- Open Access
Nonlocal nonlinear integrodifferential equations of fractional orders
© Debbouche et al.; licensee Springer 2012
- Received: 11 May 2012
- Accepted: 9 July 2012
- Published: 24 July 2012
In this paper, Schauder fixed point theorem, Gelfand-Shilov 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.
- fractional evolution equation
- nonlocal condition
- Schauder’s fixed point theorem
- uniformly continuous semigroup
where , is the Riemann-Liouville fractional derivative, , are real numbers, B and A are linear closed operators with domains contained in a Banach space X and ranges contained in a Banach space Y, , is a family of linear closed operators defined on dense sets respectively in X into X, and are given abstract functions. Here .
Fractional differential equations have attracted many authors [1, 6–8, 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  gave a novel method for the design of fractional order digital controllers.
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 [2–4, 11–13, 19, 28].
where are given constants and .
However, among the previous research on nonlocal Cauchy problems, few authors have been concerned with mild solutions of fractional semilinear differential equations .
Recently, many authors have extended this work to various kinds of nonlinear evolution equations [2, 3, 5, 11, 12, 18]. Balachandran and Uchiyama  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 Gelfand-Shilov approach in fractional calculus and are given in terms of some probability density functions such that their Laplace transforms are indicated .
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.
In this section, we mention some results obtained by Balachandran , El-Borai  and Pazy , which will be used to get our new results. Let X and Y be Banach spaces with norm and respectively. The operator satisfies the following hypotheses: (H1) = B is bijective,; (H2) = is compact.. The above fact and the closed graph theorem imply the boundedness of the linear operator . Further generates a uniformly continuous semigroup such that for every and all , see .
Let , and .
It is supposed that (H3) = f and g are continuous in t on I, Δ respectively, and there exist constants such that , for all , and ..
u is a continuous function in and ,
exists and is continuous on , , and u satisfies (1.1) on and (1.2).
where is the Gamma function.
where , , and is a positive constant, .; (H6) = The functions are uniformly Hölder continuous in for every element h in .. Suppose that is a -semigroup of operators on X such that , where δ is a positive constant and . Noting that (see , p.4]).
If , then , which achieves that ψ exists on X.
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 3.2 If the assumptions (H1)∼(H4) hold and, then the problem (1.1), (1.2) has a mild solution on I.
The right-hand side of the above inequality is independent of and tends to zero as as a consequence of the continuity of and in the uniform operator topology for . It is clear that S is bounded in Z. Thus by Arzela-Ascoli’s theorem, S is precompact. Hence by the Schauder fixed point theorem, φ has a fixed point in and any fixed point of φ is a mild solution of (1.1), (1.2) on I such that for all . □
Conditions (H 1)∼(H 6) hold,
Y is a reflexive Banach space with norm ,
for all, and all, whereand.
Then the problem (1.1), (1.2) has a unique strong solution on I.
Proof Applying Theorem 3.2, the problem (1.1), (1.2) has a mild solution . Now, we shall show that u is a unique strong solution of the considered problem on I.
According to (H6), is uniformly Hölder continuous in for every element u in combined with (iii), which implies that and are uniformly Hölder continuous on I.
for all and . It follows that for all . □
- (i)The operator is uniformly elliptic on . In other words, all the coefficients , are continuous and bounded on , and there is a positive number c such that
for all and all , , where , and .
(ii) All the coefficients , satisfy a uniform Hölder condition on . Under these conditions, the operator E with the domain of definition generates an analytic semigroup defined on , and it is well known that is dense in , see , p.438].
Lemma 4.1 The solution representation of (4.1), (4.2) can be written explicitly.
has roots which satisfy the inequality , for all and for any real vector σ, . If Θ is a matrix of order , then we introduce .
where M is a positive constant, and .
- (iii)There are numbers and such that
for all , , and all . Then applying Theorem 3.3, we deduce that (4.1), (4.2) has a unique strong solution. □
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 Arzela-Ascoli’s theorem and Schauder fixed point theorem. Further, the uniformly Hölder continuous condition has been applied for the existence of strong solution.
The authors would like to thank the referees for their valuable comments and remarks.
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