Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

Abstract

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases and

1. Introduction

The nonlocal initial conditions were introduced to extend the classical theory of initial value problems. Nonlocal conditions describe more appropriately some natural phenomena because they consider additional information in the initial conditions.

The existence of mild solutions to semilinear Cauchy problems with nonlocal conditions has been studied by several authors in the last two decades. See, for instance, [1–4] and the references cited therein.

On the other hand, many authors have studied recently the existence of mild solutions to abstract fractional differential equations with nonlocal conditions by using the theory of resolvent families of operators as well as some fixed point results. See [5–18] and the references therein for more details.

Let be a closed and linear operator defined on a Banach space , , and and suppose that , , and are suitable continuous functions. In what follows, we will denote by and the Caputo and Riemann-Liouville fractional derivatives, respectively. Now, for , we consider the following nonlinear fractional differential equations with nonlocal conditionsin case ; andin case

By using the Laplace transform, it is easy to see that the mild solutions to problems (1)–(4) are, respectively, given byin case ; andin case Here, for , is the resolvent family generated by (see definition below, Section 2).

The existence of mild solutions to problems (1)–(4) has been studied by many authors in the last years. For example, in case , we refer the reader to [8, 9, 17, 18] (for the Caputo fractional derivative) and to [10] (for the Riemann-Liouville fractional derivative), that is, problems (1) and (2), respectively. On the other hand, in case , the existence of mild solutions to the Caputo fractional Cauchy problems with nonlocal conditions (3) has been considered in [12, 19] and the references therein, and, to the best of our knowledge, nonlocal Riemann-Liouville fractional Cauchy problem (4) has not been addressed in the existing literature.

A common assumption in many of the above-mentioned papers to obtain the existence of mild solutions to problems (1)–(4) is that generates a compact analytic semigroup , or generates a compact fractional resolvent family (see the definition below) because the compactness of (or ) allows applying, for example, the Krasnoselskii fixed point theorem.

According to the variation of constants formulas (5)–(7), we observe that if we have compactness criteria of (for suitable and ), we will be able to apply some fixed point techniques to obtain the existence of mild solutions to problems (1)–(4). For example, to prove the existence of mild solutions to problem (3), the authors in [12, Theorem ] assume that the operators , , and generated by are compact for all However, there are not completely clear conditions on implying the compactness of , , and for all , because there are no compactness criteria for , when Therefore, we notice that the compactness of gives a powerful tool to obtain existence of mild solutions to problems (1)–(4).

The compactness of is well known in some special cases. For example, if , then is compact for all if and only if is norm continuos and is compact for all , because corresponds to a -semigroup. See [20, Theorem , Chapter ]. If , then is compact for all if and only if is compact , because is the sine family generated by ; see [21]. In case , the compactness of has been studied by using subordination methods; that is, the operator is supposed to be a generator of a compact semigroup; see [22]. On the other hand, if is an almost sectorial operator and the resolvent is compact for all , then is compact for all (see [23]), and, very recently, it was proved that if is norm continuous, then is compact for all if and only if is compact for all See [24, 25]. Finally, in case , the characterization of compactness asserts that is compact for all if and only if is compact for all ; see [25, Theorem ].

In this paper, we study the existence of mild solution to nonlocal fractional Cauchy problems (1)–(4). Our approach relies on the compactness of resolvent family for suitable , as well as some fixed point techniques. We remark that we study simultaneously the nonlocal fractional Cauchy problem for the Caputo and Riemann-Liouville fractional derivatives.

The paper is organized as follows. Section 2 gives the preliminaries. Section 3 is devoted to the norm continuity and compactness of for Here, we give characterizations of the compactness of for for suitable In Section 4 we study nonlocal fractional Cauchy problems for the Caputo fractional derivative. We give some results on the existence of mild solutions to problems (1) and (3). Section 5 treats nonlocal fractional Cauchy problems for the Riemann-Liouville fractional derivative. Here, we study the existence of mild solutions to problems (2) and (4). Finally, Section 6 is devoted to some applications.

2. Preliminaries

Let be a Banach space. We denote by the space of all bounded linear operators from into If is a closed linear operator on , we denote by the resolvent set of and the resolvent operator of defined for all

We recall that a strongly continuous family is said to be of type or is exponentially bounded, if there exist two constants and such that for all

Now, we review some results on fractional calculus. For , define where is the Gamma function. We define , the Dirac delta. For , denotes the smallest integer greater than or equal to As usual, the finite convolution of and is defined by .

Definition 1. Let The -order Riemann-Liouville fractional integral of is defined by

Also, we define . Because of the convolution properties, the integral operators satisfy the semigroup law: for all

Definition 2. Let The -order Caputo fractional derivative is defined as where .

Definition 3. Let The -order Riemann-Liouville fractional derivative of is defined as where .

We notice that if , then

Throughout this paper we use the notation of and to the -fractional derivative of Caputo and Riemann-Liouville, respectively.

Example 4. If , then (i),(ii)(iii)

We observe that the Riemann-Liouville derivative operator is a left inverse operator of but not a right inverse, that is, . On the other hand, the Caputo derivative operator satisfies If we denote by (or ) the Laplace transform of , we have the following properties for the fractional derivatives:where and For and , the generalized Mittag-Leffler function is defined by The Laplace transform of the Mittag-Leffler function satisfies

Definition 5. Let be closed linear operator with domain , defined on a Banach space , and We say that is the generator of an -resolvent family, if there exist and a strongly continuous function such that is exponentially bounded, , and, for all , In this case, is called the -resolvent family generated by

We notice that Definition 5 corresponds to the concept of -regularized families introduced in [26]. In fact, if and , then the function is a -regularized family. Moreover, the function satisfies the following functional equation (see [27, 28]): for all On the other hand, if an operator with domain is the infinitesimal generator of the -resolvent family , then for all we have For example, the case corresponds to a -semigroup and is a cosine family, whereas is a sine family. Finally, if , then is the -resolvent family (also called the -times resolvent family) for fractional differential equations. We notice that, in the scalar case, that is, when , where and denotes the identity operator, then by the uniqueness of the Laplace transform corresponds to the function . Finally, let and . Define by where , , and the function is defined bywhere () denotes the Wright function. Then, is an -resolvent family on the Banach space generated by See [29, Example ].

The proof of the next result follows as in [26, 27].

Proposition 6. Let and let be an -resolvent family generated by Then the following holds: (1) and for all and .(2)If and , then(3)If and , then , and In particular,

Finally, we recall the following results.

Theorem 7 (Mazur theorem). If is a compact subset of a Banach space , then its convex closure is compact.

Theorem 8 (Krasnoselskii theorem). Let be a closed convex and nonempty subset of a Banach space Let and be two operators such that (i)if , then ,(ii) is a mapping contraction,(iii) is compact and continuous.Then, there exists such that

Theorem 9 (Schauder’s fixed point theorem). Let be a nonempty, closed, bounded, and convex subset of a Banach space . Suppose that is a compact operator. Then has at least a fixed point in .

Theorem 10 (Leray-Schauder alternative theorem). Let be a convex subset of a Banach space . Suppose that . If is a completely continuous map, then either has a fixed point or the set is unbounded.

3. Continuity and Compactness of

In this section we study, for all , the norm continuity (continuity in ) and the compactness of for given

Proposition 11. Let and . Suppose that is the -resolvent family of type generated by . Then the function is continuous in for all .

Proof. Let Observe that, for all , We conclude by the uniqueness of the Laplace transform that , for all . Take . Then Since , we have and we obtain and therefore as
On the other hand, Since we obtain that the function is decreasing in and therefore , for all , obtaining Therefore, as We conclude that is norm continuous, for .
On the other hand, if , then, by the uniqueness of the Laplace transform, we obtain that for all . Take . Then for all . Therefore as

Lemma 12. Suppose that generates an -resolvent family of type If , then generates an -resolvent family of type

Proof. By hypothesis we get, for all , Therefore is Laplace transformable and, for all , we have We conclude that generates an -resolvent family of type .

Definition 13. We say that the resolvent family is compact if, for every , the operator is a compact operator.

In what follows, we will assume that is strongly continuous for all .

Theorem 14. Let , , and be an -resolvent family of type generated by . Then the following assertions are equivalent: (i) is a compact operator for all .(ii) is a compact operator for all .

Proof. Suppose that the resolvent family is compact. Let be fixed. Then we have where the integral in the right-hand side exists in the Bochner sense. Because is continuous in the uniform operator topology (by Proposition 11), we conclude that is a compact operator by [30, Corollary ].
Let be fixed. Assume that . Since , it follows that and therefore, by [31, Proposition ], we obtain in . Therefore, where is the path consisting of the vertical line By hypothesis and [30, Corollary ], we conclude that is compact for all and Now, we take Observe that in we have by [31, Proposition ], and as in case we conclude that is compact for all .

By Theorem 14 we have the following Corollary.

Corollary 15. Let and be an -resolvent family of type generated by . Then the following assertions are equivalent: (i) is a compact operator for all (ii) is a compact operator for all

Proposition 16. Let , and be the -resolvent family of type generated by . Suppose that is continuous in the uniform operator topology for all . Then the following assertions are equivalent: (i) is a compact operator for all (ii) is a compact operator for all

Proof. Suppose that that the resolvent family is compact. Let be fixed. Then we have where the integral in the right-hand side exists in the Bochner sense, because is continuous in the uniform operator topology, by hypothesis. Then, by [30, Corollary ], we conclude that is a compact operator.
Let be fixed. Since , it follows that and therefore, by [31, Proposition ], we obtain in Therefore, where is the path consisting of the vertical line By hypothesis and [30, Corollary ], we conclude that is compact.

Proposition 17. Let and be the -resolvent family of type generated by . Suppose that is continuous in the uniform operator topology for all . Then the following assertions are equivalent: (i) is a compact operator for all (ii) is a compact operator for all

Proof. It follows as in the proof of Proposition 16.
Let be fixed. Since , it follows that and therefore, by [31, Proposition ], we obtain in Therefore, where is the path consisting of the vertical line By hypothesis and [30, Corollary ], we conclude that is compact.

The proof of the next result follows similarly to Proposition 16, because for we have in and by [31, Proposition ].

Proposition 18. Let and be the -resolvent family of type generated by . Suppose that is continuous in the uniform operator topology for all . Then, the following assertions are equivalent: (i) is a compact operator for all (ii) is a compact operator for all

Remark 19. Let be fixed. If , then by [31, Proposition ] we have in Therefore, as is Proposition 18, if , where , generates the -resolvent family of type , and is norm continuous for all , then is a compact operator for all if and only if is a compact operator for all The same conclusion holds if , where is fixed and is the -resolvent family of type generated by , which is norm continuous for all .

4. Nonlocal Fractional Cauchy Problems: The Caputo Case

In this section we consider the nonlocal problem for the Caputo fractional derivative, , , and is a closed linear operator defined on which generates the -resolvent family The nonlinear function is continuous and the nonlocal conditions are also continuous functions. We recall also that the derivative denotes the Caputo fractional derivative.

The mild solution to problem (45) is given by By the uniqueness of the Laplace transform, it is easy to see that the mild solution to fractional nonlocal problem (45) can be written asfor all

We assume the following:(H1)The function satisfies the Carathéodory condition; that is, is strongly measurable for each and is continuous for each (H2)There exists a continuous function such that (H3)The functions are continuous and there exist such that

We have the following existence results.

Theorem 20. Let . Let be the generator of an -resolvent family of type . Suppose that is compact for all . If and , then, under assumptions (H1)–(H3), problem (45) has at least one mild solution.

Proof. Let , where On we define the operators , by and We shall prove that has at least one fixed point by the Krasnoselskii fixed point theorem. We will consider several steps in the proof.
Step 1. We will see that if , then . In fact, by Lemma 12 we have Hence for all
Step 2. is a contraction on In fact, if , then Since , we conclude that is a contraction.
Step 3. is completely continuous.
Firstly, we prove that is a continuous operator on Let such that in . By Lemma 12 we get We notice that the function is integrable on By Lebesgue’s dominated convergence theorem, as Since we obtain that is continuous in
Now, we will prove that is relatively compact. By the Ascoli-Arzela theorem, we need to show that the family is uniformly bounded and equicontinuous, and the set is relatively compact in for each In fact, for each we have (as in Step ) that and therefore is uniformly bounded.
In order to prove the equicontinuity, let , and take Observe that Observe that, for , by Lemma 12 we have and therefore For , we have Observe that and, by Lemma 12, for all Moreover, by Proposition 11 we have that is norm continuous and therefore if , then in We obtain by Lebesgue’s dominated convergence theorem that Therefore, is an equicontinuous family.
Finally, we prove that is relatively compact in for each Obviously, is relatively compact in Now, we take For we define on the operator The hypotheses implies the compactness of for all (by Lemma 12 and by Theorem 14) and therefore the set is compact for all Then is also a compact set by Theorem 7. By using the mean-value theorem for the Bochner integrals (see [32, Corollary , page ]), we obtain that Therefore, the set is relatively compact in for all Now, observe that