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The paper can be understood as a completion of the $q$ -Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear $q$ -difference equations. The $q$ -Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice $q^{ℕ_{0}} : = {q^{k} : k \in ℕ_{0}}$ with $q > 1$ . In addition to recalling the existing concepts of $q$ -regular variation and $q$ -rapid variation we introduce $q$ -regularly bounded functions and prove many related properties. The $q$ -Karamata theory is then...

Self-adjoint differential equations and generalized Karamata functions

J. Jaroš, T. Kusano (2004)

Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques

Selfadjoint Means and Operator Monotone Functions.

Frank Hansen (1981)

Mathematische Annalen

Semiéquicontinuité approximative et measurabilité

Zbigniew Grande (1981)

Colloquium Mathematicae

Semigroups of ...-density continuous functions.

K. Ciesielski, L. Larson, K. Ostaszewski (1992)

Semigroup forum

Semilinear fractional order integro-differential equations with infinite delay in Banach spaces

Khalida Aissani, Mouffak Benchohra (2013)

Archivum Mathematicum

This paper concerns the existence of mild solutions for fractional order integro-differential equations with infinite delay. Our analysis is based on the technique of Kuratowski’s measure of noncompactness and Mönch’s fixed point theorem. An example to illustrate the applications of main results is given.

Semilinear functional differential equations of fractional order with state-dependent delay.

Darwish, Mohamed Abdalla, Ntouyas, Sotiris K. (2009)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Separate approximate continuity and strong approximate continuity

Richard J. O'Malley (1985)

Colloquium Mathematicae

Separating sets by Darboux-like functions

Aleksander Maliszewski (2002)

Fundamenta Mathematicae

We consider the following problem: Characterize the pairs ⟨A,B⟩ of subsets of ℝ which can be separated by a function from a given class, i.e., for which there exists a function f from that class such that f = 0 on A and f = 1 on B (the classical separation property) or f < 0 on A and f > 0 on B (a new separation property).

Separation by monotonic functions.

Förg-Rob, Wolfgang, Nikodem, Kazimierz, Páles, Zsolt (1996)

Mathematica Pannonica

Separation via quadratic functions.

Zsolt Páles, Vera Zeidan (1996)

Aequationes mathematicae

Separation via quadratic functions. (Summary).

Zsolt Páles, Vera Zeidan (1995)

Aequationes mathematicae

Series solutions of time-fractional PDEs by homotopy analysis method.

Abdulaziz, O., Hashim, I., Saif, A. (2008)

Differential Equations & Nonlinear Mechanics

Set-valued fractional order differential equations in the space of summable functions

Hussein A.H. Salem (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we study the existence of integrable solutions for the set-valued differential equation of fractional type $(D^{α ₙ} - \sum_{i = 1}^{n - 1} a_{i} D^{α_{i}}) x (t) \in F (t, x (φ (t)))$ , a.e. on (0,1), $I^{1 - α ₙ} x (0) = c$ , αₙ ∈ (0,1), where F(t,·) is lower semicontinuous from ℝ into ℝ and F(·,·) is measurable. The corresponding single-valued problem will be considered first.

Several comments on the Henstock-Kurzweil and McShane integrals of vector-valued functions

Kirill Naralenkov (2011)

Czechoslovak Mathematical Journal

We make some comments on the problem of how the Henstock-Kurzweil integral extends the McShane integral for vector-valued functions from the descriptive point of view.

Sharkovskiĭ's theorem holds for some discontinuous functions

Piotr Szuca (2003)

Fundamenta Mathematicae

We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected $G_{δ}$ graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected $G_{δ}$ graph.

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