Journal of Uncertainty in Mathematics Science 2014 (2014) 1-9
Available online at www.ispacs.com/jums Volume 2014, Year 2014 Article ID jums-00011, 9 Pages doi:10.5899/2014/jums-00011 Research Article
Existence and uniqueness for fractional differential equations with uncertainty A. Armand1∗ , S. Mohammadi1 (1) Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran c A. Armand and S. Mohammadi. This is an open access article distributed under the Creative Commons Attribution License, Copyright 2014 ⃝ which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract We discuss existence, and uniqueness of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Caputo sense and the initial condition is a fuzzy number. To this aim, contraction mapping principle and the fixed point theorem are used and finally an example is given to more illustration of obtained results. Keywords: Generalized Hukuhara differentiability, Generalized fuzzy Caputo differentiability, Fuzzy fractional differential equa-
tions.
1 Introduction In the past decade, the concept of Riemann- Liouville fractional differential equations with uncertainty was introduced by Agarwal et al [1]. They have considered the Riemann-Liouville differentiability concept based on the Hukuhara differentiability to solve uncertain fractional differential equations. Consequently, Arshad et al in [9] prove some results on the existence and uniqueness of solutions of Riemann- Liouville fuzzy fractional differential equations and in [10] study the existence and uniqueness of the solution for a class of fractional differential equation with fuzzy initial value. Also, Ref. [18] is devoted to propose Riemann-Liouville differentiability by using Hukuhara difference. They adopted the fuzzy Laplace transforms method to solve fuzzy fractional differential equations. See [3] for interpretation of explicit solutions of uncertain fractional differential equations under Riemann-Liouville Hukuhara differentiability using Mittag-Leffler functions. Two new uniqueness results for fuzzy fractional differential equations involving Riemann-Liouville generalized Hukuhara differentiability have been investigated in [4] with the Nagumotype condition and the Krasnoselskii-Krein-type condition.
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Moreover the existence, uniqueness and approximate solutions of fuzzy fractional differential equations under Caputos H-differentiability are studied in [19] and [2]. In 2009, a generalization of the Hukuhara difference and also generalizations of the Hukuhara differentiability to the case of interval valued functions were presented by Stefanini and Bede in [20], and it is studied in several papers (see e.g.[8, 15]). Following this approach, the concept of fuzzy fractional differential equation under Caputo generalized Hukuhara derivative is defined in [5]. The existence and uniqueness of the solution for differential equation of noninteger order with fuzzy initial value are presented. The authors introduced fractional integro-differential equations with fuzzy initial value in [6] and also studied the existence and uniqueness results of solution of this equations. In this paper, we present a new conditions which guarantee the existence and uniqueness of solutions of differential equations of fractional order q ∈ (0, 1) by using Caputo generalized Hukuhara differential operator with fuzzy initial condition. 2 Basic concepts In this section, we recall some basic concepts which are used throughout the paper. Definition 2.1. A fuzzy number is a function such as u : R −→ [0, 1] satisfying the following properties: (i) u is normal, i. e. ∃x0 ∈ R with u(x0 ) = 1, (ii) u is a convex fuzzy set i. e. u((1 − λ )x + λ y) ≥ min{u(x), u(y)}, ∀x, y ∈ R, λ ∈ [0, 1], (iii) u is upper semi-continuous on R, (iv) {x ∈ R : u(x) > 0} is compact, where A denotes the closure of A.
{ } The set of all fuzzy real numbers is denoted by RF . For 0 < α ≤ 1 denote uα = x ∈ R u(x) ≥ α = [uα− , u+ α ]. Then from (i) to (iv), it follows that the α -level set uα is a closed interval for all α ∈ [0, 1]. For arbitrary u, v ∈ RF and k ∈ R, the addition and scalar multiplication are defined by (u + v)α = uα + vα , (k ⊙ u)α = kuα , respectively. Definition 2.2. The metric d on RF given by d : RF × RF −→ R+ ∪ {0} d(u, v) = sup dH (uα , vα ), 0≤α ≤1
(2.1)
− + + where dH is the Hausdorff metric defined as dH (uα , vα ) = max {|v− α − uα | , |vα − uα |}, such that d(u, v) satisfies the following properties:
(i) d(u + w, v + w) = d(u, v) ∀ u, v, w ∈ RF , (ii) d(λ u, λ v) = |λ |d(u, v) ∀ u, v ∈ RF , λ inR, (iii) d(u, v) ≤ d(u, w) + d(w, v) ∀ u, v, w ∈ RF , (iv) (RF , d) is a complete metric space. Definition 2.3. [11] Let u, v ∈ RF . If there exists w ∈ RF such that u = v + w, then w is called the Hukuhara difference of u and v, and it is denoted by u ⊖ v. Definition 2.4. [12] Let u, v ∈ RF . If there exists w ∈ RF such that { (i ) u = v + w , u ⊖gH v = w ⇐⇒ or (ii) v = u + (-1) w . Then w is called the generalized Hukuhara difference of u and v.
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Please note that a function f : [a, b] → RF so called fuzzy-valued function. The α -level representation of fuzzyvalued function f is expressed by fα (t) = [ fα− (t), fα+ (t)], t ∈ [a, b], α ∈ [0, 1]. Definition 2.5. [20] The generalized Hukuhara derivative of a fuzzy-valued function f : (a, b) −→ RF at t0 is defined as f (t0 + h) ⊖gH f (t0 ) ′ fgH (t0 ) = lim , h→0 h ′
if ( f )gH (t0 ) ∈ RF , we say that f is generalized Hukuhara differentiable (gH-differentiable ) at t0 . Also we say that f is c f [i − gH] - differentiable at t0 if ′
′
′
0 ≤ α ≤ 1,
(2.2)
′
′
0 ≤ α ≤ 1.
(2.3)
( fgH )α (t0 ) = [( fα− ) (t0 ), ( fα+ ) (t0 )], and that f is c f [ii − gH] - differentiable at t0 if ′
( fgH )α (t0 ) = [( fα+ ) (t0 ), ( fα− ) (t0 )],
Definition 2.6. [20]. We say that a point t0 ∈ (a, b), is a switching point for the differentiability of f , if in any neighborhood V of t0 there exist points t1 < t0 < t2 such that type (I) at t1 (2.2) holds while (2.3) does not hold and at t2 (2.3) holds and (2.2) does not hold, or type (II) at t1 (2.3) holds while (2.2) does not hold and at t2 (2.2) holds and (2.3) does not hold. Definition 2.7. [7] A fuzzy-valued function f : [a, b] → RF is said to be continuous at t0 ∈ [a, b] if for each ε > 0 there is δ > 0 such that d( f (t), f (t0 )) < ε , whenever t ∈ [a, b] and |t − t0 | < δ . We say that f is fuzzy continuous on [a, b] if f is continuous at each t0 ∈ [a, b] such that the continuity is one-sided at endpoints a, b. Definition 2.8. [14] Let f : [a, b] → RF , for each partition P = {t0 ,t1 , ...,tn } of [a, b] and for arbitrary ξi ∈ [ti−1 ,ti ], 1 ≤ i ≤ n, suppose R p = ∑ni=1 f (ξi )(ti − ti−1 ), and ∆ := max{|ti − ti−1 |, 1 ≤ i ≤ n}. The definite Riemann integral of f (t) over [a, b] is ∫ b a
f (t)dt = lim R p , ∆→0
provided that this limit exists in the metric d. Note that if the fuzzy function f (t) is continuous in the metric d, Lebesgue integral and Riemann integral yield the same value, and also (∫ b [∫ b ] ) ∫ b − + f (t)dt = fα (t)dt, fα (t)dt , 0 ≤ α ≤ 1, a
α
a
a
Definition 2.9. [3] Let f : [a, b] → RF ; the fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as follows: ∫ x 1 f (t) (Iaq f )(x) = dt Γ(q) a (x − t)1−q for a ≤ x, and 0 < q ≤ 1 . Definition 2.10. [17] Consider f : [a, b] → R, fractional derivative of f (t) in the Caputo sense is defined as (Dq∗ f )(t) = (I m−q Dm f )(t) =
1 Γ(q)
∫ t a
(t − s)(q−m−1) f (m) (s)ds
q−1 < m ≤ q , m ∈ N , s > a
where D stand for classic derivative . We denote CF [a, b] as the space of all continuous fuzzy-valued functions on [a, b]. Also, we denote the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval [a, b] ⊂ R by LF [a, b].
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∩
Definition 2.11. [18] Let f ∈ CF [a, b] LF [a, b] . The fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as following: ∫ t 1 f (s)ds (Iaq f )(t) = , a < s < t, 0 < q ≤ 1. (2.4) Γ(q) a (t − s)1−q ∩
Definition 2.12. [5] Let f ′ ∈ CF [a, b] LF [a, b]. The fractional generalized Hukuhara Caputo derivative of fuzzyvalued function f is defined as following: ′ ( gH Dq∗ f )(t) = Ia1−q ( fgH )(t) =
1 Γ(1 − q)
∫ t ( f ′ )(s)ds gH a
(t − s)q
,
a < s < t,
0 0, and h∗ > 0. Let f be continuous fuzzy function on R0 = {(t, x) : t ∈ [t0 ,t0 + h∗ ], d(x, x0 ) ≤ b} such that supt∈J d( f (t, x(t)), 0) = M and { h∗ if M=0; 1 h= q }, else. min{h∗ , ( bΓ(q+1) ) M Then there exists a function x ∈ CF [t0 ,t0 + h] solving the initial value problem (3.8). Proof. If M = 0 then x0 is a solution of the initial value problem, because d( f (t, x), 0) = 0 for all (t, x) ∈ R0 . Hence we find that x0 as a solution in this case. Now we introduce U = {x ∈ CF [t0 ,t0 + h] : d(x, x0 ) < b}, that is a closed and convex subset of the Banach space of all continuous functions on [t0 ,t0 + h∗ ]. Therefore, U is a Banach space, too. First, we suppose that the solution of problem (3.8) is c f [i − gH]-differentiable. Using Lemma 3.2, the solution of problem (3.8) for case c f [i − gH]-differentiability is equivalent to integral Eq.(3.9). So, we define a mapping A : U → U, that given by (Ax)(t) := x0 +
1 Γ(q)
∫ t t0
(t − s)q−1 f (s, x(s))ds
Now, we want to show that A is contraction. For t0 ≤ t1 ≤ t2 ≤ t0 + h ∫
∫
t1 t2 1 d( (t1 − s)q−1 f (s, x(s))ds, (t2 − s)q−1 f (s, x(s))ds) Γ(q) t0 t0 ∫ t1 ∫ t2 1 |(t1 − s)q−1 − (t2 − s)q−1 |d( f (s, x(s), 0)ds + (t2 − s)q−1 d( f (s, x(s), 0)ds Γ(q) t0 t1 ∫ t1 ∫ t2 M |(t1 − s)q−1 − (t2 − s)q−1 |ds + (t2 − s)q−1 ds := ∗ Γ(q) t0 t1
d((Ax)(t1 )), (Ax)(t2 )) ≤ ≤ ≤
Since q < 1 then q − 1 < 0, so we have ∫ t1 t0
|(t1 − s)q−1 − (t2 − s)q−1 |ds
=
∫ t1 ( t0
=
) (t1 − s)q−1 − (t2 − s)q−1 ds
) 1 1( (t2 − t1 )q + (t1 − t0 )q − (t2 − t0 )q ≤ (t2 − t1 )q q q
Therefore ∗ =
) 1 2M M (1 (t2 − t1 )q + (t2 − t1 )q = Γ(q) q q Γ(q + 1)
(3.12)
The right-hand side of Eq.(3.12) tends to zero as t2 → t1 . It means that A is continuous on U. Furthermore, for x ∈ U and t ∈ [t0 ,t0 + h], we have 1 d((Ax)(t), x0 ) ≤ Γ(q)
∫ t t0
(t − s)q−1 d( f (s, x(s)), 0)ds ≤
M bΓ(q + 1) M (t − t0 )q ≤ )=b Γ(q + 1) Γ(q + 1) M
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So if x ∈ U then Ax ∈ U, which implies that A maps the set U to itself. Now, we show that A(U) : {Au : u ∈ U} is a relatively compact set by utilize the Arzel´a-Ascoli theorem. For z ∈ A(U), we find that, for all t ∈ [t0 ,t0 + h] ∫
t 1 d(z(t), 0) = d((Ax)(t), 0) ≤ d(x0 , 0) + (t − s)q−1 d( f (s, x(s)), 0)ds Γ(q) t0 ∫ t M ≤ d(x0 , 0) + (t − s)q−1 ds Γ(q) t0 ≤ d(x0 , 0) + b
which yields that A(U) is bounded set. Moreover, by Eq.(3.12), we have d((Ax)(t1 ), (Ax)(t2 )) ≤
2M (t2 − t1 )q Γ(q + 1)
For t0 ≤ t1 ≤ t2 ≤ t0 + h, setting (t2 − t1 ) = δ gives d((Ax)(t1 )), (Ax)(t2 )) ≤
2M δ q Γ(q + 1)
(3.13)
inasmuch as the right-hand side of Eq.(3.13) is independent of, x, t1 and t2 , we conclude that the set A(U) is equicontinuous. Therefore the Arzel´a-Ascoli theorem implies that A(U), is relatively compact set. Hence Schaefer’s fixed point theorem, claim that A has a fixed point, which proves the theorem for case c f [i − gH]−differentiability. Also, for case c f [ii − gH]-differentiability by lemma 3.2 problem (3.8) is equivalent to Eq.(3.10). Define a mapping A : U → U that defined by (Bx)(t) := x0 ⊖
−1 Γ(q)
∫ t t0
(t − s)q−1 f (s, x(s))ds
Similar mentioned process can be easily show that the mapping B has a fixed point. Theorem 3.4. Let q ∈ (0, 1), b > 0 and x0 ∈ RF . If f : J × RF → RF be fuzzy continuous function and satisfies in inequality d( f (t, x), f (t, y))
