Mild Solutions for Fractional Differential Equations with Nonlocal Conditions

doi:10.1155/2010/287861

Research Article

Mild Solutions for Fractional Differential Equations with Nonlocal Conditions

Fang Li

School of Mathematics, Yunnan Normal University, Kunming 650092, China

Correspondence should be addressed to Fang Li,[email protected] Received 8 January 2010; Accepted 21 January 2010

Academic Editor: Gaston Mandata N’Guerekata

Copyrightq2010 Fang Li. 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.

This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions d^qxt/dt^q −Axt ft, xt, Gxt, t ∈ 0, T,andx0 gx x0, in a Banach space X, where 0 < q < 1. General existence and uniqueness theorem, which extends many previous results, are given.

1. Introduction

The fractional differential equations can be used to describe many phenomena arising in engineering, physics, economy, and science, so they have been studied extensivelysee, e.g., 1–8and references therein.

In this paper, we discuss the existence and uniqueness of mild solution for d^qxt

dt^q −Axt ft, xt, Gxt, t∈0, T, x0 gx x₀,

1.1

where 0 < q < 1,T > 0, and −A generates an analytic compact semigroup {St}_t≥0 of uniformly bounded linear operators on a Banach spaceX. The termGxtwhich may be interpreted as a control on the system is defined by

Gxt: _t

0

Kt, sxsds, 1.2

whereK ∈CD,Rthe set of all positive function continuous onD:{t, s∈R² : 0≤s≤ t≤T}and

G^∗ sup

t∈0,T

_t

0

Kt, sds <∞. 1.3

The functionsfandgare continuous.

The nonlocal conditionx0 gx x₀can be applied in physics with better effect than that of the classical initial condition x0 x0. There have been many significant developments in the study of nonlocal Cauchy problemssee, e.g.,6,7,9–14and references cited there.

In this paper, motivated by1–7,9–15 especially the estimating approach given by Xiao and Liang14, we study the semilinear fractional differential equations with nonlocal condition1.1in a Banach spaceX, assuming that the nonlinear mapfis defined on0, T× Xα×Xαandgis defined onC0, T, XαwhereXαDA^α, for 0< α <1, the domain of the fractional power ofA. New and general existence and uniqueness theorem, which extends many previous results, are given.

2. Preliminaries

In this paper, we set I 0, T, a compact interval in R. We denote byX a Banach space with norm · . Let−A : DA → X be the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators{St}_t≥0, that is, there existsM > 1 such thatSt ≤M; and without loss of generality, we assume that 0 ∈ρA. So we can define the fractional powerA^αfor 0< α≤ 1, as a closed linear operator on its domainDA^αwith inverseA^−α, and one has the following known result.

Lemma 2.1 see15. 1X_α DA^α is a Banach space with the normx_α : A^αx for x∈DA^α.

2 St:X → Xαfor eacht >0 andα >0.

3For everyu∈DA^αandt≥0,StA^αuA^αStu.

4For everyt >0,A^αStis bounded onXand there existsM_α>0 such that

A^αSt ≤M_αt^−α. 2.1

Definition 2.2. A continuous functionx:I → Xsatisfying the equation

xt St

x₀−gx 1

Γ q

_t

0

t−s^q−1St−sfs, xs, Gxsds 2.2

fort∈Iis called a mild solution of1.1.

In this paper, we usef_pto denote theL^pnorm offwheneverf ∈L^p0, Tfor some pwith 1 ≤ p < ∞. We denote byC_α the Banach spaceC0, T, Xαendowed with the sup norm given by

x_∞:sup

t∈I x_α, 2.3

forx∈C_α.

The following well-known theorem will be used later.

Theorem 2.3Krasnoselkii, see16. LetΩbe a closed convex and nonempty subset of a Banach spaceX. LetA, Bbe two operators such that

iAxBy∈Ωwheneverx, y∈Ω.

iiAis compact and continuous, iiiBis a contraction mapping.

Then there existsz∈Ωsuch thatzAzBz.

3. Main Results

We require the following assumptions.

H1The function f : 0, T×X_α×X_α → X is continuous, and there exists a positive functionμ·:0, T → Rsuch that

f

t, x, y≤μt, the functions −→ μs

t−s^α belongs to L^p0, t,R, γt:

t 0

μs t−s^α

p

ds

1/p

≤MT <∞, fort∈0, T,

3.1

wherep >1/q >1.

H2The functiong:C_α → X_αis continuous and there existsb >0 such that gx−gy

α≤bx−y

∞, 3.2

for anyx, y∈Cα.

Theorem 3.1. Let−Abe the infinitesimal generator of an analytic compact semigroup{St}_t≥0with St ≤M, t≥0,and 0∈ρA. If the mapsfandgsatisfy (H1), (H2), respectively, andMb <1, then1.1has a mild solution for everyx0∈Xα.

Proof. Setλsup_x∈C

αgx_αand choosersuch that r≥Mx0_αλ MαMT

Γ

q Mp,q·T^q−1/p, 3.3

whereM_p,q: p−1/pq−1^p−1/p.

LetBr {x∈C0, T, Xα| x_∞≤r}.

Define

Axt: 1 Γ

q _t

0

t−s^q−1St−sfs, xs, Gxsds, Bxt:St

x0−gx .

3.4

Letx, y∈B_r, then fort∈0, Twe have the estimates Axt Byt

α

≤ Stx0_αλ 1 Γ

q _t

0

t−s^q−1A^αSt−sfs, xs, Gxsds

≤Mx0_αλ M_α Γ

q _t

0

t−s^q−1 μs t−s^αds

≤Mx0_αλ Mα

Γ q

t 0

t−s^q−1p/p−1ds

p−1/p

· t

0

μs t−s^α

p

ds

1/p

≤Mx0_αλ M_αM_T Γ

q Mp,q·T^q−1/p

≤r.

3.5 Hence we obtainAxBy∈Br.

Now we show thatAis continuous. Let{xn}be a sequence ofBrsuch thatxn → xin B_r. Then

fs, xns, Gxns−→fs, xs, Gxs, n−→ ∞, 3.6

since the functionfis continuous onI×Xα×Xα. Fort∈0, T, using2.1, we have Axnt−Axt_α

1 Γ

q

_t

0

t−s^q−1St−s

fs, xns, Gxns−fs, xs, Gxs ds

α

≤ 1 Γ

q _t

0

t−s^q−1A^αSt−s

fs, xns, Gxns−fs, xs, Gxsds

≤ M_α Γ

q _t

0

t−s^q−1fs, xns, Gxns−fs, xs, Gxst−s^−αds.

3.7

In view of the fact that

fs, xns, Gxns−fs, xs, Gxs≤2μs, s∈0, T, 3.8 and the function s → 2μst−s^−α is integrable on 0, t, then the Lebesgue Dominated Convergence Theorem ensures that

_t

0

t−s^q−1fs, xns, Gxns−fs, xs, Gxst−s^−αds−→0 asn−→ ∞. 3.9

Therefore, we can see that

nlim→ ∞Axnt−Axt_∞0, 3.10

which means thatAis continuous.

Noting that

Axt_α 1 Γ

q

_t

0

t−s^q−1St−sfs, xs, Gxsds

α

≤ 1 Γ

q _t

0

t−s^q−1A^αSt−sfs, xs, Gxsds

≤ M_α Γ

q _t

0

t−s^q−1 μs t−s^αds

≤ M_αM_T Γ

q Mp,q·T^q−1/p,

3.11

we can see thatAis uniformly bounded onBr.

Next, we prove thatAxtis equicontinuous. Let 0 < t₂ < t₁ < T, and letε > 0 be small enough, then we have

Axt1−Axt2_α≤ 1 Γ

q

_t2

0

t1−s^q−1−t2−s^q−1

St2−sfs, xs, Gxsds

α

1 Γ

q

_t1

t2

t1−s^q−1St1−sfs, xs, Gxsds

α

1 Γ

q

_t2

0

t1−s^q−1St1−s−St2−sfs, xs, Gxsds

α

I1I2I3.

3.12

Using2.1andH1, we have

I₁ 1 Γ

q

_t2

0

t1−s^q−1−t2−s^q−1

St2−sfs, xs, Gxsds

α

≤ 1 Γ

q _t2

0

t1−s^q−1−t2−s^q−1A^αSt2−sfs, xs, Gxsds

≤ Mα

Γ q

_t2

0

t1−s^q−1−t2−s^q−1 μs t2−s^αds

≤ M_α Γ

q _t2−ε

0

t2−s^q−1−t1−s^q−1 μs t2−s^αds

M_α Γ

q _t2

t2−εt2−s^q−1 μs t2−s^αds I₁I₁.

3.13

It follows from the assumption ofμsthatI₁ tends to 0 ast₂ → t₁. ForI₁, using the H ¨older inequality, we can see thatI₁tends to 0 ast2 → t1andε → 0.

ForI₂, using2.1,H1, and the H ¨older inequality, we have

I2 1 Γ

q

_t1

t2

t1−s^q−1St1−sfs, xs, Gxsds

α

≤ 1 Γ

q _t1

t2

t1−s^q−1A^αSt1−sfs, xs, Gxsds

≤ M_α Γ

q _t1

t2

t1−s^q−1 μs

t1−s^αds−→0 ast₂−→t₁.

3.14

Moreover,

I₃≤ 1 Γ

q

_t2−ε

0

t1−s^q−1St1−s−St2−sfs, xs, Gxsds

α

1 Γ

q

_t2

t2−εt1−s^q−1St1−s−St2−sfs, xs, Gxsds

α

≤ 1 Γ

q _t2−ε

0

t1−s^q−1 S

t₁−t₂

2 t₁−s 2

−S t₂−s

2

· A^αS

t2−s 2

fs, xs, Gxs ds M_α

Γ q

_t2

t2−εt1−s^q−1

μs

t1−s^α μs t2−s^α

ds

≤ 2^αM_α Γ

q _t2−ε

0

t1−s^q−1 S

t₁−t₂

2 t₁−s 2

−S t₂−s

2

· μs t2−s^αds Mα

Γ q

_t2

t2−εt1−s^q−1

μs

t1−s^α μs t2−s^α

ds

I₃I₃. 3.15

Using the compactness of St in X implies the continuity oft → St for t ∈ 0, T; integrating withs →μs/t2−s^α∈L¹_loc0, t2,R, we see thatI₃tends to 0, ast₂ → t₁. For I₃, from the assumption ofμsand the H ¨older inequality, it is easy to see thatI₃tends to 0 ast2 → t1andε → 0.

Thus,Axt1−Axt2_α → 0, ast₂ → t₁, which does not depend onx.

So,ABris relatively compact. By the Arzela-Ascoli Theorem,Ais compact.

Now, let us prove thatBis a contraction mapping. Forx, y ∈ C0, T, Xαandt ∈ 0, T, we have

Bxt−Byt

α≤ Stgx−gy

α≤Mbx−y

∞<x−y

∞. 3.16

So, we obtain

Bxt−Byt

∞<x−y

∞. 3.17

We now conclude the result of the theorem by Krasnoselkii’s theorem.

Now we assume the following.

H3There exists a positive functionμ₁·:0, T → Rsuch that

ft, xt, Gxt−f

t, yt, Gyt≤μ₁tx−y

αGx−Gy

α

, 3.18

the functions →μ1s/t−s^α belongs to L¹0, t,Rand

γt: t

0

μ₁s t−s^α

p

ds

1/p

≤M_T <∞, fort∈0, T. 3.19

H4The functionL_α,q :I → R, 0< α, q <1 satisfies

Lα,qt MbM_αM_T Γ

q Mp,q·t^q−1/p1G^∗≤ω <1, t∈0, T. 3.20 Theorem 3.2. Let−Abe the infinitesimal generator of an analytic semigroup{St}_t≥0withSt ≤ M, t≥0 and 0∈ρA. Ifx₀∈X_αand (H2)–(H4) hold, then1.1has a unique mild solutionx∈C_α. Proof. Define the mappingF:C0, T, Xα → C0, T, Xαby

Fxt St

x0−gx 1

Γ q

_t

0

t−s^q−1St−sfs, xs, Gxsds. 3.21

Obviously,Fis well defined onC0, T, Xα. Now takex, y∈C0, T, Xα, then we have Fxt−Fyt

α

≤Stgx−gy

α

1 Γ

q _t

0

t−s^q−1St−sfs, xs, Gxs−fs, ys, Gys

αds

≤Mgx−gy

α

1 Γ

q _t

0

t−s^q−1A^αSt−s

fs, xs, Gxs−f

s, ys, Gysds

≤Mbx−y

∞ Mα

Γ q

_t

0

t−s^q−1 μ1s

t−s^αx−y

αGx−Gy

α

ds

≤Mbx−y

∞MαM_T Γ

q Mp,q·t^q−1/p1G^∗x−y

α

≤L_α,qtx−y

∞.

3.22

Therefore, we obtain

Fxt−Fyt

∞≤ωx−y

∞<x−y

∞, 3.23

and the result follows from the contraction mapping principle.

Acknowledgment

This work is supported by the NSF of Yunnan Province2009ZC054M.

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