Jiří Šremr

On a periodic-type boundary value problem for first-order nonlinear functional differential equations

Nonimprovable,in a certain sense, sufficient conditions for the unique solvability of antiperiodic type BVP for first order scalar functional differential equations are established.

ON NONNEGATIVE SOLUTIONS OF A CERTAIN BOUNDARY VALUE PROBLEM FOR FIRST ORDER LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

Unimprovable efficient conditions are established for the existence and uniqueness of a nonnegative solution of the problem u′(t) = `(u)(t) + q(t), u(a) = h(u) + c, where ` : C([a, b]; R) → L([a, b]; R) is a linear bounded operator, h : C([a, b]; R) → R is a linear bounded functional, q ∈ L([a, b]; R) and c > 0.

On oscillations of solutions to second-order linear delay differential equations

Abstract. New oscillation criteria are proved for second-order two-term linear delay differential equations with locally integrable coefficients and measurable delays. In particular, a suitable a priori lower bound of non-oscillatory solutions is found, which plays a crucial role in the proofs of the results obtained.

SOLVABILITY AND THE UNIQUE SOLVABILITY OF A PERIODIC TYPE BOUNDARY VALUE PROBLEM FOR FIRST ORDER SCALAR FUNCTIONAL DIFFERENTIAL EQUATIONS

Abstract Nonimprovable in a certain sense, sufficient conditions for the solvability and unique solvability of the problem 𝑢′ (𝑡) = 𝐹 (𝑢) (𝑡), 𝑢(𝑎) – λ𝑢(𝑏) = ℎ(𝑢) are established, where 𝐹 : 𝐶([𝑎, 𝑏]; 𝑅) → 𝐿([𝑎, 𝑏];𝑅) is a continuous operator satisfying the Carathéodory condition, ℎ : 𝐶([𝑎, 𝑏]; 𝑅) → 𝑅 is a continuous functional, and λ ∈ 𝑅+.

Linear Hyperbolic Functional-Differential Equations with Essentially Bounded Right-Hand Side

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problem 𝑢(1,1)(𝑡,𝑥)=l0(𝑢)(𝑡,𝑥)

On the characteristic initial-value problem for linear partial functional-differential equations of hyperbolic type

Theorems on the Fredholm alternative and well-posedness of the characteristic initial-value problem are established, where l : C ( ;ℝ) is a linear bounded operator, q ∈ L ( ;ℝ), ϕ:[ a,b ]→ℝ, ψ:[ c,d ]→ℝ are absolutely continuous functions and =[ a,b ]×[ c,d ]. Some solvability conditions of the problem considered are also given.

Abstract differential inequalities and the Cauchy problem for infinite-dimensional linear functional differential equations

We establish optimal, in a sense, unique solvability conditions of the Cauchy problem for a wide class of linear functional differential equations in a Banach space with a solid wedge. The conditions are formulated in terms of certain abstract functional differential inequalities.

Carathéodory solutions to a hyperbolic differential inequality with a non-positive coefficient and delayed arguments

New effective conditions are found for the validity of a theorem on differential inequalities corresponding to the Darboux problem for linear hyperbolic differential equations with argument deviations.MSC:35L10, 35L15.

On oscillation and nonoscillation of two-dimensional linear differential systems

Abstract. New oscillation and nonoscillation criteria are established for the two-dimensional system , , where are locally integrable functions, for a.e. , and .

Component-wise positivity of solutions to periodic boundary problem for linear functional differential system

The classical Ważewski theorem claims that the condition pij ≤ 0, j ≠ i, i, j =1,...,n, is necessary and sufficient for non-negativity of all the components of solution vector to a system of the inequalities x′(t)+∑j=1npij(t)x(t)≥0, xi (0) ≥ 0, i =1, ..., n. Although this result was extent on various boundary value problems and on delay differential systems, analogs of these heavy restrictions on non-diagonal coefficients pij preserve in all assertions of this sort. It is clear from formulas of the integral representation of the general solution that these theorems claim actually the positivity of all elements of Greens matrix. The method to compare only one component of the solution vector, which does not require such heavy restrictions, is proposed in this article. Note that comparison of only one component of the solution vector means the positivity of elements in a corresponding row of Greens matrix. Necessary and sufficient conditions of this fact are obtained in the form of theorems about differential inequalities. It is demonstrated that the sufficient conditions of positivity of the elements in the nth row of Greens matrix, proposed in this article, cannot be improved in corresponding cases. The main idea of our approach is to construct a first order functional differential equation for the n th component of the solution vector and then to use assertions, obtained recently for first order scalar functional differential equations. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. Note that in some cases the sufficient conditions, obtained in the article, does not require any smallness of the interval [0, ω], where the system is considered.Mathematics Subject Classification 2000: 34K06; 34K10.