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Register a new userDichotomy of Poincare map and boundedness of solutions of certain non-autonomous periodic cauchy problems
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In this paper we study the dichotomy of Poincarce map. We give a relation between the dichotomy of the Poincarce map and boundedness of solutions of certain periodic Cauchy problems
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Under a mild Lipschitz condition we prove a theorem on the existence and uniqueness of global solutions to delay fractional differential equations. Then, we establish a result on the exponential boundedness for these solutions.
We consider the set of monic real univariate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities of their positive and negative roots (all roots are distinct). For $d=6$ and for signs of the coefficients $(+,-,+,+,+,-,+)$, we prove that the set of such polynomials having two positive, two negative and two complex conjugate roots, is not connected.
Existence and spatio-temporal symmetric patterns of periodic solutions to second order reversible equivariant non-autonomous periodic systems with multiple delays are studied under the Hartman-Nagumo growth conditions. The method is based on using the Brouwer $D_1 times mathbb Z_2times Gamma$-equivariant degree theory, where $D_1$ is related to the reversing symmetry, $mathbb Z_2$ is related to the oddness of the right-hand-side and $Gamma$ reflects the symmetric character of the coupling in the corresponding network. Abstract results are supported by a concrete example with $Gamma = D_n$ -- the dihedral group of order $2n$.
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The authors use steepest descent ideas to obtain a priori $L^p$ estimates for solutions of Riemann-Hilbert Problems. Such estimates play a crucial role, in particular, in analyzing the long-time behavior of solutions of the perturbed nonlinear Schrodinger equation on the line.
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