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1
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On oscillation and nonoscillation properties of Emden-Fowler difference equations

100%
Cecchi M., Došlá Z., Marini M.
Open Mathematics
|
2009
|
tom 7
|
nr 2
322-334
EN
A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation $$ \Delta (a_n \left| {\Delta x_n } \right|^\alpha sgn\Delta x_n ) + b_n \left| {x_{n + 1} } \right|^\beta sgnx_{n + 1} = 0 $$ is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.
2
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Unbounded solutions of the max-type difference equation $$ x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\} $$

100%
Kerbert C., Radin M.
Open Mathematics
|
2008
|
tom 6
|
nr 2
307-324
EN
We investigate the boundedness nature of positive solutions of the difference equation $$ x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\},n = 0,1,..., $$ where {A n}n=0∞ and {B n}n=0∞ are periodic sequences of positive real numbers.
3
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Continuous dependence on parameters for second order discrete BVP’s

81%
Galewski M., Głąb S.
Open Mathematics
|
2012
|
tom 10
|
nr 3
1076-1083
EN
Using Fan’s Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional.
4
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Results on the deficiencies of some differential-difference polynomials of meromorphic functions

81%
Zheng X.-M., Xu H.-Y.
Open Mathematics
|
2016
|
tom 14
|
nr 1
100-108
EN
In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying lim sup r→+∞ T(r, f) T(r,  f ′ ) <+∞, $$\mathop {\lim \,{\rm sup}}\limits_{r \to + \infty } {{T(r,\,f)} \over {T(r,\,f')}}{\rm{ < }} + \infty ,$$ and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate the value distribution of some differential-difference polynomials taking small function a(z) with respect to f(z).
5
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A trichotomy result for non-autonomous rational difference equations

81%
Palladino F., Radin M.
Open Mathematics
|
2011
|
tom 9
|
nr 5
1135-1142
EN
We study non-autonomous rational difference equations. Under the assumption of a periodic non-autonomous parameter, we show that a well known trichotomy result in the autonomous case is preserved in a certain sense which is made precise in the body of the text. In addition we discuss some questions regarding whether periodicity preserves or destroys boundedness.
6
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Unboundedness results for systems

81%
Lugo G., Palladino F.
Open Mathematics
|
2009
|
tom 7
|
nr 4
741-756
EN
We study k th order systems of two rational difference equations $$ x_n = \frac{{\alpha + \sum\nolimits_{i = 1}^k {\beta _i x_{n - i} + } \sum\nolimits_{i = 1}^k {\gamma _i y_{n - i} } }} {{A + \sum\nolimits_{j = 1}^k {B_j x_{n - j} + } \sum\nolimits_{j = 1}^k {C_j y_{n - j} } }},n \in \mathbb{N}, $$ In particular we assume non-negative parameters and non-negative initial conditions. We develop several approaches which allow us to prove that unbounded solutions exist for certain initial conditions in a range of the parameters.
7
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Some boundedness results for systems of two rational difference equations

81%
Lugo G., Palladino F.
Open Mathematics
|
2010
|
tom 8
|
nr 6
1058-1090
EN
We study k th order systems of two rational difference equations $$ x_n = \frac{{\alpha + \sum\nolimits_{i = 1}^k {\beta _i x_{n - 1} + } \sum\nolimits_{i = 1}^k {\gamma _i y_{n - 1} } }} {{A + \sum\nolimits_{j = 1}^k {B_j x_{n - j} + } \sum\nolimits_{j = 1}^k {C_j y_{n - j} } }}, y_n = \frac{{p + \sum\nolimits_{i = 1}^k {\delta _i x_{n - i} + } \sum\nolimits_{i = 1}^k {\varepsilon _i y_{n - i} } }} {{q + \sum\nolimits_{j = 1}^k {D_j x_{n - j} + } \sum\nolimits_{j = 1}^k {E_j y_{n - j} } }} n \in \mathbb{N} $$. In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.
8
Content available remote

Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations

62%
Luo L.-Q., Zheng X.-M.
Open Mathematics
|
2016
|
tom 14
|
nr 1
970-976
EN
In this paper, we investigate the value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations, and obtain the results on the relations between the order of the solutions and the convergence exponents of the zeros, poles, a-points and small function value points of the solutions, which show the relations in the case of non-homogeneous equations are sharper than the ones in the case of homogeneous equations.
9
Content available remote

Triple solutions for a Dirichlet boundary value problem involving a perturbed discretep(k)-Laplacian operator

52%
Khaleghi Moghadam M., Henderson J.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1075-1089
EN
Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.
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