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February  2020, 25(2): 733-747. doi: 10.3934/dcdsb.2019264

On the approximation of fixed points for non-self mappings on metric spaces

Adrian Petruşel , Radu Precup , and  Marcel-Adrian Şerban

Babeş-Bolyai University, Department of Mathematics, 400084 Cluj-Napoca, Romania

* Corresponding author: Radu Precup

Dedicated to Professor Juan J. Nieto on the occasion of his 60th birthday

Received  November 2018 Revised  January 2019 Published  February 2020 Early access  November 2019

Starting from some classical results of R. Conti, A. Haimovici and K. Iseki, and from a more recent result of S. Reich and A.J. Zaslavski, we present several theorems of approximation of the fixed points for non-self mappings on metric spaces. Both metric and topological conditions are involved. Some of the results are generalized to the multi-valued case. An application is given to a class of implicit first-order differential systems leading to a fixed point problem for the sum of a completely continuous operator and a nonexpansive mapping.

Keywords: Fixed point, metric space, generalized contraction mapping, measure of noncompactness, condensing mapping, nonexpansive mapping, multi-valued operator, implicit differential equation.
Mathematics Subject Classification: Primary: 47H10; Secondary: 47H09.
Citation: Adrian Petruşel, Radu Precup, Marcel-Adrian Şerban. On the approximation of fixed points for non-self mappings on metric spaces. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 733-747. doi: 10.3934/dcdsb.2019264
References:
[1]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhaäuser Boston, Inc., Boston, 1990.

[2]

L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math., 12 (1971), 19-26. 

[3]

R. Conti, Un'osservazione sulle transformazioni continue di uno spazio metrico e alcume applicazioni, Matematiche (Catania), 15 (1960), 92-97. 

[4]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[5]

A. Haimovici, Un théorèe d'existence pour des équations fonctionnelles généralisant le th éorèe de Peano, An. Şti. Univ. "Al. I. Cuza" Iaşi. Secţ. I. (N.S.), 7 (1961), 65–76.

[6]

K. Iseki, A theorem on existence of solution for functional equations, Math. Japon., 7 (1962), 203-204. 

[7]

R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76. 

[8]

W. A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl., (2004), 309–316. doi: 10.1155/S1687182004406081.

[9]

W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014. doi: 10.1007/978-3-319-10927-5.

[10]

D. O'Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Series in Mathematical Analysis and Applications, 3, Gordon and Breach Science Publishers, Amsterdam, 2001.

[11]

A. Petruşel, Operatorial Inclusions, House of the Book of Science, Cluj-Napoca, 2002.

[12]

A. Petruşel, I. A. Rus and M.-A. Şerban, Fixed points, fixed sets and iterated multifunction systems for nonself multivalued operators, Set-Valued Var. Anal., 23 (2015), 223-237.  doi: 10.1007/s11228-014-0291-6.

[13]

R. Precup, On the continuation principle for nonexpansive maps, Studia Univ. Babeş-Bolyai Math., 41 (1996), 85–89.

[14]

R. Precup, Existence and approximation of positive fixed points of nonexpansive maps, Rev. Anal. Numér. Théor. Approx., 26 (1997), 203-208. 

[15]

R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9986-3.

[16]

D. Reem, S. Reich and A. J. Zaslavski, Two results in metric fixed point theory, J. Fixed Point Theory Appl., 1 (2007), 149-157.  doi: 10.1007/s11784-006-0011-4.

[17]

S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121-124.  doi: 10.4153/CMB-1971-024-9.

[18]

S. Reich, Fixed points of condensing functions, J. Math. Anal. Appl., 41 (1973), 460-467.  doi: 10.1016/0022-247X(73)90220-5.

[19]

S. Reich and A. J. Zaslavski, A fixed point theorem for Matkowski contractions, Fixed Point Theory, 8 (2007), 303-307. 

[20]

S. Reich and A. J. Zaslavski, Genericity in Nonlinear Analysis, Developments in Mathematics, 34, Springer, New York, 2014. doi: 10.1007/978-1-4614-9533-8.

[21]

I. A. Rus, Some fixed point theorems in metric spaces, Rend. Ist. Mat. Univ. Trieste, 3 (1971), 169-172. 

[22]

I. A. Rus and M.-A. Şerban, Some fixed point theorems for nonself generalized contraction, Miskolc Math. Notes, 17 (2016), 1021-1031.  doi: 10.18514/MMN.2017.1186.

[23]

M.-A. Şerban, Some fixed point theorems for nonself generalized contraction in gauge spaces, Fixed Point Theory, 16 (2015), 393-398. 

show all references

Dedicated to Professor Juan J. Nieto on the occasion of his 60th birthday

References:
[1]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhaäuser Boston, Inc., Boston, 1990.

[2]

L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math., 12 (1971), 19-26. 

[3]

R. Conti, Un'osservazione sulle transformazioni continue di uno spazio metrico e alcume applicazioni, Matematiche (Catania), 15 (1960), 92-97. 

[4]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[5]

A. Haimovici, Un théorèe d'existence pour des équations fonctionnelles généralisant le th éorèe de Peano, An. Şti. Univ. "Al. I. Cuza" Iaşi. Secţ. I. (N.S.), 7 (1961), 65–76.

[6]

K. Iseki, A theorem on existence of solution for functional equations, Math. Japon., 7 (1962), 203-204. 

[7]

R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71-76. 

[8]

W. A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl., (2004), 309–316. doi: 10.1155/S1687182004406081.

[9]

W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014. doi: 10.1007/978-3-319-10927-5.

[10]

D. O'Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Series in Mathematical Analysis and Applications, 3, Gordon and Breach Science Publishers, Amsterdam, 2001.

[11]

A. Petruşel, Operatorial Inclusions, House of the Book of Science, Cluj-Napoca, 2002.

[12]

A. Petruşel, I. A. Rus and M.-A. Şerban, Fixed points, fixed sets and iterated multifunction systems for nonself multivalued operators, Set-Valued Var. Anal., 23 (2015), 223-237.  doi: 10.1007/s11228-014-0291-6.

[13]

R. Precup, On the continuation principle for nonexpansive maps, Studia Univ. Babeş-Bolyai Math., 41 (1996), 85–89.

[14]

R. Precup, Existence and approximation of positive fixed points of nonexpansive maps, Rev. Anal. Numér. Théor. Approx., 26 (1997), 203-208. 

[15]

R. Precup, Methods in Nonlinear Integral Equations, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9986-3.

[16]

D. Reem, S. Reich and A. J. Zaslavski, Two results in metric fixed point theory, J. Fixed Point Theory Appl., 1 (2007), 149-157.  doi: 10.1007/s11784-006-0011-4.

[17]

S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull., 14 (1971), 121-124.  doi: 10.4153/CMB-1971-024-9.

[18]

S. Reich, Fixed points of condensing functions, J. Math. Anal. Appl., 41 (1973), 460-467.  doi: 10.1016/0022-247X(73)90220-5.

[19]

S. Reich and A. J. Zaslavski, A fixed point theorem for Matkowski contractions, Fixed Point Theory, 8 (2007), 303-307. 

[20]

S. Reich and A. J. Zaslavski, Genericity in Nonlinear Analysis, Developments in Mathematics, 34, Springer, New York, 2014. doi: 10.1007/978-1-4614-9533-8.

[21]

I. A. Rus, Some fixed point theorems in metric spaces, Rend. Ist. Mat. Univ. Trieste, 3 (1971), 169-172. 

[22]

I. A. Rus and M.-A. Şerban, Some fixed point theorems for nonself generalized contraction, Miskolc Math. Notes, 17 (2016), 1021-1031.  doi: 10.18514/MMN.2017.1186.

[23]

M.-A. Şerban, Some fixed point theorems for nonself generalized contraction in gauge spaces, Fixed Point Theory, 16 (2015), 393-398. 

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