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1
Content available remote

Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms

100%
Pişkin E.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
We consider the existence, both locally and globally in time, the decay and the blow up of the solution for the extensible beam equation with nonlinear damping and source terms. We prove the existence of the solution by Banach contraction mapping principle. The decay estimates of the solution are proved by using Nakao’s inequality. Moreover, under suitable conditions on the initial datum, we prove that the solution blow up in finite time.
2
Content available remote

On the nonlocal Cauchy problem for semilinear fractional order evolution equations

100%
Wang J., Zhou Y., Fečkan M.
Open Mathematics
|
2014
|
tom 12
|
nr 6
911-922
EN
In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.
3
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Periodic Solutions for Nonlinear Evolution Equations with Non-instantaneous Impulses

88%
Fečkan M., Wang J., Zhou Y.
Nonautonomous Dynamical Systems
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2014
|
tom 1
|
nr 1
EN
In this paper, we consider periodic solutions for a class of nonlinear evolution equations with non-instantaneous impulses on Banach spaces. By constructing a Poincaré operator, which is a composition of the maps and using the techniques of a priori estimate, we avoid assuming that periodic solution is bounded like in [1-4] and try to present new sufficient conditions on the existence of periodic mild solutions for such problems by utilizing semigroup theory and Leray-Schauder's fixed point theorem. Furthermore, existence of a global compact connected attractor for the Poincaré operator is derived.
4
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Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions

75%
Aqlan M. H., Alsaedi A., Ahmad B., Nieto J. J.
Open Mathematics
|
2016
|
tom 14
|
nr 1
723-735
EN
We develop the existence theory for sequential fractional differential equations involving Liouville-Caputo fractional derivative equipped with anti-periodic type (non-separated) and nonlocal integral boundary conditions. Several existence criteria depending on the nonlinearity involved in the problems are presented by means of a variety of tools of the fixed point theory. The applicability of the results is shown with the aid of examples. Our results are not only new in the given configuration but also yield some new special cases for specific choices of parameters involved in the problems.
5
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System of fractional differential equations with Erdélyi-Kober fractional integral conditions

75%
Thongsalee N., Laoprasittichok S., Ntouyas S. K., Tariboon J.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
In this paper we study existence and uniqueness of solutions for a system consisting from fractional differential equations of Riemann-Liouville type subject to nonlocal Erdélyi-Kober fractional integral conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.
6
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Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

63%
Staněk S.
Open Mathematics
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2013
|
tom 11
|
nr 3
574-593
EN
We investigate the fractional differential equation u″ + A c D α u = f(t, u, c D μ u, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.
7
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Affine compact almost-homogeneous manifolds of cohomogeneity one

63%
Guan D.
Open Mathematics
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2009
|
tom 7
|
nr 1
84-123
EN
This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.
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