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Some problems in the fixed point theory

Year 2018, Volume: 2 Issue: 1, 1 - 10, 25.03.2018
https://doi.org/10.31197/atnaa.379280

Abstract

In this paper we present some of my favorite problems, all the time open, in the fixed point theory. These
problems are in connection with the following two:
Which properties have the fixed point equations for which an iterative algorithm is convergent ?
Let us have a fixed point theorem, T, and an operator f (single or multivalued) which does not satisfy
the conditions in T. In which conditions the operator f has an invariant subset Y such that the restriction
of f to Y , fY , satisfies the conditions of T ?

References

  • O. Agratini, I.A. Rus, Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Anal. Forum, 8 (2003), 159-168.
  • O. Agratini, I.A. Rus, Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Carolinae, 44 (2003), No. 3, 555-563.
  • J. Andres, L. Górniewicz, Topological Principles for Boundary Value Problems, Kluwer, 2003.
  • J. Appell, E. De Pascale, A. Vignoli, Nonlinear Spectral Theory, Walter de Gruyter, 2004.
  • G.R. Belitskii, Yu.I. Lyubich, Matrix Norm and their Applications, Birkhäuser, 1988.
  • V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007.
  • V. Berinde, St. Maruster, I.A. Rus, An abstract point of view on iterative approximation of fixed points of nonself operators, J. Nonlinear and Convex Anal., 15 (2014), No. 5, 851-865.
  • V. Berinde, A. Petrusel, I.A. Rus, M.A. Serban, The retraction-displacement condition in the theory of fixed point equation with a convergent iterative algorithm, In: T.M. Rassias and V. Gupta (Eds.), Mathematical Analysis, Approximation Theory and Their Applications, Springer, 2016, 75-106.
  • R.F. Brown, M. Furi, L. Górniewicz, B. Jiang (Eds.), Handbook of Topological Fixed Point Theory, Springer, 2005.
  • T. Catinas, D. Otrocol, I.A. Rus, The iterates of positive linear operators with the set of constant functions as the fixed point set, Carpathian J. Math., 32 (2016), No. 2, 165-172.
  • C. Craciun, N. Lungu, Abstract and concrete Gronwall lemmas, Fixed Point Theory, 10 (2009), No. 2, 221-228.
  • K. Goebel, Problems I left behind, In: Proc. 10th IC-FPTA, 9-20, 2012, Cluj-Napoca.
  • A. Granas, J. Dugundji, Fixed Point Theory, Springer, 2003.
  • R.B. Hollmess, A formula for the spectral radius of an operator, Amer. Math. Monthly, 75 (1968), 163-166.
  • W.A. Horn, Some fixed point theorems for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc., 149 (1970), 391-404.
  • J. Jachymski, Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems, Studia Math., 195 (2009), No. 2, 99-112.
  • L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, 1982.
  • W.A. Kirk, Metric fixed point theory: old problems and new directions, Fixed Point Theory, 11 (2010), No. 1, 45-58.
  • W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer, 2001.
  • J.P. LaSalle, The Stability of Dynamical Systems, SIAM, 1976.
  • N. Lungu, S.A. Ciplea, Optimal Gronwall lemmas, Fixed Point Theory, 18 (2017), No. 1, 293-304.
  • N. Lungu, I.A. Rus, On a functional Volterra-Fredholm integral equation via Picard operators, J. Math. Ineq., 3 (2009), No. 4, 519-527.
  • N. Lungu, I.A. Rus, Gronwall inequalities via Picard operators, An. St. Univ. “Al. I. Cuza” (Iasi), Mat., 58 (2012), f.2, 269-278.
  • R.D. Nussbaum, The fixed point index and fixed point theorems for k-set-contractions, Ph.D. Dissertation, Univ. of Chicago, 1969.
  • J.M. Ortega, Numerical Analysis, Acad. Press, New York, 1972.
  • J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Acad. Press, New York, 1970.
  • A. Petrusel, I.A. Rus, An abstract point of view on iterative approximation schemes of fixed points for multivalued operators, J. Nonlinear Sci. Appl., 6 (2013), 97-107.
  • A. Petrusel, I.A. Rus, M.A. Serban, Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem for a multivalued operator, J. Nonlinear and Convex Anal., 15 (2014), No. 3, 493-513.
  • A. Petrusel, I.A. Rus, M.A. Serban, Fixed point structures, invariant operators, invariant partitions, and applications to Carathéodory integral equations, In: P.M. Pardalos and T.M. Rassias (Eds.), Contributions in Mathematics and Engineering, Springer, 2016, . . .
  • A. Petrusel, I.A. Rus, M.A. Serban, Nonexpansive operators as graphic contractions, J. Nonlinear Anal. and Convex Anal., 17 (2016), No. 7, 1409-1415.
  • A. Petrusel, I.A. Rus, Multivalued Picard and weakly Picard operators, In: Fixed Point Theory and Applications (E. Llorens Fuster, J. Garcia Falset and B. Sims (Eds.)), Yokohama Publ., 2004, 207-226.
  • A. Petrusel, I.A. Rus, J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11 (2007), No. 3, 903-914.
  • I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, Cluj-Napoca, 2001.
  • I.A. Rus, Strict fixed point theory, Fixed Point Theory, 4 (2003), No. 2, 177-183.
  • I.A. Rus, Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math., 20 (2004), No. 1, 125-134.
  • I.A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292 (2004), 259-264.
  • I.A. Rus, Fixed Point Structure Theory, Cluj Univ. Press, Cluj-Napoca, 2006.
  • I.A. Rus, Gronwall lemmas: ten open problems, Sci. Math. Jpn., 70 (2009), No. 2, 221-228.
  • I.A. Rus, Picard operators and applications, Sci. Math. Jpn., 58 (2003), No. 1, 191-219.
  • I.A. Rus, Fixed point and interpolation point set of a positive linear operator on C(D), Stud. Univ. Babes-Bolyai, Math., 55 (2010), No. 4, 243-248.
  • I.A. Rus, Five open problems in fixed point theory in terms of fixed point structures (I): singlevalued operators, In: Proc. 10th IC-FPTA, 39-60, 2012, Cluj-Napoca.
  • I.A. Rus, An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations, Fixed Point Theory, 13 (2012), No. 1, 179-192.
  • I.A. Rus, Heuristic introduction to weakly Picard operator theory, Creat. Math. Inform., 23 (2014), No. 2, 243-252.
  • I.A. Rus, Results and problems in Ulam stability of operatorial equations and inclusions, In: T.M. Rassias (Ed.), Handbook of Functional Equations: Stability Theory, Springer, 2014, 323-352.
  • I.A. Rus, Iterates of increasing linear operators, via Maia’s fixed point theorem, Stud. Univ. Babes-Bolyai, Math., 60 (2015), No. 1, 91-98.
  • I.A. Rus, Remarks on a LaSalle conjecture on global asymptotic stability, Fixed Point Theory, 17 (2016), No. 1, 159-172.
  • I.A. Rus, Some variants of contraction principle, generalizations and applications, Stud. Univ. Babes-Bolyai, Math., 61 (2016), No. 3, 343-358.
  • I.A. Rus, Relevant classes of weakly Picard operators, An. Univ. Vest Timisoara, Mat.-Inform., 54 (2016), No. 2, 3-19.
  • I.A. Rus, A. Petrusel, G. Petrusel, Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca, 2008.
  • I.A. Rus, A. Petrusel, M.A. Serban, Weakly Picard operators: equivalent definitions, applications and open problems, Fixed Point Theory, 7 (2006), No. 1, 3-22.
  • I.A. Rus, A. Petrusel, A. Sîntamarian, Data dependence of the fixed point setof some multivalued weakly Picard operators, Nonlinear Anal., 52 (2003), 1947-1959.
  • I.A. Rus, M.A. Serban, Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem, Carpathian J. Math., 29 (2013), No. 2, 239-258.
  • A. Sîntamarian, Metrical strict fixed point theorems for multivalued mappings, Seminar on Fixed Point Theory, Preprint No. 3, 1997, Babes-Bolyai Univ., Cluj-Napoca, 27-30.
  • M.A. Serban, Saturated fibre contraction principle, Fixed Point Theory, 18 (2017), No. 2, 729-740.
  • T. van der Walt, Fixed and Almost Fixed Points, Math. Centrum, Amsterdam, 1963.
  • J.Wang, Y. Zhou, M. Medved, Picard and weakly Picard operators techniques for nonlinear differential equations in Banach spaces, J. Math. Anal. Appl., 389(2012), 261-274.
  • D.Y. Zhou, Basic Theory of Fractional Differential Equations, World Sci. Publ. Co., 2014.
Year 2018, Volume: 2 Issue: 1, 1 - 10, 25.03.2018
https://doi.org/10.31197/atnaa.379280

Abstract

References

  • O. Agratini, I.A. Rus, Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Anal. Forum, 8 (2003), 159-168.
  • O. Agratini, I.A. Rus, Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Carolinae, 44 (2003), No. 3, 555-563.
  • J. Andres, L. Górniewicz, Topological Principles for Boundary Value Problems, Kluwer, 2003.
  • J. Appell, E. De Pascale, A. Vignoli, Nonlinear Spectral Theory, Walter de Gruyter, 2004.
  • G.R. Belitskii, Yu.I. Lyubich, Matrix Norm and their Applications, Birkhäuser, 1988.
  • V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007.
  • V. Berinde, St. Maruster, I.A. Rus, An abstract point of view on iterative approximation of fixed points of nonself operators, J. Nonlinear and Convex Anal., 15 (2014), No. 5, 851-865.
  • V. Berinde, A. Petrusel, I.A. Rus, M.A. Serban, The retraction-displacement condition in the theory of fixed point equation with a convergent iterative algorithm, In: T.M. Rassias and V. Gupta (Eds.), Mathematical Analysis, Approximation Theory and Their Applications, Springer, 2016, 75-106.
  • R.F. Brown, M. Furi, L. Górniewicz, B. Jiang (Eds.), Handbook of Topological Fixed Point Theory, Springer, 2005.
  • T. Catinas, D. Otrocol, I.A. Rus, The iterates of positive linear operators with the set of constant functions as the fixed point set, Carpathian J. Math., 32 (2016), No. 2, 165-172.
  • C. Craciun, N. Lungu, Abstract and concrete Gronwall lemmas, Fixed Point Theory, 10 (2009), No. 2, 221-228.
  • K. Goebel, Problems I left behind, In: Proc. 10th IC-FPTA, 9-20, 2012, Cluj-Napoca.
  • A. Granas, J. Dugundji, Fixed Point Theory, Springer, 2003.
  • R.B. Hollmess, A formula for the spectral radius of an operator, Amer. Math. Monthly, 75 (1968), 163-166.
  • W.A. Horn, Some fixed point theorems for compact maps and flows in Banach spaces, Trans. Amer. Math. Soc., 149 (1970), 391-404.
  • J. Jachymski, Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems, Studia Math., 195 (2009), No. 2, 99-112.
  • L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, 1982.
  • W.A. Kirk, Metric fixed point theory: old problems and new directions, Fixed Point Theory, 11 (2010), No. 1, 45-58.
  • W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer, 2001.
  • J.P. LaSalle, The Stability of Dynamical Systems, SIAM, 1976.
  • N. Lungu, S.A. Ciplea, Optimal Gronwall lemmas, Fixed Point Theory, 18 (2017), No. 1, 293-304.
  • N. Lungu, I.A. Rus, On a functional Volterra-Fredholm integral equation via Picard operators, J. Math. Ineq., 3 (2009), No. 4, 519-527.
  • N. Lungu, I.A. Rus, Gronwall inequalities via Picard operators, An. St. Univ. “Al. I. Cuza” (Iasi), Mat., 58 (2012), f.2, 269-278.
  • R.D. Nussbaum, The fixed point index and fixed point theorems for k-set-contractions, Ph.D. Dissertation, Univ. of Chicago, 1969.
  • J.M. Ortega, Numerical Analysis, Acad. Press, New York, 1972.
  • J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Acad. Press, New York, 1970.
  • A. Petrusel, I.A. Rus, An abstract point of view on iterative approximation schemes of fixed points for multivalued operators, J. Nonlinear Sci. Appl., 6 (2013), 97-107.
  • A. Petrusel, I.A. Rus, M.A. Serban, Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem for a multivalued operator, J. Nonlinear and Convex Anal., 15 (2014), No. 3, 493-513.
  • A. Petrusel, I.A. Rus, M.A. Serban, Fixed point structures, invariant operators, invariant partitions, and applications to Carathéodory integral equations, In: P.M. Pardalos and T.M. Rassias (Eds.), Contributions in Mathematics and Engineering, Springer, 2016, . . .
  • A. Petrusel, I.A. Rus, M.A. Serban, Nonexpansive operators as graphic contractions, J. Nonlinear Anal. and Convex Anal., 17 (2016), No. 7, 1409-1415.
  • A. Petrusel, I.A. Rus, Multivalued Picard and weakly Picard operators, In: Fixed Point Theory and Applications (E. Llorens Fuster, J. Garcia Falset and B. Sims (Eds.)), Yokohama Publ., 2004, 207-226.
  • A. Petrusel, I.A. Rus, J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems for multivalued operators, Taiwanese J. Math., 11 (2007), No. 3, 903-914.
  • I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, Cluj-Napoca, 2001.
  • I.A. Rus, Strict fixed point theory, Fixed Point Theory, 4 (2003), No. 2, 177-183.
  • I.A. Rus, Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math., 20 (2004), No. 1, 125-134.
  • I.A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292 (2004), 259-264.
  • I.A. Rus, Fixed Point Structure Theory, Cluj Univ. Press, Cluj-Napoca, 2006.
  • I.A. Rus, Gronwall lemmas: ten open problems, Sci. Math. Jpn., 70 (2009), No. 2, 221-228.
  • I.A. Rus, Picard operators and applications, Sci. Math. Jpn., 58 (2003), No. 1, 191-219.
  • I.A. Rus, Fixed point and interpolation point set of a positive linear operator on C(D), Stud. Univ. Babes-Bolyai, Math., 55 (2010), No. 4, 243-248.
  • I.A. Rus, Five open problems in fixed point theory in terms of fixed point structures (I): singlevalued operators, In: Proc. 10th IC-FPTA, 39-60, 2012, Cluj-Napoca.
  • I.A. Rus, An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations, Fixed Point Theory, 13 (2012), No. 1, 179-192.
  • I.A. Rus, Heuristic introduction to weakly Picard operator theory, Creat. Math. Inform., 23 (2014), No. 2, 243-252.
  • I.A. Rus, Results and problems in Ulam stability of operatorial equations and inclusions, In: T.M. Rassias (Ed.), Handbook of Functional Equations: Stability Theory, Springer, 2014, 323-352.
  • I.A. Rus, Iterates of increasing linear operators, via Maia’s fixed point theorem, Stud. Univ. Babes-Bolyai, Math., 60 (2015), No. 1, 91-98.
  • I.A. Rus, Remarks on a LaSalle conjecture on global asymptotic stability, Fixed Point Theory, 17 (2016), No. 1, 159-172.
  • I.A. Rus, Some variants of contraction principle, generalizations and applications, Stud. Univ. Babes-Bolyai, Math., 61 (2016), No. 3, 343-358.
  • I.A. Rus, Relevant classes of weakly Picard operators, An. Univ. Vest Timisoara, Mat.-Inform., 54 (2016), No. 2, 3-19.
  • I.A. Rus, A. Petrusel, G. Petrusel, Fixed Point Theory, Cluj Univ. Press, Cluj-Napoca, 2008.
  • I.A. Rus, A. Petrusel, M.A. Serban, Weakly Picard operators: equivalent definitions, applications and open problems, Fixed Point Theory, 7 (2006), No. 1, 3-22.
  • I.A. Rus, A. Petrusel, A. Sîntamarian, Data dependence of the fixed point setof some multivalued weakly Picard operators, Nonlinear Anal., 52 (2003), 1947-1959.
  • I.A. Rus, M.A. Serban, Basic problems of the metric fixed point theory and the relevance of a metric fixed point theorem, Carpathian J. Math., 29 (2013), No. 2, 239-258.
  • A. Sîntamarian, Metrical strict fixed point theorems for multivalued mappings, Seminar on Fixed Point Theory, Preprint No. 3, 1997, Babes-Bolyai Univ., Cluj-Napoca, 27-30.
  • M.A. Serban, Saturated fibre contraction principle, Fixed Point Theory, 18 (2017), No. 2, 729-740.
  • T. van der Walt, Fixed and Almost Fixed Points, Math. Centrum, Amsterdam, 1963.
  • J.Wang, Y. Zhou, M. Medved, Picard and weakly Picard operators techniques for nonlinear differential equations in Banach spaces, J. Math. Anal. Appl., 389(2012), 261-274.
  • D.Y. Zhou, Basic Theory of Fractional Differential Equations, World Sci. Publ. Co., 2014.
There are 57 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

İoan A. Rus

Publication Date March 25, 2018
Published in Issue Year 2018 Volume: 2 Issue: 1

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