327 research outputs found
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations
In our previous paper [12] (Rev. Math. Phys. 16, 383-420 (2004)), a general
framework was outlined to treat the approximate solutions of semilinear
evolution equations; more precisely, a scheme was presented to infer from an
approximate solution the existence (local or global in time) of an exact
solution, and to estimate their distance. In the first half of the present work
the abstract framework of \cite{uno} is extended, so as to be applicable to
evolutionary PDEs whose nonlinearities contain derivatives in the space
variables. In the second half of the paper this extended framework is applied
to theincompressible Navier-Stokes equations, on a torus T^d of any dimension.
In this way a number of results are obtained in the setting of the Sobolev
spaces H^n(T^d), choosing the approximate solutions in a number of different
ways. With the simplest choices we recover local existence of the exact
solution for arbitrary data and external forces, as well as global existence
for small data and forces. With the supplementary assumption of exponential
decay in time for the forces, the same decay law is derived for the exact
solution with small (zero mean) data and forces. The interval of existence for
arbitrary data, the upper bounds on data and forces for global existence, and
all estimates on the exponential decay of the exact solution are derived in a
fully quantitative way (i.e., giving the values of all the necessary constants;
this makes a difference with most of the previous literature). Nextly, the
Galerkin approximate solutions are considered and precise, still quantitative
estimates are derived for their H^n distance from the exact solution; these are
global in time for small data and forces (with exponential time decay of the
above distance, if the forces decay similarly).Comment: LaTeX, 84 pages. The final version published in Reviews in
Mathematical Physic
Existence of monotonic -solutions for quadratic Volterra functional-integral equations
We study the quadratic integral equation in the space of Orlicz space in the most important case when satisfies the -condition. Considered operators are not compact and then we use the technique of measure of noncompactness associated with the Darbo fixed point theorem to prove the existence of a monotonic, but discontinuous solution. Our present work allows to generalize both previously proved results for quadratic integral equations as well as that for classical equations. Due to different continuity properties of considered operators in Orlicz spaces, we distinguish different cases and we study the problem in the most important case – in such a way to cover all Lebesgue spaces ()
On quadratic integral equations in Orlicz spaces
AbstractIn this paper we study the quadratic integral equation of the formx(t)=g(t)+λx(t)∫abK(t,s)f(s,x(s))ds. Several existence theorems for a.e. monotonic solutions in Orlicz spaces are proved for strongly nonlinear functions f. The presented method of the proof can be easily extended to different classes of solutions
On solutions of some delay Volterra integral problems on a half-line
In this paper, we study the existence of a.e. monotonic solutions of some general delay integral problems for both fractional and integer orders in the space of Lebesgue integrable functions on the interval R+ = [0;1) and in the space of locally integrable functions L1loc (R+). In particular, the uniqueness of solutions for considered problems is obtained
Existence and Asymptotic Stability of Solutions of a Perturbed Quadratic Fractional Integral Equation
Mathematics Subject Classification: 45G10, 45M99, 47H09We study the solvability of a perturbed quadratic integral equation of
fractional order with linear modification of the argument. This equation is
considered in the Banach space of real functions which are defined, bounded
and continuous on an unbounded interval. Moreover, we will obtain some
asymptotic characterization of solutions. Finally, we give an example to
illustrate our abstract results
Affine Volterra processes
We introduce affine Volterra processes, defined as solutions of certain
stochastic convolution equations with affine coefficients. Classical affine
diffusions constitute a special case, but affine Volterra processes are neither
semimartingales, nor Markov processes in general. We provide explicit
exponential-affine representations of the Fourier-Laplace functional in terms
of the solution of an associated system of deterministic integral equations of
convolution type, extending well-known formulas for classical affine
diffusions. For specific state spaces, we prove existence, uniqueness, and
invariance properties of solutions of the corresponding stochastic convolution
equations. Our arguments avoid infinite-dimensional stochastic analysis as well
as stochastic integration with respect to non-semimartingales, relying instead
on tools from the theory of finite-dimensional deterministic convolution
equations. Our findings generalize and clarify recent results in the literature
on rough volatility models in finance
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