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On the approximation of fixed points for non-self mappings on metric spaces
Babeş-Bolyai University, Department of Mathematics, 400084 Cluj-Napoca, Romania |
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.
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
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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