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Mild Solutions for Fractional Differential Equations with Nonlocal Conditions

Abstract

This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions , in a Banach space , where . 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 extensively (see, e.g., [1–8] and references therein).

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

(11)

where ,  ,  and generates an analytic compact semigroup of uniformly bounded linear operators on a Banach space . The term which may be interpreted as a control on the system is defined by

(12)

where (the set of all positive function continuous on ) and

(13)

The functions and are continuous.

The nonlocal condition can be applied in physics with better effect than that of the classical initial condition . There have been many significant developments in the study of nonlocal Cauchy problems (see, e.g., [6, 7, 9–14] and references cited there).

In this paper, motivated by [1–7, 9–15] (especially the estimating approach given by Xiao and Liang [14]), we study the semilinear fractional differential equations with nonlocal condition (1.1) in a Banach space , assuming that the nonlinear map is defined on and is defined on where , for , the domain of the fractional power of . New and general existence and uniqueness theorem, which extends many previous results, are given.

2. Preliminaries

In this paper, we set , a compact interval in . We denote by a Banach space with norm . Let be the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators , that is, there exists such that ; and without loss of generality, we assume that . So we can define the fractional power for , as a closed linear operator on its domain with inverse , and one has the following known result.

Lemma 2.1 (see [15]).

is a Banach space with the norm for .

   for each and .

For every and , .

For every , is bounded on and there exists such that

(2.1)

Definition 2.2.

A continuous function satisfying the equation

(2.2)

for is called a mild solution of (1.1).

In this paper, we use to denote the norm of whenever for some with . We denote by the Banach space endowed with the sup norm given by

(2.3)

for .

The following well-known theorem will be used later.

Theorem 2.3 (Krasnoselkii, see [16]).

Let be a closed convex and nonempty subset of a Banach space . Let be two operators such that

whenever .

is compact and continuous,

is a contraction mapping.

Then there exists such that .

3. Main Results

We require the following assumptions.

  1. (H1)

    The function is continuous, and there exists a positive function such that

    (3.1)

    where .

  2. (H2)

    The function is continuous and there exists such that

    (3.2)

for any .

Theorem 3.1.

Let be the infinitesimal generator of an analytic compact semigroup with and . If the maps and satisfy (H1), (H2), respectively, and , then (1.1) has a mild solution for every .

Proof.

Set and choose such that

(3.3)

where .

Let .

Define

(3.4)

Let , then for we have the estimates

(3.5)

Hence we obtain .

Now we show that is continuous. Let be a sequence of such that in . Then

(3.6)

since the function is continuous on . For , using (2.1), we have

(3.7)

In view of the fact that

(3.8)

and the function is integrable on , then the Lebesgue Dominated Convergence Theorem ensures that

(3.9)

Therefore, we can see that

(3.10)

which means that is continuous.

Noting that

(3.11)

we can see that is uniformly bounded on .

Next, we prove that is equicontinuous. Let , and let be small enough, then we have

(3.12)

Using (2.1) and (H1), we have

(3.13)

It follows from the assumption of that tends to 0 as . For , using the Hölder inequality, we can see that tends to 0 as and .

For , using (2.1), (H1), and the Hölder inequality, we have

(3.14)

Moreover,

(3.15)

Using the compactness of in implies the continuity of for integrating with , we see that tends to , as . For , from the assumption of and the Hölder inequality, it is easy to see that tends to 0 as and .

Thus, , as , which does not depend on .

So, is relatively compact. By the Arzela-Ascoli Theorem, is compact.

Now, let us prove that is a contraction mapping. For and , we have

(3.16)

So, we obtain

(3.17)

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

Now we assume the following.

(H3) There exists a positive function such that

(3.18)

the  function  belongs and

(3.19)

(H4) The function ,   satisfies

(3.20)

Theorem 3.2.

Let be the infinitesimal generator of an analytic semigroup with and . If and (H2)–(H4) hold, then (1.1) has a unique mild solution .

Proof.

Define the mapping by

(3.21)

Obviously, is well defined on .

Now take , then we have

(3.22)

Therefore, we obtain

(3.23)

and the result follows from the contraction mapping principle.

References

  1. Agarwal RP, Belmekki M, Benchohra M: A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Advances in Difference Equations 2009, 2009:-47.

    Google Scholar 

  2. El-Borai MM, Amar D: On some fractional integro-differential equations with analytic semigroups. International Journal of Contemporary Mathematical Sciences 2009,4(25–28):1361–1371.

    MATH  MathSciNet  Google Scholar 

  3. Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3337–3343. 10.1016/j.na.2007.09.025

    Article  MATH  MathSciNet  Google Scholar 

  4. Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2677–2682. 10.1016/j.na.2007.08.042

    Article  MATH  MathSciNet  Google Scholar 

  5. Lv ZW, Liang J, Xiao TJ: Solutions to fractional differential equations with nonlocal initial condition in Banach spaces. reprint, 2009

    Google Scholar 

  6. Liu H, Chang J-C: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(9):3076–3083. 10.1016/j.na.2008.04.009

    Article  MATH  MathSciNet  Google Scholar 

  7. N'Guérékata GM: A Cauchy problem for some fractional abstract differential equation with non local conditions. Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1873–1876. 10.1016/j.na.2008.02.087

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhu X-X: A Cauchy problem for abstract fractional differential equations with infinite delay. Communications in Mathematical Analysis 2009,6(1):94–100.

    MATH  MathSciNet  Google Scholar 

  9. Liang J, van Casteren J, Xiao T-J: Nonlocal Cauchy problems for semilinear evolution equations. Nonlinear Analysis: Theory, Methods & Applications 2002,50(2):173–189. 10.1016/S0362-546X(01)00743-X

    Article  MATH  MathSciNet  Google Scholar 

  10. Liang J, Liu J, Xiao T-J: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Analysis: Theory, Methods & Applications 2004,57(2):183–189. 10.1016/j.na.2004.02.007

    Article  MATH  MathSciNet  Google Scholar 

  11. Liang J, Liu JH, Xiao T-J: Nonlocal Cauchy problems for nonautonomous evolution equations. Communications on Pure and Applied Analysis 2006,5(3):529–535. 10.3934/cpaa.2006.5.529

    Article  MATH  MathSciNet  Google Scholar 

  12. Liang J, Liu JH, Xiao T-J: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Mathematical and Computer Modelling 2009,49(3–4):798–804. 10.1016/j.mcm.2008.05.046

    Article  MATH  MathSciNet  Google Scholar 

  13. Liang J, Xiao T-J: Semilinear integrodifferential equations with nonlocal initial conditions. Computers & Mathematics with Applications 2004,47(6–7):863–875. 10.1016/S0898-1221(04)90071-5

    Article  MATH  MathSciNet  Google Scholar 

  14. Xiao T-J, Liang J: Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Analysis: Theory, Methods & Applications 2005,63(5–7):e225-e232. 10.1016/j.na.2005.02.067

    Article  MATH  Google Scholar 

  15. Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.

    Book  Google Scholar 

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

    MATH  Google Scholar 

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Acknowledgment

This work is supported by the NSF of Yunnan Province (2009ZC054M).

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Li, F. Mild Solutions for Fractional Differential Equations with Nonlocal Conditions. Adv Differ Equ 2010, 287861 (2010). https://doi.org/10.1155/2010/287861

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