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Operators that are Weakly Coercive and a Compact Perturbation

  • José María Soriano Arbizu EMAIL logo and Manuel Ordoñez Cabrera
Published/Copyright: December 18, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 6

ABSTRACT

Let X, Y be two Banach spaces over K=ℝ or ℂ , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

2020 Mathematics Subject Classification: Primary 47H14; Secondary 47A55
Keywords: Compact operator; Fredholm mapping; weakly coercive operator; index; local Cr-diffeomorphism; global inverse mapping theorem

(Communicated by Tomasz Natkaniec)

Funding statement: The authors have been partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 Grant P08-FQM-03543, and by MEC Grant MTM2015-65242-C2-1-P.

Acknowledgement

The authors are very grateful to the referee for carefully reading the manuscript and for offering substantial comments and suggestions which enabled them to improve the paper.

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Received: 2022-04-28
Accepted: 2023-01-13
Published Online: 2023-12-18

© 2023 Mathematical Institute Slovak Academy of Sciences

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  7. Generalized Anti-Symmetry Laws in Groupoids
  8. New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications
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https://doi.org/10.1515/ms-2023-0112
Keywords for this article
Compact operator; Fredholm mapping; weakly coercive operator; index; local Cr-diffeomorphism; global inverse mapping theorem

Articles in the same Issue

  1. ℒ-fuzzy Annihilators in Residuated Lattices
  2. Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications
  3. Uniform Local Connectedness and Completion of Metric σ-Frames
  4. On Transcendental Regular Continued Fractions
  5. On Repdigits Which are Sums or Differences of Two k-Pell Numbers
  6. Peirce Decompositions for Evolution Algebras
  7. Generalized Anti-Symmetry Laws in Groupoids
  8. New Bounds of Cyclic Jensen’s Differences via Weighted Hadamard Inequalities with Applications
  9. Geometric Properties of Generalized Bessel Function Associated with the Exponential Function
  10. On the Attainable Set of Iterative Differential Inclusions
  11. On the Differential-Difference Sine-Gordon Equation with an Integral Type Source
  12. A Critical p(x)-Laplacian Steklov Type Problem with Weights
  13. Distributivity of a Uni-nullnorm with Continuous and Archimedean Underlying T-norms and T-conorms Over an Arbitrary Uninorm
  14. Some Approximation Properties of Operators Including Degenerate Appell Polynomials
  15. Operators that are Weakly Coercive and a Compact Perturbation
  16. On the Structure Lie Operator of a Real Hypersurface in the Complex Quadric
  17. Paracompactness and Open Relations
  18. On (Strongly) (Θ-)Discrete Homogeneous Spaces
  19. The Alpha Power Muth-G Distributions and Its Applications in Survival and Reliability Analyses
  20. Weakly Einstein Equivalence in a Golden Space Form and Certain CR-submanifolds
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