Abstract
Let X be a real Hilbert space and \(X^{*}\) its dual space. Let A be a strongly positive operator from X to \(X^{*}\). Sufficient conditions are provided to assert that a compact perturbation of A reaches a fixed value at least once and at most finitely many times. When the compact perturbation is linear, the value is reached just once. The same conclusions are obtained when operators map the space X into itself. As an application of our results two examples are given related to some types of integral equations. The proof of results is constructive and is based upon a continuation method.
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23 June 2018
Professor Biagio Ricceri recently sent us a counterexample to a result in our paper [1]. After a revision of the paper, we have detected a non-correct use of Sard-Smale theorem in the proof of Section (aI-2-2) of Theorem 7. This leads us to add hypothesis (iv) to this theorem. In the particular case in which the operator B is linear, the hypothesis (iv) coincides with (iii) and it is non superfluous.
References
Allgower, E.L.: A survey of homotopy methods for smooths mappings. In: Allgower, E., Glashoff, K., Peitgen, H. (eds.) Proceedings of the Conference on Numerical Solution of Nonlinear Equations, pp. 2–29. Springer, Berlin (1981)
Allgower, E.L., Georg, K.: Numerical Continuation Methods, Springer Series in Computational Mathematics 13. Springer, New York (1990)
Alexander, J.C., York, J.A.: Homotopy continuation method: numerically implementable topological procedures. Trans. Am. Math. Soc. 242, 271–284 (1978)
Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)
Garcia, C.B., Li, T.I.: On the number of solutions to polynomial systems of non-linear equations. SIAM J. Numer. Anal. 17, 540–546 (1980)
Soriano, J.M.: Global minimum point of a convex function. Appl. Math. Comput. 55(2–3), 213–218 (1993)
Soriano, J.M.: Continuous embeddings and continuation methods. Nonlinear Anal. Theory Methods Appl. 70(11), 4118–4121 (2009)
Soriano, J.M., Ordoñez Cabrera, M.: Continuation methods and condensing mappings. Nonlinear Anal. Theory Methods Appl. 102, 84–90 (2014)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications I. Springer, New York (1992)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications IIA. Springer, New York (1992)
Zeidler, E.: Applied Functional Analysis, Applied Mathematical Sciences, vol. 108. Springer, New York (1995)
Zeidler, E.: Applied Functional Analysis, Applied Mathematical Sciences, vol. 109. Springer, New York (1995)
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The authors have been partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127, and by MEC Grant MTM2012-34847-C02-01.
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Soriano, J.M., Cabrera, M.O. Compact Perturbations of a Strongly Positive Operator. Mediterr. J. Math. 15, 23 (2018). https://doi.org/10.1007/s00009-018-1069-0
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DOI: https://doi.org/10.1007/s00009-018-1069-0