Existence and uniqueness theorem for state-dependent delay-differential equations of neutral type is given. This theorem generalizes previous results by Grimm and the author.

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This paper considers the existence, uniqueness and non-existence of solution for a quasi-linear fractional boundary value problems with sign-changing Green’s function. Under certain growth conditions on the nonlinear term, we employ the Leray-Schauder alternative fixed point theorem to obtain an existence result of nontrivial solution and use the Banach contraction mapping principle to obtain a...

This paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions d q xt/dt q −Axt ft, xt, Gxt, t ∈ 0, T, and x0 gx x 0 , in a Banach space X, where 0 < q < 1. General existence and uniqueness theorem, which extends many previous results, are given.

For the 1-dimensional Kuramoto–Sivashinsky equation with random forcing term, existence and uniqueness of solutions is proved. Then, the Markovian semigroup is well defined; its properties are analyzed, in order to provide sufficient conditions for existence and uniqueness of invariant measures for this stochastic equation. Finally, regularity results are obtained by means of Girsanov theorem.

An existence and uniqueness theorem for mild solutions of stochastic evolution equations is presented and proved. The diffusion coefficient is handled in a unified way which allows a unified theorem to be formulated for different cases, in particular, of multiplicative space–time white noise and trace-class noise. 2000 Mathematics subject classification: primary 60H15; secondary 35K90.