Abstract
In this paper, Schauder fixed point theorem, GelfandShilov principles combined with semigroup theory are used to prove the existence of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. To illustrate our abstract results, an example is given.
MSC: 35A05, 34G20, 34K05, 26A33.
Keywords:
fractional evolution equation; nonlocal condition; Schauder’s fixed point theorem; uniformly continuous semigroup1 Introduction
We are concerned with the nonlocal nonlinear fractional problem
where
Fractional differential equations have attracted many authors [1,68,21,24,25,30]. This is mostly because it efficiently describes many phenomena arising in engineering, physics, economics and science. In fact, we can find several applications in viscoelasticity, electrochemistry, electromagnetic, etc. For example, Machado [22] gave a novel method for the design of fractional order digital controllers.
Following Gelfand and Shilov [20], we define the fractional integral of order
also, the (RiemannLiouville) fractional derivative of the function f of order
where f is an abstract continuous function on the interval
The existence results to evolution equations with nonlocal conditions in a Banach space was studied first by Byszewski [9,10]; subsequently, many authors were pointed to the same field, see for instance [24,1113,19,28].
Deng [15] indicated that using the nonlocal condition
where
However, among the previous research on nonlocal Cauchy problems, few authors have been concerned with mild solutions of fractional semilinear differential equations [23].
Recently, many authors have extended this work to various kinds of nonlinear evolution equations [2,3,5,11,12,18]. Balachandran and Uchiyama [3] proved the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition.
In this paper, motivated by [3,13,17,19], we use Schauder fixed point theorem and the semigroup theory to investigate the existence and uniqueness of mild and strong solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces, the solutions were obtained by using GelfandShilov approach in fractional calculus and are given in terms of some probability density functions such that their Laplace transforms are indicated [17].
Our paper is organized as follows. Section 2 is devoted to the review of some essential results. In Section 3, we state and prove our main results; the last section deals with giving an example to illustrate the abstract results.
2 Preliminary results
In this section, we mention some results obtained by Balachandran [3], ElBorai [19] and Pazy [26], which will be used to get our new results. Let X and Y be Banach spaces with norm
Let
It is supposed that (H_{3}) = f and g are continuous in t on I, Δ respectively, and there exist constants
Definition 2.1 By a strong solution of the nonlocal Cauchy problem (1.1), (1.2), we mean a function u with values in X such that
(i) u is a continuous function in
(ii)
Remark 2.1 Let us take in the considered problem B is the identity, the inhomogeneous part is equal to an abstract continuous function
According to ElBorai [1719], we first apply the fractional integral on both sides of (2.1) and then using (2.2), we apply the Laplace transform on the new integral equations by considering a suitable onesided stable probability density whose Laplace transform is given. Hence we can conclude that a solution of the problem (2.1)(2.2) can be formally represented by
where
For more details, we refer to Zhou et al.[27,31], see also [14,16].
Using GelfandShilov principle [20], it is suitable to rewrite (1.1), (1.2) in the form
where
According to [1719], the equation (2.4) is equivalent to the integral equation
where
It is assumed that there exists an operator ψ on
satisfying
also (H_{4}) =
where
If
3 Main results
The following is different from [3,19,26] and represents the new result.
Lemma 3.1Ifuis a continuous solution of (2.5), thenusatisfies the integral equation
Proof Using (2.5) and (1.2), we get
Then
Thus
Hence the required result. □
Definition 3.1 A continuous solution of the integral equation (3.1) is called a mild solution of the nonlocal problem (1.1), (1.2) on I.
Theorem 3.2If the assumptions (H_{1})∼(H_{4}) hold and
Proof Let
Noting also that
We deduce that φ is continuous and maps
The righthand side of the above inequality is independent of
Theorem 3.3Assume that
(i) Conditions (H_{1})∼(H_{6}) hold,
(ii) Yis a reflexive Banach space with norm
(iii) there are numbers
for all
Then the problem (1.1), (1.2) has a unique strong solution onI.
Proof Applying Theorem 3.2, the problem (1.1), (1.2) has a mild solution
According to (H_{6}),
Set
From (3.1), the solution u of the considered problem can be written in the form
Noting that Ψ and ψ are bounded, using our assumptions, we observe that there exists a unique function
Also as in [19], p.409], we deduce that
for all
4 Example
Consider the nonlinear integropartial differential equation of fractional order
with nonlocal condition
where
and Ω is an open subset of
It is supposed that
(i) The operator
for all
(ii) All the coefficients
Lemma 4.1The solution representation of (4.1), (4.2) can be written explicitly.
Proof Let
has roots which satisfy the inequality
It is well known that there exists a fundamental matrix solution
This fundamental matrix also satisfies the inequality
where
It can be proved that
where
([17]) The nonlocal Cauchy problems (4.1), (4.2) are equivalent to the integral equation
where the explicit form of Q is given by
(iii) There are numbers
and
for all
5 Conclusion
In this article, a new solution representation for Sobolev type fractional evolution equation has been proved using Deng’s nonlocal condition, a suitable explicit form of the semigroup has been discussed. Moreover, the existence result of mild solutions for nonlinear fractional integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces has been established by using ArzelaAscoli’s theorem and Schauder fixed point theorem. Further, the uniformly Hölder continuous condition has been applied for the existence of strong solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
AD wrote the first draft, DB corrected and improved it and RPA prepared the final version. All authors read and approved the final draft.
Acknowledgements
The authors would like to thank the referees for their valuable comments and remarks.
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