Some problems in the fixed point theory

Yıl 2018, Cilt: 2 Sayı: 1, 1 - 10, 25.03.2018

İoan A. Rus

https://doi.org/10.31197/atnaa.379280

Cited By: 3

Öz

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 ?

Anahtar Kelimeler

ordered set, L-space, metric space, Banach space, Picard operator, weakly Picard operator, fixed point, fixed point structure, iterative algorithm, retraction-displacement condition, well-posedness of fixed point problem, Ostrowski property, global asymptotic stability, open problem, conjecture

Kaynakça

• 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.