Skip to main content


Existence of Mild Solutions to Fractional Integrodifferential Equations of Neutral Type with Infinite Delay

Article metrics

  • 1514 Accesses

  • 3 Citations


We study the solvability of the fractional integrodifferential equations of neutral type with infinite delay in a Banach space . An existence result of mild solutions to such problems is obtained under the conditions in respect of Kuratowski's measure of noncompactness. As an application of the abstract result, we show the existence of solutions for an integrodifferential equation.

1. Introduction

The fractional differential equations are valuable tools in the modeling of many phenomena in various fields of science and engineering; so, they attracted many researchers (cf., e.g., [16] and references therein). On the other hand, the integrodifferential equations arise in various applications such as viscoelasticity, heat equations, and many other physical phenomena (cf., e.g., [710] and references therein). Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (cf., e.g., [7, 1015] and references therein).

Neutral functional differential equations arise in many areas of applied mathematics and for this reason, the study of this type of equations has received great attention in the last few years (cf., e.g., [12, 1416] and references therein). In [12, 16], Hernández and Henríquez studied neutral functional differential equations with infinite delay. In the following, we will extend such results to fractional-order functional differential equations of neutral type with infinite delay. To the authors' knowledge, few papers can be found in the literature for the solvability of the fractional-order functional integrodifferential equations of neutral type with infinite delay.

In the present paper, we will consider the following fractional integrodifferential equation of neutral type with infinite delay in Banach space :


where , , is a phase space that will be defined later (see Definition 2.5). is a generator of an analytic semigroup of uniformly bounded linear operators on . Then, there exists such that . , , (), and defined by , for , belongs to and . The fractional derivative is understood here in the Caputo sense.

The aim of our paper is to study the solvability of (1.1) and present the existence of mild solution of (1.1) based on Kuratowski's measures of noncompactness. Moreover, an example is presented to show an application of the abstract results.

2. Preliminaries

Throughout this paper, we set and denote by a real Banach space, by the Banach space of all linear and bounded operators on , and by the Banach space of all -valued continuous functions on with the uniform norm topology.

Let us recall the definition of Kuratowski's measure of noncompactness.

Definition 2.1.

Let be a bounded subset of a seminormed linear space . Kuratowski's measure of noncompactness of is defined as


This measure of noncompactness satisfies some important properties.

Lemma 2.2 (see [17]).

Let and be bounded subsets of . Then,

  1. (1)

    if ,

  2. (2)

    , where denotes the closure of ,

  3. (3)

    if and only if is precompact,

  4. (4)

    , ,

  5. (5)


  6. (6)

    , where ,

  7. (7)

    for any ,

  8. (8)

    , where is the closed convex hull of .

For , we define


where .

The following lemmas will be needed.

Lemma 2.3 (see [17]).

If is a bounded, equicontinuous set, then


Lemma 2.4 (see [18]).

If and there exists an such that , a.e. , then is integrable and


The following definition about the phase space is due to Hale and Kato [11].

Definition 2.5.

A linear space consisting of functions from into with semi-norm is called an admissible phase space if has the following properties.

  1. (1)

    If is continuous on and , then and is continuous in and


    where is a constant.

  2. (2)

    There exist a continuous function and a locally bounded function in such that


    for and as in (1).

  3. (3)

    The space is complete.

Remark 2.6.

(2.5) in (1) is equivalent to , for all .

The following result will be used later.

Lemma 2.7 (see [19, 20]).

Let be a bounded, closed, and convex subset of a Banach space such that , and let be a continuous mapping of into itself. If the implication


holds for every subset of , then has a fixed point.

Let be a set defined by


Motivated by [4, 5, 21], we give the following definition of mild solution of (1.1).

Definition 2.8.

A function satisfying the equation


is called a mild solution of (1.1), where


and is a probability density function defined on such that




Remark 2.9.

According to [22], direct calculation gives that


where .

We list the following basic assumptions of this paper.

(H1) satisfies is measurable, for all and is continuous for a.e. , and there exist two positive functions such that


(H2) For any bounded sets , , and , there exists an integrable positive function such that


where and .

(H3) There exists a constant such that


(H4) For each , is measurable on and is bounded on . The map is continuous from to , here, .

(H5) There exists such that


where , .

3. Main Result

In this section, we will apply Lemma 2.7 to show the existence of mild solution of (1.1). To this end, we consider the operator defined by


It follows from (H1), (H3), and (H4) that is well defined.

It will be shown that has a fixed point, and this fixed point is then a mild solution of (1.1).

Let be the function defined by


Set , .

It is clear to see that satisfies (2.9) if and only if satisfies and for ,


Let . For any ,


Thus, is a Banach space. Set


Then, for , from(2.6), we have


where .

In order to apply Lemma 2.7 to show that has a fixed point, we let be an operator defined by and for ,


Clearly, the operator has a fixed point is equivalent to has one. So, it turns out to prove that has a fixed point.

Now, we present and prove our main result.

Theorem 3.1.

Assume that (H1)–(H5) are satisfied, then there exists a mild solution of (1.1) on provided that .


For , , from (3.6), we have


In view of (H3),


where .

Next, we show that there exists some such that . If this is not true, then for each positive number , there exist a function and some such that . However, on the other hand, we have from (3.8), (3.9), and (H4)


Dividing both sides of (3.10) by , and taking , we have


This contradicts (2.17). Hence, for some positive number , .

Let with in as . Since satisfies (H1), for almost every , we get


In view of (3.6), we have


Noting that


we have by the Lebesgue Dominated Convergence Theorem that


Therefore, we obtain


This shows that is continuous.



Let and , then we can see




It follows the continuity of in the uniform operator topology for that tends to 0, as . The continuity of ensures that tends to 0, as .

For , we have


Clearly, the first term on the right-hand side of (3.20) tends to 0 as . The second term on the right-hand side of (3.20) tends to 0 as as a consequence of the continuity of in the uniform operator topology for .

In view of the assumption of and (3.8), we see that


Thus, is equicontinuous.

Now, let be an arbitrary subset of such that .

Set ,


Noting that for , we have




where . Therefore, .

Moreover, for any and bounded set , we can take a sequence such that (see [23], P125). Thus, for , noting that the choice of , and from Lemmas 2.2–2.4 and (H2), we have


It follows from Lemma 2.2 that


since is arbitrary, we can obtain


Hence, . Applying now Lemma 2.7, we conclude that has a fixed point in . Let , then is a fixed point of the operator which is a mild solution of (1.1).

4. Application

In this section, we consider the following integrodifferential model:


where , , , , are continuous functions, and .

Set and define by


Then, generates a compact, analytic semigroup of uniformly bounded, linear operators, and .

Let the phase space be , the space of bounded uniformly continuous functions endowed with the following norm:


then we can see that in (2.6).

For , and , we set


Then (4.1) can be reformulated as the abstract (1.1).

Moreover, for , we can see


where , .

For , , we have


where .

Suppose further that there exists a constant such that


then (4.1) has a mild solution by Theorem 3.1.

For example, if we put


then , , . Thus, we see



  1. 1.

    Agarwal RP, de Andrade B, Cuevas C: On type of periodicity and ergodicity to a class of fractional order differential equations. Advances in Difference Equations 2010, 2010:-25.

  2. 2.

    Ahmed HM: Boundary controllability of nonlinear fractional integrodifferential systems. Advances in Difference Equations 2010, 2010:-9.

  3. 3.

    Alsaedi A, Ahmad B: Existence of solutions for nonlinear fractional integro-differential equations with three-point nonlocal fractional boundary conditions. Advances in Difference Equations 2010, 2010:-10.

  4. 4.

    El-Borai MM: Some probability densities and fundamental solutions of fractional evolution equations. Chaos, Solitons and Fractals 2002,14(3):433-440. 10.1016/S0960-0779(01)00208-9

  5. 5.

    El-Borai MM: On some stochastic fractional integro-differential equations. Advances in Dynamical Systems and Applications 2006,1(1):49-57.

  6. 6.

    Mophou GM, N'Guérékata GM: Existence of the mild solution for some fractional differential equations with nonlocal conditions. Semigroup Forum 2009,79(2):315-322. 10.1007/s00233-008-9117-x

  7. 7.

    Liang J, Xiao T-J, van Casteren J: A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. Applied Mathematics Letters 2004,17(4):473-477. 10.1016/S0893-9659(04)90092-4

  8. 8.

    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

  9. 9.

    Liang J, Liu JH, Xiao T-J: Nonlocal problems for integrodifferential equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2008,15(6):815-824.

  10. 10.

    Xiao T-J, Liang J: Blow-up and global existence of solutions to integral equations with infinite delay in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):e1442-e1447. 10.1016/

  11. 11.

    Hale JK, Kato J: Phase space for retarded equations with infinite delay. Funkcialaj Ekvacioj 1978,21(1):11-41.

  12. 12.

    Hernández E, Henríquez HR: Existence results for partial neutral functional-differential equations with unbounded delay. Journal of Mathematical Analysis and Applications 1998,221(2):452-475. 10.1006/jmaa.1997.5875

  13. 13.

    Liang J, Xiao TJ: Functional-differential equations with infinite delay in Banach spaces. International Journal of Mathematics and Mathematical Sciences 1991,14(3):497-508. 10.1155/S0161171291000686

  14. 14.

    Mophou GM, N'Guérékata GM: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Applied Mathematics and Computation 2010,216(1):61-69. 10.1016/j.amc.2009.12.062

  15. 15.

    Mophou GM, N'Guérékata GM: A note on a semilinear fractional differential equation of neutral type with infinite delay. Advances in Difference Equations 2010, 2010:-8.

  16. 16.

    Hernández E, Henríquez HR: Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay. Journal of Mathematical Analysis and Applications 1998,221(2):499-522. 10.1006/jmaa.1997.5899

  17. 17.

    Banaś J, Goebel K: Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics. Volume 60. Marcel Dekker, New York, NY, USA; 1980:vi+97.

  18. 18.

    Heinz H-P: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Analysis: Theory, Methods & Applications 1983,7(12):1351-1371. 10.1016/0362-546X(83)90006-8

  19. 19.

    Agarwal RP, Meehan M, O'Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.

  20. 20.

    Szufla S: On the application of measure of noncompactness to existence theorems. Rendiconti del Seminario Matematico della Università di Padova 1986, 75: 1-14.

  21. 21.

    Zhou Y, Jiao F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Analysis: Real World Applications 2010,11(5):4465-4475. 10.1016/j.nonrwa.2010.05.029

  22. 22.

    Mainardi F, Paradisi P, Gorenflo R: Probability distributions generated by fractional diffusion equations. In Econophysics: An Emerging Science. Edited by: Kertesz J, Kondor I. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000.

  23. 23.

    Bothe D: Multivalued perturbations of m -accretive differential inclusions. Israel Journal of Mathematics 1998, 108: 109-138. 10.1007/BF02783044

Download references


The authors are grateful to the referees for their valuable suggestions. F. Li is supported by the NSF of Yunnan Province (2009ZC054M). J. Zhang is supported by Tianyuan Fund of Mathematics in China (11026100).

Author information

Correspondence to Fang Li.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article


  • Banach Space
  • Phase Space
  • Probability Density Function
  • Positive Function
  • Mild Solution