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The Fixed Point Problem for Systems of Coordinate-Wise Uniformly Monotone Operators and Applications

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Abstract

We study the fixed point problem for a system of multivariate operators that are coordinate-wise uniformly monotone, in the setting of quasi-ordered sets. We show that this problem is equivalent to the fixed point problem for a mixed monotone operator that can be explicitly constructed. As a consequence, we obtain a criterion for the existence and uniqueness of solution to the considered problem, together with an approximating iterative scheme, in the setting of partially ordered metric spaces. As an application, we investigate a new abstract multidimensional fixed point problem. To validate our results, we also provide an application to a first-order differential system with periodic boundary value conditions.

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References

  1. Berinde V., Borcut M.: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 74, 4889–4897 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Berzig M., Samet B.: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl. 63, 1319–1334 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Guo D.J., Lakshmikantham V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11, 623–632 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  4. Karapınar E., Luong N.V.: Quadruple fixed point theorems for nonlinear contractions. Comput. Math. Appl. 64, 1839–1848 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Lakshmikantham V., Ćirić L.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341–4349 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Opoĭtsev, V.I.: Generalization of the theory of monotone and concave operators (in Russian). Trudy Moskov. Mat. Obshch. 36, 237–273 (1978). English translation in Trans. Moscow Math. Soc. 2, 243–279 (1979)

  7. O’Regan D., Petruşel A.: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 341, 1241–1252 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Roldán A., Martínez-Moreno J., Roldán C.: Multidimensional fixed point theorems in partially ordered complete metric spaces. J. Math. Anal. Appl. 396, 536–545 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Rus M.D.: Monotone iterative methods for systems of nonlinear equations involving mixed monotone operators. Fixed Point Theory 9, 309–318 (2008)

    MATH  MathSciNet  Google Scholar 

  10. Rus M.D.: Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric. Nonlinear Anal. 74, 1804–1813 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Turinici, M.: Product fixed points in ordered metric spaces (2011). ArXiv:1110.3079v1 [math.GN]

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Correspondence to Mircea-Dan Rus.

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The author is grateful for the financial support provided by the Sectoral Operational Programme Human Resources Development 2007–2013 of the Romanian Ministry of Labor, Family and Social Protection through the Financial Agreement POSDRU/89/1.5/S/62557.

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Rus, MD. The Fixed Point Problem for Systems of Coordinate-Wise Uniformly Monotone Operators and Applications. Mediterr. J. Math. 11, 109–122 (2014). https://doi.org/10.1007/s00009-013-0306-9

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  • DOI: https://doi.org/10.1007/s00009-013-0306-9

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