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Existence and dimension of the set of mild solutions to semilinear fractional differential inclusions
https://doi.org/10.1186/1687-1847-2012-74
© Agarwal et al; licensee Springer. 2012
- Received: 31 January 2012
- Accepted: 31 May 2012
- Published: 31 May 2012
Abstract
This article studies the existence and dimension of the set for mild solutions of semilinear fractional differential inclusions. We recall and prove some new results on multivalued maps to establish our main results.
MSC 2010: 34A12; 34A40.
Keywords
- nonlocal problem
- fractional differential inclusions
- topological dimension
- mild solution
- fixed point theorems
1 Introduction
The study of fractional calculus (differentiation and integration of arbitrary order) has emerged as an important and popular field of research. It is mainly due to the extensive application of fractional differential equations in many engineering and scientific disciplines such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc., [1–7]. Fractional derivatives are also regarded as an excellent tool for the description of memory and hereditary properties of various materials and processes [8]. Owing to these characteristics of fractional derivatives, fractional-order models are considered to be more realistic and practical than the classical integer-order models, in which such effects are not taken into account. A variety of results on initial and boundary value problems of fractional differential equations, ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions, have appeared in the literature, for instance, see [9–20] and references therein.
Differential inclusions arise in the mathematical modeling of certain problems in economics, optimal control, etc., and are widely studied by many authors, see [21–23] and the references therein. For some recent development on differential inclusions of fractional order, we refer the reader to the references [24–29].
where c D q denotes the Caputo fractional derivative of order q, A is a sectorial operator on ℝ n , g: C([0, T ], ℝ n ) → ℝ n , and F: [0, T ] × ℝ n → P (ℝ n ), where P(ℝ n ) is the family of all nonempty subsets of ℝ n .
2 Background material
Let us recall some basic definitions on multi-valued maps (for details, see [30, 31]).
Let (X, d) be a metric space. Define P(X) = {Y ⊆ X: Y ≠ Ø}, P cl (X) = {Y ∈ P (X): Y is closed}, P b (X) = {Y ∈ P (X): Y is bounded}, Pb, cl(X) = {Y ∈ P (X): Y is closed and bounded} and P cp (X) = {Y ∈ P (X): Y is compact}:
where d(a, B) = inf b∈B d(a, b). H is the (generalized) Pompeiu-Hausdorff functional. It is known that (Pb, cl(X), H) is a metric space and (P cl (X), H) is a generalized metric space (see [30]).
is measurable.
Let ℭ([0, T], ℝ n ]) denotes the Banach space of continuous functions from [0, T ] into ℝ n with the norm ǀǀx ǀǀ∞ = supt∈[0, T]ǀǀx(t)ǀǀ. Let L1([0, T ], ℝ n ) be the Banach space of measurable functions x: [0, T ] → ℝ n which are Lebesgue integrable and normed by . Let C be a nonempty subset of a Banach space X: = (X, ǀǀ.ǀǀ). Define Pc, cl(C) = {Y ∈ P (C): Y is convex and closed}, and Pc, cp(C) = {Y ∈ P (C): Y is compact and convex}.
Let us recall some definitions on fractional calculus. For more details, we refer to [1, 4].
where [q] denotes the integer part of the real number q and Γ denotes the gamma function.
provided the right-hand side is pointwise defined on (0, ∞).
3 Main results
where σ: [0, T ] → ℝ n .
The following lemma is discussed in [32]. However, for the sake of completeness, we outline its proof here.
where is a suitable path such that λ q ∉ μ + S θ for λ ∈ and R(λ q , A) = (λ q I - A)-1.
By taking inverse Laplace transform of (4), we obtain (3). This completes the proof.
with , and , where L(ℝ n ) is the Banach space of bounded linear operators from ℝ n into ℝ n equipped with natural topology and C, μ are appropriate constants (for more details see Equation (3.1) in [32]).
Remark 3.3. The definition of the mild solution used in [33] is not appropriate as it does not correspond to the classical case due to the failure of the Leibniz product rule for the Caputo fractional derivative. For more details, see [32].
Let denotes the set of all solutions of (1) on the interval [0, α], where 0 < α ≤ T.
To prove the existence of solutions for (1), we need the following lemma due to Nadler and Covitz [34].
Lemma 3.5. Let (X, d) be a complete metric space. If Ω: X → P cl (X) is a k-contraction, then Fix(Ω) ≠ ∅.
Theorem 3.6. Assume that
(A 1 ) F: [0, T] × ℝ n → P cp (ℝ n ) is such that F (., x): [0, T ] → Pc, cp(ℝ n ) is measurable for each x ∈ ℝ n ;
(A 2 ) H(F(t, x), for almost all t ∈ [0, T ] and with κ1 ∈ ℭ([0,T],ℝ+) and ǀǀF (t, x)ǀǀ = sup{ǀǀvǀǀ: v ∈ F (t, x)} ≤ κ1(t) for almost all t ∈ [0, T ] and x ∈ ℝ n ;
(A 3 ) g: ℭ([0, T], ℝ n )→ℝ n is continuous and ǀǀg(x) - g(y)ǀǀ ≤ κ2ǀǀx - y ǀǀ∞ for all x, y∈ℭ([0, T], ℝ n ) with some κ2 > 0.
( and are given by (5)).
Hence u ∈ Ω(x).
where . Since Ω is a contraction, it follows by Lemma 3.5 that Ω has a fixed point x which is a solution of (1). This completes the proof.
Then Ω(x)∈P c, cp (ℭ([0, T], ℝ n )) for each x ∈ ℭ([0, T], ℝ n ).
Obviously the right-hand side of the above inequality tends to zero independently of as t'' - t' → 0. By the Arzela-Ascoli Theorem, Ω:ℭ([0, T], ℝ n )→P(ℭ([0, T], ℝ n )) is completely continuous. As in Theorem 3.6, Ω is closed-valued. Consequently, Ω(x)∈P c,cp (ℭ([0, T], ℝ n )) for each x ∈ ℭ([0, T], ℝ n ).
It is well-known that and, in view of Theorem 3.6, it is nonempty for each 0 < α ≤ T.
The following results are useful in the sequel.
Lemma 3.8 (Dzedzej and Gelman [36]) Let F: [0, α] → Pc, cp(ℝ n ) be a measurable map such that the Lebesgue measure μ of the set {t: dim F(t) < 1} is zero. Then there are arbitrarily many linearly independent measurable selections x1(·), x2(·), . . . , x m (·) of F .
Lemma 3.9. (Dzedzej and Gelman [36]) (see also, [29, 37] for general versions) Let C be a nonempty closed convex subset of a Banach space X. Suppose that Ω: C → Pc, cp(C) is a k-contraction. If f: C → C is a continuous selection of Ω, then Fix(f) is nonempty.
Lemma 3.10. (Michael's selection theorem) [38] Let C be a metric space, X be a Banach space and Ω: C → Pc, cl(C) a lower semicontinuous map. Then there exists a continuous selection f: C → X of Ω.
Lemma 3.11. (Saint Raymond [39]) Let K be a compact metric space with dim K < n, X a Banach space and Ω: K → Pc, cp(X) a lower semicontinuous map such that 0 ∈ Ω(x) and dim Ω(x) ≥ n for every x ∈ K. Then there exists a continuous selection f of Ω such that f(x) ≠ 0 for each x ∈ K.
Theorem 3.12. Let F: [0, α] × ℝ n → Pc, cp(ℝ n ) satisfy (A 1 ), (A 2 ), and (A 3 ) and suppose that the Lebesgue measure μ of the set {t: dim F (t, x) < 1 for some x ∈ ℝ n } is zero. Then for each α, , the set of solutions of (1) has an infinite dimension for any x0.
Since T is lower semicontinuous, Michael's selection result (Lemma 3.10) guarantees that T admits a continuous selection f: ℝ n → ℝ n . Thus f: ℝ n → ℝ n is a continuous selection of Ω with no fixed points and f = h on Fix(Ω), which contradicts Lemma 3.9. Consequently, is infinite dimensional.
Recall that a metric space X is an AR-space if, whenever it is nonempty closed subset of another metric space Y , then there exists a continuous retraction r: Y → X, r(x) = x for x ∈ X. In particular, it is contractible (and hence connected).
Lemma 3.13. [41] Let C be a nonempty closed convex subset of a Banach space X and Ω: C → Pc, cp(C) a contraction. Then Fix(Ω) is a nonempty AR-space.
Theorem 3.12 together with Lemma 4.13 yields the following result.
Corollary 3.14. Let F: [0, α] × ℝ n → Pc, cp(ℝ n ) satisfy (A 1 ), (A 2 ), and (A 3 ) and suppose that the Lebesgue measure μ of the set {t: dim F (t, x) < 1 for some x ∈ ℝ n } is zero. Then for each , the set of solutions of (1) is an infinite dimensional AR-space.
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 08/31/Gr. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors are grateful to the reviewers for their useful comments.
Authors’ Affiliations
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