Equations In Hilbert Spaces Research Articles

Graph-convergent analysis of over-relaxed ( A , η , m ) -proximal point iterative methods with errors for general nonlinear operator equations

In this paper, we introduce and study a new class of over-relaxed (A,η,m)-proximal point iterative methods with errors for solving general nonlinear operator equations in Hilbert spaces. By using Liu’s inequality and the generalized resolvent operator technique associated with (A,η,m)-monotone operators, we also prove the existence of solution for the nonlinear operator inclusions and discuss the graph-convergent analysis of iterative sequences generated by the algorithm. Furthermore, we give some examples and an application for solving the open question (2) due to Li and Lan (Adv. Nonlinear Var. Inequal. 15(1):99-109, 2012). The numerical simulation examples are given to illustrate the validity of our results. MSC:49J40, 47H05, 65B05.

Existence and approximate controllability of Sobolev type fractional stochastic evolution equations

Abstract We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s < b. An example is provided to illustrate the application of the obtained results.

Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

We prove uniqueness for continuity equations in Hilbert spaces $$H$$ . The corresponding drift $$F$$ is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna–Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio–Figalli and Fang–Luo, we use a different approximation procedure, based on a more regularizing Ornstein–Uhlenbeck semigroup and consider Sobolev spaces of vector fields taking values in $$H$$ rather than the Cameron–Martin space of the Gaussian measure. This leads to different conditions on the derivative of $$F$$ , which are incompatible with previous work on the subject. Furthermore, we can drop the usual exponential integrability conditions on the Gaussian divergence of $$F$$ , thus improving known uniqueness results in this respect.

Wiener-Itô Chaos Expansion of Hilbert Space Valued Random Variables

The notion of n-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.

Convergence rates for Kaczmarz-type regularization methods

This article is devoted to the convergence analysis of a special family of iterative regularization methods for solving systems of ill--posed operator equations in Hilbert spaces, namely Kaczmarz-type methods. The analysis is focused on the Landweber--Kaczmarz (LK) explicit iteration and the iterated Tikhonov--Kaczmarz (iTK) implicit iteration. The corresponding symmetric versions of these iterative methods are also investigated (sLK and siTK). We prove convergence rates for the four methods above, extending and complementing the convergence analysis established originally in [22,13,12,8].

Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term

We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $$B$$ and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato et al. in Ann Probab 41:3306–3344, 2013) which generalized Veretennikov’s fundamental result to infinite dimensions assuming boundedness of the drift term. As in Da Prato et al. (Ann Probab 41:3306–3344, 2013), pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift $$B$$ is assumed only to be measurable and bounded and grow more than linearly.

Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

Stepanov-like weighted asymptotic behavior of solutions to some stochastic differential equations in Hilbert spaces

In this paper, we first introduce the notation and properties of -weighted pseudo almost automorphy for stochastic processes. And then, we apply the results obtained to consider the existence and uniqueness of -weighted pseudo almost automorphic solutions to some stochastic differential equations in a real separable Hilbert space under global Lipschitz conditions. Moreover, we also investigate asymptotic behavior of solutions to a stochastic differential equation under -weighted pseudo almost automorphic coefficients without global Lipschitz conditions. Our main results extend some known ones in the sense of square-mean weighted pseudo almost automorphy or -pseudo almost automorphy for stochastic processes.

Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integrodifferential Equations in Hilbert Spaces

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

Some results for pathwise uniqueness in Hilbert spaces

An abstract evolution equation in Hilbert spaces with Holder continuous drift is considered. By proceeding as in [3], we transform the equation in another equation with Lipschitz continuous coefficients.Then we prove existence and uniqueness of this equation by a fixed point argument.

Control problems for semilinear neutral differential equations in Hilbert spaces.

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied.

On smooth solutions of operator-differential equation in Hilbert space

On smooth solutions of operator-differential equation in Hilbert space

Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces

We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlin- ear differential equations of Sobolev type in Hilbert spaces. We use Holder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted di- rectly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.

Expanding the applicability of Tikhonov's regularization and iterative approximation for ill-posed problems

Abstract Recently, Vasin [J. Inverse Ill-Posed Probl. 21 (2013), 109–123] considered a new iterative method for approximately solving nonlinear ill-posed operator equation in Hilbert spaces. In this paper we introduce a modified form of the method considered by Vasin. This paper weakens the conditions needed in the existing results. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in [J. Inverse Ill-Posed Probl. 21 (2013), 109–123]. This way a tighter convergence analysis is obtained and under less computational cost, since the more precise and easier to compute center-Lipschitz instead of the Lipschitz constant is used in the convergence analysis. Order optimal error bounds are given in case the regularization parameter is chosen a priori and by the adaptive method of Pereverzev and Schock [SIAM J. Numer. Anal. 43 (2005), 2060–2076]. A numerical example of a nonlinear integral equation proves the efficiency of the proposed method.

Periodic forcing for some difference equations in Hilbert spaces

Let H be a real Hilbert space and let A : D(A) ⊂ H → H be a (possibly multivalued) maximal monotone operator. We are concerned with the difference equation ∆un + cnAun+1 ∋ fn, n = 0, 1, ..., where (cn) ⊂ (0,+∞), ( fn) ⊂ H are p-periodic sequences for a positive integer p. We investigate the existence of periodic solutions to this equation as well as the weak or strong convergence of solutions to p-periodic solutions. The first result of this paper (Theorem 1) is a discrete analogue of the 1977 result by Baillon and Haraux (on the periodic forcing problem for the continuous counterpart of the above equation) and was essentially stated by Djafari Rouhani and Khatibzadeh in a recent paper [5]. Here we provide a simpler proof of this result that is based on old existing results due to Browder and Petryshyn [4] and Opial (see, e.g., [6], p.5). A strong convergence result is also given and some examples are discussed to illustrate the theoretical results.

Local convergence of quasi-Newton methods under metric regularity

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results.

Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\mathbb{R}^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.

Harnack inequalities for stochastic equations driven by Lévy noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces.

Long-time behavior of nonlinear integro-differential evolution equations

We study the long-time behavior as time tends to infinity of globally bounded strong solutions to certain integro-differential equations in Hilbert spaces. Based on an appropriate new Lyapunov function and the Łojasiewicz–Simon inequality, we prove that any globally bounded strong solution converges to a steady state in a real Hilbert space.

Optimal secant-updates of rank 1

We consider a class of secant update iterative methods for solving nonlinear operator equations in Hilbert spaces. Our goal is to characterize optimal inverse secant-updates of rank 1. As the optimality criterion we use the factor, by which one iteration of a secant update iterative method reduces the entropy of solution’s position within the set of its guaranteed existence and uniqueness. Having got the optimality condition, we select among optimal updates of rank 1 one that minimizes the condition number of the updated operator. The resulting update determines a new secant update iterative method.