Existence theorems for solutions to random fuzzy differential equations

Nonlinear Analysis 73 (2010) 1515–1532

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Existence theorems for solutions to random fuzzy differential equations Marek T. Malinowski Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

article

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Article history: Received 27 May 2009 Accepted 24 April 2010 MSC: 34F05 34G20 34A12 26E50

abstract We examine a random fuzzy initial value problem with two kinds of fuzzy derivatives. For both cases we establish the results of existence and uniqueness of local solutions to random fuzzy differential equations. The existence of global solutions is also obtained. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Random fuzzy differential equations Initial value problem for fuzzy mappings Fuzzy derivative Fuzzy stochastic process Fuzzy random variable

1. Introduction Fuzzy differential equations (FDEs) is a topic very important from the theoretical point of view (see e.g. [1,2] and the references therein) as well as of their applications, for example, in civil engineering [3], in modeling hydraulic [4] and in population models [5]. There are many suggestions on how to define a fuzzy derivative and consequently several ways to study FDEs. Various types of derivatives of fuzzy functions were compared, and the solutions of FDEs related to them were investigated in [6]. Fuzzy differential equations were first formulated by Kaleva [7,8]. He used the concept of H-differentiability which was introduced by Puri and Ralescu [9], and obtained the existence and uniqueness theorem for a solution of FDE under the Lipschitz condition, whereas in [8] he characterized those subsets of the fuzzy set space in which the Peano theorem is valid. Since then there appeared a lot of papers concerning the theory and applications of fuzzy differential equations, fuzzy dynamics and fuzzy differential inclusions (see e.g. [10,11,5,12–30]). For a significant collection of results from the theory of FDEs we refer to the monograph of Lakshmikantham and Mohapatra [1], and to Diamond and Kloeden [2]. In this paper we will consider random fuzzy differential equations (RFDEs) as they can provide good models of dynamics of real phenomena which are subjected to two kinds of uncertainties: randomness and fuzziness, simultaneously. The first source of uncertainty is connected with the uncertainty in prediction of the outcome of an experiment. Randomness intends to break the law of causality and the probabilistic methods are applied in its analysis. Fuzziness means nonstatistical inexactness that is due to subjectivity and imprecision of human knowledge rather than to the occurrence of random events. It is caused by the lack of sharply defined criteria of membership in the sets of some considered space (a simple example could be a class of all real numbers which are much greater than 1). Fuzziness intends to break the law of excluded middle and is appropriately treated by fuzzy set theory. The probability and fuzzy set theories team up in the concept of fuzzy random variable. This notion is a crucial one in the analysis of RFDEs.

E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.04.049

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In the literature one can find various definitions of fuzzy random variables. For the first time the concept of fuzzy random variable was proposed by Kwakernaak [31]. Further it was used by Kruse and Meyer [32]. In [33–35,2] there appear two notions of measurability of fuzzy mappings. The relations between different concepts of measurability for fuzzy random variables are contained in the papers of Colubi et al. [36], Terán Agraz [37], López-Diaz and Ralescu [38]. In this paper we will use a definition of fuzzy random variable which was introduced by Puri and Ralescu [35]. This definition is currently the most often used in probabilistic and statistical aspects of the theory of fuzzy random variables. In papers [39–41] one can find the studies of differential equations where the two kinds of uncertainties are incorporated. Feng [40] considered fuzzy stochastic differential systems using a notion of mean-square derivative (which is different from our derivative) and mean-square integral of second-order fuzzy stochastic processes introduced by himself in [42]. The fuzzy stochastic process is viewed there as a mapping acting from an interval to the space of second-order fuzzy random variables. In his setting, the fuzzy random variable comes from a narrower class than ours. The existence and uniqueness of a Cauchy problem is then obtained under an assumption that the coefficients satisfy a condition with the Lipschitz constant. The proof is based on the application of the Banach fixed point theorem. In [39] the existence and uniqueness of the solution for RFDEs with non-Lipschitz coefficients is proven. The values of fuzzy mappings are in the space of fuzzy sets of a reflexive separable Banach space. However only the autonomous case is treated, where right-hand side is non-random and initial value is a constant fuzzy set. The behaviour of solutions to the Cauchy problem such as existence, uniqueness, lifetime, dependence on initial values and non-confluence is studied. The author uses a concept of the support function of a fuzzy set which allows him to consider the random scalar differential equations instead of RFDEs. The main idea in the proofs is to construct a family of positive increasing functions so that the Gronwall lemma can be applied to the composition of these functions with appropriate processes. Malinowski [41] considered RFDEs with fuzzy derivative defined as in [9]. The coefficients of the equation were random fuzzy functions, also the initial condition was treated as a fuzzy random variable. With an assumption that the right-hand side of the equation satisfies a global Lipschitz-type condition the existence and uniqueness of the solution to RFDEs was proven. Random fuzzy initial value problem can be viewed as non-random one but with a parameter. However the rich theory of non-random FDEs cannot be straightforward applied. As the examples show (see [41]), the existence of solution to the deterministic version of random fuzzy initial value problem does not determine the existence of a random solution. A solution to RFDE is a fuzzy stochastic process, i.e. a family of fuzzy random variables. Therefore in the proving of the existence of a solution we apply the method of successive approximations as opposed to the non-random case where the method of fixed point is frequently used. We examine RFDEs with two kinds of fuzzy derivatives and obtain parallel results for both settings. Supposing that the Lipschitz condition holds on bounded sets, we establish the existence and uniqueness of a local solution to RFDEs. The existence of at least one local solution is obtained under assumption that the right-hand side of the equation satisfies some integrability condition. This result is then applied in a demonstration of existence of a global solution to RFDEs. The paper is organized as follows. In Section 2 we collect the fundamental notions and facts which will be used in the rest of the article, the formulation of the main problem is also contained. In Section 3 we discuss RFDEs where the fuzzy derivative is understood in the sense of Puri and Ralescu [9]. The theorems on existence of local and global solutions are presented. Section 4 is a parallel one to the Section 3. We consider there RFDEs with second type of fuzzy derivative which was proposed in [43,13]. Theorems similar to those in Section 3 are stated. In Section 5 we present some examples which illustrate the theory of RFDEs. 2. Preliminaries In this section our aim is to give a background of the fuzzy set space, and an overview of properties used by us of integration and differentiation of fuzzy set-valued mappings. Let A, B be nonempty compact subsets of Rd . The Hausdorff metric is defined as follows dH (A, B) = max d∗H (A, B), d∗H (B, A) ,





where d∗H (A, B) = supx∈A infy∈B kx − yk, and k · k denotes usual Euclidean norm in Rd . We have d∗H (A, B) = 0 if and only if A ⊂ B, and d∗H (A, B) ≤ d∗H (A, C ) + d∗H (C , B) for nonempty compact subsets A, B, C of Rd . Let K (Rd ) denote a family of all nonempty compact convex subsets of Rd and define addition and scalar multiplication in K (Rd ) as usual, i.e. for A, B ∈ K (Rd ) and λ ∈ R A + B = {a + b | a ∈ A, b ∈ B},

λA = {λa | a ∈ A}.

Denote E d = {u: Rd → [0, 1] | u satisfies (i)–(iv) below}, (i) (ii) (iii) (iv)

u is normal, i.e. there exists x0 ∈ Rd such that u(x0 ) = 1, u is fuzzy convex, i.e. u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ Rd and λ ∈ [0, 1], u is upper semicontinuous, [u]0 = cl{x ∈ Rd | u(x) > 0} is compact, where cl denotes the closure in (Rd , k · k).

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

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For α ∈ (0, 1], denote [u]α = {x ∈ Rd | u(x) ≥ α}. We will call this set an α -cut (α -level set) of u. For u ∈ E d one has that [u]α ∈ K (Rd ) for every α ∈ [0, 1]. If g: Rd × Rd → Rd is a function then according to Zadeh’s extension principle we can extend g to E d × E d → E d by the formula g (u, v)(z ) = sup min{u(x), v(y)}. z =g (x,y)

It is well known that if g is continuous then [g (u, v)]α = g ([u]α , [v]α ) for all u, v ∈ E d , α ∈ [0, 1]. Especially for addition and scalar multiplication in fuzzy number space E d we have:

[u + v]α = [u]α + [v]α ,

[λu]α = λ[u]α ,

where u, v ∈ E , λ ∈ R and α ∈ [0, 1]. Define D: E d × E d → [0, ∞) by the expression d

D(u, v) = sup dH ([u]α , [v]α ), 0≤α≤1

where dH is the Hausdorff metric in K (Rd ). It is easy to see that D is a metric in E d . In fact (E d , D) is a complete metric space, and for every u, v, w, z ∈ E d , λ ∈ R one has D(u + w, v + w) = D(u, v), D(u + v, w + z ) ≤ D(u, w) + D(v, z ), D(λu, λv) = |λ|D(u, v) (see e.g. [35]). We define θˆ ∈ E d as θˆ = χ{0} , where for x ∈ Rd we have χ{x} (y) = 1 if y = x and χ{x} (y) = 0 if y 6= x. Let [a, b] ⊂ R be a compact interval, −∞ < a, b < +∞. A fuzzy valued mapping F : [a, b] → E d is called strongly measurable if for all α ∈ [0, 1] the set-valued mapping [F (·)]α : [a, b] → K (Rd ) is measurable, i.e. the set

{t ∈ [a, b] | [F (t )]α ∩ C 6= ∅} for each closed set C ⊂ Rd is Lebesgue measurable. A fuzzy mapping F : [a, b] → E d is called integrably bounded if there exists an integrable function h: [a, b] → R such that kxk ≤ h(t ) for all x ∈ [F (t )]0 . Definition 1 (Puri and Ralescu [35]). Let F : [a, b] → E d . The integral of F over [a, b], denoted by levelwise by the expression b

Z

F (t )dt

α

b

Z =

a

Rb a

F (t )dt, is defined

[F (t )]α dt

a b

Z =



f (t )dt | f : [a, b] → Rd is a measurable selection for [F (·)]α ,

a

for all α ∈ (0, 1]. By virtue of Remark 4.1 in [7] we have that

hR b a

F (t )dt

i0

=

Rb a

[F (t )]0 dt.

A strongly measurable and integrably bounded mapping F : [a, b] → E d is said to be integrable over [a, b] if We recall (see [7]) some properties of integrability for fuzzy mappings.

Rb a

F (t )dt ∈ E d .

(I1) Let FR, G: [a, b] → E d
be integrable and Rλ ∈ R. Then Rb b b (i) a F (t ) + G(t ) dt = a F (t )dt + a G(t )dt,

Rb

Rb

(ii) a λF (t )dt = λ a F (t )dt, (iii) D(F , G) is integrable,  (iv) D

Rb a

F (t )dt ,

Rb a

G(t )dt

≤

Rb a

D F (t ), G(t ) dt.




(I2) If F : [a, b] → E d is continuous then it is integrable. Rb Rc Rb (I3) If F : [a, b] → E d is integrable and c ∈ [a, b] then a F (t )dt = a F (t )dt + c F (t )dt. Let u, v ∈ E d . If there exists w ∈ E d such that u = v + w then we call w the H-difference of u and v and we denote it by u v . Note that u v 6= u + (−1)v . Definition 2 (Puri and Ralescu [9]). A mapping F : [a, b] → E d is differentiable at t0 ∈ [a, b] if there exists F 0 (t0 ) ∈ E d such that the limits lim

h→0+

1 h

(F (t0 + h) F (t0 )) ,

lim

h→0+

1 h

(F (t0 ) F (t0 − h))

exist and are equal to F 0 (t0 ). The limits are taken in the metric space (E d , D), and at the boundary points we consider only the one-sided derivatives. All the following results are due to Kaleva [7]. (D1) If F : [a, b] → E d is differentiable then it is continuous.

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M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

(D2) If F , G: [a, b] → E d are differentiable and λ ∈ R then

(F + G)0 (t ) = F 0 (t ) + G0 (t ),

(λF )0 (t ) = λF 0 (t ).

(D3) If F : [a, b] → E d is continuous then for every t ∈ [a, b] the mapping t 7→

Rt a

F (s)ds is differentiable and

R t a

F (s)ds

0

=

F (t ). (D4) Let F : [a, b] →R E d is differentiable and assume that F 0 is integrable over [a, b]. Then for every t ∈ [a, b] we have t F (t ) = F (a) + a F 0 (s)ds. Let (Ω , F , P) be a complete probability space. A function x0 : Ω → E d is called fuzzy random variable, if for all α ∈ [0, 1] the set-valued mapping [x0 (·)]α : Ω → K (Rd ) is a measurable multifunction, i.e.

{ω ∈ Ω | [x0 (ω)]α ∩ C 6= ∅} ∈ F

for every closed set C ⊂ Rd .

For set-valued analysis we refer to [44–46]. A mapping x: [a, ∞) × Ω → E d is said to be a fuzzy stochastic process if x(·, ω) is a fuzzy set-valued function with any fixed ω ∈ Ω (this function will be called a trajectory), and x(t , ·) is a fuzzy random variable for any fixed t ∈ [a, ∞). A fuzzy stochastic process x: [a, ∞) × Ω → E d is called continuous if for almost all ω ∈ Ω the trajectory x(·, ω) is a continuous function on [a, ∞) with respect to the metric D. Assume that f : Ω × [a, ∞) × E d → E d satisfies: (f1) f· (t , u): Ω → E d is a fuzzy random variable for every t ∈ [a, ∞) and every u ∈ E d , (f2) with P.1 the mapping fω (·, ·): [a, ∞) × E d → E d is a continuous fuzzy mapping at every point (t0 , u0 ) ∈ [a, ∞) × E d , i.e. there exists Ω0 ∈ F with P(Ω0 ) = 1 such that for every ω ∈ Ω0 the following is true: for every ε > 0 there exists δ > 0 such that for every t ∈ [a, ∞), every u ∈ E d it holds max {|t − t0 |, D(u, u0 )} < δ H⇒ D fω (t , u), fω (t0 , u0 ) < ε,




(f3) there exists τ ∈ (a, ∞) for which the following two conditions hold: - for every r > 0 there exists a constant Lτ ,r > 0 such that with P.1 D fω (t , u), fω (t , v) ≤ Lτ ,r D(u, v) ∀t ∈ [a, τ ] ∀u, v ∈ Br ,




where Br = {u ∈ E d | D(u, θˆ ) ≤ r }, - there exists a constant Sτ > 0 such that with P.1 D fω (t , θˆ ), θˆ ≤ Sτ




∀t ∈ [a, τ ].

A mapping f : Ω × [a, ∞) × E d → E d satisfying the first condition in (f3) we will call Lipschitzian on bounded sets. For convenience, from now on, the fact that with P.1 it holds x(ω) = y(ω), where x, y are random elements, we will P.1

often write as x(ω) = y(ω), and similarly for the inequalities. Also if we will have: with P.1 it holds x(t , ω) = y(t , ω) for A P.1

every t ∈ A ⊂ [a, ∞), where x, y are the stochastic processes, then we will write x(t , ω) = y(t , ω) in short, and similarly for the inequalities. In this paper we will consider a fuzzy initial value problem of the form:

(

x0 (t , ω)

[a,∞) P.1

=

P.1

fω (t , x(t , ω))

(1)

x(a, ω) = x0 (ω).

Definition 3. By a local solution to the RFDE (1) on the interval [a, b] ⊂ [a, ∞) we mean a continuous fuzzy stochastic process x: [a, b] × Ω → E d that satisfies x0 (t , ω) is unique if it holds

[a,b] P.1

=

P.1

fω (t , x(t , ω)), and x(a, ω) = x0 (ω). A local solution x to (1) on [a, b]


[a,b] P.1 = 0,

D x(t , ω), y(t , ω)

for any fuzzy stochastic process y: [a, b] × Ω → E d that is a local solution to (1) on [a, b]. The local solution of the RFDE (1) on the interval [a, b] can be viewed as a continuous fuzzy stochastic process x: [a, b] × Ω → E d whose trajectories are almost all (with respect to the measure P) the deterministic solutions of nonrandom (with fixed ω ∈ Ω ) fuzzy initial value problems x0 (t ) = fω (t , x(t )), x(a) = x0 (ω).



t ∈ [a, b],

However solving these problems separately for every fixed ω ∈ Ω does not lead to the solution of (1) in general. There are examples (see [41]) showing that a common interval (independent of ω) might not exist on which the deterministic solutions are defined for almost all ω ∈ Ω , or even if such a common interval does exist it might happen that the deterministic solutions are not the trajectories of any fuzzy stochastic process.

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

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Definition 4. A continuous fuzzy stochastic process x: [a, ∞) × Ω → E d that satisfies conditions in (1) is called a global solution to (1). 3. Existence of solutions to RFDEs with H -derivative In this section we will consider problem (1) in which the symbol 0 denotes the derivative for fuzzy mappings described in Definition 2. We begin with a very useful assertion about the equivalence between solutions of differential and integral equations. The theorems (I2), (D2), (D3) and (D4) are applied in its proof. Lemma 1. A fuzzy stochastic process x is a local solution to the RFDE (1) on the interval [a, b] if and only if x is a continuous fuzzy stochastic process and it satisfies the following random fuzzy integral equation x(t , ω)

[a,b] P.1

=

x0 (ω) +

t

Z

fω (s, x(s, ω))ds.

(2)

a

The following theorem gives a result that assures the uniqueness of the local solution to (1). The Lipschitz condition imposed in theorem is weaker than the global one in [41]. The proof is similar to that in [41] but there are some differences. For completeness we present the proof. Theorem 1. Assume that f : Ω × [0, ∞) × E d → E d satisfies conditions (f1)–(f3), and a fuzzy random variable x0 : Ω → E d is such that D x0 (ω), θˆ

P.1

≤ M, where M is a positive constant. Then there exists a constant T > 0 such that the RFDE (1) has a
P.1 unique local solution x on [0, T ] and supt ∈[0,T ] D x(t , ω), θˆ ≤ 2M.


Proof. We divide the proof into several steps. Step 1. By virtue of assumption (f3) there exists τ > 0 and there exists a constant Lτ ,2M > 0 such that with P.1 it is satisfied D fω (t , u), fω (t , v) ≤ Lτ ,2M D(u, v)




for every t ∈ [0, τ ] and every u, v ∈ B2M .

(3)

Further it follows that there is a constant Wτ ,2M > 0 such that with P.1 it holds D fω (t , u), θˆ ≤ Wτ ,2M with any t ∈ [0, τ ]




and any u ∈ B2M . Indeed, by (3) we have that D(fω (t , u), fω (t , θˆ ))

[0,τ ] P.1

≤

[0,τ ] P.1

2MLτ ,2M . Assumption (f3) implies also that there

exists a constant Sτ > 0 such that D(fω (t , θˆ ), θˆ ) ≤ Sτ . Thus with P.1 for every (t , u) ∈ [0, τ ] × B2M it holds


D fω (t , u), θˆ ≤ D fω (t , u), fω (t , θˆ ) + D fω (t , θˆ ), θˆ ≤ Wτ ,2M ,

(4)

where Wτ ,2M = 2ML nτ ,2M + Sτo. Define T = min τ ,

M Wτ ,2M

.

Let V denote the set of all the fuzzy stochastic processes x: [0, T ] × Ω → E d such that: P.1

(v1) x(0, ω) = x0 (ω),


[0,T ] P.1

(v2) D x(t , ω), θˆ ≤ 2M, (v3) x(ω, ·): [0, T ] → E d is a D-continuous fuzzy set-valued mapping with P.1. n d Step 2. We define a sequence {xn }∞ n=0 , x : [0, T ] × Ω → E of successive approximations as follows:

x0 (t , ω) = x0 (ω) for t ∈ [0, T ], ω ∈ Ω , xn (t , ω) = x0 (ω) +

t

Z

fω (s, xn−1 (s, ω))ds

for t ∈ [0, T ], ω ∈ Ω .

0

Obviously, every approximation xn verifies condition (v1). In the further steps we show that xn ’s satisfy also (v2) and (v3). Step 3. Note that for every t ∈ [0, T ] the functions xn (t , ·): Ω → E d , defined above, are fuzzy random variables. Indeed, since [x0 (·)]α is a measurable multifunction for every α ∈ [0, 1] it remains to show the same for the mapping ω 7→ h i

Rt

f 0 ω

α

(s, xn−1 (s, ω))ds , with α ∈ [0, 1], t ∈ [0, T ], n ∈ N. Let α ∈ [0, 1] be fixed. By virtue of the definition of fuzzy

integral, continuity of f and theorem of Nguyen [47] we obtain

Z 0

t

fω (s, xn−1 (s, ω))ds

α

t

Z =

fω (s, [xn−1 (s, ω)]α )ds.

0

As is a multifunction continuous in s and measurable in ω, with any t ∈ [0, T ], the mapping ω 7→ R t the integrand f (s, [xn−1 (s, ω)]α )ds is a measurable multifunction for n ∈ N. Therefore for every t ∈ [0, T ] the sequence {xn (t , ·)}∞ n=0 is 0 ω a sequence of fuzzy random variables and consequently {xn }∞ is a sequence of fuzzy stochastic processes. n=0

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M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

Step 4. By the definition of x0 we have directly that D x0 (t , ω), θˆ n −1

process x




[0,T ] P.1

≤

M < 2M. Assuming that the fuzzy stochastic

satisfies condition (v2), we have accordingly to (4) that D fω (t , xn−1 (t , ω)), θˆ

D xn (t , ω), θˆ




D x0 (ω), θˆ +




≤


[0,T ] P.1 ≤ Wτ ,2M . Therefore

t

Z

D fω (s, xn−1 (s, ω)), θˆ ds




0

[0,T ] P.1

M + Wτ ,2M T ≤ 2M ,

≤

n

which means that x satisfies (v2) too. Step 5. Observe that for every n ∈ {0, 1, . . .} the functions xn (·, ω): [0, T ] → E d are continuous with P.1. Indeed, x0 (t , ω) does not depend on t, and for the right-sided continuity of xn (·, ω) with n ≥ 1 we notice that for every (t , ω) ∈ [0, T ] × Ω t +h

Z

D xn (t + h, ω), xn (t , ω) = D




fω (s, xn−1 (ω))ds,

t

Z

fω (s, xn−1 (ω))ds



0

0 t +h

Z

fω (s, x

=D

n −1

(ω))ds, θˆ



t t +h

Z

D fω (s, xn−1 (ω)), θˆ ds.




≤ t

Since xn−1 satisfies (v2), using (4) once again we have


[0,T ) P.1 ≤ Wτ ,2M h −→ 0,

D xn (t + h, ω), xn (t , ω)

as h & 0.

Similarly for the left-sided continuity one obtains


(0,T ] P.1 ≤ Wτ ,2M h −→ 0,

D xn (t − h, ω), xn (t , ω)

as h & 0.

Hence the functions x (·, ω) are right-sided and left-sided continuous with P.1. Therefore the fuzzy stochastic processes xn satisfy (v3), n ∈ {0, 1, 2 . . .} and in view of the results obtained in the preceding steps we claim that xn ∈ V for every n ∈ {0, 1, 2 . . .}. n

Step 6. Observe that for every (t , ω) ∈ [0, T ] × Ω it holds D x1 (t , ω), x0 (t , ω) = D




t

Z

fω (s, x0 (ω))ds, θˆ



0 t

Z

D fω (s, x0 (ω)), θˆ ds.




≤ 0

By virtue of (4) we have D fω (t , x0 (ω)), θˆ


[0,T ] P.1 ≤ Wτ ,2M . Hence


[0,T ] P.1 ≤ Wτ ,2M t ≤ Wτ ,2M T .

D x1 (t , ω), x0 (t , ω)

For every n ≥ 2 and every (t , ω) ∈ [0, T ] × Ω we can write D x (t , ω), x n

n −1


(t , ω) ≤

t

Z

D fω (s, xn−1 (s, ω)), fω (s, xn−2 (s, ω)) ds.




0

Since for x

n −1

,x

n−2

D x (t , ω), x n

the condition (v2) holds and the function f satisfies (3), we get n −1


[0,T ] P.1 (t , ω) ≤ Lτ ,2M

t

Z

D xn−1 (s, ω), xn−2 (s, ω) ds.




0

Thus, by mathematical induction, for every n ∈ N we obtain n n
[0,T ] P.1 1 t 1 T ≤ Wτ ,2M Lnτ − ≤ Wτ ,2M Lτn− . ,2M ,2M n! n!

D xn (t , ω), xn−1 (t , ω)

(5)

Step 7. In what follows we shall show that for the sequence {xn (t , ω)}∞ n=0 the Cauchy convergence condition is satisfied uniformly on the variable t with P.1, and as a consequence {xn (·, ω)}∞ n=0 is uniformly convergent with P.1. Denote Fn (t , ω) = D xn+1 (t , ω), xn (t , ω) ,




n ∈ N.

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

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Observe that Fn (t + h, ω) − Fn (t , ω) ≤ D

t +h

Z

fω (s, xn (s, ω))ds,

t +h

Z

fω (s, xn−1 (s, ω))ds

t

t t +h

Z

D fω (s, x (s, ω)), θˆ ds + n

≤






t +h

Z

D fω (s, xn−1 (s, ω)), θˆ ds.




t

t

Because xn ∈ V for every n ∈ N, we have Fn (t + h, ω) − Fn (t , ω)

[0,T ) P.1

2Wτ ,2M h,

≤

where h > 0. Similarly one obtains Fn (t , ω) − Fn (t − h, ω)

(0,T ] P.1

2Wτ ,2M h.

≤

Hence Fn is a continuous function in the variable t with P.1. Notice now that for n > m > 0 using (5) one obtains n −1 X

sup D xn (t , ω), xm (t , ω) ≤




sup Fk (t , ω)

k=m t ∈[0,T ]

t ∈[0,T ]

P.1

≤ Wτ ,2M

n −1 X

Lkτ ,2M

k=m

The convergence of the series n, m ≥ n0

P∞

n =1

T k+1

(k + 1)!

.

1 n Lnτ − ,2M T /n! implies that for any ε > 0 one can find n0 ∈ N, large enough, such that for


P.1

sup D xn (t , ω), xm (t , ω) ≤ ε.

(6)

t ∈[0,T ]

As (E d , D) is a complete metric space and (6) holds, there exists Ωc ∈ F such that P(Ωc ) = 1 and for every ω ∈ Ωc the sequence {xn (·, ω)} is uniformly convergent. For ω ∈ Ωc denote its limit by x˜ (·, ω). Define now x: [0, T ] × Ω → E d in the following way: x(·, ω) = x˜ (·, ω) if ω ∈ Ωc , and x(·, ω) as a freely chosen fuzzy function in the case ω ∈ Ω \ Ωc . Notice that x(·, ω): [0, T ] → E d is D-continuous with P.1. Observing that


n→∞

dH [xn (t , ω)]α , [x(t , ω)]α −→ 0 for every α ∈ [0, 1], t ∈ [a, b] with P.1 we infer that [x(t , ·)]α : Ω → K (Rd ), with any α and t, is a measurable multifunction. Therefore x is a continuous fuzzy stochastic process. As D xn (t , ω), θˆ




[0,T ] P.1

≤ 2M for
[0,T ] P.1 every n ∈ {0, 1, 2, . . .}, we have by triangle inequality that D x(t , ω), θˆ ≤ 2M. Thus x ∈ V . Step 8. We shall show that x is a solution to the random fuzzy integral equation (2) with [a, b] = [0, T ]. For n ∈ {0, 1, . . .} let us denote by yn (t , ω) the expression fω (t , xn (t , ω)), t ∈ [0, T ], ω ∈ Ω . The functions yn (·, ·): [0, T ] × Ω → E d are the
[0,T ] P.1
continuous fuzzy stochastic processes. By (3) for n, m ≥ 0 one has D yn (t , ω), ym (t , ω) ≤ Lτ ,2M D xn (t , ω), xm (t , ω) . Hence


P.1

sup D yn (t , ω), ym (t , ω) ≤ Lτ ,2M sup D xn (t , ω), xm (t , ω) .




t ∈[0,T ]

t ∈[0,T ]

Together with (6) it means that the sequence {yn (·, ω)}∞ n=0 is uniformly convergent with P.1. Note that for any ε > 0 there is n0 , large enough, such that for every n ≥ n0


P.1


P.1

sup D yn (t , ω), fω (t , x(t , ω)) ≤ Lτ ,2M sup D xn (t , ω), x(t , ω) ≤ ε.

t ∈[0,T ]

t ∈[0,T ]

Hence


n→∞

D yn (t , ω), fω (t , x(t , ω)) −→ 0 ∀t ∈ [0, T ] with P.1. Notice also that t

Z

y (s, ω)ds, n

sup D

t ∈[0,T ]

0

t

Z

fω (s, x(s, ω))ds



0

P.1

T

Z

D xn (s, ω), x(s, ω) ds.




≤ Lτ ,2M 0

Thus, by Lebesgue dominated convergence theorem, we infer that t

Z

yn (s, ω)ds,

D 0

t

Z

fω (s, x(s, ω))ds 0



n→∞

−→ 0 ∀t ∈ [0, T ] with P.1.

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M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

Further note that



sup D x(t , ω), x0 (ω) +

t ∈[0,T ]

t

Z

fω (s, x(s, ω))ds




≤ sup D x(t , ω), xn (t , ω) t ∈[0,T ]

0



+ sup D x (t , ω), x0 (ω) + n

t ∈[0,T ]

Z

t

fω (s, x

n−1



(s, ω))ds + sup D t ∈[0,T ]

0

t

Z

fω (s, x

n −1

(s, ω))ds,

0

t

Z



fω (s, x(s, ω))ds . 0

Thus, in view of the previous convergences and the fact that the second term of the right-hand side is equal to zero, one obtains



D x(t , ω), x0 (ω) +

t

Z

fω (s, x(s, ω))ds



[0,T ] P.1

=

0.

0

Due to Lemma 1 the fuzzy process x is a local solution to the RFDE (1) on the interval [0, T ]. Step 9. For the uniqueness of the solution let us assume that x, y: [0, T ] × Ω

→ E d are two continuous fuzzy
P.1 stochastic processes which are the solutions to (2) on the interval [0, T ]. Because supt ∈[0,T ] D x(t , ω), θˆ ≤ 2M and
P.1 supt ∈[0,T ] D y(t , ω), θˆ ≤ 2M, we get by (3) that Z
[0,T ] P.1 t
D x(t , ω), y(t , ω) ≤ Lτ ,2M D x(s, ω), y(s, ω) ds. 0

Hence, applying the Gronwall inequality, we get


[0,T ] P.1 ≤ 0,

D x(t , ω), y(t , ω)

which completes the proof.



Let a be a finite real number. The next theorem deals with the problem of existence of at least one local solution to (1). Theorem 2. Let f : Ω × [a, ∞) × E d → E d satisfy (f1)–(f2) and let x0 : Ω → E d be a fuzzy random variable such that D x0 (ω), θˆ

P.1

≤ M, where M is a positive constant. Assume that there exists a real-valued stochastic process g: [a, ∞) × Ω → Rb P.1 [0, ∞) satisfying a g (t , ω)dt ≤ K with some constants b > a, K ≥ 0, and such that with P.1
D fω (t , u), θˆ ≤ g (t , ω) for every t ∈ [a, b] and every u ∈ E d .



P.1 ≤ M + K. S n Proof. Denote for n ∈ N the intervals Ikn = [a +(k − 1)(b − a)/n, a + k(b − a)/n], k = 1, 2, 3, . . . , n. We have k=1 Ikn = [a, b] ∞ d for every n ∈ N. Let us define a sequence {xn }n=1 , xn : [a, b] × Ω → E as follows: Then there exists at least one local solution x to the RFDE (1) on the interval [a, b], and supt ∈[a,b] D x(t , ω), θˆ

x1 (t , ω) = x0 (ω)

for every (t , ω) ∈ [a, b] × Ω

and for n ≥ 2 xn (t , ω) = x0 (ω),

xn (t , ω) = x0 (ω) +

if (t , ω) ∈ I1n × Ω , t −(b−a)/n

Z

fω (s, xn (s, ω))ds,

if (t , ω) ∈ Ikn × Ω , k = 2, 3, . . . , n.

(7)

a

Similarly as in the proof of Theorem 1 we can show that xn : [a, b] × Ω → E d , n ∈ N, are the continuous fuzzy stochastic processes. Let us observe that D xn (t , ω), θˆ




≤

D x0 (ω), θˆ +




t −(b−a)/n

Z

D fω (s, xn (s, ω)), θˆ ds




a

[a,b] P.1

≤

M + K.

Therefore the sequence {xn } is uniformly bounded with P.1. For every n ∈ N let us define yn : [a, b] × [0, 1] × Ω → K (Rd ) as yn (t , α, ω) = [xn (t , ω)]α . Since the above inequality holds, we claim that {yn (·, ·, ω)} is uniformly bounded with P.1.

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

1523

For (t1 , α1 ), (t2 , α2 ) ∈ [a, b] × [0, 1] and ω ∈ Ω we obtain dH yn (t1 , α1 , ω), yn (t2 , α2 , ω) ≤ dH [xn (t1 , ω)]α1 , [xn (t2 , ω)]α1 + dH [xn (t2 , ω)]α1 , [xn (t2 , ω)]α2












≤ D xn (t1 , ω), xn (t2 , ω) + dH [xn (t2 , ω)]α1 , [xn (t2 , ω)]α2 . Without loss of generality we can assume that t1 ≤ t2 and α1 ≤ α2 . Then we get dH yn (t1 , α1 , ω), yn (t2 , α2 , ω) ≤




Z

t2 −(b−a)/n

t1 −(b−a)/n

g (s, ω)ds + dH [xn (t2 , ω)]α2 +(α1 −α2 ) , [xn (t2 , ω)]α2 .




If (t1 − t2 ) % 0 and (α1 − α2 ) % 0 then the right-hand side of the above inequality converges to zero with P.1. This implies that {yn (·, ·, ω)} is equicontinuous with P.1. Accordingly to the Blaschke selection
principle for every (t , α) ∈ [a, b] × [0, 1] the set {yn (t , α, ω) | n ∈ N} is relatively compact in the metric space K (Rd ), dH with P.1. Thus, by Arzela’s theorem, there is a subsequence {ynm (·, ·, ω)} ⊂ {yn (·, ·, ω)} which is uniformly convergent to some y˜ (·, ·, ω): [a, b] × [0, 1] → K (Rd ) with P.1, i.e. there exists Ωc ∈ F with P(Ωc ) = 1 such that for every ω ∈ Ωc it holds dH ynm (t , α, ω), y˜ (t , α, ω) −→ 0,




sup

(t ,α)∈[a,b]×[0,1]

as m → ∞.

(8)

Define y: [a, b] × [0, 1] × Ω → K (Rd ) as y(·, ·, ω) = y˜ (·, ·, ω) if ω ∈ Ωc and for ω ∈ Ω \ Ωc as any chosen K (Rd )-valued multifunction. Due to the convergence (8) we claim that the mapping y(t , α, ·): Ω → K (Rd ) is a measurable multifunction for every fixed t ∈ [a, b] and every fixed α ∈ [0, 1]. We shall show that y(t , α, ·) is the α -cut of some fuzzy random variable x(t , ·): Ω → E d . Assume 0 ≤ β ≤ α ≤ 1. Then for every t ∈ [a, b]




d∗H y(t , α, ω), y(t , β, ω) ≤ d∗H y(t , α, ω), ynm (t , α, ω) + d∗H ynm (t , α, ω), ynm (t , β, ω)
+ d∗H ynm (t , β, ω), y(t , β, ω)

≤ dH y(t , α, ω), ynm (t , α, ω) + dH ynm (t , β, ω), y(t , β, ω) −→ 0, as m → ∞, with P.1. Hence it follows that y(t , β, ω) ⊃ y(t , α, ω) with P.1. Now suppose that {αk }∞ k=1 is a nondecreasing sequence which converges to α ∈ (0, 1]. The inequality dH y(t , αk , ω), y(t , α, ω) ≤ dH ynm (t , αk , ω), y(t , αk , ω) + dH ynm (t , α, ω), y(t , α, ω)










leads to dH y(t , αk , ω), y(t , α, ω) −→ 0,




as k → ∞, with P.1.

By virtue of Theorem II-2 in [45] we infer that y(t , α, ω) = k=1 cl n=k y(t , αn , ω) with P.1. The facts that y(t , αm , ω) ∈ d K (R ) for every m ∈ N, and y(t , α1 , ω) ⊃ y(t , α2 , ω) ⊃ . . . with P.1 lead us to the conclusion

T∞

∞ \

y(t , α, ω) =

S∞




y(t , αm , ω) with P.1.

m=1

Hence by Stacking Theorem [48] there variable x(t , ·): Ω → E d such that [x(t , ω)]α = y(t , α, ω) for
Sexists a fuzzy random 0 d every α ∈ (0, 1], and [x(t , ω)] = cl α∈(0,1] y(t , α, ω) ⊂ y(t , 0, ω). It follows that x: [a, b]× Ω → E is a fuzzy stochastic process. Since xnm are the continuous fuzzy stochastic processes and with P.1 the following convergence holds dH [xnm (t , ω)]α , [x(t , ω)]α = sup D xnm (t , ω), x(t , ω) −→ 0,




sup

(t ,α)∈[a,b]×[0,1]




t ∈[a,b]

as m → ∞,

we infer that the fuzzy stochastic process x is continuous. We shall show that x: [a, b] × Ω → E d is a solution to (2). Let {nm } ⊂ N be the sequence defined in preceding steps. Due

R t −(b−a)/n

m to (7) we have xnm (t , ω) = x0 (ω) + a fω (s, xnm (s, ω))ds for (t , ω) ∈ [a + (b − a)/nm , b] × Ω . Let us notice the following: for every (t , ω) ∈ [a, b] × Ω there exists m0 ∈ N such that for every m ≥ m0 we can write



D x(t , ω), x0 (ω) +

t

Z

fω (s, x(s, ω))ds



a t −(b−a)/nm

 Z
≤ D x(t , ω), xnm (t , ω) + D xnm (t , ω), x0 (ω) + a t −(b−a)/nm

Z +D a


fω s, xnm (s, ω) ds,

t

Z



fω s, x(s, ω) ds .




a

fω s, xnm (s, ω) ds






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M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

The first term of the right-hand side of this inequality uniformly converges to zero with P.1, and the second is equal to zero. It remains to show that the third one also converges to zero. Note that t −(b−a)/nm

Z D

fω s, xnm (s, ω) ds,




t

Z



fω s, x(s, ω) ds




a

a t −(b−a)/nm

Z ≤D

fω (s, xnm (s, ω))ds,

a

t −(b−a)/nm

Z


D fω (s, xnm (s, ω)), fω (s, x(s, ω)) ds +

≤



Z

t −(b−a)/nm

fω (s, x(s, ω))ds, θˆ



t

Z

t −(b−a)/nm

a

t

+D

a

t

Z

fω (s, x(s, ω))ds

g (s, ω)ds.


[a,b] P.1 ≤ 2g (t , ω), we have by the Lebesgue dominated convergence theorem that
m→∞ D f ( s , x ( s , ω)), f ( s , x ( s , ω)) ds −→ 0 for every t ∈ [a, b] with P.1. Integrability of g implies that also the second ω nm ω a Rt [a,b] P.1 summand has limit zero. Hence we obtain x(t , ω) = x0 (ω) + a fω (s, x(s, ω))ds which means by Lemma 1 that x is the
[a,b] P.1 ≤ M + K . Hence we infer that local solution to RFDE (1) on the interval [a, b]. Lastly let us recall that D xnm (t , ω), θˆ
[a,b] P.1 the inequality D x(t , ω), θˆ ≤ M + K holds.  As D fω (t , xnm (t , ω)), fω (t , x(t , ω))

Rt

This result we shall apply to obtain the existence of a global solution to (1). Theorem 3. Let f : Ω × [a, ∞) × E d → E d and x0 : Ω → E d be such as in Theorem 2. Assume that there exists a real-valued stochastic process g: [a, ∞)×Ω → [0, ∞) satisfying

R∞ a

P.1

g (t , ω)dt ≤ K with a non-negative constant K , and such that with P.1

D fω (t , u), θˆ ≤ g (t , ω) for every t ∈ [a, ∞) and every u ∈ E d .




Then there exists at least one global solution to the RFDE (1). Proof. By our assumptions we have RFDE x0 (t , ω)

[a,a+1] P.1

=

R a+1 a

P.1

g (t , ω)dt ≤ K . Applying Theorem 2 we can state that there exists a solution x to P.1

fω (t , x(t , ω)) with initial condition x(a, ω) = x0 (ω) or equivalently we can write that the solution

x satisfies the equation x(t , ω)

[a,a+1] P.1

Rt

x0 (ω) + a fω (s, x(s, ω))ds. Denote x1 (ω) = x(a + 1, ω). By Theorem 2 there is a
P.1 constant W > 0 such that D x1 (ω), θˆ ≤ W . Note that Z a+1 P.1 x1 (ω) = x0 (ω) + fω (s, x(s, ω))ds. (9)

=

a

Next we shall extend the above solution x: [a, a + 1] × Ω → E d to a solution on the interval [a, a + 2] = [a, a + 1] ∪ [a + 1, a + 2]. To this end let us notice once again that x(t , ω)

[a+1,a+2] P.1

=

R a+2 a +1

P.1

g (t , ω)dt ≤ K . Thus, accordingly to Theorem 2, the equation

Rt

f (s, x(s, ω))ds has a solution which we denote by x again. Using (9) we a+1 ω Rt [a+1,a+2] P.1 = x0 (ω) + a fω (s, x(s, ω))ds. A spline of the solutions on [a, a + 1] and [a

x1 (ω) +

see that x is

a solution to x(t , ω) + 1, a + 2] is a continuous fuzzy stochastic process x: [a, a + 2] × Ω → E d which satisfies the random fuzzy integral equation x(t , ω)

[a,a+2] P.1

=

x0 (ω) +

Rt a

fω (s, x(s, ω))ds. Repeating of this procedure allows us to obtain a global solution to (1).



4. Random fuzzy initial value problem with a generalized derivative In the deterministic studies of FDEs where the concept of H-derivative was exploited (see e.g. [7,8,20,30]) it has been noticed that the solutions possess a disadvantage. Namely, the diameter (length of the support) of any solution to FDE is an increasing function of time. As a consequence formulation based on H-differentiability cannot fully reflect the rich behaviour of solutions to corresponding ordinary differential equations. A new approach proposed by Hüllermeier [15] is based on considering FDE as a family of differential inclusions. This allowed to consider some interesting aspects of FDEs such as periodicity, Lyapunov stability, regularity of solution sets, attraction, variation of constants formula (see [49–52]). Further research that has been made in this spirit one can find in [10,11,18,19]. However Bede and Gal pointed out that this second approach has the following shortcomings: one cannot talk about the derivative of a fuzzy set-valued function and the solutions are not necessarily fuzzy set-valued functions. They introduced a concept of strongly generalized differentiability for functions with values in the space of fuzzy sets of the real line and examined, in a traditional fashion, some type of FDEs. For the traditional treatment of FDEs argued also Bhaskar et al. [12]. Recently, in the case of fuzzy sets of a real line, Kaleva [16] gave a sufficient condition under which the solutions of FDEs obtained by the two described approaches are identical. Following the concept of differentiability proposed by Bede and Gal [43], Chalco-Cano and Román-Flores defined the fuzzy lateral H-derivative for fuzzy mappings which have values in E d . The definition is as follows (see [13]).

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

1525

Definition 5. Let F : [a, b] → E d . We say that F is differentiable at t0 ∈ [a, b] if there exists an element F 0 (t0 ) ∈ E d such that (L1) for all h > 0 sufficiently small there exist F (t0 + h) F (t0 ), F (t0 ) F (t0 − h) and lim

h→0+

1 h

(F (t0 + h) F (t0 )) = lim

h→0+

1 h

(F (t0 ) F (t0 − h)) = F 0 (t0 ),

or (L2) for all h < 0 sufficiently near to zero there exist F (t0 + h) F (t0 ), F (t0 ) F (t0 − h) and lim

h→0−

1 h

(F (t0 + h) F (t0 )) = lim

h→0−

1 h

(F (t0 ) F (t0 − h)) = F 0 (t0 ).

The limits are taken in the metric space (E d , D), and at the boundary points we consider only the one-sided derivatives. The definition (L1) coincides with Definition 2. Due to (L2) the class of differentiable fuzzy mappings becomes larger. Both derivatives (L1) and (L2) are different. It is known (see Remark 6 in [43], Remark 2 in [13]) that if there exists F 0 (t0 ) in the (L1) form ((L2) form) and F 0 (t0 ) 6= χ{x} for every x ∈ Rd then does not exist F 0 (t0 ) in the (L2) form ((L1) form, respectively). The usage of two kinds of derivatives in the fuzzy initial value problem x0 (t ) = f (t , x(t )), x(0) = x0 leads to different solutions. These obtained by application of (L2) derivative have now the property that the length of the solution support decreases as time increases (see Example 27 in [43], Example 2 in [13]). This is the main advantage of the new approach in FDEs. In this section we consider random fuzzy initial value problem (1) where the symbol 0 denotes now the (L2) derivative of trajectories of a fuzzy stochastic process x. It is clear that currently considered RFDE (1) will not correspond to random fuzzy integral equation (2). Taking into account the results in [43,7,8,53,9], one can notice that the following equivalence between differential and integral equations takes place. Lemma 2. A fuzzy stochastic process x is a local solution to the RFDE (1) on the interval [a, b] if and only if x is a continuous fuzzy stochastic process and it satisfies the following random fuzzy integral equation x0 (ω)

[a,b] P.1

=

x(t , ω) + (−1)

t

Z

fω (s, x(s, ω))ds.

(10)

a

In what follows we write the counterparts of theorems from the Section 3. Although the proofs are similar to the preceding ones we present them for clarity. In these proofs we will use the following facts. Remark 1. If for fuzzy sets u, v, w ∈ E d there exist H-differences u v , u w then D(u v, θˆ ) = D(u, v) and D(u v, u w) = D(v, w). Theorem 4. Let f : Ω × [0, ∞) × E d → E d and x0 : Ω → E d be such as in Theorem 1. Assume that there exists a constant γ > 0 d such that the sequence {xn }∞ n=0 , xn : [0, γ ] × Ω → E given by x0 (t , ω) = x0 (ω),

(t , ω) ∈ [0, γ ] × Ω ,

and for n = 1, 2, . . . xn (t , ω) = x0 (ω) (−1)

t

Z

fω (s, xn−1 (s, ω))ds,

(t , ω) ∈ [0, γ ] × Ω

0

is well defined, i.e. the foregoing H-differences do exist. Then there exists a constant T > 0 such that the RFDE (1) has a unique local solution x on [0, T ] and similarly as in Theorem 1 the local solution x is bounded. Proof. Similarly as in the proof of Theorem 1 we obtain the inequalities (3) and (4) which hold for some positive constants τ , Lτ ,2M , Wτ ,2M . Let us define the constant T as T = min{τ , γ , W M } and the set V as before. The successive approximations τ ,2M

xn are now as follows: xn (t , ω) = xn (t , ω) for t ∈ [0, T ], ω ∈ Ω ,

n ∈ {0, 1, 2 . . .}. The mappings xn : [0, T ] × Ω → E d are the fuzzy stochastic processes. Indeed, note first that if F , G: Ω → K (Rd ) are the measurable multifunctions and the Hukuhara difference F − G exists with P.1 then F − G is Rt a measurable multifunction. Since the H-differences x0 (ω) (−1) 0 fω (s, xn−1 (s, ω))ds exist for every (t , ω) ∈ [0, T ] × Ω and every n ∈ N, we have that for every α ∈ [0, 1]



x0 (ω) (−1)

Z 0

t

fω (s, xn−1 (s, ω))ds

α

 α Z t = [x0 (ω)]α − (−1) fω (s, xn−1 (s, ω))ds .

(11)

0

Proceeding similarly as in the proof of Theorem 1 one can notice that ω 7→

hR

t f 0 ω

iα (s, xn−1 (s, ω))ds , where t ∈ [0, T ]

is fixed, is a measurable multifunction for every n ∈ N and every α ∈ [0, 1]. Hence for every fixed t the right-hand side

1526

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

Rt

of (11) is a measurable multifunction. Therefore the mapping ω 7→ xn (t , ω) = x0 (ω) (−1) 0 fω (s, xn−1 (s, ω))ds is a fuzzy random variable for every t ∈ [0, T ]. In what follows we shall note that xn ∈ V for n ∈ {0, 1, 2, . . .}. This is clear for n = 0. By Remark 1 we have for n ∈ N D xn (t , ω), θˆ




  Z t fω (s, xn−1 (s, ω))ds = D x0 (ω), (−1) 0  Z t
n −1 ˆ ˆ fω (s, x (s, ω)), θ . ≤ D x0 (ω), θ + D 0


[0,T ] P.1

≤ 2M. Hence, similarly as in the proof of Theorem 1, we obtain D xn (t , ω), θˆ In order to see that xn ’s are the continuous fuzzy stochastic processes notice that by Remark 1 we have for n ∈ N t +h

 Z
D x (t + h, ω), x (t , ω) = D x0 (ω) (−1) n

fω (s, x

n

n −1

(s, ω))ds, x0 (ω) (−1)

t +h

fω (s, xn−1 (s, ω))ds,

=D 0

fω (s, x

n −1

(s, ω))ds



0

0

Z

t

Z

t

Z



fω (s, xn−1 (s, ω))ds . 0

Proceeding as in the proof of Theorem 1 one obtains right-sided continuity and then left-sided continuity of xn .


[0,T ] P.1 1 n ≤ Wτ ,2M Lτn− ,2M T /n! and the mapping

Similarly as before, by the usage of Remark 1, we have D xn (t , ω), xn−1 (t , ω)

t → 7 D xn (t , ω), xn−1 (t , ω) is continuous with P.1. Thus, for the same reasons as in the proof of Theorem 1, the sequence {xn (·, ω)} is uniformly convergent with P.1. On the base of its limit x˜ one constructs a continuous fuzzy stochastic process x: [0, T ] × Ω → E d that will be the desired solution to (1) with (L2) derivative. Indeed, note that




  Z t ≤ D x0 (ω), xn (t , ω) + (−1) fω (s, xn−1 (s, ω))ds 0 0   Z t Z t n n −1 + D x (t , ω) + (−1) fω (s, x (s, ω))ds, x(t , ω) + (−1) fω (s, x(s, ω))ds 0 0 Z t

≤ D xn (t , ω), x(t , ω) + D fω (s, xn−1 (s, ω)), fω (s, x(s, ω)) ds.



D x0 (ω), x(t , ω) + (−1)

Z

t

fω (s, x(s, ω))ds



0

Analogously as in the proof of Theorem 1, the last sum converges to zero for every t ∈ [0, T ] with P.1. This implies that x0 (ω)

[0,T ] P.1

x(t , ω) + (−1)

Rt

(s, x(s, ω))ds and by virtue of Lemma 2 the existence of solution to (1) with (L2) derivative
[0,T ] P.1 is proved. The boundedness of the solution is clear and it holds D x(t , ω), θˆ ≤ 2M. Now, let x, y be the solutions to (1) with (L2) derivative. According to Lemma 2 we can write Z t [0,T ] P.1 x(t , ω) = x0 (ω) (−1) fω (s, x(s, ω))ds, =

f 0 ω

0

and y(t , ω)

[0,T ] P.1

=

x0 (ω) (−1)

t

Z

fω (s, y(s, ω))ds. 0

Considering of D x(t , ω), y(t , ω) (applying Remark 1 and Gronwall’s inequality) leads us to the conclusion on uniqueness of the solution. 




The following result is a counterpart of Theorem 2. Theorem 5. Let f : Ω × [a, ∞) × E d → E d , x0 : Ω → E d and g: [a, ∞) × Ω → [0, ∞) satisfy the conditions of Theorem 2. d Assume that the sequence {xn }∞ n=1 , xn : [a, b] × Ω → E given by x1 (t , ω) = x0 (ω),

(t , ω) ∈ [a, b] × Ω ,

and for n = 2, 3, . . .

 x0 (ω),

(t , ω) ∈ I1n × Ω , Z t −(b−a)/n xn (t , ω) = fω (s, xn (s, ω))ds, x0 (ω) (−1)

(t , ω) ∈ (I2n ∪ I3n ∪ · · · ∪ Inn ) × Ω ,

(12)

a

is well defined (partition I1n , I2n , . . . , Inn of interval [a, b] is defined as in the proof of Theorem 2). Then there exists at least one local solution x to the RFDE (1) on the interval [a, b] and similarly as in Theorem 2 the local solution x is bounded.

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

1527

Proof. Similarly as in the proof of Theorem 4 one can show that xn ’s are the continuous fuzzy stochastic processes. We define a sequence of mappings yn : [a, b] × [0, 1] × Ω → K (Rd ) as before, i.e. yn (t , α, ω) := [xn (t , ω)]α . Using arguments analogous to those contained in the proof of Theorem 2 and with a help of Remark 1 we come to the claim: there exists a subsequence {ynm } of {yn }, that uniformly converges to y˜ with P.1. Then, in the identical way as in the proof of Theorem 2, we define a mapping y: [a, b] × [0, 1] × Ω → K (Rd ) and show that there exists a continuous fuzzy stochastic process x: [a, b] × Ω → E d such that [x(t , ω)]α = y(t , α, ω) for each α ∈ (0, 1]. In order to see that x is a solution to (1) with (L2) derivative notice that for m ∈ N (large enough) we can write



D x0 (ω), x(t , ω) + (−1)

t

Z

fω (s, x(s, ω))ds



a



+ D xnm (t , ω) + (−1)

t −(b−a)/nm

 Z ≤ D x0 (ω), xnm (t , ω) + (−1)

fω (s, xnm (s, ω))ds



a t −(b−a)/nm

Z

fω (s, xnm (s, ω))ds, x(t , ω) + (−1)

fω (s, x(s, ω))ds



a

a


≤ D xnm (t , ω), x(t , ω) + D

t

Z

t −(b−a)/nm

Z

fω (s, xnm (s, ω))ds,

a

t

Z



fω (s, x(s, ω))ds . a

The first summand, in the last sum, uniformly converges to zero with P.1. For the same reasons as those in the proof of Theorem 2, also the second summand converges to zero for every t ∈ [a, b] with P.1. Hence the desired property follows, [a,b] P.1

i.e. x0 (ω) = M + K. 

x(t , ω)+(−1)

Rt a

fω (s, x(s, ω))ds. Finally we observe that the solution x is bounded and D x(t , ω), θˆ


[a,b] P.1 ≤

There is also a counterpart of Theorem 3 and we shall only note that there exists a global solution to (1) with (L2) derivative. A derivation is similar to that contained in the proof of Theorem 3. We present it in the following outline form. Let f : Ω × [a, ∞) × E d → E d and x0 : Ω → E d be such as in Theorem 2. Let a real-valued stochastic process g: [a, ∞) × Ω → [0, ∞) satisfy conditions of Theorem 3. Assume that there exists a1 > a such that for n = 2, 3, . . . x1n

(t , ω) ∈ I11,n × Ω , Z t −(a1 −a)/n (t , ω) = x0 (ω) (−1) fω (s, x1n (s, ω))ds,  x0 (ω),

(t , ω) ∈ (I21,n ∪ I31,n ∪ · · · ∪ In1,n ) × Ω ,

a

1,n

are well defined, where Ik = [a + (k − 1)(a1 − a)/n, a + k(a1 − a)/n], k = 1, 2, . . . , n. Then, by Theorem 5, there is a local solution x1 to (1) on the interval [a, a1 ], initial condition was x0 (ω). Denote x20 (ω) = x1 (a1 , ω) for ω ∈ Ω . Now suppose there is a2 > a1 such that for n = 2, 3, . . . x2n

(t , ω) ∈ I12,n × Ω , Z t −(a2 −a1 )/n (t , ω) = 2 x0 (ω) (−1) fω (s, x2n (s, ω))ds,  2 x0 (ω),

(t , ω) ∈ (I22,n ∪ I32,n ∪ · · · ∪ In2,n ) × Ω ,

a

2,n

are well defined, where Ik = [a1 + (k − 1)(a2 − a1 )/n, a1 + k(a2 − a1 )/n], k = 1, 2, . . . , n. Considering (1) with initial condition x20 (ω) we state, by Theorem 5 once again, that there is a local solution x2 on the interval [a1 , a2 ]. If we denote x1,2 (t , ·) =

x1 (t , ·) x2 (t , ·)



for t ∈ [a, a1 ], for t ∈ [a1 , a2 ],

then we see that x1,2 is a continuous fuzzy stochastic process which is a local solution to (1) with initial condition x0 (ω) and this solution is defined on [a, a2 ]. For the next step we would denote x30 (ω) = x2 (a2 , ω). Assuming that there exist a3 , a4 , . . . for which the foregoing procedure can be applied we obtain the existence of a global solution provided that ak −→ ∞ as k → ∞. 5. Illustrations This part presents some examples being simple illustrations of the theory of RFDEs. We shall consider the RFDE (1) with (L1) and (L2) derivative, respectively, where f : Ω × [0, ∞) × E 1 → E 1 and x0 : Ω → E 1 . In this case, for a solution x we can denote its α -cuts (α ∈ [0, 1]) as [x(t , ω)]α = [Lα (t , ω), Uα (t , ω)], where Lα (t , ω), Uα (t , ω) are some real-valued stochastic processes. It is known that for (L1) differentiation of x(t , ω) we have [x0 (t , ω)]α = [L0α (t , ω), Uα0 (t , ω)], whereas for (L2) derivative it holds [x0 (t , ω)]α = [Uα0 (t , ω), L0α (t , ω)]. By diam(x(t , ω)) := U0 (t , ω) − L0 (t , ω) we denote the diameter of x. Denote also the α -cuts of x0 as [x0 (ω)]α = [xL0,α (ω), xU0,α (ω)]. For convenience, from now on, we will denote the solutions of RFDE (1) with (L1) derivative by x1 , and the solutions of (1) with (L2) derivative by x2 . The α -cuts (α ∈ [0, 1]) of u ∈ E 1 are the nonempty compact convex subsets of R, so we denote [u]α = [uα− , uα+ ]. In the sequel we will often consider the triangular fuzzy numbers from E 1 . Therefore we introduce a notation for such a fuzzy set of the real line. For a, b, c ∈ R, a < b < c, we can define the triangular fuzzy number u = (a, b, c ) as u ∈ E 1 such that [u]α = [a + (b − a)α, c − (c − b)α], α ∈ [0, 1]. Since u1− = u1+ = b, we denote u1 = b. The set of all triangular fuzzy numbers in E 1 we denote by T . There exists a nice sufficient condition of existence of H-difference u v , if u, v ∈ T . This criterion will be useful in the further considerations and we formulate it as the following lemma.

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M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

0 0 Lemma 3 (Lemma 23 of Bede and Gal [43]). If u, v ∈ T and diam(v) ≤ min{u1 − u0− , u0+ − u1 }, where diam(v) = v+ − v− , then H-difference u v exists.

Let us start the illustrations with a classical problem. Example 1. Let Ω = (0, 1), F -Borel σ -field of subsets of Ω , P-Lebesgue measure on (Ω , F ). Let us consider the equation

(

x0 (t , ω)

[0,∞) P.1

−x(t , ω) x(0, ω) = x0 (ω), =

(13)

P.1

where x0 is as follows: x0 (ω) = (−ω, 0, ω).

(14)

We have then xL0,α (ω) = (α − 1)ω,

xU0,α (ω) = (1 − α)ω

and it is clear that x0 is a fuzzy random variable. At first, let us examine problem (13) taking (L1) derivative. Using the α -cuts of x, x0 and x0 we obtain a system

(

L0α (t , ω)

[0,∞) P.1

=

[0,∞) P.1 Uα0 (t , ω) =

−Uα (t , ω), −Lα (t , ω),

with initial conditions P.1

Lα (0, ω) = (α − 1)ω,

P.1

Uα (0, ω) = (1 − α)ω.

The solution of this system is as follows: Lα (t , ω)

[0,∞) P.1

=

(α − 1)et ω and Uα (t , ω)

[0,∞) P.1

=

(1 − α)et ω,

where α ∈ [0, 1]. Hence the fuzzy stochastic process x1 : [0, ∞) × Ω → E 1 defined as x1 (t , ω) = (−et ω, 0, et ω) for every fixed (t , ω) ∈ [0, ∞) × Ω is a solution to (13) with (L1) derivative. Considering (13) with (L2) derivative we get the following system

(

L0α (t , ω)

[0,∞) P.1

=

[0,∞) P.1 Uα0 (t , ω) =

−Lα (t , ω), −Uα (t , ω),

with initial conditions as before. Hence for every α ∈ [0, 1] Lα (t , ω)

[0,∞) P.1

=

(α − 1)e−t ω and Uα (t , ω)

[0,∞) P.1

=

(1 − α)e−t ω,

and consequently x2 (t , ω) = (−e−t ω, 0, e−t ω) for every (t , ω) ∈ [0, ∞) × Ω . One can observe that for every ω ∈ Ω t →∞

diam(x1 (t , ω)) = 2ωet −→ ∞

and

t →∞

diam(x2 (t , ω)) = 2ωe−t −→ 0.

Let us consider now the same equation as in Example 1, but with another initial value. Example 2. Let Ω = [1, 3], F -Borel σ -field of subsets of Ω , P-normed Lebesgue measure on (Ω , F ). Let us put fuzzy random variable x0 : Ω → E 1 as x0 (ω) = (ω, ω + 1, ω + 2).

(15)

Hence xL0,α (ω) = α + ω

and xU0,α (ω) = 2 − α + ω.

For the solution x1 : [0, ∞) × Ω → E 1 of (13) we obtain that for every α ∈ [0, 1] Lα (t , ω)

[0,∞) P.1

=

(1 + ω)e−t + (α − 1)et ,

Uα (t , ω)

[0,∞) P.1

=

(1 + ω)e−t + (1 − α)et .

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

1529

This implies that for every (t , ω) ∈ [0, ∞) × Ω x1 (t , ω) = ((1 + ω)e−t − et , (1 + ω)e−t , (1 + ω)e−t + et ) and t →∞

diam(x1 (t , ω)) = 2et −→ ∞,

for every ω ∈ Ω .

For the solution x2 : [0, ∞) × Ω → E 1 of (13) with initial value x0 as in (15) we have Lα (t , ω)

[0,∞) P.1

=

(α + ω)e−t ,

Uα (t , ω)

[0,∞) P.1

=

(2 − α + ω)e−t ,

where α ∈ [0, 1]. Hence we infer that for every (t , ω) ∈ [0, ∞) × Ω x2 (t , ω) = (ωe−t , (1 + ω)e−t , (2 + ω)e−t ), and t →∞

diam(x2 (t , ω)) = 2e−t −→ 0

for every ω ∈ Ω .

Example 3. Consider the following RFDE with (L2) derivative

(

x0 (t , ω)

[0,∞) P.1

−x5 (t , ω) x(0, ω) = x0 (ω), =

(16)

P.1

where (Ω , F , P), x0 are as in Example 2. Proceeding as in Example 1 we obtain for the α -cuts of solution x2 that Lα (t , ω)

1

[0,∞) P.1

=

q 4

1

(ω+α)4

,

Uα (t , ω)

+ 4t

1

[0,∞) P.1

=

q 4

.

1

(2+ω−α)4

+ 4t

Hence we get a global solution x2 : [0, ∞) × Ω → E 1 such that for every (t , ω) ∈ [0, ∞) × Ω

x2 (t , ω)(y) =

4



0

1

ω4

4

otherwise. 4

+ 4t

(1+ω)4

1 (2+ω)4

1

4

(1+ω)4

+ 4t

, 

1

1

1

,q

1

if y ∈  q

For every ω ∈ Ω we have that diam(x2 (t , ω)) = r t →∞

1

if y ∈  q

1    2+ω− q   1 4  − 4t  y4  

diam(x2 (t , ω)) −→ 0





   1   q − ω,   4 1  − 4t  4  y

+ 4t

− +4t

,q

q 1 4

1

ω4

4

+4t

1 1

(2+ω)4

+ 4t

,

is a decreasing function and

for every ω ∈ Ω .

Example 4. Let us take Ω = [0, b], where b ∈ (0, ∞), F -Borel σ -field of subsets of Ω , P-normed Lebesgue measure on P.1

(Ω , F ). Let a fuzzy random variable x0 : Ω → E 1 be such that D(x0 (ω), θˆ ) ≤ M, where M > 0. Consider the RFDE (1) with (L1) derivative, and with the following data: a = 0,

fω (t , u) = ωt · u2n−1 + χ{ωt }

for every (ω, t , u) ∈ Ω × [0, ∞) × E 1 ,

where n ∈ N. Let us observe that for such an x0 and f the assumptions of Theorem 1 are satisfied. Indeed, it is obvious that f satisfies the conditions (f1)–(f2). For (f3) let us note that for every fixed (ω, τ ) ∈ Ω × (0, ∞) it holds: for every r > 0, every u, v ∈ Br , every t ∈ [0, τ ] D(fω (t , u), fω (t , v)) = D(ωt · u2n−1 , ωt · v 2n−1 )

 = ωt sup max |(u2n−1 )α− − (v 2n−1 )α− |, |(u2n−1 )α+ − (v 2n−1 )α+ | α∈[0,1]

 α α ≤ (2n − 1)r 2n−2 ωt sup max |uα− − v− |, |uα+ − v+ | α∈[0,1]

= (2n − 1)r ωtD(u, v) ≤ (2n − 1)r 2n−2 bτ D(u, v). 2n−2

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M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

For the second condition in (f3) we have D(fω (t , θˆ ), θˆ ) = D(χ{ωt } , θˆ ) = ωt ≤ bτ

for every (ω, t ) ∈ Ω × [0, τ ],

where τ ∈ (0, ∞). Hence, applying Theorem 1, we infer that there exists a unique local solution (on some interval [0, T ]) to RFDE x0 (t , ω)

[0,∞) P.1

=

ωt · x2n−1 (t , ω) + χ{ωt } ,

P.1

x(0, ω) = x0 (ω),

(17)

where n ∈ N. In what follows we want to present explicit solution to (17) in the case n = 1. Let us consider x0 : Ω → E 1 as in (15). Proceeding as in Example 1 we get Lα (t , ω)

[0,∞) P.1

=

(1 + ω + α)eωt

2 /2

− 1,

Uα (t , ω)

[0,∞) P.1

=

(3 + ω − α)eωt

2 /2

− 1.

Hence we obtain a global solution x1 : [0, ∞) × Ω → E such that for every (t , ω) ∈ [0, ∞) × Ω 1

2 2 2 x1 (t , ω) = ((1 + ω)eωt /2 − 1, (2 + ω)eωt /2 − 1, (3 + ω)eωt /2 − 1).

For the diameter of x1 we have 2 diam(x1 (t , ω)) = 2eωt /2

and

t →∞

P diam(x1 (t , ω)) −→ ∞ = 1.




Example 5. Let (Ω , F , P) be the probability space defined as in Example 1 and let fuzzy random variable x0 : Ω → E 1 be


P.1

such that D x0 (ω), θˆ ≤ M, where M > 0. Consider problem (1) with a = 1, where the derivative is understood in the sense of (L1) and f : Ω × [1, ∞) × E 1 → E 1 is described as follows: fω (t , u) =

ω
· u. t D(u, θˆ ) + 1

(18)

Straightforward calculations show that f satisfies conditions (f1)–(f2). We have also that D fω (t , u), θˆ ≤




ω

for every (ω, t , u) ∈ Ω × [1, ∞) × E 1 .

t

For the mapping g (t , ω) = ω/t one can see that g (·, ·) is a stochastic process with values in (0, ∞) and such that for every b ∈ (1, ∞) it holds b

Z

P.1

g (t , ω)dt ≤ ln b. 1

Thus, in view of Theorem 2, there exists at least one local solution (to the considered problem) on [1, b], where b is any fixed number from (1, ∞). If we change the nonlinearity f for the following one: f : Ω × [1, ∞) × E 1 → E 1 defined as fω (t , u) =

ω

(1 +

t2


· u, ) D(u, θˆ ) + 1

(19)

then one can see that D fω (t , u), θˆ ≤




ω 1 + t2

for every (ω, t , u) ∈ Ω × [1, ∞) × E 1 .

P.1

R∞

Since 1 ω/(1 + t 2 )dt ≤ π /4, we infer (accordingly to Theorem 3) that there exists at least one global solution to the problem x0 (t , ω)

ω

[1,∞) P.1

=

(1 +

t2


· x(t , ω), ) D(x(t , ω), θˆ ) + 1

P.1

x(1, ω) = x0 (ω).

Until now we presented applicable character of Theorems 1–3. In the sequel we want to give some examples in which Theorems 4 and 5 are applied. A main difficulty seems to be: to justify (in a simple way) that the appropriate H-differences, from assumptions of these theorems, exist. Using some ideas from the paper on generalized differentiability for fuzzy mappings [43], where FDEs were also investigated, we are able to formulate the conditions which imply the existence of desired H-differences.

M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

1531

Remark 2. Let us assume that f : Ω × [0, ∞) × T → T satisfies conditions of (f1)–(f3) type and let fuzzy random variable x0 : Ω → T be such that

(x0 (ω))1 − (x0 (ω))0− = const. and (x0 (ω))0+ − (x0 (ω))1 = const.

(20)

Denote m = min{(x0 (ω))1 − (x0 (ω))0− , (x0 (ω))0+ − (x0 (ω))1 }. One can infer that D x0 (ω), θˆ with P.1


P.1 ≤ M, where M is some positive constant. Similarly as in the proof of Theorem 1 we get that

D fω (t , u), θˆ ≤ Wτ ,2M

for every (t , u) ∈ [0, τ ] × (B2M ∩ T ),




where Wτ ,2M is a constant defined as in the proof of Theorem 1. We want to note that for m

n γ = min τ ,

M

,

o

2Wτ ,2M Wτ ,2M

the sequence {xn }, xn : [0, γ ] × Ω → T given by x0 (t , ω)

[0,γ ] P.1

x0 (ω),

=

and for n = 1, 2, . . . xn (t , ω)

[0,γ ] P.1

x0 (ω) (−1)

=

t

Z

fω (s, xn−1 (s, ω))ds, 0

is well defined. Indeed, proceeding similarly as in the proof of Corollary 24 in [43] we have that with P.1 it holds: for every t ∈ [0, γ ]



diam (−1)

t

Z

fω (s, xn−1 (s, ω))ds 0



Z t  = diam fω (s, xn−1 (s, ω))ds 0 Z t  ≤ 2D fω (s, xn−1 (s, ω))ds, θˆ 0

≤ 2Wτ ,2M t ≤ m. Hence, applying Lemma 3, we infer that with P.1 for every t ∈ [0, γ ] there exists H-difference x0 (ω) Rt (−1) 0 fω (s, xn−1 (s, ω))ds. Now, proceeding similarly as in the proof of Theorem 4 we obtain the existence and uniqueness of local solution x2 to (1) on [0, T ], where T = min{τ ,

m 2Wτ ,2M

, WτM,2M }.

Example 6. Let Ω = (0, 1), F -Borel σ -field of subsets of Ω , P-Lebesgue measure on (Ω , F ). Let fuzzy random variable x0 : Ω → T satisfy (20). Denote, as before, m = min{(x0 (ω))1 − (x0 (ω))0− , (x0 (ω))0+ − (x0 (ω))1 }. Consider f : Ω × [1, ∞) × T → T which is defined as in (18). Then for every fixed b > 1 we have b

Z

D(fω (t , u), θˆ )dt ≤

b

Z

1

1

ω t

dt ≤ ln b

for every (ω, t , u) ∈ Ω × [1, b] × T .

In what follows we shall note that if b ∈ (1, em/2 ] then there exists at least one local solution x2 to the RFDE (1) on the interval [1, b]. To this end, due to Theorem 5, it is enough to show that for every n ∈ {2, 3, . . .} it holds: the H-difference in (12), i.e. x0 (ω) (−1)

t −(b−1)/n

Z

fω (s, xn (s, ω))ds,

1

exists for every (t , ω) ∈ [1 + (b − 1)/n, b] × Ω . Obviously we have that xn (t , ω) = x0 (ω) for (t , ω) ∈ [1, 1 + (b − 1)/n] × Ω . Notice that for every (t , ω) ∈ [1 + (b − 1)/n, b] × Ω it holds



diam (−1)

t −(b−1)/n

Z 1

fω (s, xn−1 (s, ω))ds



t −(b−1)/n

Z ≤2

(ω/t )dt ≤ 2 ln b ≤ m,

1

provided that 1 < b ≤ em/2 . By Lemma 3 and Theorem 5 our considerations are completed. Acknowledgements The author wishes to thank the anonymous referees for their suggestions that improved the paper presentation.

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M.T. Malinowski / Nonlinear Analysis 73 (2010) 1515–1532

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