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A new approach for solving nonlinear BVP's on the half-line for second order equations and applications

Serena Matucci (2015)

Mathematica Bohemica

We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications,...

Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential.

Charles L. Fefferman, Luis A. Seco (1993)

Revista Matemática Iberoamericana

In [FS1] we announced a precise asymptotic formula for the ground-state energy of a non-relativistic atom. The purpose of this paper is to establish an elementary inequality that plays a crucial role in our proof of that formula. The inequality concerns the Thomas-Fermi potentialVTF = -y(ar) / r, a > 0, where y(r) is defined as the solution of⎧   y''(x) = x-1/2y3/2(x),⎨   y(0) = 1,⎩   y(∞) = 0.

Bifurcation of periodic solutions to variational inequalities in κ based on Alexander-Yorke theorem

Milan Kučera (1999)

Czechoslovak Mathematical Journal

Variational inequalities U ( t ) K , ( U ˙ ( t ) - B λ U ( t ) - G ( λ , U ( t ) ) , Z - U ( t ) ) 0 for all Z K , a.a. t [ 0 , T ) are studied, where K is a closed convex cone in κ , κ 3 , B λ is a κ × κ matrix, G is a small perturbation, λ a real parameter. The assumptions guaranteeing a Hopf bifurcation at some λ 0 for the corresponding equation are considered and it is proved that then, in some situations, also a bifurcation of periodic solutions to our inequality occurs at some λ I λ 0 . Bifurcating solutions are obtained by the limiting process along branches of solutions to penalty problems starting at λ 0 constructed...

Boundary layer phenomenon for three -point boundary value problem for the nonlinear singularly perturbed systems

Robert Vrabel (2011)

Kybernetika

This paper deals with the three-point boundary value problem for the nonlinear singularly perturbed second-order systems. Especially, we focus on an analysis of the solutions in the right endpoint of considered interval from an appearance of the boundary layer point of view. We use the method of lower and upper solutions combined with analysis of the integral equation associated with the class of nonlinear systems considered here.

Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

Ahmet Yantir, Ireneusz Kubiaczyk, Aneta Sikorska-Nowak (2015)

Open Mathematics

In this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s...

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