- Open Access
Mild solutions for abstract fractional differential equations with almost sectorial operators and infinite delay
© Li; licensee Springer. 2013
- Received: 2 July 2013
- Accepted: 8 October 2013
- Published: 19 November 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
- mild solution
- infinite delay
- measure of noncompactness
- fixed point theorem
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).
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 .
- (2)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 .
- (i)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 .
- (1)If is continuous on J and , then and is continuous in , and(2.1)
for a positive constant M.
- (2)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.
Theorem 2.4 ()
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)
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 .
This measure of noncompactness satisfies some basic properties as follows.
Lemma 2.9 ()
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 ).
integrable if it admits a Bochner integrable selection , for a.e. ;
- (ii)integrably bounded if there exists a function such that
where . Then for all .
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.
for almost all ;
Theorem 3.1 Suppose that hypotheses (H1) and (H2) hold. Then, for every with , there exists a mild solution of (1.1) on .
From Theorems 2.4-2.5 and (H1), we infer that .
thus, is a Banach space.
Clearly, the operator has a fixed point is equivalent to ℱ has one. So, it turns out to prove that has a fixed point.
where , , .
Let be a sequence such that as . Obviously, the Lebesgue dominated convergence theorem enables us to prove that is continuous.
It results that by (3.1). Hence, for some positive number ρ, .
We will show that each term on the right-hand side of (3.4) uniformly converges to zero.
This together with Theorem 2.5 shows that the right-hand side tends to zero as and .
Therefore, the set is equicontinuous.
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. □
() 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).
then we can see that in (2.2).
Thus, problem (4.1) has at least a mild solution by Theorem 3.1 for every ().
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).
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