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 Open Access
Measure of noncompactness and application to stochastic differential equations
 Abdelkader Dehici^{1, 2} and
 Nadjeh Redjel^{1, 2}Email author
https://doi.org/10.1186/s136620160748z
© Dehici and Redjel 2016
 Received: 3 August 2015
 Accepted: 10 January 2016
 Published: 27 January 2016
Abstract
In this paper, we study the existence and uniqueness of the solution of stochastic differential equation by means of the properties of the associated condensing nonexpansive random operator. Moreover, by taking account of the results of Diaz and Metcalf, we prove the convergence of Kirk’s process to this solution for small times.
Keywords
 Wiener process
 Itô integral
 Banach space
 fixed point
 existence
 uniqueness
 measure of noncompactness
 condensing operators
 Kirk’s process
MSC
 47H10
 47H08
 60H10
1 Introduction and notations
It has been found over the years that the fixed point theory is a powerful tool for the resolution of nonlinear problems (differential equations, integrodifferential equations, …). The roots of this theory go back to the famous works of Brouwer (1912) and Banach (1922), the latter author gave an abstract formulation of the successive approximations method, systematically used by Liouville (1837) in his results. We note that Banach’s work was established in the case of normed spaces and extended in metric spaces by Caccioppoli (1930). Since then this theory has become a burgeoning field for several authors who have contributed in the elaboration by thousands of papers of the subject. The development of this theory has been heavily linked to that of the functional analysis in the 1950s. The Italian mathematician Darbo has published a result which ensures the existence of fixed point for a type of operators so called condensing operators generalizing the Schauder fixed point and Banach contraction principle. This discovery was the subject of several applications both in linear and nonlinear analysis (integral equations with singular kernels, differential equations defined on unbounded domains, neutral differential equations, differential operators having nonempty essential spectra, boundary value problems in Banach spaces and others). A condensing (or densifying) mapping is a mapping for which the image of any set is in a certain sense more compact than the set itself, the degree of noncompactness of a set is measured by means of functions called measures of noncompactness. Among the application areas of these tools, the theory of probabilistic operators, which is a branch of stochastic analysis which deals with random operators and their properties, is seen as an extension of operators theory (determinist case). This axis of research has emerged in the 1950s, thanks to the works of East European school of probabilities whose main purpose was the resolution of stochastic differential equations and stochastic partial differential equations, modeling the trajectories of random phenomena, studied and developed for the first time by Itô in 1946. A stochastic differential equation is an ordinary differential equation perturbed by a white noise (involving the Brownian motion). The history of this direction goes back to the works of the English botanist Brown who described in 1827 this motion as that of an organic fine particle in suspension in a gas or a fluid. In the late 19 century, scientists (Bachelier, Smoluchowski) addressed the study of this type of motions. Afterwards, and more precisely in 1905, Einstein published a paper in which he showed that the probability density of the Brownian motion satisfies the heat equation. The first rigorous mathematical treatment is due to Wiener in the 1920s of the previous century who has proven the existence of the Brownian motion. For more details of these equations, we can quote for example [1–5].
Recall that this equation models for example the motion of a particle subjected to the infinity of shocks at the time t. Here \(a_{1}\) is a coefficient of transfer while \({a_{2}}\) is a diffusion coefficient. In the case where \(a_{1}\) and \(a_{2}\) satisfy the Lipschitz condition with respect to the second variable, the result for \(f(\theta) \equiv\theta\) was established by Gikhman and Skorohod [6] showing that the associated mapping to (1.1) is a contraction and the solution was obtained by means of the successive approximations method.
Our goal here is to investigate the problem (1.1) by imposing general conditions on the functions \(a_{1}\) and \(a_{2}\), therefore, we show that the associated mapping T is nonexpansive and condensing mapping having a unique fixed point which is the solution of (1.1). On the other hand, we prove that if this solution satisfies the metric property of Diaz and Metcalf, the convergence of Kirk’s process to this solution is ensured.
Definition 1.1
A probability space \((\Omega, \mathcal{F}, \mathbb{P})\) is a triplet for which Ω is a nonempty set, \(\mathcal{F}\) is a σalgebra of Ω and \(\mathbb{P}\) is a probability measure defined on \(\mathcal{F}\) (\(\mathbb {P}(\Omega) = 1\)).
Notice that some results concerning the existence of fixed point theorems involving probabilistic metric spaces can be found for example in [7–9].
A real random variable X is a \(\mathbb{P}\)measurable function defined on Ω with values in \({\mathbb{R}}\). A family of random variables \(X_{t}(\omega)\) (\(t \geq0\)) (denoted also by \(X(t,\omega )\) or simply \(X_{t}\)) is called a stochastic process. For \(\omega\in \Omega\), the function \(t \longrightarrow X(t,\omega)\) is the path of the stochastic process \(X(t,\omega)\).
Definition 1.2
 (a)
\(w_{0}=0\);
 (b)
for \(0< t_{1} <\cdots< t_{n}\) the random variables \(w_{t_{2}} w_{t_{1}},w_{t_{3}} w_{t_{2}},\ldots,w_{t_{n}} w_{t_{n1}} \) are independent;
 (c)
the random variables \(w_{t+s}w_{t}\) have a normal distribution with zero expectation and 1 as a variance.
Remark 1.1
Definition 1.3
A random variable Y is said to be \(\mathcal{F}_{t}\)measurable if knowledge of Y depends only on the information known up to time t.
Definition 1.4
 (i)
Itô integral is linear;
 (ii)if \(\int_{0}^{T} \mathbb {E}(f(t)^{2})\,dt < \infty\), thenand$$ \mathbb{E}\biggl( \int_{0}^{T}f(t)\,dw(t)\biggr) = 0, $$(1.2)$$ \mathbb{E}\biggl( \sup_{0 \leq s \leq\mu }\biggl \int_{0}^{s}f(t)\,dw(t)\biggr^{2}\biggr) \leq4 \int _{0}^{\mu}\mathbb{E}\bigl(\biglf(t)\bigr^{2} \bigr)\,dt \quad(0 \leq\mu\leq T). $$(1.3)
Definition 1.5
2 Main results
We denote by \(X_{T}\) the vector space of measurable random functions \(\xi(t, \omega)\) with respect to the σalgebra \(\mathcal{F}_{t}\) for any \(t \in[0, T]\) such that \(\mathbb{P}(\{\omega \in\Omega \mid t\longrightarrow\xi(t, \omega) \mbox{ is continuous}\}) = 1\). We put \(\\xi\_{X_{T}} = \sqrt{\mathbb{E}( \sup_{0 \leq s \leq T}\xi(s, \omega))^{2}}\). It is easy to show that \(\\cdot\_{X_{T}}\) defines a norm on \(X_{T}\).
Theorem 2.1
\((X_{T}, \\cdot\_{X_{T}})\) is a Banach space.
Proof
Since \(\xi_{n}\) is a Cauchy sequence in \(X_{T}\) and contains a subsequence \(\xi_{m_{k}}\) which converges to ξ, \(\xi_{n}\) converges to ξ in \(X_{T}\), and by this we achieve the proof. □
Proposition 2.1
Proof
If \(X(t)\) is a solution of the equation (1.1) on the interval \([0, s]\), \(0 \leq s \leq T\), we denote \(\varphi(s) = \X\^{2}_{X_{s}}\).
Lemma 2.1
The function φ is bounded on the interval \([0, T]\).
Proof
We introduce here the concept of measure of noncompactness of Hausdorff which is a real positive function measuring the degree of noncompactness of sets.
Let X be a complex Banach space and let \(\mathcal{P}(X)\) be the set of all subsets of X, we denote by \(B(x, r)\) and \(\overline{B}(x, r)\), respectively, the open and closed ball of center x and radius \(r > 0\).
Definition 2.1
The Hausdorff measure of noncompactness \(\alpha(A)\) of \(A \in\mathcal{P}(X)\) is defined as the infimum of the numbers \(\epsilon> 0\) such that A has a finite ϵnet in X. Recall that a set \(S \subseteq X\) is called an ϵnet of A if \(A \subseteq S + \epsilon \overline{B}(0, 1) = \{s + \epsilon b: s \in S, b \in \overline{B}(0, 1) \}\).
 (a)
regularity: \(\alpha(A) = 0\) if and only if A is totally bounded;
 (b)
nonsingularity: α is equal to zero on every oneelement set;
 (c)
monotonicity: \(A_{1} \subseteq A_{2}\) implies \(\alpha (A_{1}) \leq\alpha(A_{2})\);
 (d)
semiadditivity: \(\alpha(A_{1} \cup A_{2}) = \max\{ \alpha(A_{1}), \alpha(A_{2}) \}\);
 (e)
Lipschitzianity: \( \alpha(A_{1})  \alpha(A_{2}) \leq \rho(A_{1}, A_{2}) \); here ρ denotes the Hausdorff semimetric: \(\rho(A_{1}, A_{2}) = \inf\{ \epsilon> 0: A_{1} + \epsilon\overline{B}(0, 1) \supset A_{2}, A_{2} + \epsilon\overline{B}(0, 1) \supset A_{1} \}\);
 (f)
continuity: for any \(A_{1} \in\mathcal{P}(X)\) and any ϵ, there exists \(\delta> 0\) such that \(\alpha(A_{1})  \alpha(A) < \epsilon\) for all A satisfying \(\rho(A_{1}, A) < \delta\);
 (g)
semihomogeneity: \(\alpha(t A) = t \alpha(A)\) for any number t;
 (h)
algebraic semiadditivity: \(\alpha(A_{1} + A_{2}) \leq\alpha(A_{1}) + \alpha(A_{2})\);
 (i)
invariance under translations: \(\alpha(A + x_{0}) = \alpha(A)\) for any \(x_{0} \in X\).
Theorem 2.2
([1], Theorem 1.1.5)
The Hausdorff measure of noncompactness is invariant under passage to the closure and to the convex hull: \(\alpha(A) = \alpha(\overline{A}) = \alpha (co A)\).
We note that the measure of noncompactness has many applications in mathematics. On this topic, we refer to [1, 10–13].
Definition 2.2
Let X be a Banach space. A function ψ defined on \(\mathcal{P}(X)\) with values in some partially ordered set \((\Gamma, \leq)\) is called a measure of noncompactness in the general sense if \(\psi(A) = \psi (\overline{co} A)\) for all \(A \in\mathcal{P}(X)\).
Definition 2.3
Let \((X, \\cdot\)\) be a normed space and ϑ one of measure of noncompactness given above. A continuous mapping \(G: X \longrightarrow X\) is said to be densifying or condensing, if, for every bounded subset of X such that \(\vartheta(A) > 0\), we have \(\vartheta(G(A)) < \vartheta(A)\).
Lemma 2.2
The function γ defines a measure of noncompactnesss in the general sense on \(X_{T}\) which is additively nonsingular (i.e., \(\gamma(A\cup\{X\}) = \gamma (A)\) for all \(A\subset X_{T}\) and \(X \in X_{T}\)).
Proof
Definition 2.4
Let \((X, \\cdot\)\) be a normed space and K a nonempty bounded subset of X. A selfmapping T on K is called a nonexpansive mapping if \(\T(x)  T(y)\ \leq\ x  y \\) for all \(x, y \in K\).
 (\(\mathcal{H}_{1}\)):

\(a_{1}(s, u_{1})  a_{1}(s, u_{2})^{2}\leq h(s) g(\ u_{1} u_{2}\^{2})\),
 (\(\mathcal{H}_{2}\)):

For all \(A > 0\), the inequalitycannot admit nontrivial solutions.$$\widetilde{h}(s) \leq A \int_{0}^{s}h(\theta ) g\bigl(\widetilde{h}\bigl(f( \theta)\bigr)\bigr)\,d\theta,\quad 0 \leq s \leq T $$
 (\(\mathcal{H}_{3}\)):

\(\lambda(f^{1}(B)) \longrightarrow0\) when \(\lambda(B) \longrightarrow0\); here λ is the Lebesgue measure and \(B \subset[0, t] \) (\(0 \leq t \leq T\)).
Remark 2.1
We note that if we take \(h(t) = \alpha\) (\(\alpha> 0\)), \(g(u) = u\), \(f(x) = x\), and \(T = 2 + \sqrt{4 + \frac{1}{2\alpha}}\), then the assumptions given in (\(\mathcal{H}_{1}\)), (\(\mathcal{H}_{2}\)), and (\(\mathcal{H}_{3}\)) are satisfied.
Proposition 2.2
Under the assumption (\(\mathcal{H}_{1}\)), the mapping C defined by (2.1) is nonexpansive on every \(X_{t}\) (\(0 \leq t \leq T\)).
Proof
Remark 2.2
We note that nonexpansive selfmappings on bounded subsets of Banach spaces do not necessarily have fixed points, we can refer for example to the famous work of Alspach [14] who gave an example of a weakly compact convex subset M of the space \(L_{1}([0, 1])\) and a fixed point free isometry on M.
In the sequel, we will need the following two lemmas; the first lemma is one of the classical results in measure theory.
Lemma 2.3
Let \(\epsilon> 0\) and let \(\phi: [0, T] \longrightarrow\mathbb{R}\) a monotone function, then the set of discontinuity points of ϕ having a magnitude ≥ϵ is a finite set in \([0, T]\).
Lemma 2.4
Proof
Let \(t_{1}, t_{2} \in[0, T]\) such that \(t_{1} \leq t_{2}\); then \(\Lambda _{t_{1}} \subset\Lambda_{t_{2}}\). The monotonicity of the measure of noncompactness of Hausdorff gives \({\gamma}_{t_{1}}(\Lambda_{t_{1}}) \leq {\gamma}_{t_{2}}(\Lambda_{t_{2}})\) and implies that \(\gamma(\Lambda)(t_{1}) \leq\gamma(\Lambda)(t_{2})\), which proves the first assertion. The second assertion follows directly from the first one, indeed by the monotonicity, we deduce that \(\gamma(\Lambda)(t) \leq\gamma(\Lambda )(T)\) for all \(t \in[0, T]\). □
Theorem 2.3
Under the assumptions (\(\mathcal{H}_{1}\)), (\(\mathcal{H}_{2}\)), (\(\mathcal{H}_{3}\)) together with the growth polynomial conditions given in (2.2), C is a condensing mapping with respect to the measure of noncompactness γ on \(X_{t}\) for all \(t \in[0, T]\).
Proof
Theorem 2.4
([1], p.26)
Let \(A: K \longrightarrow K\) be a mapping defined on a closed, bounded, convex subset K of a Banach space X. Assume that A is condensing with respect to the additively nonsingular measure of noncompactness in the general sense Ψ. Then A has at least one fixed point in K.
Theorem 2.5
The mapping \(C: X_{T} \longrightarrow X_{T}\) defined by (2.1) has a unique fixed point in \(X_{T}\).
Proof
The existence follows from Theorem 2.4, more precisely, for τ belonging to the interval \([0, 2 + \sqrt{4 + \frac{1}{2M} \frac{H}{H + 1}}]\), the inequality (2.8) shows that the random mapping \(C: X_{\tau}\longrightarrow X_{\tau}\) leaves \(\overline{B}_{X_{T}}(0, \sqrt {H})\) (the ball of center 0 and radius \(\sqrt{H}\)) invariant, which implies the existence of the solution \(X(t)\) in \(X_{\tau}(\tau\in[0, 2 + \sqrt{4 + \frac{1}{2M} \frac{H}{H + 1}}])\), the result in \(X_{T}\) follows from an extension to the whole interval \([0, T]\).
3 Application to the convergence of Kirk’s iterative process
Let us recall the following theorem due to Diaz and Metcalf [15].
Theorem 3.1
 (i)
\(\operatorname{Fix}(B) \neq\emptyset\) (\(\operatorname{Fix}(B)\) is the set of fixed point of B);
 (ii)for each \(y \in X\) such that \(y \notin\operatorname {Fix}(B)\), and for each \(z \in\operatorname{Fix}(B)\) we have$$d\bigl(B(y), z\bigr) < d( y, z). $$
 (a)
for each \(x_{0} \in X\) the Picard sequence \(\{B^{n}(x_{0})\}\) contains no convergent subsequences;
 (b)
for each \(x_{0} \in X\) the sequence \(\{B^{n}(x_{0})\}\) converges to a point belonging to \(F(B)\).
Theorem 3.2
[16]
Let K be a convex subset of a Banach space and \(A: K \longrightarrow K\) be a nonexpansive mapping. Then \(S(x) = x\) if and only if \(A(x) = x\).
Let \(T > 0\), and \(H = K e^{K'T}\) a real positive which is an upper bound of the function \(\varphi(t) = \X\_{X_{t}}^{2} \) (\(0 \leq t \leq T\)). We denote by \(\overline{B}_{X_{t}}(0, \sqrt{H})\) the closed ball of center 0 and radius \(\sqrt{H}\) in \(X_{t}\).
Theorem 3.3
If there exists \(\tau_{0} \in [0, 2 + \sqrt{4+ \frac{1}{2M} \frac{H}{H + 1}}]\) such that \(C:\overline {B}_{X_{\tau_{0}}}(0, \sqrt{H}) \longrightarrow\overline{B}_{X_{\tau _{0}}}(0, \sqrt{H})\) satisfies DiazMetcalf’s condition, then, for each \(X \in\overline{B}_{X_{\tau_{0}}}(0, \sqrt{H})\), Kirk’s process \(\{ S^{n}(X)\}\) (associated to the mapping C) converges to the unique fixed point of T.
Proof
Remark 3.1
Let K be a closed, bounded, and convex subset of a Banach space X and let \(A: K \longrightarrow K\) be a mapping. For each \(x \in K\), some sufficient conditions on A are given to ensure the convergence or the weak convergence of Kirk’s process \(\{S^{n}(x)\}\) to the fixed point of A (see for example [16–18]) with the additional condition that the space X is uniformly convex or strictly convex. Unfortunately, in our case the Banach space \(X_{T}\) is not strictly convex or uniformly convex, indeed, it suffices to take its subspace of functions \(\zeta(t)\) independent of ω equipped with its norm sup, to see that this does not have these properties.
Declarations
Acknowledgements
The authors would like to thank the editor and the anonymous referees for their remarks and valuable comments, which helped to improve this paper.
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Authors’ Affiliations
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