, Hxsds.

3.15

Firstly, we show that the operatorΨmapsBrinto itself. By usingA1and triangle inequality, we have

Ψxt ≤ eM

1−μx 1 Γα

_t

0

t−s^α−1fs, xs, Kxs, Hxsds

≤ eM

1−μx 1 Γα

_t

0

t−s^α−1 fs, xs, Kxs, Hxs ds

≤ eM 1−μx

1 Γα

_t

0

t−s^α−1fs, xs, Kxs, Hxs−fs,0,0,0 fs,0,0,0ds

≤ eM

1−μx 1 Γα

_t

0

t−s^α−1fs, xs, Kxs, Hxs−fs,0,0,0ds 1

Γα _t

0

t−s^α−1fs,0,0,0ds.

3.16 Now, ifA2is satisfied, then

Ψxt ≤ eM

1−μx L₁ Γα

_t

0

t−s^α−1xs Kxs Hxsds L2

Γα _t

0

t−s^α−1ds

≤ eM

1−μx L1

Γα _t

0

t−s^α−1xsds L1

Γα _t

0

t−s^α−1Kxsds L₁

Γα _t

0

t−s^α−1Hxsds L₂ Γα

_t

0

t−s^α−1ds.

3.17

UsingLemma 3.2andA3, we have

Ψxt ≤ eM

1−μx L1

Γα 1t^αx L1

ΓαN1x N2 _t

0

t−s^α−1sds L₁

Γα 1C1x C₂t^α L₂ Γα 1t^α

≤ eM

1−μx L1

Γα 1t^αx L1

Γα 2t^{α 1}N1x N₂ L₁

Γα 1C1x C2t^α L₂ Γα 1t^α eM

1−μx L₁N₂

Γα 2t^{α 1} L₁C₂

Γα 1t^α L₂ Γα 1t^α L1

Γα 1t^α

1 N1

α 1t C1

x,

3.18

ifx∈Br, we have

Ψxt ≤ eM

1−μx L

Γα 1

1 N

α 1 C

Lr

Γα 1

1 N

α 1 C

≤r.

3.19

Thus ΨBr ⊂ B_r. Next, we prove that Ψis a contraction mapping. For this, let x₁, x₂ ∈ Y. ApplyingA2, we have

Ψx1t−Ψx2t

Qτ, α 1−μ

Γα ₁

0

fs, x1τ, Kx1τ, Hx1τ−fs, x2τ, Kx2τ, Hx2τ dτ

1

Γα _t

0

t−s^α−1fs, x1s, Kx1s, Hx1sds

− 1 Γα

_t

0

t−s^α−1fs, x2s, Kx2s, Hx2sds

≤ e 1−μ

₁

0

L1x1τ−x₂τ Kx1τ−Kx₂τ Hx1τ−Hx₂τdτ L₁

Γα _t

0

t−s^α−1x1s−x₂sds _t

0

t−s^α−1Kx1s−Kx₂sds _t

0

t−s^α−1Hx1s−Hx₂sds

3.20 then usingA3,A4andLemma 3.2, one gets

Ψx1t−Ψx2t

≤ L₁e

1−μx1−x_{2 1}

0

dτ N_{1 1}

0

τdτ C_{1 1}

0

dτ

L1

Γαx1−x2 t

0

t−s^α−1ds N1

_t

0

t−s^α−1sds C1

_t

0

t−s^α−1ds

≤ L₁e 1−μ

1 N₁

2 C₁

x1−x₂ L₁ Γα

t^α α

N₁Γαt^{α 1} Γα 2

C₁t^α α

x1−x₂

L1e

1−μ

1 N1

2 C₁

L1

Γα 1

1 C₁ N1

α 1t

t^α

x1−x₂

≤ Le

1−μ

1 N

2 C1

L Γα 1

1 C N

α 1

x1−x2

≤ Le

1−μ

1 N

2 C

p

x1−x₂

≤ΩC,N,α,μ,px1−x2,

3.21

whereΩC,N,α,μ,p Le/1−μ1 N/2 C p < 1 depends on the parameter of the problem. ThereforeΨhas a unique fixed-pointx Ψx ∈ B_r, which is a solution of3.5, and hence is a solution of1.2.

Theorem 3.4. Assume (A1)–(A4) holds. IfeM <1−μ, then1.2has at least one solution onI.

Proof. Chooser ≥ eM/1−μ pfL¹/Γα 1and considerB_r : {x ∈C :x ≤ r}. Now define onB_r the operatorsA, Bby

Axt:φαt∗ft, xt, Kxt, Hxt, Bxt: 1

1−μ Γα

₁

0

Qτ, αfτ, xτ, Kxτ, Hxτdτ. 3.22

Let us observe that ifx, y∈B_rthenAx By∈B_r. Indeed it is easy to check the inequality

Ax By ≤ eM

1−μ p_f

L¹

Γα 1 ≤r. 3.23

We can easily show that thatBis a contraction mapping. Letu, v∈Br. Then But−Bvt

: 1

1−μ Γα

₁

0

Qτ, α fτ, uτ, Kuτ, Huτ−fτ, vτ, Kvτ, Hvτ dτ

≤ e 1−μ

₁

0

L1uτ−vτ Kuτ−Kvτ Huτ−Hvτdτ

≤ L₁e 1−μ

1 N₁

2 C₁

u−v

≤ΛN1,C1,L1u−v,

3.24 whereΛN1,C1,L1 L1e/1−μ1 N1/2 C₁ <1 depends only on the parameter of the problem and henceBis contraction. Sincexis continuous, thenAxtis continuous in view ofA1. Let us now note thatAis uniformly bounded onBr. This follows from the inequality

Axt ≤ pf

L¹

Γα 1. 3.25

Now let us prove thatAxtis equicontinuous.

Lett1, t2∈Jandx∈Br. Using the fact thatfis bounded on the compact setJ×Br thus sup_t,s∈J×B

rft, xs, Kxs, Hxs:c₀<∞, we will get Axt1−Axt2 1

Γ q

_t1

0

t1−s^q−1 _s

0

fs, xs, Kxs, Hxsds

− _t2

0

t2−s^q−1 _s

0

fs, xs, Kxs, Hxsds

1 Γ

q _t1

t2

t1−s^q−1 _s

0

fs, xs, Kxs, Hxsds

− _t2

0

t2−s^q−1−t1−s^{q−1 s}

0

fs, xs, Kxs, Hxsds

≤ 1 Γ

q _t1

t2

t1−s^q−1 _s

0

fs, xs, Kxs, Hxsds

1 Γ

q _t2

0

t2−s^q−1−t1−s^q−1 _s

0

fs, xs, Kxs, Hxsds

≤ c0

Γ

q 12t1−t2^q t^q₂−t^q₁

≤ 2c0

Γ

q 1|t1−t2|^q,

3.26 which does not depend onx. SoABris relatively compact. By the Arzela-Ascoli Theorem, Ais compact. We now conclude the result of the theorem based on the Krasnoselkii’s theorem above.

4. Example

Consider the following fractional integrodifferential equation:

cD^αxt e^−txt

9 e^t1 xt 1 10

_t

0

e^−1/2xsds 1 10

_t

0

e^−1/49xsds, t∈J 0,1,

x0 ₁

0

1

2xsds,

4.1

where 0 < α ≤ 1. Take E R . Set Kxt _t

0e^−1/2xsds, Hxt _t

0e^−1/49xsds, ft, x, Kx, Hx e^−tx/9 e^t1 x Kx Hx,gs 1/2,pft e^−txt/9 e^t. Then it is clear that

f∈C0,1×E×E×E, E, pft≤ 1 10 M, p_f ∈L0,1, R , ft, x, Kx, Hx ≤p_fx.

4.2

So,A1is satisfied. Letx, y∈Eandt∈J. Then we have Kx−Ky

_t

0

e^−1/2xsds− _t

0

e^−1/2ysds ≤ 1

2x−y, Hx−Hy

_t

0

e^−1/49xsds− _t

0

e^−1/49ysds ≤ 1

49x−y, ft, x, Kx, Hx−f

t, y, y, Hy

e^−tx

9 e^t1 x− e^−ty 9 e^t

1 y 1

10

Kx−Ky

Hx−Hy

≤ e^−tx−y 9 e^t1 x

1 y 1

10

Kx−Ky Hx−Hy

≤ e^−t

9 e^tx−y 1 10

Kx−Ky Hx−Hy

≤ 1

10x−y Kx−Ky Hx−Hy .

4.3

Hence the conditions A1–A4 hold withM 1/10,L₁ 1/10,N₁ 1/2, C₁ 1/49.

Chooser1 andμ1/2. Indeed L1e 1−μ

1 N1

2 C1

149e

490 <1, eM

1−μ e 5 <1.

4.4

FurtherA5is satisfied by a suitable choice of α. Then byLemma 3.2the problem1.2has a unique solution on0,1.

Acknowledgment

This paper has been partially supported by MICINNproject MTM2010-16499to which the authors are very thankful.

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