Robert Hakl

On radial stationary solutions to a model of non-equilibrium growth

We present the formal geometric derivation of a nonequilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of nonequilibrium statistical mechanics.

On periodic solutions of first order linear functional differential equations

New optimal sufficient conditions are established for the existence of a unique periodic solution of first order scalar functional differential equations

Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth

The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. It turns out that the existence or not of such solutions depends on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter, we prove the existence of solutions to this boundary value problem. For large values of the same parameter, we prove the nonexistence of solutions. We also provide rigorous bounds for the values of this parameter, which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions applied to an associated two-point boundary value problem. Copyright

A PERIODIC BOUNDARY VALUE PROBLEM FOR FUNCTIONAL DIFFERENTIAL EQUATIONS OF HIGHER ORDER

Abstract On the interval [0,ω], consider the periodic boundary value problem where 𝑛 ≥ 2, 𝑙𝑖 : 𝐶([0,ω];𝑅) → 𝐿([0,ω];𝑅) (𝑖 = 0,…,𝑛 – 1) are linear bounded operators, 𝑞 ∈ 𝐿([0,ω];𝑅), 𝑐𝑗 ∈ 𝑅 (𝑗 = 0,…,𝑛 – 1). The effective sufficient conditions guaranteeing the unique solvability of the considered problem are established.

On One Estimate for Periodic Functions

Abstract For , the estimate is derived, where and 𝑑𝑛 are defined by a certain recurrent formula.

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.

Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign

New criteria for the existence of a maximum or antimaximum principle of a general second order operator with periodic conditions, as well as conditions for nonresonance, are provided and compared with the related literature.

On a Boundary Value Problem for -th Order Linear Functional Differential Systems

Abstract In this paper, theorems on the Fredholm alternative and wellposedness of the linear boundary value problem 𝑢′(𝑡) = ℓ(𝑢)(𝑡) + 𝑞(𝑡), ℎ(𝑢) = 𝑐, where ℓ : 𝐶([𝑎, 𝑏]; 𝑅𝑛) → 𝐿([𝑎, 𝑏]; 𝑅𝑛) and ℎ : 𝐶([𝑎, 𝑏]; 𝑅𝑛) → 𝑅𝑛 are linear bounded operators, 𝑞 ∈ 𝐿([𝑎, 𝑏]; 𝑅𝑛), and 𝑐 ∈ 𝑅𝑛, are established even when ℓ is not a strongly bounded operator.

On the Periodic Boundary Value Problem for First-Order Functional-Differential Equations

V clanku jsou nalezeny efektivni podminky zarucujici řesitelnost periodicke ulohy pro funkcionalni diferencialni rovnice prvniho řadu.

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.