A relationship between the periodic and the dirichlet bvps of singular differential equations
01 Jan 1998-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences (Cambridge University Press)-Vol. 128, Iss: 5, pp 1099-1114
TL;DR: In this article, a relationship between the periodic and Dirichlet boundary value problems for second-order ODEs with singularities is established, which may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations.
Abstract: In this paper, a relationship between the periodic and the Dirichlet boundary value problems for second-order ordinary differential equations with singularities is established. This relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations. Using this relationship, we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity.
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TL;DR: In this article, the existence of periodic solutions of the second-order Caratheodory problem is studied, by combining some new properties of Green's function together with Krasnoselskii fixed point theorem on compression and expansion of cones.
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TL;DR: In this paper, the authors generalize the results reached by M. Nkashama, J. Santanilla and L. Sanchez and present estimates for nonnegative and nonpositive solutions of the boundary value problem.
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TL;DR: In this article, it was proved that the singular perturbation problem has at least two positive periodic solutions when the anti-maximum principle holds for the Hill operator and the perturbations are superlinear at infinity.
136 citations
Cites background or result from "A relationship between the periodic..."
...See for example [23,24], where the existence of one positive 2 -periodic solution is proved when 1 and a > 0 is less than the first weighted anti-periodic eigenvalue of x′′ + (1+ cost)x = 0....
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...In some sense, the results in this paper unify some previous works such as in [20,23,24] ....
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...Such a type of singular equations appears in many problems of applications such as the Brillouin focusing system[1,6,23,24]and nonlinear elasticity[4,5]....
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TL;DR: In this paper, a weak force condition enables the achievement of new existence criteria through a basic application of Schauder's fixed point theorem in periodically forced semilinear differential equations with singular nonlinearity.
123 citations
Cites background from "A relationship between the periodic..."
...Since then, the strong force condition became standard in the related works, see, for instance, [3,5,6,10,11,13–16,18,24,27], the recent review [17] and their bibliographies....
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01 Jan 2009
81 citations
Cites background from "A relationship between the periodic..."
...by Habets and Sanchez [101], Mawhin [135], del Pino, Manásevich and Montero [68], Omari and Ye [146], Zhang [202] and [204], Ge and Mawhin [95], Rach̊unková and Tvrdý [168] or Rach̊unková, Tvrdý and Vrkoč [172]....
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References
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TL;DR: The best constant for the simplest Sobolev inequality was proved in this paper by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus of variations.
Abstract: The best constant for the simplest Sobolev inequality is exhibited. The proof is accomplished by symmetrizations (rearrangements in the sense of Hardy-Littlewood) and one-dimensional calculus of variations.
2,011 citations
Book•
31 Dec 1979
TL;DR: In this article, the index of isolated zeros of some mappings is defined as a measure of the number of isolated zero points in a set of mappings, and Bifurcation theory is applied to periodic solutions of autonomous ODEs around an equilibrium.
Abstract: Introduction Suggestions for the readerSuggestions for the reader Fredholm mappings of index zero and linear boundary value problems Degree theory for some classes of mappings Duality theorems for several fixed point operators associated to periodic problems for ordinary differential equations Existence theorems for equations in normed spaces Boundary value problems for second order nonlinear vector differential equations Periodic solutions of ordinary differential equations with one-sided growth restrictions Bound sets for functional differential equations The index of isolated zeros of some mappings Bifurcation theory Periodic solutions of autonomous ordinary differential equations around an equilibrium References.
809 citations
01 Jan 1993
306 citations
01 Jan 1987
TL;DR: In this article, the authors present conditions necessaires et suffisantes d'existence de solutions periodiques d'equations differentielles comportant des singularites.
Abstract: On donne des conditions necessaires et suffisantes d'existence de solutions periodiques d'equations differentielles comportant des singularites. On applique les resultats a l'equation u''+1/uα=h(t)=h(t+T) pour tout α>0 et a u''−1/uα=h/t) si α≥1
241 citations
TL;DR: In this paper, it was shown that the strong force (SF) condition is closely related to the completeness of certain Jacobi metrics associated with the potential V, and this fact permits the use of the standard results of riemannian geometry in the analysis of SF systems.
Abstract: We consider conservative dynamical systems associated with potentials V which have singularities at a set S: V(x) a as x -:+ S. It is shown that various "action" integrals satisfy Condition C of Palais and Smale provided that the potential satisfy a certain strong force (SF) condition. Hence, e.g., we establish the existence in SF systems of periodic trajectories which wind around S and have arbitrary given topological (homotopy) type and which have arbitrary given period, and also periodic trajectories which make arbitrarily tight loops around S. Similar results are also obtained concerning the existence of trajectories which wind around S and join two given points. The SF condition is shown to be closely related to the completeness (in the riemannian sense) of certain Jacobi metrics associated with the potential V, and this fact permits the use of the standard results of riemannian geometry in the analysis of SF systems. The SF condition excludes the gravitational case, and the action integrals do not satisfy the Palais-Smale condition in the gravitational case. The Jacobi metrics associated with gravitational potentials are not complete. For SF systems there exist trajectories which join two given points and make arbitrarily tight loops around S, and this is not the case in the gravitational two body problem. On the other hand, for SF systems any smooth family of X-periodic trajectories (A fixed) is bounded away from S, and this also is not the case for gravitational systems. Thus the definition of the SF condition is "well motivated", and leads to the disclosure of certain differences between the behavior of SF systems and gravitational (and other weak force) systems.
233 citations