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### Approximate iterative method and the existence of solutions of nonlinear parabolic differential-functional equations

Annales Polonici Mathematici

### Constructing One-Parameter Transformations on White Noise Functions in Terms of Equicontinuous Generators.

Monatshefte für Mathematik

### Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Czechoslovak Mathematical Journal

Let $A=\mathrm{d}/\mathrm{d}\theta$ denote the generator of the rotation group in the space $C\left(\Gamma \right)$, where $\Gamma$ denotes the unit circle. We show that the stochastic Cauchy problem $\mathrm{d}U\left(t\right)=AU\left(t\right)+f\mathrm{d}{b}_{t},\phantom{\rule{1.0em}{0ex}}U\left(0\right)=0,\phantom{\rule{2.0em}{0ex}}\left(1\right)$ where $b$ is a standard Brownian motion and $f\in C\left(\Gamma \right)$ is fixed, has a weak solution if and only if the stochastic convolution process $t↦{\left(f*b\right)}_{t}$ has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all...

### Convergence estimate for second order Cauchy problems with a small parameter

Czechoslovak Mathematical Journal

We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter $\epsilon .$ The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.

### Differential operators with generalized constant coefficients.

Portugaliae Mathematica

### Dirichlet problem for parabolic equations on Hilbert spaces

Studia Mathematica

We study a linear second order parabolic equation in an open subset of a separable Hilbert space, with the Dirichlet boundary condition. We prove that a probabilistic formula, analogous to one obtained in the finite-dimensional case, gives a solution to this equation. We also give a uniqueness result.

### Dissipative sets and nonlinear perturbated equations in Banach spaces

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Embedding theorems in Banach-valued $B$-spaces and maximal $B$-regular differential-operator equations.

Journal of Inequalities and Applications [electronic only]

### Equazioni alle derivate parziali con infinite variabili

Bollettino dell'Unione Matematica Italiana

### Existence of solutions and monotone iterative method for infinite systems of parabolic differential-functional equations

Annales Polonici Mathematici

We consider the Fourier first boundary value problem for an infinite system of weakly coupled nonlinear differential-functional equations. To prove the existence and uniqueness of solution, we apply a monotone iterative method using J. Szarski's results on differential-functional inequalities and a comparison theorem for infinite systems.

### Existence of solutions of the Cauchy problem for semilinear infinite systems of parabolic differential-functional equations.

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

### Garding's inequality for elliptic differential operator with infinite number of variables.

International Journal of Mathematics and Mathematical Sciences

### Generalized Fractional Evolution Equation

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20In this paper we study the generalized Riemann-Liouville (resp. Caputo) time fractional evolution equation in infinite dimensions. We show that the explicit solution is given as the convolution between the initial condition and a generalized function related to the Mittag-Leffler function. The fundamental solution corresponding to the Riemann-Liouville time fractional evolution equation does not admit a probabilistic...

### Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications

Annales de l'I.H.P. Analyse non linéaire

### Irregularities in the numerical treatment of nonlinear initial value problems

Banach Center Publications

### ${L}^{P}$-uniqueness for infinite dimensional symmetric Kolmogorov operators : the case of variable diffusion coefficients

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

### Maximal regular boundary value problems in Banach-valued weighted space.

Boundary Value Problems [electronic only]

### Monotone iteration for infinite systems of parabolic equations with functional dependence

Annales Polonici Mathematici

We consider the initial value problem for an infinite system of differential-functional equations of parabolic type. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. The solutions are obtained by the monotone iterative method. We prove theorems on weak partial differential-functional inequalities as well the existence and uniqueness theorems in the class of continuous bounded functions and in the class of functions satisfying...

### New optimal regularity results for infinite-dimensional elliptic equations

Bollettino dell'Unione Matematica Italiana

In questo articolo si ottengono stime di Schauder di tipo nuovo per equazioni ellittiche infinito-dimensionali del secondo ordine con coefficienti Hölderiani a valori nello spazio degli operatori Hilbert-Schmidt. In particolare si mostra che la derivata seconda delle soluzioni è Hilbert-Schmidt.

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