Volume 2010, Article ID 603819,16pages doi:10.1155/2010/603819
Research Article
Existence of Solutions of Nonlinear Stochastic
Volterra Fredholm Integral Equations of
Mixed Type
K. Balachandran
1and J.-H. Kim
21Department of Mathematics, Bharathiar University, Coimbatore 641 046, India
2Department of Mathematics, Yonsei University, Seoul 120-749, South Korea
Correspondence should be addressed to K. Balachandran,balachandran [email protected]
Received 13 August 2009; Accepted 19 January 2010
Academic Editor: Jewgeni Dshalalow
Copyrightq2010 K. Balachandran and J.-H. Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.
1. Introduction
Random or stochastic integral equations are important in the study of many physical phenomena in life sciences, engineering, and technology 1–13. Currently there are two basic versions of stochastic integral equations being studied by mathematical statisticians and probabilists namely, those integral equations involving Ito-Doob type of stochastic integrals and those which can be formed as probabilistic analogues of classical deterministic integral equations whose formulation involves the usual Lebesgue integral. Equations of the later category have been studied extensively by several authors 4, 10, 14–40. Many papers have been appeared on the problem of existence of solutions of nonlinear random integral equations and the results are established by applying various fixed point techniques. These methods are broadly classified into three categories:
iadmissibility theory,2,7,24,27,41–47,
iirandom contractor method,17,21,35,47–52,
All these methods are effectively used to study the existence of solutions for stochastic integral equations. Further asymptotic behaviour and stability of solutions of stochastic integral equations are discussed in the papers 33,42,50, 54,55,59,61–63. In this paper we will study the existence of random solutions of nonlinear stochastic integral equations of mixed type.
Consider a nonlinear stochastic integral equation of the form
xt;w ht, xt;w
t
0
k1t, τ;wf1τ, xτ;wdτ ∞
0
k2t, τ;wf2τ, xτ;wdτ t
0
k3t, τ;wf3τ, xτ;wdβτ,
1.1
wheret∈R , βtis a stochastic process and
a-iw∈Ω, the supporting set of the complete probability measure spaceΩ, A, μ, with theσ-algebraAand probability measureμ,
a-iixt;wis the unknown random function fort∈R , the nonnegative real numbers,
a-iiiht, xis a scalar function defined fort∈R andx∈R, the real line,
a-ivk1t, τ;wandk3t, τ;ware stochastic kernels defined fortandτsatisfying 0≤τ ≤
t <∞,
a-vk2t, τ;wis the stochastic kernel defined fortandτinR ,
a-vif1t, x, f2t, x, f3t, xare scalar functions defined fort ∈R andx∈R, the real
line.
The first and the second part of the stochastic integral 1.1 are to be understood as an ordinary Lebesque integral with probabilistic characterization, while the third part is an Ito-Doob stochastic integral. Our aim is to investigate the existence as well as uniqueness of random solutions of the stochastic integral equation 1.1by making use of “admissibility theory” that was first introduced by Tsokos40and fixed point theorems due to Krasnoselskii and Banach. The results generalize the previous results of2,7,24,27,41–46.
2. Preliminaries
Letβt;wbe the random process. We will assume that for eacht∈R , a minimalσ-algebra
At,At⊂A, is such thatβt;wis measurable with respect toAt. In addition, we will assume
that the minimalσ-algebraAtis an increasing family such that H1the random process{βt;w, At:t∈R }is a real martingale
H2there is a real continuous nondecreasing function,Ft, such that fors < twe have
E{|βt;w−βs;w|2} E{|βt;w−βs;w|2 : At} Ft−Fsμ- a.e. whereE
denotes the expected value of the random process.
In the definitions that follow, we will assume that xt;w is At measurable and that
E|xt;w|2<∞, for eacht∈R . Also we denote
E|xt;w|21/2xt;wL2Ω,A,μ
Ω|xt:w|
2dμw 1/2
Definition 2.1. Denote byCcthe linear space of all mean square continuous mapsxt;won
R and define a topology onCcby means of the following family of seminorms.
xt;wn sup
0≤t≤n
E|xt;w|21/2. 2.2
It is known that such a topology is metrizable and that the metric spaceCcis complete.
Definition 2.2. DefineCg⊂Ccto be the space of all mapsxt;wonR such that
E|xt;w|21/2≤agt, 2.3
wherea >0, a constant andgt>0, a continuous function onR . The norm in the spaceCg
is defined by
xt;wCg sup
t≥0
1
gt
E|xt;w|21/2
. 2.4
Definition 2.3. LetC ⊂Ccbe the space of mapsxt;wonR with{E|xt;w|2}1/2 < M, for
someM >0. The norm in spaceCis defined by
xt;wCsup
t≥0
E|xt;w|21/2. 2.5
Definition 2.4. The pair of Banach spaces B, D with B, D ⊂ Cc is called admissible with
respect to the operatorT :Cc → CcifTB⊂D.
Definition 2.5. We will callxt;wa random solution of the stochastic integral equation1.1
ifxt;w∈Ccfor eacht∈R and satisfies equation1.1μ-a.e.,for allt >0.
Definition 2.6. The Banach space Bis said to be stronger than Cg, if every sequence which
converges in the topology ofBconverges also in the topology ofCg.
Finally, letB, D ⊂ Cg be Banach spaces andT a linear operator fromCgintoCc. The
following lemma is well known13.
Lemma 2.7. LetT be a continuous operator fromCg intoCc. IfBandD are Banach spaces inCg
stronger thanCgand if the pairB, Dis admissible with respect toT, thenTis a continuous operator
Let us define the operators
T1xt;w t
0
k1t, τ;wxτ;wdτ, 2.6
T2xt;w ∞
0
k2t, τ;wxτ;wdτ, 2.7
T3xt;w t
0
k3t, τ;wxτ;wdβτ, 2.8
forxt;w∈Cg.
We state the following assumptions for our use.
a1The functionsf1t, xt;w, f2t, xt;w, andf3t, xt;ware continuous functions of
t∈R with values inL2Ω, A, μ.
a2For eachtandτinR , k2t, τ;whas values in the spaceL∞Ω, A, μand the functions
k1t, τ;wandk3t, τ;wfor eachtandτsuch that 0≤τ ≤t <∞has values in the space
L∞Ω, A, μ.
a3The stochastic kernelsk1t, τ;wandk3t, τ;ware essentially a bounded function with
respect to μfor every tand τ such that 0 ≤ τ ≤ t < ∞and continuous as maps from
{t, τ: 0≤τ≤t <∞}intoL∞Ω, A, μ.
a4The stochastic kernelk2t, τ;wis essentially a bounded function with respect toμfor every
tandτinR and continuous as maps from{t, τ: 0≤τ ≤t <∞}intoL∞Ω, A, μ.
Define for 0≤τ ≤t <∞,
|k1t, τ;w|μ−ess sup
w∈Ω|
k1t, τ;w|,
|k2t, τ;w|μ−ess sup
w∈Ω|k2t, τ;w|,
|k3t, τ;w|μ−ess sup
w∈Ω
|k3t, τ;w|.
2.9
The assumptionsa1–a4imply that ifxt;w∈Cc, then for eacht∈R ,
E|k3t, τ;wxτ;w|2≤ |k3t, τ;w|2E|xt;w|2. 2.10
Because of the continuity assumptions on |k3t, τ;w| and E|xτ;w|2 it follows from the above
inequality that
t
0
E|k3t, τ;wxτ;w|2dFτ<∞, 2.11
Lemma 2.8. Under the assumptionsa1–a4, (H1) and (H2),T1, T2, andT3are continuous linear
operators fromCgintoCcprovided
∞
0
|k3t, τ;w|2g2τdτ ≤N <∞ for some N >0. 2.12
Proof. It is easy to show thatT1, T2andT3are linear maps fromCgintoCc. The continuity of
T1andT2are also easy to prove8,13. We will prove thatT3is continuous.
Letxt;w∈Cg. Then
E|T3xt;w|2E
t
0
k3t, τ;wxτ;wdβτ 2
t
0
E|k3t, τ;wxτ;w|2dFτ
≤ t
0
|k3t, τ;w|2E|xt;w|2dFτ
≤ xt;w2Cg
t
0
|k3t, τ;w|2g2τdFτ, t < n.
2.13
Hence, on compact intervals0, n
sup
0≤t≤n
T3xt;wL2Ω,A,μ≤ xt;wCg ⎧ ⎨ ⎩0sup≤t≤n
t
0
|k3t, τ;w|2g2τdFτ 1/2⎫⎬
⎭
≤N1xt;wCg,
2.14
whereN1is a constant depends uponn. This proves the continuity ofT3. The linearity ofT3
is obvious.
To show thatT2mapsCgintoCc. Letyt;w
∞
0k2t, τ;wxτ;wdτ. Then
yt1;w−yt2;wL2Ω,A,μxt;wCg ∞
0
|k2t1, τ;w−k2t2, τ;w|2g2τdτ. 2.15
The right-hand side of the above inequality goes to zero ast2 → t1, sincek2t, τ;wgτ ∈
L2Ω, A, μ. Thus, this proves thatT2 mapsCg intoCc. The proof of the continuity ofT2 is
Let the operators T1, T2, and T3 be as defined in 2.6, 2.7, and 2.8 and let the
assumptions ofLemma 2.8hold. Then it follows fromLemma 2.7that, ifBandDare Banach spaces stronger thanCg and the pairB, Dis admissible with respect to the operatorsT1, T2
andT3, thenT1, T2, andT3are continuous fromBintoD. Thus, there exist positive constants
K1, K2, andK3such that
T1xt;wD≤K1xt;wB,
T2xt;wD≤K2xt;wB,
T3xt;wD≤K3xt;wB.
2.16
The constantsK1, K2, K3are the bounds of the operatorT1, T2, T3.
Theorem 2.9Krasnoselskii Theorem. LetSbe a closed, bounded and convex subset of a Banach spaceXand letU1andU2be operators onSsatisfying the following conditions:
iU1x U2y∈Swheneverx, y∈S,
iiU1is a contraction operator onS,
iiiU2is completely continuous.
Then there is at least one pointx∗∈Ssuch thatU1x∗ U2x∗ x∗.
3. Main Results
In this section we will prove the main result of this paper.
Theorem 3.1. For the stochastic integral equation1.1assume the following conditions
iBandDare Banach spaces inCg, stronger thanCg, such thatB, Dis admissible with
respect to the operatorsT1, T2, andT3defined by2.6,2.7, and2.8;
ii∞0 |k2t, τ;w|2g2τdτ≤N <∞for someN >0;
iiixt;w → f1t, xt;wis a continuous map from
Sxt;w:xt;w∈D, xt;wD≤ρ 3.1
with values inBsatisfying
f1t, xt;w−f1
t, yt;wB≤λ1xt;w−yt;wD 3.2
forxt;w, yt;w∈Sandλ1≥0 a constant;
ivxt;w → f2t, xt;wis a completely continuous map fromSintoB;
vxt;w → f3t, xt;wis a continuous map fromSwith values inBsatisfying f3t, xt;w−f3
t, yt;wB≤λ3xt;w−yt;wD 3.3
vixt;w → ht, xt;wis a continuous map fromSintoDsuch that
ht, xt;w−h
t, yt;wD≤γxt;w−yt;wD 3.4
forxt;w, yt;w∈Sandγ >0 a constant.
Then there exists a unique random solution of 1.1inSprovided
γ K1λ1 K3λ3<1,
γht,0D K1f1t,0B K2f2t, xt;wB K3f3t,0B ≤ρ1−γ−K1λ1−K3λ3
,
3.5
whereK1, K2, andK3are defined by2.16.
Proof. The setSclosed, bounded, and convex inD. Letxt;w, yt;w∈S. Then define the operatorU1:S → Dby
U1xt;w ht, xt;w t
0
k1t, τ;wf1τ, xτ;wdτ t
0
k3t, τ;wf3τ, xτ;wdβτ.
3.6
We will show thatU1 is a contraction mapping and thatU1S ⊂ S. Letxt;w, yt;w ∈ S.
Then
U1xt;w−
U1y
t;w ht, xt;w−ht, yt;w
t
0
k1t, τ;w
f1τ, xτ;w−f1
τ, yτ;wdτ
t
0
k3t, τ;w
f3τ, xτ;w−f3
τ, yτ;wdβτ.
3.7
From our assumption it is clear that U1xt;w− U1yt;w ∈ D and f1τ, xτ;w −
f1τ, yτ;w, f3τ, xτ;w−f3τ, yτ;w∈B. Furthermore
U1xt;w−
U1y
t;wD≤ht, xt;w−ht, yt;wD
K1f1τ, xτ;w−f1
τ, yτ;wB
K3f3τ, xτ;w−f3
τ, yτ;wB
≤γ K1λ1 K3λ3xt;w−yt;w.
Sinceγ K1λ1 K3λ3<1,U1is a contraction operator. Next we show thatU1S⊂S. From
3.6, we have
U1xt;wDht, xt;wD
t
0
k1t, τ;wf1τ, xτ;wdτ
t
0
k3t, τ;wf3τ, xτ;wdβτ
≤ ht,0D γ K1λ1 K3λ3
xt;w
λ1f1t,0B λ3ft,0B.
3.9
Sincext;w∈S, by hypothesis, we haveU1xt;wD≤ρwhich implies thatU1S⊂S.
Let us define the operatorU2:S → Das
U2xt;w ∞
0
k2t, τ;wf2τ, xτ;wdτ. 3.10
It is clear thatU2 is composition of continuous mapT2and completely continuous mapf2.
HenceU2is completely continuous. Furthermore, ifxt;w, yt;w∈S, we have
U1xt;w U1yt;wD≤ ht, xt;wD
K1f1τ, xτ;wB K2f2τ, yτ;wB
K3f3τ, xτ;wB
≤ ht,0D γ K1λ1 K3λ3
ρ K1f1t,0B
K2f2t, xt;wB K3f3t,0B
≤ρ.
3.11
This shows that ifxt;w, yt;w ∈S, thenU1xt;w U2yt;w∈ S. Hence, applying
Krasnoselskii’s fixed point theorem, we can conclude that there exists a random solution of
1.1in the setS.
We will now consider the case under which the stochastic integral equation 1.1
Theorem 3.2. For the stochastic integral equation1.1assume the following conditions
iBandDare Banach spaces inCg, stronger thanCg, such thatB, Dis admissible with
respect to the operatorsT1, T2andT3defined by2.6,2.7, and2.8;
ii∞0 |k2t, τ;w|2g2τdτ≤N <∞for someN >0;
iiixt;w → f1t, xt;wis a continuous map from
Sxt;w: xt;w∈D, xt;wD≤ρ 3.12
with values inBsatisfying
f1t, xt;w−f1
t, yt;wB≤λ1xt;w−yt;wD 3.13
forxt;w, yt;w∈Sandλ1≥0 a constant;
ivxt;w → f2t, xt;wis a continuous map fromSwith values inBsatisfying f2t, xt;w−f2
t, yt;wB≤λ2xt;w−yt;wD 3.14
forxt;w, yt;w∈Sandλ2≥a constant;
vxt;w → f3t, xt;wis a continuous map fromSwith values inBsatisfying f3t, xt;w−f3
t, yt;wB≤λ3xt;w−yt;wD 3.15
forxt;w, yt;w∈Sandλ3a constant;
vixt;w → ht, xt;wis a continuous map fromSintoDsuch that
ht, xt;w−h
t, yt;wD≤γxt;w−yt;wD 3.16
forxt;w, yt;w∈Sandγ >0 a constant.
Then there exists a unique random solution of 1.1inSprovided
γ K1λ1 K2λ2 K3λ3<1,
γht,0D K1f1t,0B K2f2t,0B K3f3t,0B ≤ρ1−γ−K1λ1−K2λ2−K3λ3
,
3.17
whereK1, K2, andK3are defined by2.16.
Proof. Define the operatorU:S → Das follows
Uxt;w ht, xt;w
t
0
k1t, τ;wf1τ, xτ;wdτ ∞
0
k2t, τ;wf2τ, xτ;wdτ t
0
k3t, τ;wf3τ, xτ;wdβτ.
We will show that U is a contraction operator on S and that US ⊂ S. Let
xt;w, yt;w ∈S. ThenUxt;w−Uyt;w ∈DasUS ⊂ DandDis a Banach space. Also
Uxt;w−Uyt;w
D
≤ht, xt;w−ht, yt;wD
t
0
k1t, τ;w
f1τ, xτ;w−f1
τ, yτ;wdτ
D
∞
0
k2t, τ;w
f2τ, xτ;w−f2
τ, yτ;wdτ
D t 0
k3t, τ;w
f3τ, xτ;w−f3
τ, yτ;wdβτ
D
.
3.19
Thus, in view of2.16, we have
Uxt;w−Uyt;wD
≤γxt;w−yt;wD K1f1t, xt;w−f1t, yt;wB
K2f2t, xt;w−f2
t, yt;wB
K3f3t, xt;w−f3
t, yt;wB
≤γ K1λ1 K2λ2 K3λ3xt;w−yt;wD.
3.20
Sinceγ K1λ1 K2λ2 K3λ3<1,Uis a contraction operator onS.
We will now show thatUS⊂S. For anyxt;w∈S, we have
Uxt;wD≤ ht, xt;wD
t
0
k1t, τ;wf1τ, xτ;wdτ D ∞ 0
k2t, τ;wf2τ, xτ;wdτ D t 0
k3t, τ;wf3τ, xτ;wdβτ
D
≤ ht, xt;wD K1f1t, xt;wB
K2f2t, xt;wB K3f3t, xt;wB
≤γxt;wD γht,0D λ1K1xt;wD K1f1t,0B
λ2K2xt;wD K2f2t,0B
λ3K3xt;wD K3f3t,0B.
Sincext;wD≤ρ, it follows that
Uxt;wD≤γht,0D ργ K1λ1 K2λ2 K3λ3
K1f1t,0B K2f2t,0B K3f3t,0B.
3.22
Using the condition that
γht,0D K1f1t,0B K2f2t,0B K3f3t,0B ≤ρ1−γ−K1λ1−K2λ2−K3λ3
,
3.23
we have from3.18
Uxt;wD≤ρ. 3.24
HenceUxt;w∈Sfor allxt;w∈SorUS⊂S. Thus the condition of Banach’s fixed point theorem is satisfied and hence there exists a fixed pointxt;w ∈ Ssuch that Uxt;w xt;w. That is,
Uxt;w ht, xt;w
t
0
k1t, τ;wf1τ, xτ;wdτ ∞
0
k2t, τ;wf2τ, xτ;wdτ t
0
k3t, τ;wf3τ, xτ;wdβτ
xt;w.
3.25
4. Applications
In this section we will give some application ofTheorem 3.2.
Theorem 4.1. Suppose the stochastic integral equation1.1satisfies the following conditions:
ithere exists a constantA >0 and a continuous functiongt, such that
t
0
|k1t, τ;w|2g2τdτ ∞
0
|k2t, τ;w|2g2τdτ t
0
|k3t, τ;w|2g2τdτ < A; 4.1
iifit, x, i1,2,3 are continuous functions onR ×R, such thatfit,0∈CgR , Rand
|fit, x−fit, y| ≤λigt|x−y|,forx, y∈Rand 0≤λi<1, i1,2,3;
iiiht, xis a continuous functions onR ×R, such that|ht, x−ht, y| ≤ γ|x−y|,for x, y∈Rand 0≤γ <1.
Then there exists a unique random solutionxt;wof1.1such that
xt;wC ≤ρ 4.2
Proof. It is easy to show that the hypothesis ofTheorem 3.2are satisfied by simply showing the pair of spaces Cg, Cc is admissible with respect to the operators T1, T2, andT3. This
follows fromLemma 2.8.
Corollary 4.2. Suppose the stochastic integral equation1.1satisfies the following conditions:
it0|k1t, τ;w|2dτ ∞
0 |k2t, τ;w|2dτ t
0|k3t, τ;w|2dτ < A;
iifit, x, i1,2,3 are continuous functions onR ×R, such thatfit,0∈CgR , Rand
|fit, x−fit, y| ≤λigt|x−y|,forx, y∈Rand 0≤λi<1, i1,2,3;
iiiht, xis a continuous functions onR ×R, such that|ht, x−ht, y| ≤ γ|x−y|,for x, y∈Rand 0≤γ <1.
Then there exists a unique random solutionxt;wof1.1such that
xt;wC ≤ρ 4.3
providedht,0,fit,0Cg, i1,2,3 are small enough.
Proof. Takegt 1 inTheorem 4.1.
Corollary 4.3. Suppose the stochastic integral equation1.1satisfies the following conditions:
i|kit, τ;w| ≤A,i1,2,3 andt0g2ττ <∞;
iisame as conditionsiv,v, andviinTheorem 3.2.
Then there exists a unique random solution of 1.1providedγ, ht,0C andfit,0Cg fori1,2,3 small enough.
Proof. We will show that the pair isCg, Ccadmissible with respect to the operatorT2. Let
xt;w∈Cg. Then
sup
0≤t
T2xt;wCg ≤sup
0≤t
∞
0
|k2t, τ;w|2xτ;w2L2dτ 1/2
≤ xt;wCgA
∞
0
g2τdτ
4.4
which implies that the pairCg, Ccis admissible. Similarly we can show that the pairCg, Cc
is admissible with respect to the operatorsT1, T3. It is easy to check the other conditions of
Theorem 3.2and hence there exists a unique random solution of equation of the stochastic integral equation1.1.
Remark 4.4. Using the same argument one can establish the existence of a unique random
solution of the following general stochastic integral equation
xt;w ht, xt;w
n
i1 t
0
ait, τ;wfiτ, xτ;wdτ
n
i1 ∞
0
bit, τ;wgiτ, xτ;wdτ
n
i1 t
0
cit, τ;wkiτ, xτ;wdβτ,
whereh, ki, ai, bi, ci, gi, fi, andβsatisfy appropriate conditions. This general case is treated in
a separate paper.
5. Example
Consider the following nonlinear stochastic integral equation:
xt;w 1
4sinxt;w
t
0
sint
4 e
−s−x2s;w
ds
∞
0
e−t−s
1 |xs;w|ds
1 8
t
0
ln1 |xs;w|dβs, t∈R ,
5.1
where βt is a stochastic process. This equation is a particular case of general stochastic integral equation occurring in mathematical biology and chemotherapy10–13. The above equation takes the form of1.1with
k1t, s, w
sint
4 e
−s, k
2t, s, w e−t−s, k3t, s, w
1
4, ht, xt;w
sinxt;w
4
f1s, xs;w e−x 2s;w
, f2s, xs;w 1
1 |xs;w|,
f3s, xs;w
1
2ln1 |xs;w|.
5.2
TakeBDCg CcCandgt 1. It is easy to see thatγ 1/4,K1 K3 1/4,K2 1,
λ11,λ21/4, andλ3 1/2. Furtherγ K1λ1 K2λ2 K3λ3 7/8<1 and by takingρ≥10,
the other condition ofTheorem 3.2is satisfied. It is clear that5.1satisfies assumptionsito
viofTheorem 3.2. Hence there exists a unique random solution for5.1.
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9 W. J. Padgett and C. P. Tsokos, “A new stochastic formulation of a population growth problem,”
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10 R. Subramaniam, K. Balachandran, and J. K. Kim, “Existence of solutions of a stochastic integral equation with an application from the theory of epidemics,” Nonlinear Functional Analysis and
Applications, vol. 5, no. 1, pp. 23–29, 2000.
11 D. Szynal and S. We¸drychowicz, “On solutions of a stochastic integral equation of the Volterra type with applications for chemotherapy,” Journal of Applied Probability, vol. 25, no. 2, pp. 257–267, 1988.
12 C. P. Tsokos and W. J. Padgett, Random Integral Equations with Applications to Stochastic Sytems, vol. 233 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1971.
13 C. P. Tsokos and W. J. Padgett, Random Integral Equations with Applications to Life Sciences and
Engineering, vol. 10 of Mathematics in Science and Engineering, Academic Press, London, UK, 1974. 14 A. T. Bharucha-Reid, “On random solutions of Fredholm integral equations,” Bulletin of the American
Mathematical Society, vol. 66, pp. 104–109, 1960.
15 S. T. Hardiman and C. P. Tsokos, “Existence theory for nonlinear random integral equations using the Banach-Steinhaus theorem,” Mathematische Nachrichten, vol. 63, pp. 311–316, 1974.
16 S. T. Hardiman and C. P. Tsokos, “Existence theorems for non-linear random integral equations with time lags,” International Journal of Systems Science, vol. 7, no. 8, pp. 879–900, 1976.
17 H. H. Kuo, “On integral contractors,” Journal of Integral Equations, vol. 1, pp. 35–46, 1979.
18 A. C. H. Lee and W. J. Padgett, “On a heavily nonlinear stochastic integral equation,” Utilitas
Mathematica, vol. 9, pp. 123–138, 1976.
19 A. C. H. Lee and W. J. Padgett, “Some approximate solutions of random operator equations,” Bulletin
of the Institute of Mathematics. Academia Sinica, vol. 5, no. 2, pp. 345–358, 1977.
20 A. C. H. Lee and W. J. Padgett, “On random nonlinear contractions,” Mathematical Systems Theory, vol. 11, no. 1, pp. 77–84, 1977.
21 A. C. H. Lee and W. J. Padgett, “Random contractors and the solution of random nonlinear equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 1, no. 2, pp. 175–185, 1976/77.
22 A. C. H. Lee and W. J. Padgett, “On a class of stochastic integral equations of mixed type,” Information
and Computation, vol. 34, no. 4, pp. 339–347, 1977.
23 M. N. Manougian, A. N. V. Rao, and C. P. Tsokos, “On a nonlinear stochastic integral equation with application to control systems,” Annali di Matematica Pura ed Applicata, vol. 110, pp. 211–222, 1976.
24 J. S. Milton, W. J. Padgett, and C. P. Tsokos, “On the existence and uniqueness of a random solution to a perturbed random integral equation of the Fredholm type,” SIAM Journal on Applied Mathematics, vol. 22, pp. 194–208, 1972.
25 J. S. Milton and C. P. Tsokos, “On a random solution of a nonlinear perturbed stochastic integral equation of the Volterra type,” Bulletin of the Australian Mathematical Society, vol. 9, pp. 227–237, 1973.
26 J. S. Milton and C. P. Tsokos, “On a class of nonlinear stochastic integral equations,” Mathematische
Nachrichten, vol. 60, pp. 71–78, 1974.
27 J. S. Milton and C. P. Tsokos, “On the existence of random solutions of a non-linear perturbed random integral equation,” International Journal of Systems Science, vol. 9, no. 5, pp. 483–491, 1978.
28 J. S Milton and C. P. Tsokos, “On a nonlinear perturbed stochastic integral equation,” Journal of
Mathematical and Physical Sciences, vol. 5, pp. 361–374, 1971.
29 J. S. Milton, C. P. Tsokos, and S. T. Hardiman, “A stochastic model for metabolizing systems with computer simulation,” Journal of Statistical Physics, vol. 8, pp. 79–101, 1973.
30 H. Onose, “On the boundedness of random solutions of nonlinear stochastic integral equations,”
Bulletin of the Faculty of Science. Ibaraki University. Series A, no. 18, pp. 49–53, 1986.
31 W. J. Padgett, “On a random Volterra integral equation,” Mathematical Systems Theory, vol. 7, pp. 164– 169, 1973.
32 W. J. Padgett and C. P. Tsokos, “Random solution of a stochastic integral equation: almost sure and mean square convergence of successive approximations,” International Journal of Systems Science, vol. 4, pp. 605–612, 1973.
33 A. N. V. Rao and C. P. Tsokos, “On the existence and stability behavior of a stochastic integral equation in a Banach space,” Problems of Control and Information, vol. 5, no. 1, pp. 87–95, 1976.
34 A. N. V. Rao and C. P. Tsokos, “Existence and boundedness of random solutions to stochastic functional integral equations,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 29, no. 3-4, pp. 283–288, 1977.
35 A. N. V. Rao and W. J. Padgett, “On the solution of a class of stochastic integral systems,” Journal of
36 A. N. V. Rao and C. P. Tsokos, “On a class of stochastic functional integral equations,” Colloquium
Mathematicum, vol. 35, no. 1, pp. 141–146, 1976.
37 V. Sree Hari Rao, “Topological methods for the study of nonlinear mixed stochastic integral equations,” Journal of Mathematical Analysis and Applications, vol. 74, no. 1, pp. 311–317, 1980.
38 V. Sree Hari Rao, “On random solutions of Volterra-Fredholm integral equations,” Pacific Journal of
Mathematics, vol. 108, no. 2, pp. 397–405, 1983.
39 D. Szynal and S. We¸drychowicz, “On solutions of some nonlinear stochastic integral equations,”
Yokohama Mathematical Journal, vol. 41, no. 1, pp. 31–37, 1993.
40 C. P. Tsokos, “On a stochastic integral equation of the Volterra type,” Mathematical Systems Theory, vol. 3, pp. 222–231, 1969.
41 W. J. Padgett, “On non-linear perturbations of stochastic Volterra integral equations,” International
Journal of Systems Science, vol. 4, pp. 795–802, 1973.
42 W. J. Padgett, “Almost surely continuous solutions of a nonlinear stochastic integral equation,”
Mathematical Systems Theory, vol. 10, no. 1, pp. 69–75, 1976.
43 W. J. Padgett and C. P. Tsokos, “On a stochastic integral equation of the Fredholm type,” Zeitschrift f ¨ur
Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 23, pp. 22–31, 1972.
44 W. J. Padgett and C. P. Tsokos, “On stochastic integro-differential equation of Volterra type,” SIAM
Journal on Applied Mathematics, vol. 23, pp. 499–512, 1972.
45 A. N. V. Rao and C. P. Tsokos, “On the existence of a random solution to a nonlinear perturbed stochastic integral equation,” Annals of the Institute of Statistical Mathematics, vol. 28, no. 1, pp. 99– 109, 1976.
46 A. N. V. Rao and C. P. Tsokos, “Existence and boundedness of solutions of a stochastic integral system,” Bulletin of the Calcutta Mathematical Society, vol. 69, no. 1, pp. 1–12, 1977.
47 J. Turo, “Existence and uniqueness of random solutions of nonlinear stochastic functional integral equations,” Acta Scientiarum Mathematicarum, vol. 44, no. 3-4, pp. 321–328, 1982.
48 A. C. H. Lee and W. J. Padgett, “Random contractors with random nonlinear majorant functions,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 3, no. 5, pp. 707–715, 1979.
49 A. C. H. Lee and W. J. Padgett, “Solution of random operator equations by random step-contractors,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 4, no. 1, pp. 145–151, 1980.
50 W. J. Padgett and A. N. V. Rao, “Solution of a stochastic integral equation using integral contractors,”
Information and Control, vol. 41, no. 1, pp. 56–66, 1979.
51 A. N. V. Rao and C. P. Tsokos, “On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation,” Information and
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52 R. Subramaniam and K. Balachandran, “Existence of solutions of stochastic integral equations using integral contractors,” Libertas Mathematica, vol. 17, pp. 89–100, 1997.
53 K. Balachandran, K. Sumathy, and H. H. Kuo, “Existence of solutions of general nonlinear stochastic Volterra Fredholm integral equations,” Stochastic Analysis and Applications, vol. 23, no. 4, pp. 827–851, 2005.
54 J. Bana´s, D. Szynal, and S. Wedrychowicz, “On existence, asymptotic behaviour and stability of solutions of stochastic integral equations,” Stochastic Analysis and Applications, vol. 9, no. 4, pp. 363– 385, 1991.
55 H. Gacki, T. Szarek, and S. Wedrychowicz, “On existence, and stability of solutions of stochastic integral equations,” Indian Journal of Pure and Applied Mathematics, vol. 29, no. 2, pp. 175–189, 1998.
56 R. Subramaniam and K. Balachandran, “Existence of solutions of general nonlinear stochastic integral equations,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 6, pp. 775–789, 1997.
57 R. Subramaniam and K. Balachandran, “Existence of solutions of a class of stochastic Volterra integral equations with applications to chemotherapy,” Journal Australian Mathematical Society. Series B, vol. 41, no. 1, pp. 93–104, 1999.
58 R. Subramaniam, K. Balachandran, and J. K. Kim, “Existence of random solutions of a general class of stochastic functional integral equations,” Stochastic Analysis and Applications, vol. 21, no. 5, pp. 1189– 1205, 2003.
59 D. Szynal and S. We¸drychowicz, “On existence and asymptotic behaviour of solutions of a nonlinear stochastic integral equation,” Annali di Matematica Pura ed Applicata, vol. 142, pp. 105–119, 1985.
60 D. Szynal and S. We¸drychowicz, “On existence and an asymptotic behavior of random solutions of a class of stochastic functional-integral equations,” Colloquium Mathematicum, vol. 51, pp. 349–364, 1987.
62 S. T. Hardiman and C. P. Tsokos, “Existence and stability behavior of random solutions of a system of nonlinear random equations,” Information Sciences, vol. 9, no. 4, pp. 299–313, 1975.
