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Title and Author(s) |
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On orbital topological equivalence of cubic ODEs in two-dimensional algebras
Zolman Balanov, Wieslaw Krawcewicz and Shira Zur
| ABSTRACT.
Cubic differential systems in real commutative two-dimensional
algebras are classified up to orbital topological equivalence
via the solubility of polynomial equations in algebras.
As a by-product, existence of bounded solutions in such systems is studied via complex structures in
the algebras. Application to the existence of periodic solutions
to n-dimensional differential systems "cubic at infinity" is given.
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205
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Geodesics in conical manifolds
Marco Ghimenti
| ABSTRACT.
The aim of this paper is to extend the definition of geodesics to
conical manifolds, defined as submanifolds of Rn with a
finite number of singularities. We look for an
approach suitable both for the local geodesic problem and for the
calculus of variation in the large. We give a definition
which links the local solutions of the Cauchy problem (1.1) with variational
geodesics, i.e. critical points of the energy functional.
We prove a deformation lemma (Theorem 2.2)
which leads us to extend the Lusternik--Schnirelmann theory to
conical manifolds, and to estimate the number of geodesics
(Theorem 3.4 and Corollary 3.5).
In Section 4, we provide some applications in which conical manifolds arise naturally: in
particular, we focus on the brachistochrone problem for a
frictionless particle moving in Sn or in Rn in the presence of
a potential U(x)$ unbounded from below. We conclude with an
appendix in which the main results are presented in a general framework.
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235
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Twin positive periodic solutions of second order differential systems
Xiaoning Lin, Daqing Jiang, Donal O'Regan and Ravi P. Agarwal
| ABSTRACT.
In this paper, we study positive periodic solutions to
singular second order differential systems. It
is proved that such a problem has at least two
positive periodic solutions. The proof relies on a nonlinear
alternative of Leray-Schauder type and on Krasnosel'skii fixed
point theorem on compression and expansion of cones.
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263
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Non-collision periodic solutions of prescribed energy problem for a class of singular Hamiltonian systems
Shinji Adachi
ABSTRACT.
We study the existence of non-collision periodic solutions
with prescribed energy for the following singular Hamiltonian systems:
\cases
\ddot q + \nabla V(q) = 0,
\Half|\dot q|^2 + V(q) = H.
\endcases
In particular for the potential
V(q)\sim -1/\dist (q,D)^\alpha, where the singular set D
is a non-empty compact subset of RN,
we prove the existence of a non-collision periodic solution for
all H > 0 and \alpha\in (0,2).
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275
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Fixed point approaches to the solution of integral inclusions
Daniel Biles, Mark Robinson and John Spraker
| ABSTRACT.
Solutions to generalizations of the Volterra and
Hammerstein integral inclusions are found by using the fixed point theorems
of Covitz-Nadler and Bohnenblust-Karlin.
Several illustrative examples are presented.
Some conditions are given which also allow Lipschitz solutions to be obtained.
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297
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Topological structure of solution sets to parabolic problems
Vladimir Durikovic and Monika Durikovicova
| ABSTRACT.
In this paper we deal with the Peano phenomenon for general initial-boundary value problems
of quasilinear parabolic equations with arbitrary even order space derivatives.
The nonlinearity is assumed to be a continuous or continuously Frechet differentiable function. Using a method of transformation to an operator equation and employing the theory of proper, Fredholm (linear and nonlinear)
and Nemitskii operators, we study the existence of solution of the given problem and qualitative and quantitative structure of its solution and bifurcation sets. These results can be applied to the different technical and natural science models.
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313
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Existence results for impulsive neutral functional differential inclusions
Bapurao C. Dhage and Sotiris K. Ntouyas
| ABSTRACT.
In this paper we prove existence results for first and second order impulsive neutral functional differential inclusions under the mixed Lipschitz and Caratheodory conditions.
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349
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The Invariance of Domain Theorem for Condensing Vector Fields
In-Sook Kim
| ABSTRACT.
Using degree theory for countably condensing maps due
to Vath, we give an invariance of domain theorem
for countably condensing vector fields. The key tool
is Borsuk's theorem for odd countably condensing maps.
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363
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On generalized Sobolev algebras and their applications
Severine Bernard and Silvere P. Nuiro
| ABSTRACT.
In the last two decades, many algebras of generalized functions
have been constructed, particularly the so-called generalized
Sobolev algebras. Our goal is to study the latter and some of
their main properties. In this framework, we pose and solve a
nonlinear degenerated Dirichlet problem with irregular data such
as Dirac generalized functions.
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375
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On random coincidence point theorems
Naaser Shahzad
| ABSTRACT.
Some random coincidence point theorems are proved. The
results of Benavides et. el. [2], Itoh [8],
Shahzad and Latif [23], Tan and Yuan [24] and Xu [25] are either
extended or improved.
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391
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