Convergence results for unbounded solutions of first order non-linear differential-functional equations

Henryk Leszczyński

Displaying similar documents to “Convergence results for unbounded solutions of first order non-linear differential-functional equations”

Comparison of solutions and successive approximations in the theory of the equation $\partial^{2} z / \partial x \partial y = f (x, y, z, \partial z / \partial x, \partial z / \partial y)$

J. Kisyński, A. Pelczar

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CONTENTSIntroduction........................................................................................................................................................................................................... 5I. THE CAUCHY-DARBOUX PROBLEM IN FUNCTION CLASSES $C^{1}' * (Δ_{a, b}; E)$ AND $L_{1}^{1, *} (Δ_{a, b}; E)$ ......................... 71. Basic function classes ......................................................................................................................................................................................

Positive solutions for a class of non-autonomous second order difference equations via a new functional fixed point theorem

Lydia Bouchal, Karima Mebarki, Svetlin Georgiev Georgiev b (2022)

Archivum Mathematicum

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In this paper, by using recent results on fixed point index, we develop a new fixed point theorem of functional type for the sum of two operators $T + S$ where $I - T$ is Lipschitz invertible and $S$ a $k$ -set contraction. This fixed point theorem is then used to establish a new result on the existence of positive solutions to a non-autonomous second order difference equation.

Global behavior of the difference equation $x_{n + 1} = \frac{a x_{n - 3}}{b + c x_{n - 1} x_{n - 3}}$

Raafat Abo-Zeid (2015)

Archivum Mathematicum

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In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $x_{n + 1} = \frac{a x_{n - 3}}{b + c x_{n - 1} x_{n - 3}}, n = 0, 1, \dots$ where $a, b, c$ are positive real numbers and the initial conditions $x_{- 3}$ , $x_{- 2}$ , $x_{- 1}$ , $x_{0}$ are real numbers.

On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations

Malkhaz Ashordia (2016)

Mathematica Bohemica

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The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $d x (t) = d A_{0} (t) \cdot x (t) + d f_{0} (t)$ , $x (t_{0}) = c_{0}$ $(t \in I)$ with a unique solution $x_{0}$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $d x (t) = d A_{k} (t) \cdot x (t) + d f_{k} (t)$ , $x (t_{k}) = c_{k}$ $(k = 1, 2, \dots)$ to have a unique solution $x_{k}$ for any sufficiently large $k$ such that $x_{k} (t) \to x_{0} (t)$ uniformly on $I$ . Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems....

On the completeness of $ℒ_{p}$ -spaces over a charge

Sreela Gangopadhyay (1990)

Colloquium Mathematicae

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Convergence of greedy approximation I. General systems

S. V. Konyagin, V. N. Temlyakov (2003)

Studia Mathematica

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We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{ε} (x) : = \sum_{j \in D_{ε} (x)} e *_{j} (x) e_{j}$ , where $D_{ε} (x) : = j : | e *_{j} (x) | \geq ε$ . We study a generalized version of $T_{ε}$ that we call the weak thresholding approximation. We modify the $T_{ε} (x)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t, ε} (x) : = j : t ε \leq | e *_{j} (x) | < ε$ and consider...

Lower bounds for Schrödinger operators in H¹(ℝ)

Ronan Pouliquen (1999)

Studia Mathematica

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We prove trace inequalities of type $| | u^{'} {| |}_{L^{2}}^{2} + \sum_{j \in ℤ} k_{j} | u (a_{j}) |^{2} {\geq λ | | u | |}_{L^{2}}^{2}$ where $u \in H^{1} (ℝ)$ , under suitable hypotheses on the sequences ${a_{j}}_{j \in ℤ}$ and ${k_{j}}_{j \in ℤ}$ , with the first sequence increasing and the second bounded.

Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let $\sum_{n = 1}^{\infty} a_{n}$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_{n})$ is non-increasing, then $lim_{n \to \infty} n a_{n} = 0$ . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $lim_{n \to \infty} n a_{n} = 0$ ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$ -convergence, that is a convergence according to an ideal $ℐ$ of subsets of $ℕ$ . Again, Olivier’s theorem is a consequence...

Some limit theorems for $m$ -pairwise negative quadrant dependent random variables

Yongfeng Wu, Jiangyan Peng (2018)

Kybernetika

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The authors first establish the Marcinkiewicz-Zygmund inequalities with exponent $p$ ( $1 \leq p \leq 2$ ) for $m$ -pairwise negatively quadrant dependent ( $m$ -PNQD) random variables. By means of the inequalities, the authors obtain some limit theorems for arrays of rowwise $m$ -PNQD random variables, which extend and improve the corresponding results in [Y. Meng and Z. Lin (2009)] and [H. S. Sung (2013)]. It is worthy to point out that the open problem of [H. S. Sung, S. Lisawadi, and A. Volodin (2008)] can be...

Convergence in the dual of certain $K M_{p}$ -spaces

Larry Kitchens, Charles Swartz (1974)

Colloquium Mathematicae

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