Electronic Journal of Qualitative Theory of Differential Equations, 2013
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional dif... more Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family and are necessary for the solvability of the Cauchy problem for all equations from the family. In contrast to many known articles, we consider equations with functional operators acting into the space of essentially bounded functions.
We obtain necessary and sufficient conditions for the unique solvability of the periodic boundary... more We obtain necessary and sufficient conditions for the unique solvability of the periodic boundary value problem for a family of nth-order linear functional differential equations with pointwise constraints on the functional operators.
Some statements about oscillation properties of the spectrum of the Dirichlet boundary value prob... more Some statements about oscillation properties of the spectrum of the Dirichlet boundary value problem for ordinary differential equations are extended to the case of functional-differential equations. It is shown, in particular, that in contrast to ordinary differential equations, even the first three eigenfunctions cannot form a Chebyshev system in the case of functional-differential equations.
Employing the contemporary theory of functional differential equations, we propose an effective t... more Employing the contemporary theory of functional differential equations, we propose an effective test on the existence of a minimum for a wide class of functionals in various Banach spaces.
It is proved that if a linear operator l : C((a;b);R)!L((a;b);R) is nonpositive and for the Cauch... more It is proved that if a linear operator l : C((a;b);R)!L((a;b);R) is nonpositive and for the Cauchy problem u00(t) = l(u)(t) + q(t), u(a) = c the theorem on dierential inequalities is valid, then l is a Volterra operator.
We obtain necessary and sufficient conditions for the integro-differential inequality $$ int_a^bd... more We obtain necessary and sufficient conditions for the integro-differential inequality $$ int_a^bdot x^2(t),dtgeqgammaint_a^bq(t),|dot x(t)x(t)|,dt $$ to be valid with one of the three boundary conditions: $x(a)=0$, or $x(a)=0$, or $x(a)=x(b)=0$. For a power functions $q$, the best constants $gamma$ are found.
Necessary and sufficient conditions for the solvability of boundary value problems for a family o... more Necessary and sufficient conditions for the solvability of boundary value problems for a family of functional differential equations with a non-integrable singularity are obtained.
The Cauchy problem for a system of two scalar non-Volterra linear functional differential equatio... more The Cauchy problem for a system of two scalar non-Volterra linear functional differential equations is considered. Necessary and sufficient conditions for the solvability are obtained. The method which is used here can be applied to many other boundary value problems for functional differential equations with monotone operators. Some new results on the solvability are announced in the last part of the article.
Inequalities for periodic solutions of first-order functional differential equations are obtained... more Inequalities for periodic solutions of first-order functional differential equations are obtained. These inequalities are best possible in a certain sense. MSC: 34K06, 34K10, 34K13.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Abstract. Systems of two linear functional differential equations of the first order with regular... more Abstract. Systems of two linear functional differential equations of the first order with regular operators are considered. General necessary and sufficient conditions for the unique solvability of the periodic problem are obtained. For one system with monotone operators ...
Consider boundary value problems for a functional differential equation $ $\ begin {cases} x^{(n)... more Consider boundary value problems for a functional differential equation $ $\ begin {cases} x^{(n)}(t)=(T^+ x)(t)-(T^-x)(t)+ f (t), &t\ in [a, b],\\ lx= c,\ end {cases} $ $ where $ T^{+}, T^{-}\ colon\ bold C [a, b]\ to\ bold L [a, b] $ are positive linear operators; $ l\ colon\ bold {AC}^{n-...
We consider first-order systems of linear functional differential equations with regular operator... more We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.
Electronic Journal of Qualitative Theory of Differential Equations, 2013
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional dif... more Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family and are necessary for the solvability of the Cauchy problem for all equations from the family. In contrast to many known articles, we consider equations with functional operators acting into the space of essentially bounded functions.
We obtain necessary and sufficient conditions for the unique solvability of the periodic boundary... more We obtain necessary and sufficient conditions for the unique solvability of the periodic boundary value problem for a family of nth-order linear functional differential equations with pointwise constraints on the functional operators.
Some statements about oscillation properties of the spectrum of the Dirichlet boundary value prob... more Some statements about oscillation properties of the spectrum of the Dirichlet boundary value problem for ordinary differential equations are extended to the case of functional-differential equations. It is shown, in particular, that in contrast to ordinary differential equations, even the first three eigenfunctions cannot form a Chebyshev system in the case of functional-differential equations.
Employing the contemporary theory of functional differential equations, we propose an effective t... more Employing the contemporary theory of functional differential equations, we propose an effective test on the existence of a minimum for a wide class of functionals in various Banach spaces.
It is proved that if a linear operator l : C((a;b);R)!L((a;b);R) is nonpositive and for the Cauch... more It is proved that if a linear operator l : C((a;b);R)!L((a;b);R) is nonpositive and for the Cauchy problem u00(t) = l(u)(t) + q(t), u(a) = c the theorem on dierential inequalities is valid, then l is a Volterra operator.
We obtain necessary and sufficient conditions for the integro-differential inequality $$ int_a^bd... more We obtain necessary and sufficient conditions for the integro-differential inequality $$ int_a^bdot x^2(t),dtgeqgammaint_a^bq(t),|dot x(t)x(t)|,dt $$ to be valid with one of the three boundary conditions: $x(a)=0$, or $x(a)=0$, or $x(a)=x(b)=0$. For a power functions $q$, the best constants $gamma$ are found.
Necessary and sufficient conditions for the solvability of boundary value problems for a family o... more Necessary and sufficient conditions for the solvability of boundary value problems for a family of functional differential equations with a non-integrable singularity are obtained.
The Cauchy problem for a system of two scalar non-Volterra linear functional differential equatio... more The Cauchy problem for a system of two scalar non-Volterra linear functional differential equations is considered. Necessary and sufficient conditions for the solvability are obtained. The method which is used here can be applied to many other boundary value problems for functional differential equations with monotone operators. Some new results on the solvability are announced in the last part of the article.
Inequalities for periodic solutions of first-order functional differential equations are obtained... more Inequalities for periodic solutions of first-order functional differential equations are obtained. These inequalities are best possible in a certain sense. MSC: 34K06, 34K10, 34K13.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
Abstract. Systems of two linear functional differential equations of the first order with regular... more Abstract. Systems of two linear functional differential equations of the first order with regular operators are considered. General necessary and sufficient conditions for the unique solvability of the periodic problem are obtained. For one system with monotone operators ...
Consider boundary value problems for a functional differential equation $ $\ begin {cases} x^{(n)... more Consider boundary value problems for a functional differential equation $ $\ begin {cases} x^{(n)}(t)=(T^+ x)(t)-(T^-x)(t)+ f (t), &t\ in [a, b],\\ lx= c,\ end {cases} $ $ where $ T^{+}, T^{-}\ colon\ bold C [a, b]\ to\ bold L [a, b] $ are positive linear operators; $ l\ colon\ bold {AC}^{n-...
We consider first-order systems of linear functional differential equations with regular operator... more We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.
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