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Nonlinear iteration semigroups of fuzzy Cauchy problems
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Fuzzy Sets and Systems 209 (2012) 104 – 110 www.elsevier.com/locate/fss
Nonlinear iteration semigroups of fuzzy Cauchy problems Osmo Kaleva Tampere University of Technology, Department of Mathematics, P.O. Box 553, FI-33101 Tampere, Finland Received 15 August 2011; received in revised form 28 April 2012; accepted 28 April 2012 Available online 8 May 2012
Abstract In this paper we introduce an iteration semigroup of a nonlinear fuzzy-valued function. We show that the iterates converge and the limit function, called the fuzzy exponential function, is a solution to an autonomous fuzzy Cauchy problem. © 2012 Elsevier B.V. All rights reserved. Keywords: Fuzzy derivative; Fuzzy differential equations; Fuzzy exponential function; Fuzzy semigroup; Iteration of fuzzy-valued function
1. Introduction The theory of transformation semigroups, as well as linear [1] and nonlinear [9], is a well-studied topic in functional analysis. The semigroup generated by a set-valued function was developed among others by Wolenski [16], Smajdor [14,15] and Frankowska [2]. The semigroup generated by linear operators of a fuzzy-valued function was introduced by Gal and Gal [3]. Here we introduce a nonlinear semigroup generated by a nonlinear function f : F → F, where F denotes a space of fuzzy sets. Then we show that this semigroup is a solution to an autonomous fuzzy Cauchy problem x (t) = f (x(t)), x(0) = x0 . We also show that if f : F → F is a bounded linear operator, then the semigroup generated by f equals to the fuzzy semigroup introduced by Gal and Gal in [3]. The main result of the paper is to show that the iterates f n i+ (x) n converge for all x ∈ F under some assumptions on f. Here i : F → F denotes the identity function of F and (i + f /n)n is the n-fold composition of i + f /n. For obvious reasons we call the limit the fuzzy exponential function, denoted by e f (x). We show that T (t)(x0 ) = t f e (x0 ), t ⱖ 0, is a solution to the Cauchy problem. The paper is organized as follows. Section 2 is devoted to some preliminaries. In Section 3 we show the convergence of the iterates. Next we give some relationships between a fuzzy semigroup and a solution to an autonomous Cauchy problem. Finally in Section 5 we show that the fuzzy exponential function is a solution of the problem.
E-mail address: [email protected] 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.04.016
O. Kaleva / Fuzzy Sets and Systems 209 (2012) 104 – 110
105
2. Preliminaries Denote F = {u : R → [0, 1]|u satisfies (i)–(iv) below}, where (i) (ii) (iii) (iv)
u is normal, i.e. there exists an x0 ∈ R such that u(x0 ) = 1, u is fuzzy convex, u is upper semicontinuous, cl{x ∈ R|u(x) > 0}, denoted by [u]0 , is compact.
For 0 < ⱕ 1 let [u] = {x ∈ R|u(x) ⱖ }. Then from (i) to (iv) it follows that a fuzzy set u belongs to F if and only if the -level set [u] is a non-empty compact interval for all 0 ⱕ ⱕ 1. In addition we define a metric in F by the equation D(u, v) =
sup d([u] , [v] ),
0ⱕⱕ1
where d is the Hausdorff metric for non-empty compact sets in R. The metric has the following properties: D(u + v, u + w) = D(v, w) and D(u, v) = ||D(u, v) for all u, v, w ∈ F and ∈ R. For the proof see for example Kaleva [4]. As obtained by Puri and Ralescu [11] (F, D) is a complete metric space. Let u, v ∈ F. If there is such a w ∈ F that u = v + w, then w is the Hukuhara difference of u and v denoted by w = u H v. Using this difference we define a differentiability of a fuzzy function as follows. Let a > 0 and denote T = [t0 , t0 + a]. Definition 2.1. A mapping g : T → F is differentiable at t ∈ T if there exists a g (t) ∈ F such that the limits lim
h→0+
g(t + h) H g(t) and h
lim
h→0+
g(t) H g(t − h) h
exist and equal to g (t). Here the limit is taken in the metric space (F, D). At the end points of T we consider only the one-sided derivatives. In the sequel we need the following generalization of the classical Rådström embedding theorem. The proof of the theorem is given by Kaleva [5]. Theorem 2.1 (Embedding theorem). There exists a real Banach space X such that F can be embedded as a convex cone C with vertex 0 in X. Furthermore the following conditions hold true: (i) (ii) (iii) (iv) (v)
the embedding j is isometric, the addition in X induces the addition in F, the multiplication by a nonnegative real number in X induces the corresponding operation in F, C − C = {a − b|a, b ∈ C} is dense in X, C is closed.
3. Fuzzy exponential function Let f : F → F, i : F → F the identity mapping, x ∈ F and j : F → C the embedding given in the embedding Theorem 2.1. First we show that embedding the iterates (i + f /n)n (x) into C gives us iterates in a Banach space. Then we show that these iterates converge and hence the original iterates converge. Theorem 3.1. For all k ⱖ 1 we have f k f1 k j i+ (x) = I + ( j(x)), n n where I is the identity mapping of X and f 1 = j f j −1 : C → C.
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Proof. We prove the theorem by induction. Now let k = 1. By the properties of j we have f j( f (x)) j( f ( j −1 ( j(x)))) f1 j i+ (x) = j(x) + = j(x) + = I+ ( j(x)). n n n n Now assume that the identity holds true for some k. Then f f f1 k f k f k+1 i+ j i+ (x) = I + (x) (x) = j i+ j i+ n n n n n f1 k f1 f 1 k+1 = I+ ( j(x)), I+ ( j(x)) = I + n n n which proves the theorem. In the sequel we assume that f satisfies the following Osgood conditions: 1. the real function (r ) = sup D(x,y) ⱕ r D( f (x), f (y)) < ∞ for all r ⱖ 0, r 2. the integral r12 (1/(t)) dt exists for all 0 < r1 < r2 < ∞, ∞ r 3. the integrals 0 (1/(t)) dt and r (1/(t)) dt diverge for all 0 < r < ∞. 1 1 Clearly is increasing and (0) = 0. Since r (1/(t)) dt < ∞ and 0 (1/(t)) dt diverges, it follows that limr →0 (r ) = 0. Hence, by the definition of , we see that f is uniformly continuous. Since j is isometry then also j −1 is isometry and hence f 1 (u) − f 1 (v) = D( f ( j −1 (u)), f ( j −1 (v)) and D( j −1 (u), j −1 (v)) = u − v. It follows that sup
D(x,y) ⱕ r
D( f (x), f (y)) =
sup
u−v ⱕ r
f 1 (u) − f 1 (v))
and consequently f 1 also satisfies the Osgood conditions. Theorem 3.2. If f : F → F satisfies the Osgood conditions then the iterates (i + f /n)n (x) converge for all x ∈ F. Proof. By Lemma 1 in Seppälä [12] the iterates (I + f 1 /n)n ( j(x)) ∈ C converge and by Theorem 3.1 the iterates j((i + f /n)n (x)) converge. Since j −1 is continuous then f n f n f n j −1 lim j i+ i+ (x) = lim j −1 j (x) = lim i + (x) n→∞ n→∞ n→∞ n n n exists for all x ∈ F. Definition 3.1. Let f : F → F satisfy the Osgood conditions. The fuzzy exponential function e f is defined by f n e f (x) = lim i + (x). n→∞ n From Theorem 3.1 it follows that j(e f (x)) = e f 1 ( j(x)). 4. Fuzzy semigroups related to autonomous Cauchy problems Definition 4.1. A family of functions {T (t)}t ⱖ 0 , with T (t) : F → F, is a (one-parameter, strongly continuous, nonlinear) fuzzy semigroup if (i) T (t + s) = T (t)T (s) for all t, s ⱖ 0,
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(ii) T (0) = i, the identity mapping on F, (iii) the function g : [0, ∞) → F, defined by g(t) = T (t)(x), is continuous for all x ∈ F. Let f : F → F be continuous and consider the autonomous initial value problem x (t) = f (x(t)), x(t0 ) = x0 .
(1)
It is well known, see for example Kaleva [4], that instead of the differential equation (1) it is possible to study an equivalent integral equation. t x(t) = x0 + f (x(s)) ds. t0
A solution x(t) of (1) is independent of the initial time t0 . In fact, let s0 < a and denote y(t) = x(s0 + t). Then y (t) = x (s0 + t) = f (x(s0 + t)) = f (y(t)) and y(t0 ) = x(t0 + s0 ) = y0 . Hence y(t) and x(t) are solutions of the same differential equation with a different initial value. Theorem 4.1. If x(t) is a solution to the fuzzy initial value problem x (t) = f (x(t)), x(0) = x0 . Then T (t)(x0 ) = x(t) is a fuzzy semigroup. Furthermore T (t)(x0 ) is differentiable w.r.t. t and T (t)(x0 ) = f (x(t)) = f (T (t)(x0 )). Proof. Now let s > 0 be fixed. As obtained above y(t) = x(t + s) is a solution of the initial value problem y (t) = f (y(t)), y(0) = x(s). Hence T (t + s)(x0 ) = x(t + s) = y(t) = T (t)(x(s)) = T (t)T (s)(x0 ) and T (0)(x0 ) = x(0) = x0 . Being a solution to a differential equation T (t)(x0 ) is differentiable w.r.t. t and T (t)(x0 ) = x (t) = f (x(t)). Theorem 4.2. Suppose that a fuzzy semigroup T (t)(x) is differentiable w.r.t. t for all x ∈ F. Then T (t)(x0 ) is a solution to the initial value problem x (t) = f (x(t)), x(0) = x0 ,
(2)
where f (x) = T (0)(x). Proof. By the semigroup property we obtain T (t)(x0 ) = lim
h→0+
T (t + h)(x0 ) H T (t)(x0 ) T (h)T (t)(x0 ) H T (t)(x0 ) = lim h h h→0+
T (h)T (t)(x0 ) H T (0)T (t)(x0 )) = T (0)T (t)(x0 ) h h→0+ and T (0)(x0 ) = x0 . = lim
Hence every solution to the initial value problem (2) defines a differentiable fuzzy semigroup and conversely a differentiable semigroup is a solution to (2) with generator f (x) = T (0)(x). 5. Fuzzy semigroup defined by fuzzy exponential function Let f : F → F satisfy the Osgood conditions and t > 0. Then t f : F → F also satisfies the Osgood conditions. Define T (t)(x0 ) = et f (x0 ), where x0 ∈ F. Then T (0)(x0 ) = x0 . The mapping t f 1 , t ⱖ 0, can be seen as a function of two arguments F(t, x), which is linear in the first argument. Now by Seppälä [13] the iterates (I + F(t, ·)/n)n ( j(x)) converge uniformly in Bt × B j(x) , where Bt is an arbitrary open ball in R+ and B j(x) is a suitably chosen open ball centered at j(x). It follows that et f1 ( j(x)) and consequently et f (x) = T (t)(x) is continuous in t for all x ∈ F. Since f (et f (x0 )) is continuous, it is Riemann as well as Aumann integrable and the integrals coincide. For the Riemann integral of fuzzy-valued function see [10,17].
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For the Riemann integral we have t tf e (x0 ) = x0 + f (es f (x0 )) ds. 0
Let 0 ⱕ k ⱕ n and denote Onk (t, x) = (i + t f /n)k (x). Then et f (x) = lim Onn (t, x),
On0 (t, x) = x,
Onk+1 (t, x) = Onk (t, x) +
t f (Onk (t, x)). n
n→∞
Onk (t, x) = Okk
k t, x n
and
We immediately obtain et f (x) = lim Onn (t, x) = lim On0 (t, x) + n→∞
n→∞
t n→∞ n
= x + lim
n−1
n−1 t f (Onk (t, x)) n k=0
n−1 t k t, x . f Okk n→∞ n n
f (Onk (t, x)) = x + lim
k=0
k=0
Approximating the integral by a Riemann sum we have x+
t
n−1 t f (e(k/n)t f (x)). n→∞ n
f (es f (x)) ds = x + lim
0
k=0
Since D(u +w, v +w) = D(u, v) for all u, v, w ∈ F, then using the triangle inequality it is easy to prove the inequality n n n uk , vk ⱕ D(u k , vk ). D k=0
k=0
k=0
It follows that D et f (x), x +
t
f (es f (x)) ds
0
n−1 t k t, x , f (e(k/n)t f (x)) . D f Okk n→∞ n n
ⱕ lim
k=1
Proceeding exactly as in [6], when Banach space distances z − w are replaced by distances D(z, w), we see that the limit of the right hand side equals to zero, which gives us the desired result. t In conclusion T (t)(x0 ) is a continuous solution of the integral equation x(t) = x0 + 0 f (x(s)) ds and consequently a solution of the initial value problem (2). Now by Theorem 4.1 T (t)(x0 ) is a fuzzy semigroup. Example 5.1. Consider the initial value problem x (t) = x(t), x(0) = x0 ∈ F. Now for t ⱖ 0 we have (i + t f /n)(x0 ) = x0 + (t/n)x0 = (1 + t/n)x0 . Hence tf n t n i+ (x0 ) = 1 + x0 −→ et x0 , n→∞ n n which is a solution to the Cauchy problem. If x0 is an even function then −x0 = x0 . It follows that for the problem x (t) = −x(t), x(0) = x0
O. Kaleva / Fuzzy Sets and Systems 209 (2012) 104 – 110
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we get the same iterates as above. Hence et x0 , t ⱖ 0, solves also this problem. The same solution is obtained by level set method for solving fuzzy differential equations, see Lakshmikantham and Mohapatra [7]. Finally we show that the fuzzy exponential function is a generalization of the fuzzy semigroup introduced in [3]. Theorem 5.1. If A : F → F be a bounded linear operator then the fuzzy exponential function has a power series representation et A (x) =
∞ k t k A x, t ⱖ 0. k! k=0
Proof. Let A : F → F be a bounded linear operator as defined by Gal and Gal in [3]. Then (r ) =
sup
D(x,y) ⱕ r
D(Ax, Ay) = r A
and hence A satisfies the Osgood condition. Consequently tA n tA (x0 ) e (x0 ) = lim i + n→∞ n is a solution to the Cauchy problem x (t) = Ax(t), x(0) = x0 . Define S(t) by a power series as S(t) =
∞ k t k A . k! k=0
Now by Theorem 3.9 in [3] S(t) is a fuzzy semigroup and hence by Theorem 4.2 S(t)(x0 ) is a solution to the problem x (t) = Ax(t), x(0) = x0 . Since a bounded linear operator is Lipschitzian it follows by Theorem 6.1 in [4] that the problem x (t) = Ax(t), x(0) = x0 , has a unique solution. Hence et A (x0 ) = S(t)(x0 ) for all x0 ∈ F. 6. Conclusions We have introduced a fuzzy exponential function generated by a fuzzy-valued function f. By embedding the fuzzy numbers into a Banach space we are able to show that the iterates (i + f /n)n (x) converge for all x ∈ F. The limit is denoted by e f (x). In addition we show that et f (x0 ) is a fuzzy semigroup and consequently a solution to the fuzzy initial value problem x (t) = f (x(t)), x(0) = x0 . For further study we propose the following topic. Let t > 0, m ∈ N and t = t/m. Suppose we have a good approximation of et f for small t. Then by the semigroup property we have et f (x0 ) = (et f )m (x0 ), which gives us a numerical method for solving a fuzzy Cauchy problem. References [1] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000. [2] H. Frankowska, Local controllability and infinitesimal generators of semigroups of set-valued maps, SIAM J. Control Optim. 25 (1987) 412–432. [3] C.S. Gal, S.G. Gal, Semigroups of operators os spaces of fuzzy-number-valued functions with applications to fuzzy differential equations, J. Fuzzy Math. 13 (2005) 647–682. [4] O. Kaleva, Fuzzy differential equations, Fuzzy Set Syst. 24 (1987) 301–317. [5] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Set Syst. 35 (1990) 366–389. [6] O. Kaleva, V. Seppälä, On exponential function as the unique solution of autonomous differential equations in Banach spaces, Rend. Mat. 4 1 (Serie VII) (1982) 647–658. [7] V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, 2003. [9] I. Miyadera, Nonlinear Semigroups, American Mathematical Society, 1992. [10] S. Nanda, On integration of fuzzy mappings, Fuzzy Set. Syst. 32 (1989) 95–101. [11] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558.
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[12] V. Seppälä, A Banach space exponential function concept, Rend. Mat. 2 4 (Serie VI) (1971) 191–212. [13] V. Seppälä, Solution of a family of initial condition problems in terms of the Banach space exponential function, Rend. Mat. 2 5 (Serie VI) (1972) 329–347. [14] A. Smajdor, Hukuhara’s derivative and concave iteration semigroups of linear set-valued functions, J. Appl. Anal. 8 (2002) 297–305. [15] A. Smajdor, Hukuhara’s differentiable iteration semigroups of linear set-valued functions, Ann. Polon. Math. 83 (2004) 1–10. [16] P.R. Wolenski, The exponential formula for the reachable set of a Lipschitz differentiable inclusion, SIAM J. Control Optim. 28 (1990) 1148–1161. [17] C. Wu, Z. Gong, On Henstock integral of fuzzy-number-valued functions, Fuzzy Set Syst. 120 (2001) 523–532.
Fuzzy Sets and Systems 209 (2012) 104 – 110 www.elsevier.com/locate/fss
Nonlinear iteration semigroups of fuzzy Cauchy problems Osmo Kaleva Tampere University of Technology, Department of Mathematics, P.O. Box 553, FI-33101 Tampere, Finland Received 15 August 2011; received in revised form 28 April 2012; accepted 28 April 2012 Available online 8 May 2012
Abstract In this paper we introduce an iteration semigroup of a nonlinear fuzzy-valued function. We show that the iterates converge and the limit function, called the fuzzy exponential function, is a solution to an autonomous fuzzy Cauchy problem. © 2012 Elsevier B.V. All rights reserved. Keywords: Fuzzy derivative; Fuzzy differential equations; Fuzzy exponential function; Fuzzy semigroup; Iteration of fuzzy-valued function
1. Introduction The theory of transformation semigroups, as well as linear [1] and nonlinear [9], is a well-studied topic in functional analysis. The semigroup generated by a set-valued function was developed among others by Wolenski [16], Smajdor [14,15] and Frankowska [2]. The semigroup generated by linear operators of a fuzzy-valued function was introduced by Gal and Gal [3]. Here we introduce a nonlinear semigroup generated by a nonlinear function f : F → F, where F denotes a space of fuzzy sets. Then we show that this semigroup is a solution to an autonomous fuzzy Cauchy problem x (t) = f (x(t)), x(0) = x0 . We also show that if f : F → F is a bounded linear operator, then the semigroup generated by f equals to the fuzzy semigroup introduced by Gal and Gal in [3]. The main result of the paper is to show that the iterates f n i+ (x) n converge for all x ∈ F under some assumptions on f. Here i : F → F denotes the identity function of F and (i + f /n)n is the n-fold composition of i + f /n. For obvious reasons we call the limit the fuzzy exponential function, denoted by e f (x). We show that T (t)(x0 ) = t f e (x0 ), t ⱖ 0, is a solution to the Cauchy problem. The paper is organized as follows. Section 2 is devoted to some preliminaries. In Section 3 we show the convergence of the iterates. Next we give some relationships between a fuzzy semigroup and a solution to an autonomous Cauchy problem. Finally in Section 5 we show that the fuzzy exponential function is a solution of the problem.
E-mail address: [email protected] 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.04.016
O. Kaleva / Fuzzy Sets and Systems 209 (2012) 104 – 110
105
2. Preliminaries Denote F = {u : R → [0, 1]|u satisfies (i)–(iv) below}, where (i) (ii) (iii) (iv)
u is normal, i.e. there exists an x0 ∈ R such that u(x0 ) = 1, u is fuzzy convex, u is upper semicontinuous, cl{x ∈ R|u(x) > 0}, denoted by [u]0 , is compact.
For 0 < ⱕ 1 let [u] = {x ∈ R|u(x) ⱖ }. Then from (i) to (iv) it follows that a fuzzy set u belongs to F if and only if the -level set [u] is a non-empty compact interval for all 0 ⱕ ⱕ 1. In addition we define a metric in F by the equation D(u, v) =
sup d([u] , [v] ),
0ⱕⱕ1
where d is the Hausdorff metric for non-empty compact sets in R. The metric has the following properties: D(u + v, u + w) = D(v, w) and D(u, v) = ||D(u, v) for all u, v, w ∈ F and ∈ R. For the proof see for example Kaleva [4]. As obtained by Puri and Ralescu [11] (F, D) is a complete metric space. Let u, v ∈ F. If there is such a w ∈ F that u = v + w, then w is the Hukuhara difference of u and v denoted by w = u H v. Using this difference we define a differentiability of a fuzzy function as follows. Let a > 0 and denote T = [t0 , t0 + a]. Definition 2.1. A mapping g : T → F is differentiable at t ∈ T if there exists a g (t) ∈ F such that the limits lim
h→0+
g(t + h) H g(t) and h
lim
h→0+
g(t) H g(t − h) h
exist and equal to g (t). Here the limit is taken in the metric space (F, D). At the end points of T we consider only the one-sided derivatives. In the sequel we need the following generalization of the classical Rådström embedding theorem. The proof of the theorem is given by Kaleva [5]. Theorem 2.1 (Embedding theorem). There exists a real Banach space X such that F can be embedded as a convex cone C with vertex 0 in X. Furthermore the following conditions hold true: (i) (ii) (iii) (iv) (v)
the embedding j is isometric, the addition in X induces the addition in F, the multiplication by a nonnegative real number in X induces the corresponding operation in F, C − C = {a − b|a, b ∈ C} is dense in X, C is closed.
3. Fuzzy exponential function Let f : F → F, i : F → F the identity mapping, x ∈ F and j : F → C the embedding given in the embedding Theorem 2.1. First we show that embedding the iterates (i + f /n)n (x) into C gives us iterates in a Banach space. Then we show that these iterates converge and hence the original iterates converge. Theorem 3.1. For all k ⱖ 1 we have f k f1 k j i+ (x) = I + ( j(x)), n n where I is the identity mapping of X and f 1 = j f j −1 : C → C.
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O. Kaleva / Fuzzy Sets and Systems 209 (2012) 104 – 110
Proof. We prove the theorem by induction. Now let k = 1. By the properties of j we have f j( f (x)) j( f ( j −1 ( j(x)))) f1 j i+ (x) = j(x) + = j(x) + = I+ ( j(x)). n n n n Now assume that the identity holds true for some k. Then f f f1 k f k f k+1 i+ j i+ (x) = I + (x) (x) = j i+ j i+ n n n n n f1 k f1 f 1 k+1 = I+ ( j(x)), I+ ( j(x)) = I + n n n which proves the theorem. In the sequel we assume that f satisfies the following Osgood conditions: 1. the real function (r ) = sup D(x,y) ⱕ r D( f (x), f (y)) < ∞ for all r ⱖ 0, r 2. the integral r12 (1/(t)) dt exists for all 0 < r1 < r2 < ∞, ∞ r 3. the integrals 0 (1/(t)) dt and r (1/(t)) dt diverge for all 0 < r < ∞. 1 1 Clearly is increasing and (0) = 0. Since r (1/(t)) dt < ∞ and 0 (1/(t)) dt diverges, it follows that limr →0 (r ) = 0. Hence, by the definition of , we see that f is uniformly continuous. Since j is isometry then also j −1 is isometry and hence f 1 (u) − f 1 (v) = D( f ( j −1 (u)), f ( j −1 (v)) and D( j −1 (u), j −1 (v)) = u − v. It follows that sup
D(x,y) ⱕ r
D( f (x), f (y)) =
sup
u−v ⱕ r
f 1 (u) − f 1 (v))
and consequently f 1 also satisfies the Osgood conditions. Theorem 3.2. If f : F → F satisfies the Osgood conditions then the iterates (i + f /n)n (x) converge for all x ∈ F. Proof. By Lemma 1 in Seppälä [12] the iterates (I + f 1 /n)n ( j(x)) ∈ C converge and by Theorem 3.1 the iterates j((i + f /n)n (x)) converge. Since j −1 is continuous then f n f n f n j −1 lim j i+ i+ (x) = lim j −1 j (x) = lim i + (x) n→∞ n→∞ n→∞ n n n exists for all x ∈ F. Definition 3.1. Let f : F → F satisfy the Osgood conditions. The fuzzy exponential function e f is defined by f n e f (x) = lim i + (x). n→∞ n From Theorem 3.1 it follows that j(e f (x)) = e f 1 ( j(x)). 4. Fuzzy semigroups related to autonomous Cauchy problems Definition 4.1. A family of functions {T (t)}t ⱖ 0 , with T (t) : F → F, is a (one-parameter, strongly continuous, nonlinear) fuzzy semigroup if (i) T (t + s) = T (t)T (s) for all t, s ⱖ 0,
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107
(ii) T (0) = i, the identity mapping on F, (iii) the function g : [0, ∞) → F, defined by g(t) = T (t)(x), is continuous for all x ∈ F. Let f : F → F be continuous and consider the autonomous initial value problem x (t) = f (x(t)), x(t0 ) = x0 .
(1)
It is well known, see for example Kaleva [4], that instead of the differential equation (1) it is possible to study an equivalent integral equation. t x(t) = x0 + f (x(s)) ds. t0
A solution x(t) of (1) is independent of the initial time t0 . In fact, let s0 < a and denote y(t) = x(s0 + t). Then y (t) = x (s0 + t) = f (x(s0 + t)) = f (y(t)) and y(t0 ) = x(t0 + s0 ) = y0 . Hence y(t) and x(t) are solutions of the same differential equation with a different initial value. Theorem 4.1. If x(t) is a solution to the fuzzy initial value problem x (t) = f (x(t)), x(0) = x0 . Then T (t)(x0 ) = x(t) is a fuzzy semigroup. Furthermore T (t)(x0 ) is differentiable w.r.t. t and T (t)(x0 ) = f (x(t)) = f (T (t)(x0 )). Proof. Now let s > 0 be fixed. As obtained above y(t) = x(t + s) is a solution of the initial value problem y (t) = f (y(t)), y(0) = x(s). Hence T (t + s)(x0 ) = x(t + s) = y(t) = T (t)(x(s)) = T (t)T (s)(x0 ) and T (0)(x0 ) = x(0) = x0 . Being a solution to a differential equation T (t)(x0 ) is differentiable w.r.t. t and T (t)(x0 ) = x (t) = f (x(t)). Theorem 4.2. Suppose that a fuzzy semigroup T (t)(x) is differentiable w.r.t. t for all x ∈ F. Then T (t)(x0 ) is a solution to the initial value problem x (t) = f (x(t)), x(0) = x0 ,
(2)
where f (x) = T (0)(x). Proof. By the semigroup property we obtain T (t)(x0 ) = lim
h→0+
T (t + h)(x0 ) H T (t)(x0 ) T (h)T (t)(x0 ) H T (t)(x0 ) = lim h h h→0+
T (h)T (t)(x0 ) H T (0)T (t)(x0 )) = T (0)T (t)(x0 ) h h→0+ and T (0)(x0 ) = x0 . = lim
Hence every solution to the initial value problem (2) defines a differentiable fuzzy semigroup and conversely a differentiable semigroup is a solution to (2) with generator f (x) = T (0)(x). 5. Fuzzy semigroup defined by fuzzy exponential function Let f : F → F satisfy the Osgood conditions and t > 0. Then t f : F → F also satisfies the Osgood conditions. Define T (t)(x0 ) = et f (x0 ), where x0 ∈ F. Then T (0)(x0 ) = x0 . The mapping t f 1 , t ⱖ 0, can be seen as a function of two arguments F(t, x), which is linear in the first argument. Now by Seppälä [13] the iterates (I + F(t, ·)/n)n ( j(x)) converge uniformly in Bt × B j(x) , where Bt is an arbitrary open ball in R+ and B j(x) is a suitably chosen open ball centered at j(x). It follows that et f1 ( j(x)) and consequently et f (x) = T (t)(x) is continuous in t for all x ∈ F. Since f (et f (x0 )) is continuous, it is Riemann as well as Aumann integrable and the integrals coincide. For the Riemann integral of fuzzy-valued function see [10,17].
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O. Kaleva / Fuzzy Sets and Systems 209 (2012) 104 – 110
For the Riemann integral we have t tf e (x0 ) = x0 + f (es f (x0 )) ds. 0
Let 0 ⱕ k ⱕ n and denote Onk (t, x) = (i + t f /n)k (x). Then et f (x) = lim Onn (t, x),
On0 (t, x) = x,
Onk+1 (t, x) = Onk (t, x) +
t f (Onk (t, x)). n
n→∞
Onk (t, x) = Okk
k t, x n
and
We immediately obtain et f (x) = lim Onn (t, x) = lim On0 (t, x) + n→∞
n→∞
t n→∞ n
= x + lim
n−1
n−1 t f (Onk (t, x)) n k=0
n−1 t k t, x . f Okk n→∞ n n
f (Onk (t, x)) = x + lim
k=0
k=0
Approximating the integral by a Riemann sum we have x+
t
n−1 t f (e(k/n)t f (x)). n→∞ n
f (es f (x)) ds = x + lim
0
k=0
Since D(u +w, v +w) = D(u, v) for all u, v, w ∈ F, then using the triangle inequality it is easy to prove the inequality n n n uk , vk ⱕ D(u k , vk ). D k=0
k=0
k=0
It follows that D et f (x), x +
t
f (es f (x)) ds
0
n−1 t k t, x , f (e(k/n)t f (x)) . D f Okk n→∞ n n
ⱕ lim
k=1
Proceeding exactly as in [6], when Banach space distances z − w are replaced by distances D(z, w), we see that the limit of the right hand side equals to zero, which gives us the desired result. t In conclusion T (t)(x0 ) is a continuous solution of the integral equation x(t) = x0 + 0 f (x(s)) ds and consequently a solution of the initial value problem (2). Now by Theorem 4.1 T (t)(x0 ) is a fuzzy semigroup. Example 5.1. Consider the initial value problem x (t) = x(t), x(0) = x0 ∈ F. Now for t ⱖ 0 we have (i + t f /n)(x0 ) = x0 + (t/n)x0 = (1 + t/n)x0 . Hence tf n t n i+ (x0 ) = 1 + x0 −→ et x0 , n→∞ n n which is a solution to the Cauchy problem. If x0 is an even function then −x0 = x0 . It follows that for the problem x (t) = −x(t), x(0) = x0
O. Kaleva / Fuzzy Sets and Systems 209 (2012) 104 – 110
109
we get the same iterates as above. Hence et x0 , t ⱖ 0, solves also this problem. The same solution is obtained by level set method for solving fuzzy differential equations, see Lakshmikantham and Mohapatra [7]. Finally we show that the fuzzy exponential function is a generalization of the fuzzy semigroup introduced in [3]. Theorem 5.1. If A : F → F be a bounded linear operator then the fuzzy exponential function has a power series representation et A (x) =
∞ k t k A x, t ⱖ 0. k! k=0
Proof. Let A : F → F be a bounded linear operator as defined by Gal and Gal in [3]. Then (r ) =
sup
D(x,y) ⱕ r
D(Ax, Ay) = r A
and hence A satisfies the Osgood condition. Consequently tA n tA (x0 ) e (x0 ) = lim i + n→∞ n is a solution to the Cauchy problem x (t) = Ax(t), x(0) = x0 . Define S(t) by a power series as S(t) =
∞ k t k A . k! k=0
Now by Theorem 3.9 in [3] S(t) is a fuzzy semigroup and hence by Theorem 4.2 S(t)(x0 ) is a solution to the problem x (t) = Ax(t), x(0) = x0 . Since a bounded linear operator is Lipschitzian it follows by Theorem 6.1 in [4] that the problem x (t) = Ax(t), x(0) = x0 , has a unique solution. Hence et A (x0 ) = S(t)(x0 ) for all x0 ∈ F. 6. Conclusions We have introduced a fuzzy exponential function generated by a fuzzy-valued function f. By embedding the fuzzy numbers into a Banach space we are able to show that the iterates (i + f /n)n (x) converge for all x ∈ F. The limit is denoted by e f (x). In addition we show that et f (x0 ) is a fuzzy semigroup and consequently a solution to the fuzzy initial value problem x (t) = f (x(t)), x(0) = x0 . For further study we propose the following topic. Let t > 0, m ∈ N and t = t/m. Suppose we have a good approximation of et f for small t. Then by the semigroup property we have et f (x0 ) = (et f )m (x0 ), which gives us a numerical method for solving a fuzzy Cauchy problem. References [1] K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000. [2] H. Frankowska, Local controllability and infinitesimal generators of semigroups of set-valued maps, SIAM J. Control Optim. 25 (1987) 412–432. [3] C.S. Gal, S.G. Gal, Semigroups of operators os spaces of fuzzy-number-valued functions with applications to fuzzy differential equations, J. Fuzzy Math. 13 (2005) 647–682. [4] O. Kaleva, Fuzzy differential equations, Fuzzy Set Syst. 24 (1987) 301–317. [5] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Set Syst. 35 (1990) 366–389. [6] O. Kaleva, V. Seppälä, On exponential function as the unique solution of autonomous differential equations in Banach spaces, Rend. Mat. 4 1 (Serie VII) (1982) 647–658. [7] V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, 2003. [9] I. Miyadera, Nonlinear Semigroups, American Mathematical Society, 1992. [10] S. Nanda, On integration of fuzzy mappings, Fuzzy Set. Syst. 32 (1989) 95–101. [11] M.L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558.
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[12] V. Seppälä, A Banach space exponential function concept, Rend. Mat. 2 4 (Serie VI) (1971) 191–212. [13] V. Seppälä, Solution of a family of initial condition problems in terms of the Banach space exponential function, Rend. Mat. 2 5 (Serie VI) (1972) 329–347. [14] A. Smajdor, Hukuhara’s derivative and concave iteration semigroups of linear set-valued functions, J. Appl. Anal. 8 (2002) 297–305. [15] A. Smajdor, Hukuhara’s differentiable iteration semigroups of linear set-valued functions, Ann. Polon. Math. 83 (2004) 1–10. [16] P.R. Wolenski, The exponential formula for the reachable set of a Lipschitz differentiable inclusion, SIAM J. Control Optim. 28 (1990) 1148–1161. [17] C. Wu, Z. Gong, On Henstock integral of fuzzy-number-valued functions, Fuzzy Set Syst. 120 (2001) 523–532.