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Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay
Advances in Difference Equations volume 2013, Article number: 327 (2013)
Of concern is the existence of mild solutions to delay fractional differential equations with almost sectorial operators. Combining the techniques of operator semigroup, noncompact measures and fixed point theory, we obtain a new existence theorem without the assumptions that the nonlinearity f satisfies a Lipschitz-type condition, and the resolvent operator associated with A is compact. An example is presented.
MSC:34A08, 34K30, 47D06, 47H10.
Fractional differential equations have been increasingly used for many mathematical models in probability, engineering, physics, astrophysics, economics, etc., so the theory of fractional differential equations has in recent years been an object of investigations with increasing interest [1–15].
Most of the previous research on the fractional differential equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, a compact semigroup, or an analytic semigroup, or is a Hille-Yosida operator (see, e.g., [2, 3, 7, 11, 12]). However, as presented in Example 1.1 and Example 1.2 in , the resolvent operators do not satisfy the required estimate to be a sectorial operator. In , W. von Wahl first introduced examples of almost sectorial operators which are not sectorial. To the author’s knowledge, there are few papers about the fractional evolution equations with almost sectorial operators.
Moreover, equations with delay are often more useful to describe concrete systems than those without delay. So, the study of these equations has attracted so much attention (cf., e.g., [7, 11, 17–21] and references therein).
In this paper, we pay our attention to the investigation of the existence of mild solutions to the following fractional differential equations with almost sectorial operators and infinite delay on a separable complex Banach space X:
where , . The fractional derivative is understood here in the Caputo sense. is a phase space that will be defined later (see Definition 2.1). A is an almost sectorial operator to be introduced later. Here, , and is defined by for .
Let us recall the following definition of almost sectorial operator; for more details, we refer the readers to [22, 23].
Definition 1.1 Let and . By we denote the family of all linear closed operators which satisfy
for every , there exists a constant such that
A linear operator A will be called an almost sectorial operator on X if .
Remark 1.2 Let , then the definition implies that .
We denote the semigroup associated with A by . For ,
forms an analytic semigroup of growth order , here , the integral contour is oriented counter-clockwise [15, 17, 23]. Moreover, satisfies the following properties.
There exists a constant such that
If , then ;
The functional equation for all holds. However, it is not satisfied for or .
We refer the readers to  and references therein for more details on .
In this paper, we construct a pair of families of operators and ((2.3)-(2.4)) associated with and use the fixed point theorem (Theorem 2.11) to study the existence of a mild solution of Equation (1.1). We obtain the existence theorem based on the theory on measures of noncompactness without the assumptions that the nonlinearity f satisfies a Lipschitz-type condition, and the resolvent operator associated with A is compact. An example is given to show the application of the abstract result.
Throughout this paper, we set and denote by X a separable complex Banach space with the norm , by the Banach space of all linear and bounded operators on X, and by the Banach space of all X-valued continuous functions on J with the supremum norm. We abbreviate with for any .
We will employ an axiomatic definition of the phase space from [18–21] which is a generalization of that given by Hale and Kato .
Definition 2.1 A linear space consisting of functions from into X, with the semi-norm , is called an admissible phase space if has the following properties.
If is continuous on J and , then and is continuous in , and(2.1)
for a positive constant M.
There exist a continuous function and a locally bounded function in such that(2.2)
for and u as in (1).
The space is complete.
Remark 2.2 Equation (2.1) in (1) is equivalent to .
Based on the work in , we give the following definition.
Definition 2.3 Let with be a function of Wright type (cf., e.g., )
For any , we define operator families and by the semigroup associated with A as follows:
Theorem 2.4 ()
For each fixed , and are linear and bounded operators on X. Moreover, for all , , ,
where and .
Theorem 2.5 (, Theorem 3.2)
For , and are continuous in the uniform operator topology. Moreover, for every , the continuity is uniform on .
Remark 2.6 (, Theorem 3.4)
Let . Then, for all ,
Let be a set defined by
Motivated by [3, 15], when with , we give the following definition of a mild solution of Equation (1.1).
Definition 2.7 A function satisfying the equation
is called a mild solution of Equation (1.1).
Remark 2.8 In general, since the operator is singular at , solutions to problem (1.1) are assumed to have the same kind of singularity at as the operator . When with , it follows from Remark 2.6 that the mild solution is continuous at .
Next, we recall that the Hausdorff measure of noncompactness on each bounded subset Ω of a Banach space X is defined by
This measure of noncompactness satisfies some basic properties as follows.
Lemma 2.9 ()
Let Y be a Banach space, and let be bounded. Then
if and only if U is precompact;
, where and convU mean the closure and convex hull of U, respectively;
, where ;
for any .
Definition 2.10 A continuous map is said to be a χ-contraction if there exists a positive constant such that for any bounded closed subset .
Theorem 2.11 () (Darbo-Sadovskii)
If is bounded closed and convex, the continuous map is a χ-contraction, then the map ℱ has at least one fixed point in U.
In Section 3, we use the above fixed point theorem to obtain main result. To this end, we present the following assertion about χ-estimates for a multivalued integral (Theorem 4.2.3 of ).
Let be the family of all nonempty subsets of Y, and let be a multifunction. It is called:
integrable if it admits a Bochner integrable selection , for a.e. ;
integrably bounded if there exists a function such that
Proposition 2.12 For an integrable, integrably bounded multifunction , where X is a separable Banach space, let
where . Then for all .
3 Main result
Throughout this section, let with , . We will use fixed point techniques to establish a result on the existence of mild solutions for Equation (1.1). For this purpose, we consider the following hypotheses.
(H1) satisfies is measurable for all and is continuous for a.e. , and there exists a function () such that
for almost all ;
(H2) For any bounded sets , , there exists a nondecreasing function such that
Theorem 3.1 Suppose that hypotheses (H1) and (H2) hold. Then, for every with , there exists a mild solution of (1.1) on .
Proof Define the map ℱ on the space by and
From Theorems 2.4-2.5 and (H1), we infer that .
Let be the function defined by
Write , . It is clear that u satisfies (2.6) if and only if y satisfies and for ,
Set . For any ,
thus, is a Banach space.
In order to apply Theorem 2.11 to show that ℱ has a fixed point, we let be an operator defined by , and for ,
Clearly, the operator has a fixed point is equivalent to ℱ has one. So, it turns out to prove that has a fixed point.
For , let us introduce in the space the equivalent norm defined as
since for any ,
we can take the appropriate L to satisfy
Consider the set
here ρ is a constant chosen so that
where , , .
Let be a sequence such that as . Obviously, the Lebesgue dominated convergence theorem enables us to prove that is continuous.
In what follows, we prove that . From (2.2), it follows that
Moreover, we see from the Hölder inequality that
For , , by (2.5), (H1) and (3.2)-(3.3), we have
It results that by (3.1). Hence, for some positive number ρ, .
For , let , such that , we get
We will show that each term on the right-hand side of (3.4) uniformly converges to zero.
Combining with (3.2), we have
Taking and using (3.5), we conclude
For small enough, noting that (2.5) and (3.5), we obtain
This together with Theorem 2.5 shows that the right-hand side tends to zero as and .
Therefore, the set is equicontinuous.
For a bounded set , we define the Hausdorff measure of noncompactness on as follows:
where is a constant chosen so that
For any , we set
We consider the multifunction ,
Obviously, H is integrable, and from (2.5), (H1) and (3.5) it follows that H is integrably bounded. Moreover, noting that (H2), we have the following estimate for a.e. :
Applying Proposition 2.12, we obtain
Hence is a -contraction on by Definition 2.10. According to Theorem 2.11, the operator has at least one fixed point y in . Let , , then is a fixed point of the operator ℱ which is a mild solution of Equation (1.1). This ends the proof. □
Let , we consider the following integrodifferential problem:
() and is nondecreasing, , , are continuous functions, and .
In Example 6.3 of , the authors demonstrate that for some and . We denote the semigroup associated with by and ( is a constant).
Let the phase space be , the space of bounded uniformly continuous functions endowed with the following norm:
then we can see that in (2.2).
For , and , we set
Then we can rewrite Equation (4.1) above as abstract Equation (1.1).
Moreover, we have
For any , ,
which implies that for any bounded sets , ,
Thus, problem (4.1) has at least a mild solution by Theorem 3.1 for every ().
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The author is grateful to the referees for their valuable suggestions. This work was partly supported by the NSF of China (11201413), the Educational Commission of Yunnan Province (2012Z010).
The author has no competing interests.
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Li, F. Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay. Adv Differ Equ 2013, 327 (2013). https://doi.org/10.1186/1687-1847-2013-327
- fractional differential equations
- mild solution
- infinite delay
- measure of noncompactness
- fixed point theorem