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Fractional neutral evolution equations with nonlocal conditions
Advances in Difference Equations volume 2013, Article number: 117 (2013)
In the present paper, we deal with the fractional neutral differential equations involving nonlocal initial conditions. The existence of mild solutions are established. The results are obtained by using the fractional power of operators and the Sadovskii’s fixed point theorem. An application to a fractional partial differential equation with nonlocal initial condition is also considered.
MSC:26A33, 34K30, 34K37, 34K40.
The nonlocal condition, which is a generalization of the classical condition, was motivated by physical problems. The pioneering work on nonlocal conditions is due to Byszewski (see [1–3]). Existence results for semilinear evolution equations with nonlocal conditions were investigated in [4–6]. Neutral differential equations arises in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature for neutral functional differential equations is the Hale book .
Fractional differential equations describe many practical dynamical phenomena arising in engineering, physics, economy and science. In particular, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, seepage flow in porous media and in fluid dynamic traffic models (see [8–10]). The result obtained is a generalization and a continuation of some results reported in [11–15].
The main purpose of this paper is to study the existence of mild solutions of semilinear neutral fractional differential equations with nonlocal conditions in the following form
where −A is the infinitesimal generator of an analytic semigroup and the functions F, G and g are given functions to be defined later. The fractional derivative , is understood in the Caputo sense.
Throughout this paper, X will be a Banach space with the norm and is the infinitesimal generator of an analytic compact semigroup of uniformly bounded linear operators . This means that there exists a such that . We assume without loss of generality that . This allows us to define the fractional power , for , as a closed linear operator on its domain with inverse .
We will introduce the following basic properties of .
Theorem 2.1 (see )
is a Banach space with the norm , .
for each and for each and .
For every , is bounded on X and there exists a positive constant such that(2.1)
If , then and the embedding is compact whenever the resolvent operator of A is compact.
Let us recall the following known definitions.
The fractional integral of order with the lower limit zero for a function f can be defined as
provided the right-hand side is pointwise defined on , where is the Gamma function.
The Caputo derivative of order α with the lower limit zero for a function f can be written as
If f is an abstract function with values in X, then the integrals appearing in the above definitions are taken in Bochner’s sense.
We list the following basic assumptions of this paper.
(H1) is a continuous function, and there exists a constant and such that the function satisfies the Lipschitz condition:
for , , , and the inequality
holds for .
(H2) The function satisfies the following conditions:
for each , the function is continuous and for each the function is strongly measurable;
for each positive number , there is a positive function such that
the function and there exists a such that
(H3) , , . , here and hereafter , and g satisfies that:
There exist positive constants and such that for all ;
g is a completely continuous map.
At the end of this section, we recall the fixed-point theorem of Sadoviskii , which is used to establish the existence of the mild solution of the nonlocal Cauchy problem (1.1).
Theorem 2.2 (Sadovskii’s fixed-point theorem)
Let Φ be a condensing operator on a Banach space X, that is, Φ is continuous and takes bounded sets into bounded sets, and for every bounded set B of X with . If for a convex, closed and bounded set ϒ of X, then Φ has a fixed point in X (where denotes Kuratowski’s measure of noncompactness).
3 Main result
In this section, we study the existence of mild solutions for the neutral fractional differential equations with nonlocal conditions (1.1), so we introduce the concept of a mild solution.
A continuous function is said to be a mild solution of the nonlocal Cauchy problem (1.1) if the function , is integrable on and the following integral equation is verified:
with is a probability density function defined on , that is , and .
Lemma 3.1 (see )
The operators and have the following properties:
for any fixed , , ;
and are strongly continuous;
for every , and are also compact operators;
for any , and , we have and , .
Theorem 3.1 If the assumptions (H1)-(H3) are satisfied and , then the nonlocal Cauchy problem (1.1) has a mild solution provided that
For the sake of brevity, we rewrite that
Define the operator Φ on E by
For each positive integer q, let .
Then for each q, is clearly a bounded closed convex set in E.
From Lemma 3.1 and (2.2) yield
it follows that is integrable on J, by Bochner’s theorem  so Φ is well defined on . Similarly, from (H2)(ii), we obtain
We claim that there exists a positive number q such that . If it is not true, then for each positive number q, there is a function , but , but for some , where denotes that t is dependent of q. However, from equations (2.2), (3.4) and (3.5) and (H3)(i), we have
Dividing both sides of (3.6) by q and taking the lower limit as , we get
This contradicts (3.3). Hence, for positive q, .
Next, we will show that the operator Φ has a fixed point on , which implies that equation (1.1) has a mild solution. We decompose Φ as , where the operators and are defined on , respectively, by
for . We will show that verifies a contraction condition while is a compact operator.
To prove that satisfies a contraction condition, we take , then for each and by condition (H1) and (3.2), we have
and by assumption , we see that is a contraction.
To prove that is compact, firstly we prove that is continuous on .
Let with in , then for each , , and by (H2)(i), we have , as .
By the dominated convergence theorem, we have
as , that is continuous.
Next, we prove that the family is a family equicontinuous functions. To do this, let small, , then
We see that tends to zero independently of as , with ϵ sufficiently small since the compactness of for (see ) implies the continuity of for in t in the uniform operator topology. Similarly, using the compactness of the set we can prove that the function , are equicontinuous at . Hence, maps into a family of equicontinuous functions.
It remains to prove that is relatively compact in X. Obviously, by condition (H3), is relatively compact in X.
Let be fixed, , arbitrary , for , we define
Since , is a compact operator, then the set is relatively compact in X for every ϵ, and for all .
Moreover, for every , we have
Therefore, there are relative compact sets arbitrary close to the set , . Hence, the set , is also relatively compact in X.
Thus, by Arzela-Ascoli theorem is a compact operator. Those arguments enable us to conclude that is a condensing map on , and by the fixed-point theorem of Sadovskii there exists a fixed point for Φ on . Therefore, the nonlocal Cauchy problem (1.1) has a mild solution, and the proof is completed. □
Let , we consider the following fractional neutral partial differential equations
where is a Caputo fractional partial derivative of order , , , p is a positive integer, .
We define an operator A by with the domain
Then −A generates a strongly continuous semigroup which is compact, analytic, and self-adjoint. Furthermore, −A has a discrete spectrum, the eigenvalues are , , with the corresponding normalized eigenvectors . We also use the following properties:
If , then .
For each , . In particular, .
The operator is given by
on the space .
The system (4.1) can be reformulated as the following nonlocal Cauchy problem in X:
where that is , , .
The function is given by
holds for and .
The function is given by
holds for and , and the function is given by
where , for .
We can take and , then (H2) is satisfied. Furthermore, assume that . Then (H3) is satisfied (noting that is completely continuous).
Moreover, we assume the following conditions hold:
The function , is measurable and
The function is measurable, , and let
Therefore, the conditions (H1)-(H3) are all satisfied. Hence, according to Theorem 3.1, system (4.1) has a mild solution provided that (3.2) and (3.3) hold.
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I would like to thank the referees and Professor Ravi Agarwal for their valuable comments and suggestions.
The author declares that he has no competing interests.
About this article
- fractional calculus
- semilinear neutral differential equations
- nonlocal conditions
- mild solutions
- Sadovskii fixed-point theorem