Differential equations with causal operators in a Banach space

In this paper, differential equations with causal operators in the framework of an arbitrary Banach space are studied. Basic results such as existence, uniqueness and global existence are proved. The theory of cones is employed to investigate the extremal solutions of such equations.     

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.     

Hale functional differential equations in a Banach space

Existence, uniqueness and continuous dependence of solutions on initial conditions theorems are proved for the Hale functional differential equations in a Banach space. Concrete examples of the use of the obtained theorems are given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 918–924, July, 1990.     

Linear autonomous neutral differential equations in a Banach space

The existence of solutions for a class of linear functional differential equations defined on a general Banach space is established; the solutions are shown to generate a semigroup of class C₀; a representation of the solutions in terms of a particular family of linear transformations is developed.     

Invariant tori of differential equations in a Banach space

We establish conditions for the existence, uniqueness, and smoothness of the toroidal manifolds of differential equations with unbounded operator coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 13–21, January, 1998.     

A connected system of differential equations in a Banach space

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 7, pp. 898–905, July, 1989.     

Differential inequalities and equations in Banach spaces with a cone

A class of second-order abstract dissipative evolution differential operators D$D$ with 0∈kerD$0\in kerD$ is shown for which the fact that a non-zero t?u(t)$t?u\left(t\right)$ belongs to a cone and −Du$-Du$ to a dual cone may hold only on time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du=0$Du=0$ are given.     

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.     

Monotone method for equations describing transport phenomena in a Banach space

In this paper monotone methods have been developed for equations arising in transport processes in a Banach space. This method proves the existence of extremal solutions for such equations. The advantage of the method for such type of equations is that the successive iterates are solutions of the corresponding initial value problems.     

Second-order linear differential equations in a Banach space and splitting operators

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.     

Optimal control of a class of nonlinear equations in Banach space

The question considered is whether the optimal control problem for systems described by operator equations of the Hammerstein type in a Banach space is well-posed.Translated from Ukrainskii Matematicheskii Zhurnal, Voi. 42, no. 4 pp. 539–541, April, 1990.