Manuel Zamora

On the open problems connected to the results of Lazer and Solimini

A well-known theorem proved by A. C. Lazer and S. Solimini claims that the singular equation u 00 + 1 u = h(t); > 0; has a periodic solution if and only if the mean value of the continuous external force is positive. In this paper, we show that this result cannot be extended to the case when h is an integrable function, unless the additional assumptions are introduced. In addition, for each p 1 and h integrable function in the p-th power, we give a sharp condition guaranteeing the existence of periodic solutions to the above-mentioned equation, showing that there is a close relation between p and the order of singularity, . 2010 MSC: 34C25

PERIODIC SOLUTIONS FOR SINGULAR PERTURBATIONS OF THE SINGULAR ϕ-LAPLACIAN OPERATOR

In this paper, using Leray–Schauder degree arguments and the method of lower and upper solutions, we give existence and multiplicity results for periodic problems with singular nonlinearities of the type where r, n, e : [0, T] → ℝ are continuous functions and λ > 0. We also consider some singular nonlinearities arising in nonlinear elasticity or of Rayleigh–Plesset type.

Periodic solutions to the Liénard type equations with phase attractive singularities

AbstractSufficient conditions are established guaranteeing the existence of a positive ω-periodic solution to the equation u″+f(u)u′+g(u)=h(t,u), where f,g:(0,+∞)→R are continuous functions with possible singularities at zero and h:[0,ω]×R→R is a Carathéodory function. The results obtained are rewritten for the equation of the type u″+cu′uμ+g1uν−g2uγ=h0(t)uδ, where g1, g2, δ are non-negative constants, c, μ, ν, γ are real numbers, and h0∈L([0,ω];R). The last equation also covers the so-called Rayleigh-Plesset equation, frequently used in fluid mechanics to model the bubble dynamics in liquid. In the paper, the case when ν>γ, i.e., the case which covers the attractive singularity of the function g, is studied. The results obtained assure that there exists a positive ω-periodic solution to the above-mentioned equation if the power μ or ν is sufficiently large.MSC:34C25, 34B16, 34B18, 76N15.

Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations

A Fredholm-type theorem for boundary value problems for systems of nonlinear functional differential equations is established. The theorem generalizes results known for the systems with linear or homogeneous operators to the case of systems with positively homogeneous operators.MSC:34K10.

Existence and Multiplicity of Periodic Solutions to Indefinite Singular Equations Having a Non-monotone Term with Two Singularities

Abstract Efficient conditions guaranteeing the existence and multiplicity of T-periodic solutions to the second order differential equation u ′′ = h ⁢ ( t ) ⁢ g ⁢ ( u ) {u^{\prime\prime}=h(t)g(u)} are established. Here, g : ( A , B ) → ( 0 , + ∞ ) {g\colon(A,B)\to(0,+\infty)} is a positive function with two singularities, and h ∈ L ⁢ ( ℝ / T ⁢ ℤ ) {h\in L(\mathbb{R}/T\mathbb{Z})} is a general sign-changing function. The obtained results have a form of relation between multiplicities of zeros of the weight function h and orders of singularities of the nonlinear term. Our results have applications in a physical model, where from the equation u ′′ = h ⁢ ( t ) sin 2 ⁡ u {u^{\prime\prime}=\frac{h(t)}{\sin^{2}u}} one can study the existence and multiplicity of periodic motions of a charged particle in an oscillating magnetic field on the sphere. The approach is based on the classical properties of the Leray–Schauder degree.

Existence of a solution to the Dirichlet problemassociated to a second-order differential equation withsingularities: The method of lower and upper functions

Abstract. This paper is devoted to the study of the existence of a solution to the Dirichlet boundary value problem for the second-order differential equation where the functions satisfy the local Carathéodory conditions and may have singularities both in the time (for and ) and the phase (for and ) variables. Sufficient conditions for the solvability of the above-mentioned problem are established.