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The Arzelà–Ascoli theorem by means of ideal convergence

  • Emre Taş EMAIL logo and Tugba Yurdakadim
Published/Copyright: November 30, 2017
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 25 Issue 3

Abstract

In this paper, using the concept of ideal convergence, which extends the idea of ordinary convergence and statistical convergence, we are concerned with the I-uniform convergence and the I-pointwise convergence of sequences of functions defined on a set of real numbers D. We present the Arzelà–Ascoli theorem by means of ideal convergence and also the relationship between I-equicontinuity and I-continuity for a family of functions.

Keywords: Pointwise and uniform convergence; ideal convergence; classical Arzelà–Ascoli theorem
MSC 2010: 40A30; 40A35

Acknowledgements

The authors wish to thank the referee for several helpful suggestions that have improved the exposition of these results.

References

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Received: 2016-04-07
Revised: 2016-12-29
Accepted: 2017-06-26
Published Online: 2017-11-30
Published in Print: 2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Semilinear fractional order integro-differential inclusions with infinite delay
  3. Quasi-nilpotent perturbations of the generalized Kato spectrum
  4. Construction and numerical resolution of high-order accuracy decomposition scheme for a quasi-linear evolution equation
  5. Relations between BV*(q;α) and Λ*BVp classes of functions
  6. On the constancy of signs and order of magnitude of Fourier–Haar coefficients
  7. On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations
  8. Optimal control problem for the equation of vibrations of an elastic plate
  9. Generalized Hausdorff capacities and their applications
  10. Approximation by bivariate (p,q)-Baskakov–Kantorovich operators
  11. A hydromagnetic flow through porous medium near an accelerating plate in the presence of magnetic field
  12. A note on the Borel types of some small sets
  13. On some boundary value problems for the heat equation in a non-regular type of a prism of ℝN+1
  14. Some weighted integral inequalities for differentiable h-preinvex functions
  15. Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras
  16. On Baer invariants of pairs of groups
  17. The Arzelà–Ascoli theorem by means of ideal convergence
  18. Absolute convergence of multiple Fourier series of a function of p(n)-Λ-BV
https://doi.org/10.1515/gmj-2017-0052
Keywords for this article
Pointwise and uniform convergence; ideal convergence; classical Arzelà–Ascoli theorem

Articles in the same Issue

  1. Frontmatter
  2. Semilinear fractional order integro-differential inclusions with infinite delay
  3. Quasi-nilpotent perturbations of the generalized Kato spectrum
  4. Construction and numerical resolution of high-order accuracy decomposition scheme for a quasi-linear evolution equation
  5. Relations between BV*(q;α) and Λ*BVp classes of functions
  6. On the constancy of signs and order of magnitude of Fourier–Haar coefficients
  7. On the oscillatory behavior of solutions of higher order nonlinear fractional differential equations
  8. Optimal control problem for the equation of vibrations of an elastic plate
  9. Generalized Hausdorff capacities and their applications
  10. Approximation by bivariate (p,q)-Baskakov–Kantorovich operators
  11. A hydromagnetic flow through porous medium near an accelerating plate in the presence of magnetic field
  12. A note on the Borel types of some small sets
  13. On some boundary value problems for the heat equation in a non-regular type of a prism of ℝN+1
  14. Some weighted integral inequalities for differentiable h-preinvex functions
  15. Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras
  16. On Baer invariants of pairs of groups
  17. The Arzelà–Ascoli theorem by means of ideal convergence
  18. Absolute convergence of multiple Fourier series of a function of p(n)-Λ-BV
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