On a class of nonlinear stochastic integral equations

Nonlinear Analysis: Real World Applications 54 (2020) 103104

Contents lists available at ScienceDirect

Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

On a class of nonlinear stochastic integral equations R. Negrea Department of Mathematics, Politehnica University of Timisoara, 300006, Romania

article

info

Article history: Received 19 October 2019 Received in revised form 16 January 2020 Accepted 17 January 2020 Available online xxxx

abstract We prove some existence and uniqueness results for a nonlinear stochastic integral equation using fixed-point theory methods to ensure the convergence of the successive approximations to the unique random solution. © 2020 Elsevier Ltd. All rights reserved.

Keywords: Stochastic equation Integral equation Fixed point methods Renormalization of Banach space

1. Introduction We will investigate the existence, uniqueness and continuous dependence of the random solution for the stochastic integral equation ∫ t x(t, ω) = h(t, ω) + k(t, s, ω)f (s, x(s, ω))ds, t ≥ 0, (1) 0

where ω ∈ Ω , Ω being the support of a complete probability measure space (Ω , A, P ). We assume that the stochastic process x(t, ω) and the stochastic free term h(t, ω) are continuous real-valued stochastic processes with x(t, ·), h(t, ·) ∈ L2 (Ω , A, P ), t ∈ R+ . The values of the stochastic kernel k(t, s, ω) for fixed t and s, 0 ≤ s ≤ t < ∞ are in L∞ (Ω , A, P ) so that the product of k(t, s, ω) and f (t, x(t, ω)) will always be in L2 (Ω , A, P ) and the mapping (t, s) ↦→ k(t, s, ω), 0 ≤ s ≤ t < ∞, is continuous into the Banach space L∞ (Ω , A, P ). The stochastic process x(t, ω) is called a random solution of the stochastic integral equation (1) if, for each t ∈ R+ , x(t, ω) is a random variable which satisfies (1) P -almost surely. We introduce now a useful class of functions that permits us to go beyond contractions. E-mail address: [email protected]. URL: http://www.mat.upt.ro. https://doi.org/10.1016/j.nonrwa.2020.103104 1468-1218/© 2020 Elsevier Ltd. All rights reserved.

R. Negrea / Nonlinear Analysis: Real World Applications 54 (2020) 103104

2

Definition 1.1 ([1,2]). Let M be the class of all continuous nondecreasing functions φ : [0, ∞) → [0, ∞) ∑∞ such that x ↦→ x − φ(x) is nonnegative and strictly increasing on [0, ∞) and satisfies n=1 φ∗(n) (x0 ) < ∞ for some x0 > 0, where φ∗(n) is the nth iterate of φ. Let us discuss some basic properties of the functions in the class M. Firstly, if φ ∈ M, then φ(0) = 0 ∑∞ and 0 ≤ φ(x) < x for x > 0. Moreover, n=1 φ∗(n) (x) < ∞ for all x ∈ [0, x0 ]. Note that the condition ∑∞ ∗(n) (x0 ) < ∞ for some x0 > 0 is not implied by the other properties defining the class M (see the n=1 φ discussion in [2]). Examples of functions φ ∈ M are φ(x) = αx for α ∈ (0, 1). However, there are also functions in this class for which there does not exist a constant α ∈ (0, 1) such that φ(x) ≤ αx for all x ≥ 0. To illustrate this we consider a positive strictly nonincreasing sequence {an }n≥1 with properties: a a −an+2 limn→∞ an = 0, an − an+1 ≥ an+1 − an+2 , for all n ≥ 1, limn→∞ n+1 = 1 and limn→∞ n+1 = an an −an+1 2 ∑ an an+2 −(an+1 ) an+1 −an+2 , n ≥ 1. 1 and n≥1 an < ∞ . Also we consider the sequences αn = an −an+1 , and βn = an −an+1 Now define the function φ : [0, ∞) → [0, ∞) by φ(0) = 0 and { a2 , a1 ≤ x , φ(x) = αn x + βn , an+1 ≤ x < an , n ≥ 1 . The piecewise linear function φ is continuous on [0, ∞) and φ ([an+1 , an ]) = [an+2 , an+1 ] for all n ≥ 1. Indeed, we have that a1 a3 − (a2 )2 a2 − a3 + = a2 = lim φ(x) = φ(a1 ) , x↘a1 x↗a1 x↗a1 a1 − a2 a1 − a2 an+1 − an+2 an an+2 − (an+1 )2 lim φ(x) = lim (αn x + βn ) = an + = an+1 = φ(an ) , x↗an x↗an an − an+1 an − an+1 an − an+1 an−1 an+1 − (an )2 lim φ(x) = lim (αn−1 x + βn−1 ) = an + = an+1 , x↘an x↘an an−1 − an an−1 − an an+2 − an+3 an+1 an+3 − (an+2 )2 lim φ(x) = lim (αn+1 x + βn+1 ) = an+1 + = an+2 , x↗an+1 x↗an+1 an+1 − an+2 an+1 − an+2 an an+2 − (an+1 )2 an+1 − an+2 + = an+2 . lim φ(x) = lim (αn x + βn ) = an+1 x↘an+1 x↘an+1 an − an+1 an − an+1 lim φ(x) = lim (α1 x + β1 ) = a1

Since the sequence {an }n≥1 is strictly decreasing with limit zero, this implies that x > φ(x) for x > 0. On the other hand, limx→0 (x − φ(x)) = 0 since φ(x) ≤ an+1 for all x ∈ [an+1 , an ], n ≥ 1. Also, limx→∞ (x − φ(x)) = ∞ since φ(x) = a2 for x ≥ a1 . Furthermore, since by construction φ(an ) = an+1 ∑ ∑ for all n ≥ 1, we obtain for the iterate function φ∗(n) (x) that n≥2 φ∗(n) (a1 ) = n≥2 an+1 < ∞ , and therefore φ ∈ M. Note that there does not exist α ∈ (0, 1) such that φ(x) ≤ αx for all x ≥ 0 since a φ(an ) = an+1 ≤ α an for all n ≥ 1 is impossible as limn→∞ n+1 an = 1. From the point of view of applications, it is important to note that the existence of the derivative φ′ on the complement of the countable subset {an }n≥1 of [0, ∞), and the fact that φ′ (x) = αn for x ∈ (an+1 , an ) means that φ′ is decreasing which ensures that φ is concave (see [3]). Also the following relation (∗) is verified and it is important from the point of view of applications ∫ x ∫ x−y (∗) φ(x) − φ(y) = φ′ (s) ds ≤ φ′ (s) ds = φ(x − y) , 0 ≤ y < x . y

0

{ } As an example for a sequence {an }n≥1 with the above properties we define {an }n≥1 = n12 n≥1 for which 2n+3 we have that an − an+1 > an+1 − an+2 is equivalent with n22n+1 − (n+1) 2 (n+2)2 ≥ 0 and limn→∞ an = 0, (n+1)2 limn→∞

an+1 an

= 1 and αn =

n2 (2n+3) , (2n+1)(n+2)2

αn − αn+1 = −

βn =

2n2 +4n+1 , (n+2)2 (n+1)2 (2n+1)

n ≥ 1,

4(5 + 27n + 36n2 + 18n3 + 3n4 ) < 0, (n + 2)2 (n + 3)2 (3 + 8n + 4n2 )

n≥1.

R. Negrea / Nonlinear Analysis: Real World Applications 54 (2020) 103104

3

Fig. 1. Depiction of the first 20 terms of the sequence {αn }n≥1 .

Fig. 2. The graphs of φ(x) (thick line) and of the identity map f (x) = x (thin line), with some of the points (an , φ(an )) located (dots).

We see that the sequence {αn }n≥1 (depicted in Fig. 1) is strictly increasing with limn→∞ αn = 1, while clearly limn→∞ βn = 0. We observe that limx→0 φ(x) = 1 which assures that φ is not a contraction. Moreover for the choice of x {1} {an }n≥1 = n2 n≥1 one can have a look at Fig. 2, which compares the function φ with the identity map for x ∈ [a11 , a1 + 0.1]. Definition 1.2 ([4]). For a given continuous function g : [0, ∞) → (0, ∞), we denote by Cg (R+ , L2 (Ω , A, P )) the space of all continuous L2 (Ω , A, P )-valued random variables x(t, ω) such that }1/2 { [ ]}1/2 {∫ 2 2 E |x(t, ω)| = |x(t, ω)| dω ≤ M g(t), t ∈ R+ , Ω

for some constant M > 0. The norm of a function in the Banach space Cg (R+ , L2 (Ω , A, P )) is } { ∥x(t, ω)∥L2 (Ω,A,P ) ∥x(t, ω)∥Cg = sup . g(t) t≥0 To see that Cg (R+ , L2 (Ω , A, P )) is a Banach space, note that if {un }n≥1 is a Cauchy sequence of random variables, then for every T > 0 the fact that the space C([0, T ], L2 (Ω , A, P )) of L2 (Ω , A, P )-valued random { } variables is a Banach space if endowed with the norm supt∈[0,T ] ∥x(t, ω)∥L2 (Ω,A,P ) ensures the existence of the uniform limit uT (t) of {un (t)}n≥1 on [0, T ], with the random variable uT1 representing the restriction of uT2 to the time-interval [0, T1 ] if T2 > T1 > 0, due to the uniqueness of the limit. Thus we obtain as a limit a random variable u(t, ω), defined for all t ≥ 0. Passing to the pointwise time-limit in the relevant inequality, one can easily check that u ∈ Cg (R+ , L2 (Ω , A, P )) and ∥u(t, ω) − un (t, ω)∥Cg → 0 as n → ∞.

R. Negrea / Nonlinear Analysis: Real World Applications 54 (2020) 103104

4

2. Main results We now prove the main result of this paper. Theorem 2.1. Assume that there exist ε > 0, φ ∈ M and continuous functions g0 , g1 : [0, ∞) → (0, ∞) such that (i)

f (t, 0) ∈ Cg1 (R+ , L2 (Ω , A, P )) ;

(ii)

∥k(s, t, ω)∥L∞ (Ω,A,P ) ≤ g0 (s),

(iii) (iv)

s, t ∈ R+ ;

∫ t g0 (s)g1 (s) ds ; h(t, ω) ∈ Cg (R+ , L2 (Ω , A, P )) for g(t) = ε + 0 ( ) |x − y| |f (t, x) − f (t, y)| ≤ g1 (t)φ , t ∈ R+ , x, y ∈ R . g(t)

Then there exists a unique random solution x ∈ Cg (R+ , L2 (Ω , A, P )) of the integral equation (1), obtained as the limit of the sequence of successive approximations. Moreover, this solution is stable with respect to small perturbations of the stochastic free term h. Proof . We define the operator T : Cg (R+ , L2 (Ω , A, P )) → Cg (R+ , L2 (Ω , A, P )) , ∫ t T x(t, ω) = h(t, ω) + k(t, s, ω)f (s, x(s, ω))ds . 0

Let us first show that the operator T is well defined. For this, let M > 0 be such that ∥f (t, 0)∥L2 (Ω,A,P ) ≤ M g1 (t) ,

t ≥ 0.

For x(t, ω) ∈ Cg (R+ , L2 (Ω , A, P )) we have that ∥T x(t, ω)∥L2 (Ω,A,P ) ∥h(t, ω)∥L2 (Ω,A,P ) ≤ g(t) g(t) ∫ t ] [ 1 + ∥k(s, t, ω)∥L∞ (Ω,A,P ) ∥f (s, x(s, ω)) − f (s, 0)∥L2 (Ω,A,P ) + ∥f (s, 0)∥L2 (Ω,A,P ) ds g(t) 0 ) ( ∫ t ∫ t ∥h(t, ω)∥L2 (Ω,A,P ) ∥x(s, ω)∥L2 (Ω,A,P ) 1 M ≤ + ds + g1 (s)g0 (s)φ g1 (s)g0 (s) ds g(t) g(t) 0 g(s) g(t) 0 ( ) ∫ t { ∥h(t, ω)∥ } { ∥x(s, ω)∥ } 1 L2 (Ω,A,P ) L2 (Ω,A,P ) ≤ sup + g1 (s)g0 (s)φ sup ds g(t) g(t) 0 g(s) s≥0 t≥0 ∫ t M + g1 (s)g0 (s) ds g(t) 0 ∫ t ∫ t 1 M ≤ ∥h(t, ω)∥Cg + φ(∥x(t, ω)∥Cg ) · g1 (s)g0 (s) ds + g1 (s)g0 (s) ds g(t) 0 g(t) 0 ≤ ∥h(t, ω)∥Cg + φ(∥x(t, ω)∥Cg ) + M ≤ ∥h(t, ω)∥Cg + ∥x(t, ω)∥Cg + M since 0 ≤ φ(x) ≤ x, x ∈ R+ . This proves that T is well-defined on the space Cg (R+ , L2 (Ω , A, P )). In a similar way we see that if x(t, ω), y(t, ω) ∈ Cg (R+ , L2 (Ω , A, P )), then ∥T x(t, ω) − T y(t, ω)∥L2 (Ω,A,P ) g(t) ∫ t 1 ≤ ∥k(s, t, ω)∥L∞ (Ω,A,P ) ∥f (s, x(s, ω)) − f (s, y(s, ω))∥L2 (Ω,A,P ) ds g(t) 0

R. Negrea / Nonlinear Analysis: Real World Applications 54 (2020) 103104

5

( ) ∫ t ∥x(s, ω) − y(s, ω)∥L2 (Ω,A,P ) 1 g1 (s)g0 (s)φ ds ≤ g(t) 0 g(s) ( ∫ t { ∥x(s, ω) − y(s, ω)∥ }) 1 L2 (Ω,A,P ) ≤ g1 (s)g0 (s)φ sup ds g(t) 0 g(s) s≥0 ∫ ( ( ) t ) 1 ≤ φ ∥x(t, ω) − y(t, ω)∥Cg g1 (s)g0 (s) ds ≤ φ ∥x(t, ω) − y(t, ω)∥Cg . g(t) 0 Consequently ( ) ∥T x(t, ω) − T y(t, ω)∥Cg ≤ φ ∥x(t, ω) − y(t, ω)∥Cg .

(2)

In particular, (2) shows that T is continuous on Cg (R+ , L2 (Ω , A, P )), without being necessarily a contraction. Let us now fix x0 ∈ Cg (R+ , L2 (Ω , A, P )) and set dn = ∥T n x0 − T n−1 x0 ∥Cg

(3)

where T n is the nth iterate of the operator T . From (2) we infer that dn ≤ φ(dn−1 ), n ≥ 1.

(4)

In view of the properties of the function φ ∈ M, we deduce that the sequence {dn }n≥1 of nonnegative numbers is nonincreasing. Therefore it will converge to some limit d = limn→∞ dn ≥ 0. Actually d = 0 since letting n → ∞ in (4) we get d ≤ φ(d), which is only possible for d = 0. We claim that {T n x0 }n≥1 is a Cauchy sequence in the Banach space ∑∞ Cg (R+ , L2 (Ω , A, P )). Indeed, if x1 > 0 is such that n=1 φ∗(n) (x1 ) < ∞, since limn→∞ dn = 0, there exists some integer n0 ≥ 1 such that dn0 < x1 . Given ε > 0, for n > m > N + n0 , where N ≥ 1 is an integer to be chosen later on, we have ∥T n x0 − T m x0 ∥Cg = ∥ ≤

n−m ∑ k=0

φ(dm+k ) ≤

n−m−1 ∑

(T m+k+1 x0 − T m+k x0 )∥Cg ≤

k=0 n−m ∑ ∗(N +k)

φ

(dn0 ) ≤

N +n−m ∑ j=N

k=0

n−m−1 ∑

∥T m+k+1 x0 − T m+k x0 ∥Cg

k=0 ∞ ∑

φ∗(j) (x1 ) ≤

φ∗(j) (x1 ) → 0 for N → ∞ .

j=N

In the last part of this chain of inequalities we used the fact that (4) yields φ(dm+k ) ≤ φ(dm+k−1 ) ≤ φ∗(2) (dm+k−2 ) ≤ · · · ≤ φ∗(N +k) (dm−N ) ≤ · · · ≤ φ∗(N +k) (dn0 ) . The completeness of the Banach space Cg (R+ , L2 (Ω , A, P )) ensures the existence of x ¯ = limn→∞ T n x0 in n n+1 Cg (R+ , L2 (Ω , A, P )). Letting n → ∞ in the relation T (T x0 ) = T x0 , the continuity of T yields T x ¯=x ¯. Note that if y¯ were another fixed point of T , then by (2) we would have ( ) ∥¯ x − y¯∥Cg = ∥T x ¯ − T y¯∥Cg ≤ φ ∥¯ x − y¯∥Cg and this is only possible if ∥¯ x − y¯∥Cg = 0, that is, if x ¯ = y¯ (see [5]]. We have proved that T has a unique fixed point x ¯, so that the stochastic integral equation (1) has a unique solution x ¯ in Cg (R+ , L2 (Ω , A, P )) (see [6]). Let now hn → h in Cg (R+ , L2 (Ω , A, P )) as n → ∞, and let x ¯n be the unique solution of the stochastic integral equation ∫ t

k(t, s, ω)f (s, x(s, ω))ds, n ≥ 1 ,

x(t, ω) = hn (t, ω) + 0

(5)

R. Negrea / Nonlinear Analysis: Real World Applications 54 (2020) 103104

6

in Cg (R+ , L2 (Ω , A, P )). We can write the fact that x ¯n solves equation (5) as the nonlinear operator equation x ¯n = hn − h + T x ¯n

(6)

in Cg (R+ , L2 (Ω , A, P )). Given ε > 0, let δ = ε − φ(ε) > 0. If n0 ≥ 1 is such that ∥hn − h∥Cg < δ, n ≥ n0 , we will show that ∥¯ xn − x ¯∥Cg < ε, n ≥ n0 .

(7)

Indeed, (2) and (6) yield ( ) ∥¯ xn − x ¯∥Cg = ∥ (hn − h + T x ¯n ) − T x ¯∥Cg ≤ ∥hn − h∥Cg + ∥T x ¯n − T x ¯∥Cg ≤ ∥hn − h∥Cg + φ ∥¯ xn − x ¯∥Cg so that ( ) ∥¯ xn − x ¯∥Cg − φ ∥¯ xn − x ¯∥Cg ≤ ∥hn − h∥Cg < δ, n ≥ n0 . Since δ = ε − φ(ε) > 0 and φ ∈ M, the above inequality ensures (7). This proves the stability of the solution to (1) with respect to small perturbations of the stochastic free term (see [1,4,5]). Remark 2.1. The renormalization of the space C(R+ , L2 (Ω , A, P )) made in the proof of Theorem 2.1. enables us to improve some results in the same direction of Tsokos [7,8] and Tsokos and Padgett [9] where they suppose that ∥h(t, ω)∥L2 (Ω,A,P ) to be “small enough”, which in our hypotheses is not necessary. This renormalization represents the stochastic counterpart of Bielecki’s approach to deterministic differential equations (see the discussion in [5,10]). Particular choices of the functions g0 and g1 comprise the following examples 1. to g0 (t) = K0 > 0 and g1 (t) = K1 > 0 corresponds the linear function g(t) = ε + K0 K1 t on R+ ; 2. to g0 (t) = g1 (t) = e−βt with β > 0 corresponds the bounded and increasing function g(t) = −2βt 1 ≤ ε + 2β on R+ ; ε + 1−e2β K 1 3. to g0 (t) = g1 (t) = (1+t) α with K > 0 and α > 2 corresponds the bounded and increasing function g(t) = ε +

K 2 [1−(1+t)1−2α ] 2α−1

on R+ .

Note that when g is bounded and increasing on R+ , then g(0) > 0 ensures that the norm on C(R+ , L2 (Ω , A, P )) is equivalent to the supremum norm (this being a manifestation of the renormalization pioneered by 1 (t) Bielecki [10]). This is relevant for the choices of f of the form f (t, x) = αgg(t) x with α ∈ (0, 1), so that hypothesis (iv) holds for the function φ(x) = αx in the class M. On the other hand, if φ ∈ M is the function constructed in Section 1, then f (t, x) = φ(x) satisfies (iv) with g(t) = g1 (t) = 1 for all t ∈ R+ , showing that our considerations go beyond the contraction case. Note that the existence theory of solutions for deterministic integral equations is based on either contraction-type arguments or on Schauder-type compactness arguments. For stochastic integral equations results of Arzela–Ascoli type are typically not available, so that there is a greater emphasis on contractions. Our approach permits an extension of this method. Acknowledgment The author is grateful to the referee for constructive suggestions and comments.

R. Negrea / Nonlinear Analysis: Real World Applications 54 (2020) 103104

7

References [1] A. Constantin, Some existence results for nonlinear integral equations, in: C. Corduneanu (Ed.), Qualitative Problems for Differential Equations and Control Theory, World Scientific Publishing, 1995, pp. 105–111. [2] R. Negrea, On a nonlinear stochastic integral equation, Monatsh. Math. (2020) https://doi.org/10.1007/s00605-019-013 53-y. [3] E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York-Heidelberg, 1975. [4] A. Constantin, A random integral equation and applications, Indian J. Math. 37 (2) (1995) 151–163. [5] W.A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath and Co, Boston, Mass, 1965. [6] Gh. Constantin, R. Negrea, An application of Schauder’s fixed point theorem in stochastic McShane modeling, J. Fixed Point Theory 5 (2004) 37–52. [7] C.P. Tsokos, On a stochastic integral equation of the Volterra type, Math. Syst. Theory 3 (1969) 222–231. [8] J. Padgett, C.P. Tsokos, Random solution of atochastic integral equation: almost sure and mean square convergence of successive approximations, Internat. J. Systems Sci. 4 (1973) 605–612. [9] C.P. Tsokos, W.J. Padgett, Random Integral Equations with Applications to Life Sciences and Engineering, Academic Press, New York, 1974. [10] A. Bielecki, Une remarque sur la m´ ethode de Banach-Cacciopoli-Tikhonov dans la th´ eorie des ´ equations diff´ erentielles ordinaires, Bull. Acad. Polon. Sci. Cl. 4 (1956) 261–264.