Nonlinear perturbations and abstract evolution equations in general banach spaces

Summary Part I deals with the problem of determining sufficient conditions under which the sum of two m-accretive operators on a closed convex set Q₁ is m-accretive on Q₁. Part II is concerned with the initial value problem: u′+Au+g(u)=v, u(0)=u₀. Applications are given to the Boltzmann equation. Entrata in Redazione il 2 luglio 1975.     

Nonlinear evolution equations in an arbitrary Banach space

This paper proves the existence of an evolution operatorU(t, s)x ₀ corresponding to a weak or generalized solution of the differential equation:du (t)/dt +A (t)u(t) ? f(t), u(s) =x ₀,t ≧ s; the operatorsA(t) are eachm-accretive in a Banach spaceX and, loosely speaking, have an “L¹ modulus of continuity” int. The continuity and differentiability properties ofU(t, s)x₀ are investigated, and some simple examples are presented.     

On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space $ \mathfrak{B} $ \mathfrak{B} over the field Ω of complex p-adic numbers having an inverse operator defined on the whole $ \mathfrak{B} $ \mathfrak{B} , and f be a locally holomorphic at 0 $ \mathfrak{B} $ \mathfrak{B} -valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y ^(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} $ \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.     

Solvability of nonlinear equations in a cone of a banach space

The solvability conditions for the equation Tu+F(u)=0 are found in the case where the operator [T+F′(u)]⁻¹ exists only for u∈K, where K is a cone in a Banach space X. An application concerning the solvability of boundary-value problems for systems of second-order differential equations is provided. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 225–230. Translated by L. Yu. Kolotilina.     

Method of successive approximations for abstract volterra equations in a banach space

We apply the method of successive approximations to abstract Volterra equations of the formx=f+a*Ax, whereA is a closed linear operator. The assumption is made that a kernela is continuous but is not necessarily of bounded variation. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 376–382, March, 1999.     

Periodic solutions of nonlinear differential equations with pulse influence in a banach space

Rank conditions for control of linear pulse systems are established. An example of control synthesis in a problem for linear pulse systems is given.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 370–380, March, 1995.This work was partially supported by the Ukrainian State Committee on Science and Technology.     

Weak invariance principle for solutions of stochastic recurrence equations in a banach space

We show that, in a Banach space, continuous random processes constructed by using solutions of the difference equationX _n =A _n X _n+1+V _n , n=1, 2,..., converge in distribution to a solution of the corresponding operator equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 114–117, January, 1995.     

Spectrum of a linear operator in a family of banach spaces containing Hilbert space

The mutual relation is established between the spectra of a bounded linear operator acting in a family of Banach spaces. It is assumed in addition that one of the spaces is a Hilbert space and that the operator acting on it is self-adjoint. An example is presented illustrating the properties proved.Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 495–498, October, 1977.     

Existence and comparison results for nonlinear volterra integral equations in a banach space

Let L ⁰(ΩA,P) be the space of equivalent classes of random variables defined on a probability space (Ω,A,P). Let H be the closed subspace of L ⁰(Ω,A,P) spanned by a sequence of i.i.d. (independent and identically distributed) random variables having the symmetric nondegenerate law F. It is proved that H is linearly homeomorphic to l _p for 0<p≤2 if F belongs to the domain of normal attraction of symmetric stable law withexponent p.     

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D _s), whereD _s is the differentiation operator, withF bounded linear andK andD _sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.     

Existence of solutions of abstract volterra equations in a banach space and its subsets

We consider a criterion and sufficient conditions for the existence of a solution of the equation

Iterative methods for solving nonlinear irregular operator equations in banach space

We present an approach to constructing stable methods for solving nonlinear operator equations in Banach space without anyassumptions on the regularity of the operator. The approach is based on the linearization of the equation and the use of aregularization algorithm to find an approximate solution of the linearized equation at each iteration. The local convergence of proposed methods is proved and the estimations of the rate of convergence are established, provided that solution satisfies a sourcewise representation condition. The case of noisy data is also analysed.     

Nonlinear Volterra equations on a Hilbert space

We consider equations of the form, u(t) = ? ∝₀^tA(t ? τ)g(u(τ)) dτ + ?(t) (I) on a Hubert space $\text{H}$. A(t) is a family of bounded, linear operators on $\text{H}$ while g is a transformation on $\text{D}$_g ?$\text{H}$ which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators e^Mt and ?e^Mtsin Mt where M is a bounded, negative self-adjoint operator on $\text{H}$ satisfy these conditions. This is shown to yield stability results for differential equations of the form, $\text{Q (}\text{d}\text{dt}\text{) u = ? P (}\text{d}\text{dt}\text{) g(u(t)) + χ(t)}$, on $\text{H}$.