On the Existence of Solutions of Some Stochastic Functional Integral Equations

Monika Pasławska-Południak

On the Existence of Solutions of Some Stochastic Functional Integral Equations

Abstract. The integral equation of Urysohn type is considered, for the determini- stic and stochastic cases . We show, using the fixed point theorem of Darbo type that under some assumptions the equations have solutions belonging to the space of continuous functions. The main tool used in our paper is the technique associated with measures of noncompactness.

2000 Mathematics Subject Classification: 34K50, 45G10, 45R05, 47H09, 60H20.

Key words and phrases: measures of noncompactness, functional integral equations, stochastic functional integral equations.

1. Introduction. We introduce some spaces of continuous functions C = C([ −r, 0], ℝ ^{n ), C 1} = C([−r, T ], ℝ ^{n ) and C 2 = C(I,} ℝ ⁿ ), for I = [0, T ], endowed with the standard supremum norms ||·|| ^C , ||·|| ^C

and ||·|| ^C

, respectively. Moreover, we fix r > 0 for the whole paper. For further considerations we shall need the notion of a segment of a trajectory.

Let y be a function of t ∈ [−r, T ], r > 0, T < ∞. For a fixed t ∈ I the function y ^t defined by the formula

y t (τ) = y(t + τ), τ ∈ [−r, 0]

is called the segment of the trajectory of y on [t−r, t], so observe that t+τ ∈ [t−r, t].

First, the aim of the paper is to discuss the solvability of the stochastic nonlinear functional integral equation of Urysohn type having the form

x(t, ω) = h(t, x(t, ω)) + Z

I

u(t, s, x(s, ω), x s (·, ω))ds

(1.1) + Z

I

v(t, s, x(s, ω), x s (·, ω))dβ(s) ,

with the initial condition

(1.2) x(t, ω) = ϕ(t, ω) for t e ∈ [−r, 0] . We recall that

x(0, ω) = x 0 (ω) = x(0 + τ, ω) for τ ∈ [−r, 0] ,

where ω denotes the sample point of the complete probability space (Ω, F, P ) . With respect to the random process β(t, ω) we shall assume that for each t ∈ I a minimal σ-algebra F ^t in F (being an increasing family) is defined such that β(t, ω) is measurable with respect to F ^t . Moreover, {β(t, ω); F ^t , t ∈ I} is a real martingale.

The second integral in equation (1.1) is a stochastic integral of the Itˆo - Doob type.

The functions h : I × ℝ ⁿ → ℝ ⁿ , u, v : I ×I × ℝ ⁿ ×C → ℝ ^{n ,} ϕ : [ e −r, 0]×Ω → ℝ ^{n ,}

and a number r > 0 are given, while x : I × Ω → ℝ ⁿ is an unknown function. Note, that e ϕ(0) = h(0) .

By L ² (Ω, F, P ) we denote the space of F - measurable square integrable random functions x(t, ·) with

||x(t)|| L

= E

|x(t)| ² =



 Z

Ω

|x(t, ω)| ² dP (ω)





12

.

Definition 1.1 The function x(t, ω) is a random solution of equation (1.1) with initial condition (1.2) if for every fixed t ∈ [−r, T ], x(t, ω) ∈ L ² (Ω, F, P ) and it satisfies (1.1) - (1.2) P -a.e. on I and is equal to the given initial function on [−r, 0].

Using the technique associated with measures of noncompactness we prove in Section 3 the existence of solutions to problem (1.1) - (1.2) in the space C = C([ −r, T ], L ² (Ω, F, P )) of continuous functions x(t, ·) on [−r, T ] with the topology defined by the norm

||x|| 1 = sup{||x(t)|| L

: t ∈ [−r, T ]} < ∞ . The space C with the norm || · || ¹ is a real Banach space.

In the last section we consider the corresponding deterministic case. Similar problems concerning stochastic integral equations have been treated in papers [4], [6], [7], [8]. The deterministic case without delay was considered e.g. in papers [1], [3].

2. On some auxiliary results. First, we present a few facts concerning me- asures of noncompactness from book [2].

Let E be a Banach space with the norm || · || ^E and zero element Θ. For a set

X ⊂ E denote by X the closure of X and by X ^w the weak closure of X. Introduce

the symbol ConvX for the convex closure of a set X. The symbol B(x, r) denotes

the closed ball centered at x and with radius r. Put B r = B(Θ, r).

Moreover, let 𝔐 E be the family of all nonempty and bounded subsets of E and let 𝔑 E , 𝔑 ^w _E be its subfamilies consisting of all relatively compact and relatively weakly compact sets, respectively.

Definition 2.1 ([2]) A nonempty family 𝔓 _⊂ 𝔑 E is said to be the kernel (of a measure noncompactness), provided it satisfies the following conditions:

1 ^o X ∈ 𝔓 _{⇒ X ∈} 𝔓 ,

2 ^o X ∈ 𝔓 , Y ⊂ X, Y 6≡  ⇒ Y ∈ 𝔓 ,

3 ^o X, Y ∈ 𝔓 ⇒ λX + (1 − λ)Y ∈ 𝔓 , λ ∈ [0, 1], 4 ^o X ∈ 𝔓 _{⇒ ConvX ∈} 𝔓 ,

5 ^o 𝔓 ^c (the subfamily of 𝔓 consisting of closed sets) is closed in 𝔑 ^c _E (the family of all closed subsets of E) with respect to the topology generated by Hausdorff metric.

Definition 2.2 ([2]) A function µ : 𝔐 E → ℝ ^{+ is a} measure of noncompactness if the following conditions are fullfield:

1 ^o The kernel of the measure µ, kerµ = {X ∈ 𝔐 E : µ(X) = 0}, is nonempty and kerµ ⊂ 𝔑 E ,

2 ^o X ⊂ Y =⇒ µ(X) ¬ µ(Y ), 3 ^o µ(ConvX) = µ(X), 4 ^o µ(X) = µ(X)

5 ^o µ(λX + (1 − λ)Y ) ¬ λµ(X) + (1 − λ)µ(Y ), λ ∈ [0, 1],

6 ^o If {X ⁿ } is a sequence of closed sets from 𝔐 E such that X n+1 ⊂ X ⁿ for n = 1, 2, ... and if lim

n →∞ µ(X n ) = 0 then the set X _∞ = T ^∞

n=1 X n is nonempty.

If measure of noncompactness µ satisfies in addition the following two conditions:

7 ^o µ(U + V ) ¬ µ(u) + µ(V ), 8 ^o µ(λU ) = |λ|µ(U), λ ∈ ℝ ^,

it will be called sublinear.

To verify that any function µ fulfilles conditions of definition of a measure of non- compactness one can use the following convienient lemma.

Lemma 2.3 ([4]) Let µ : 𝔐 E → ℝ ⁺ be a function satisfying the following codi-

tions:

1 ^o µ(X) = 0 ⇔ X is relatively compact, 2 ^o X ⊂ Y ⇒ µ(x) ¬ µ(Y ),

3 ^o µ(X S Y ) = max {µ(x), µ(Y )}.

Then µ fulfilles the axiom 6 ⁰ of Definition 2.2.

Now we recall the fixed point theorem of Darbo type.

Theorem 2.4 ([2]) Let Ω be a nonempty, bounded, closed and convex subset of E and let F : Ω → Ω be a continuous mapping which is a contraction with respect to a measure of noncompactness µ, i.e. there exists a constant k ∈ [0, 1) such that µ(F X) ¬ kµ(X) for any nonempty subset X of Ω. Then F has at least one fixed point in the set Ω and the set Fix F = {x ∈ Ω : F x = x} of fixed points of F belongs to ker µ.

Now, we introduce the formula for a measure of noncompactness. Namely, fix x ∈ C, X ∈ 𝔐 C , T > 0 and ε > 0.

Definition 2.5 ([2]) The following function (2.1) µ(X) = lim

ε→0 sup{sup{||x(t) − x(s)|| ^L

: t, s ∈ [−r, T ], |t − s| ¬ ε}; x ∈ X}, is a measure of noncompactness in the space (C, 𝔐 C ).

It is also known from Arzela - Ascoli Theorem ([2]) that ker µ is the collection of all sets X ∈ 𝔐 C such that all maps belonging to X are equicontinuous. Further properties of µ can be found in [2].

3. Main results. We shall follow similiary as in [6]. Let t ∈ I be fixed. We shall also assume that the random process β(t, ω) is a real martingale and there is a real continuous non-decreasing function G(t) such that for s < t we have

E {|β(t, ω) − β(s, ω)| ² } = E{|β(t, ω) − β(s, ω)| ² |F ^t } = G(t) − G(s) , P − a.e., where E denotes the expected value of the random process.

We will often use the following estimation [6]

||

Z

I

v(t, s, x(s, ω), x s (·, ω))dβ(s)|| ^L

(3.1) ¬



 Z

I

||v(t, s, x(s, ω), x ^s (·, ω))|| ² L

dG(s)





12

.

Remark 3.1 In particular case when β(t) = W (t), W (t) is a Brownian motion for t ∈ I adapted to an increasing family of sub-σ-algebras F ^t in F then for s < t we have ([5], Ch. 4)

E {|W (t, ω) − W (s, ω)| ² } = E{|W (t, ω) − W (s, ω)| ² |F ^t } = t − s , P − a.e., and also

||

Z

I

v(t, s, x(s, ω), x s (·, ω))dW (s)|| L

(3.1 ⁰ ) ¬



 Z

I

||v(t, s, x(s, ω), x ^s (·, ω))|| ² L

ds





12

.

We assume that the functions involved in problem (1.1) - (1.2) satisfy the following hypotheses:

(i) the following inequality

|u(t, s, x(s, ω), x ^s (·, ω))| ¬ q ¹ (t, s)|x(s, ω)| + q ² (t, s)|x ^s (·, ω)| + q ³ (t, s) P − a.s.

is satisfied for all t, s ∈ I and for nonnegative, continuous functions q ¹ , q 2 and q 3 defined for t, s ∈ I,

(ii) the following inequality

|v(t, s, x(s, ω), x ^s (·, ω))| ¬ p 1 (t, s)|x(s, ω)|+p 2 (t, s)|x ^s (·, ω))|+p 3 (t, s) P −a.s.

is satisfied for all t, s ∈ I and for nonnegative, continuous functions p 1 , p 2 and p 3 defined for t, s ∈ I,

(iii) the mapping x(t, ω) → u(t, s, x(s, ω), x ^s (·, ω)) from C(I, L ² (Ω, F, P )) into C(I, L ² (Ω, F, P )) is continuous in the topology generated by the norm ||x|| ² = sup{||x(t)|| ^L

: t ∈ I},

(iv) the mapping x(t, ω) → v(t, s, x(s, ω), x ^s (·, ω)) from C(I, L ² (Ω, F, P )) into C(I, L ² (Ω, F, P )) is continuous in the topology generated by the norm ||x|| ² = sup{||x(t)|| ^L

: t ∈ I},

(v) define A = sup

t ∈I { Z

I

(q 1 (t, s)+q 2 (t, s))ds+ √ 2( Z

I

p ² ₁ (t, s)dG(s))

+2( Z

I

p ² ₂ (t, s)dG(s))

},

B(t) = |h(t, 0)| + R

I

q 3 (t, s)ds + 2( R

I

p ² ₃ (t, s)dG(s))

for t ∈ I,

B = sup {B(t) : t ∈ I}.

(vi) |h(t, x(t, ω)) − h(t, y(t, ω))| ¬ k|x(t, ω) − y(t, ω)| P −a.s., k ∈ [0, 1), (vii) M = A + k < 1,

(viii) for any fixed T > 0

ε→0 lim sup{||h(t, x(s)) − h(s, x(s))|| ^L

: s, t ∈ [0, T ], |t − s| ¬ ε} = 0 uniformly with respect to x ∈ U ⊂ K(Θ, R), where R = B/(1 − M).

Moreover, let

||x ^s || − = sup{||x(s + τ)|| ^L

: τ ∈ [−r, 0], s + τ ∈ [s − r, s]}

and

||x|| ¹ = sup{||x(t)|| ^L

: t ∈ [−r, T ]}.

Now we can formulate our main result and prove it similary as in [6].

Theorem 3.2 Let assumptions (i) - (viii) be saisfied. Then there exists at least one solution x ∈ C to problem (1.1) - (1.2).

Proof Observe that using assumption (ii) and the fact that for any norm|| · || it holds ||a + b + c|| ² ¬ 2||a|| ² + 4||b|| ² + 4||c|| ² we can rewrite (3.1) as follows

||

Z

I

v(t, s, x(s, ω), x s (·, ω))dβ(s)|| ^L

¬ ( Z

I

||v(t, s, x(s, ω), x ^s (·, ω))|| ² L

dG(s))

¬ √ 2||x|| ²



 Z

I

p ² ₁ (t, s)dG(s)





12

+

(3.2) 2||x ^s || −



 Z

I

p ² ₂ (t, s)dG(s)





12

+ 2



 Z

I

p ² ₃ (t, s)dG(s)





12

.

For x ∈ C we define the operator H on C by (Hx)(t, ω) = h(t, x(t, ω)) +

Z

I

u(t, s, x(s, ω), x s (·, ω))ds

(3.3) + Z

I

v(t, s, x(s, ω), x s (·, ω))dβ(s).

Observe that from (vi) we get

|h(t, x(t, ω))| = |h(t, x(t, ω)) − h(t, 0) + h(t, 0)| ¬ |h(t, x(t, ω)) − h(t, 0)| + |h(t, 0)|

¬ k|x(t, ω)| + |h(t, 0)|

so |h(t, x(t, ω))| ¬ k|x(t, ω)| + |h(t, 0)|.

Now, using (i), (ii), (vi) and (3.2) we have

||Hx(t)|| ^L

¬ k||x(t)|| ^L

+ |h(t, 0)| + Z

I

||u(t, s, x(s, ω), x ^s (·, ω))|| ^L

ds

+



 Z

I

||v(t, s, x(s, ω), x ^s (·, ω))|| ² L

dG(s)





12

¬ k||x|| 2 + |h(t, 0)| + Z

I

q 1 (t, s)||x(s)|| L

ds + Z

I

q 2 (t, s)||x ^s || L

ds + Z

I

q 3 (t, s)ds

+ √ 2



 ||x|| ²



 Z

I

p ² ₁ (t, s)dG(s)





12

+ √ 2||x ^s || −



 Z

I

p ² ₂ (t, s)||x ^s || ² L

dG(s)





12

+ √ 2



 Z

I

p ² ₃ (t, s)dG(s)





12



 

¬ |h(t, 0)| + Z

I

q 3 (t, s)ds + 2



 Z

I

p ² ₃ (t, s)dG(s)





12

+||x|| 1

 



  k + Z

I

q 1 (t, s)ds + Z

I

q 2 (t, s)ds + √ 2



 Z

I

p 1 (t, s) ² dG(s)





12

+2



 Z

I

p ² ₂ (t, s)dG(s)





12



 

  . Hence, using assumptions (v) and (vii), we obtain estimation

||Hx|| ¹ ¬ B + M||x|| ¹ ,

which proves that H transforms the space C into itself and moreover it maps the

ball K(Θ, R) into the same ball for radius R = B/(1 − M). Indeed, then ||Hx|| ¬

B + M ||x|| ¹ ¬ B + M · _1−M ^B = _1−M ^B = R.

Now, we shall prove the continuity of H in the ball K(Θ, R), i.e. we shall show that for any ε > 0 there exists δ such that for all t ∈ I

||Hx − Hy|| 1 < ε, whenever ||x − y|| 2 < δ.

Let x, y ∈ K(Θ, R), then from (vi) putting kδ ¹ = ε 1 we have

||h(t, x(t)) − h(t, y(t))|| ^L

< ε 1 , whenever ||x − y|| ² < δ 1 . Next, from (iii) we have

||u(t, s, x(s), x ^s (·)) − u(t, s, y(s), y ^s (·))|| ^L

< ε 2 , whenever ||x − y|| ² < δ 2 . From (iv) we get

||v(t, s, x(s), x ^s (·)) − v(t, s, y(s), y ^s (·))|| L

< ε 3 , whenever ||x − y|| 2 < δ 3 . Hence, we have for any given ε > 0, using (3.1)

||Hx(t) − Hy(t)|| ^L

¬ ||h(t, x(t)) − h(t, y(t))|| ^L

+ Z

I

||u(t, s, x(s), x ^s (·)) − u(t, s, y(s), y ^s (·))|| ^L

ds

+



 Z

I

||v(t, s, x(s), x ^s (·)) − v(t, s, y(s), y ^s (·))|| ² L

dG(s)





12

¬ ε ¹ + Z

I

ε 2 ds +



 Z

I

ε ² ₃ dG(s)





12

= ε ₁ + ε ₂ Z

I

ds + ε 3



 Z

I

dG(s)





12

(3.4) = ε ₁ + ε ₂ · T + ε 3 (G(T ) − G(0))

. Thus, by (3.4), δ = max{δ ¹ , δ 2 , δ 3 } for any given ε > 0 we get

||Hx − Hy|| ¹ < ε whenever ||x − y|| ² < δ, x, y ∈ K(Θ, R), so we get the continuity of operator H.

Next, we shall show that µ(HU) ¬ kµ(U). Namely, fix an arbitrary ε > 0 and

t, z ∈ I, |t − z| < ε. By, (3.3) for 0 ¬ z < t and x ∈ U ⊂ K(Θ, R) we obtain

||Hx(t) − Hx(z)|| ^L

¬ ||h(t, x(t)) − h(z, x(z)|| ^L

+ ||

Z

I

u(t, s, x(s, ω), x s (·, ω))ds

+ Z

I

v(t, s, x(s, ω), x s (·, ω))dβ(s) − Z

I

u(z, s, x(s, ω), x s (·, ω))ds

− Z

I

v(z, s, x(s, ω), x s (·, ω))dβ(s)|| L

¬ ||h(t, x(t)) − h(z, x(z)|| ^L

+|| Z

I

(u(t, s, x(s, ω), x s (·, ω)) − u(z, s, x(s, ω), x ^s (·, ω)))ds|| ^L

(3.5) +|| Z

I

(v(t, s, x(s, ω), x s (·, ω)) − v(z, s, x(s, ω), x ^s (·, ω)))dβ(s)|| ^L

.

Now, using assumptions (i), (ii) and (viii) and inequality (3.2) we get from (3.5)

||Hx(t) − Hx(z)|| ^L

¬ ||h(t, x(t)) − h(z, x(z)|| ^L

+

Z

I

(q 1 (t, s) − q ¹ (z, s))||x(s)|| ^L

ds + Z

I

(q 2 (t, s) − q ² (z, s))||x ^s || ^L

ds

+ Z

I

(q 3 (t, s) − q ³ (z, s))ds

+ √ 2



 



 Z

I

(p 1 (t, s) − p ¹ (z, s)) ² ||x(s)|| ² L

dG(s)





12

+ √ 2



 Z

I

(p ₂ (t, s) − p 2 (z, s)) ² ||x ^s || ² L

dG(s)





12

+ √ 2



 Z

I

(p 3 (t, s) − p ³ (z, s)) ² dG(s)





12



 

¬ ||h(t, x(t)) − h(z, x(z)|| L

+ T · R max _s {(q 1 (t, s) − q 1 (z, s))}

+T · R max _s {(q ² (t, s) − q ² (z, s))} + T max _s {(q ³ (t, s) − q ³ (z, s))}

+ √

2(G(T ) − G(0))

R {max _s {(p ¹ (t, s) − p ¹ (z, s)) ² }

+2(G(T ) − G(0))

R {max _s (p 2 (t, s) − p ² (z, s)) ² }

+2(G(T ) − G(0))

{max _s (p 3 (t, s) − p ³ (z, s)) ² }

.

Now, we recall the definition of the modulus of continuity, which is defined for a real function w as

ν T (w, ε) = sup{|w(t) − w(z)| : t, z ∈ I, |t − z| ¬ ε}, ε > 0.

Since we have assumed, that |t − z| < ε then using the continuity of functions q 1 , q 2 , q 3 , p 1 , p 2 , p 3 and G and letting ε → 0 we can see that

µ(HU ) = 0.

But 0 ¬ µ(U), therefore this proves that H is µ- contraction. To complete the proof

we shall apply Theorem 2.4. _■

Remark 3.3 We can generalize Theorem 3.2 putting both integrals in (1.1) on [0, t), t ∈ [0, ∞). The proof will be similar as in [6], [7].

4. Deterministic case. In the same way we can consider solutions of the following deterministic functional integral equation

(4.1) x(t) = h(t) + Z

I

u(t, s, x(s), x s (·))ds for I = [0, T ], T > 0

with the initial condition

(4.2) x(t) = ϕ(t) for t ∈ [−r, 0] ,

where the functions h : I → ℝ ⁿ , u : I × I × ℝ ⁿ × C → ℝ ^{n ,} ϕ : [ e −r, 0] → ℝ ^{n and}

the number r > 0 are given, while x : I → ℝ ⁿ is an unknown function.

Definition 4.1 The function x(t) ∈ C ¹ is a solution to problem (4.1) - (4.2) for t ∈ [−r, T ] if it satisfies (4.1) - (4.2) a.e. on I and is equal to the given initial function on [−r, 0].

Now, we introduce the formula for a measure of noncompactness . Namely, fix

x ∈ C ¹ , X ∈ 𝔐 C

, T > 0 and ε > 0.

Definition 4.2 ([2]) (4.3) µ(X) = lim

ε →0 sup{sup{|x(t) − x(s)| ℝ

: t, s ∈ [−r, T ], |t − s| ¬ ε}; x ∈ X}, where |a − b| ℝ

= p

(a 1 − b ¹ ) ² + ... + (a n − b ⁿ ) ² is a measure of noncompactness in the space C 1 .

It is also known from Arzela - Ascoli Theorem ([2]) that ker µ is the collection of all sets X ∈ 𝔐 C

such that all maps belonging to X are equicontinuous.

Let the following conditions ([7]) be fullfield:

(i) he following inequality

|u(t, s, x(s), x ^s (·)| ¬ q ¹ (t, s)|x(s)| + q ² (t, s)|x ^s (·)| + q ³ (t, s) a.e.

is satisfied for all t, s ∈ I and for nonnegative, continuous functions q 1 , q 2 and q 3 defined for t, s ∈ I,

(ii) the mapping x(t) → u(t, s, x(s), x ^s (·)) from C ² into C 2 is continuous, (iii) A = sup{ R ^t

0

(q 1 (t, s) + q 2 (t, s))ds

B(t) = |h(t)| + R t 0

q 3 (t, s)ds for t ∈ I, B = sup {B(t) : t ∈ I} < ∞.

(iv) for any fixed T > 0

ε→0 lim sup{|h(t) − h(s)| ℝ

: s, t ∈ I, |t − s| ¬ ε} = 0

uniformly with respect to x ∈ U ⊂ K(Θ, R), where R = (||h|| ^C

+ B)/(1 − A), (compare [7], p. 353)

Then using similar argumentation as in Section 3 we prove the following theorem.

Theorem 4.3 Let assumptions (i) - (iv) be satisfied. Then there exists at least one solution x ∈ C ¹ to problem (4.1) - (4.2).

References

[1] J. Banaś, Integrable solutions of Hammerstein and Urysohn integral equation, J. Austral.

Math. Soc, 46 (1989), 61–68.

[2] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in

Pure and Applied Mathematics, vol. 60, Marcel Dekker, New York and Basel, 1980.

[3] J. Banaś and M. Pasławska-Południak, Monotonic solutions of Urysohn integral equation on unbounded interval, Int. J. Comput. Math. Appl., 47 (2004), 1947–1954.

[4] J. Banaś, D. Szynal and S. Wędrychowicz, On existence, asymptotic behavior and stability of solutions of stochastic integral equations, Stoch. Annal. Appl., 9, (4) (1991), 363–385.

[5] R. Liptser and A. Shiryayev, Statistics of Random Processes, Springer, Berlin, 1977.

[6] D. Szynal and S. Wędrychowicz, On existence and asymptotic behavior of solutions of a non- linear stochastic integral equation, Ann. Mat. Pura Appl., 142, (4) (1985), 105–119.

[7] D. Szynal and S. Wędrychowicz, On existence and asymptotic behavior of random solutions of a class of stochastic functional - integral equations, Coll. Mat, 51 (1987), 349–364.

[8] D. Szynal and S. Wędrychowicz, On solutions of a stochastic integral equations of the Volterra - Fredholm type, Ann. Univ. Mariae Curie - Skłodowska, Sectio A, 53, (2) (1989), 107–122.

Monika Pasławska-Południak

Department of Mathematics, Rzeszów University of Technology W. Pola 2, 35-959 Rzeszów, Poland

E-mail: mpol@prz.edu.pl

(Received: 28.12.2009)