Systems of Inclusions Involving Fredholm Operators and Noncompact Maps

Dorota Gabor

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 1, page 119-158
  • ISSN: 0392-4041

Abstract

top
We consider the existence of solutions to the system of two inclusions involving Fredholm operators of nonnegative index and the so-called fundamentally restrictible maps with not necessarily convex values. We apply the technique of a solution map and, since the assumptions admit a 'dimension defect', we use the coincidence index, i.e. the homotopy invariant based on the cohomotopy theory. Two examples of applications to boundary value problems are included.

How to cite

top

Gabor, Dorota. "Systems of Inclusions Involving Fredholm Operators and Noncompact Maps." Bollettino dell'Unione Matematica Italiana 10-B.1 (2007): 119-158. <http://eudml.org/doc/290432>.

@article{Gabor2007,
abstract = {We consider the existence of solutions to the system of two inclusions involving Fredholm operators of nonnegative index and the so-called fundamentally restrictible maps with not necessarily convex values. We apply the technique of a solution map and, since the assumptions admit a 'dimension defect', we use the coincidence index, i.e. the homotopy invariant based on the cohomotopy theory. Two examples of applications to boundary value problems are included.},
author = {Gabor, Dorota},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {119-158},
publisher = {Unione Matematica Italiana},
title = {Systems of Inclusions Involving Fredholm Operators and Noncompact Maps},
url = {http://eudml.org/doc/290432},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Gabor, Dorota
TI - Systems of Inclusions Involving Fredholm Operators and Noncompact Maps
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/2//
PB - Unione Matematica Italiana
VL - 10-B
IS - 1
SP - 119
EP - 158
AB - We consider the existence of solutions to the system of two inclusions involving Fredholm operators of nonnegative index and the so-called fundamentally restrictible maps with not necessarily convex values. We apply the technique of a solution map and, since the assumptions admit a 'dimension defect', we use the coincidence index, i.e. the homotopy invariant based on the cohomotopy theory. Two examples of applications to boundary value problems are included.
LA - eng
UR - http://eudml.org/doc/290432
ER -

References

top
  1. AKHMEROV, R. R. - KAMENSKII, M. I. - POTAPOV, A. S. - RODKINA, A. E. - SADOVSKII, B. N., Measures of noncompactness and condensing operators, Birkhäuser Verlag, Basel-Boston-Berlin1992. Zbl0748.47045MR1153247DOI10.1007/978-3-0348-5727-7
  2. Y BORISOVICH, U. G., Modern approach to the theory of topological characteristics of nonlinear operators, Lecture Notes in Math.1334, Springer, Berlin, New York1988. Zbl0666.47037MR964702DOI10.1007/BFb0080430
  3. BORSUK, K., Theory of retracts, PWN, Warszawa1967. Zbl0153.52905MR216473
  4. BREZIS, H., Analyse Fonctionelle, Masson, Paris1983. MR697382
  5. BRYSZEWSKI, J. - GÓRNIEWICZ, L., Multivalued maps of subsets of Euclidean spaces, Fund. Math.90 (1976), 233-251. Zbl0355.55013MR402726DOI10.4064/fm-90-3-233-251
  6. BRYSZEWSKI, J., On a class of multi-valued vector fields in Banach spacesFund. Math.97 (1977), 79-94. Zbl0369.47034MR515321
  7. BORISOVICH, YU. G. - GELMAN, B. D. - MYSHKIS, A. D. - OBUKHOVSKII, V. V., Topological methods in the fixed point theory of multivalued mappings, Russian Math. Surveys35 (1980), 65-143. MR565568
  8. CONTI, G. - KRYSZEWSKI, W. - ZECCA, P., On the solvability of systems of non-convex inclusions in Banach spaces, Ann. Mat. pura Appl.CLX (1991), 371-408. Zbl0754.47039MR1163216DOI10.1007/BF01764135
  9. ERBE, L. H. - Krawcewicz, W. - WU, J. H., A composite coincidence degree with applications to boundary value problems of neutral equations, Trans. Amer. Math. Soc.335, 2 (1993), 459-478. Zbl0770.34053MR1169080DOI10.2307/2154389
  10. GABOR, D., The coincidence index for fundamentally contractible multivalued maps with nonconvex values, Ann. Polon. Math.75 (2), (2000), 143-166. Zbl0969.47041MR1821162DOI10.4064/ap-75-2-143-166
  11. GABOR, D., Coincidence points of Fredholm operators and noncompact set-valued maps, (in Polish), PhD Thesis, Torun2001. 
  12. GABOR, D. - KRYSZEWSKI, W., On the solvability of systems of nonconvex and noncompact inclusions in Banach spaces, Diff. Equations and Dynamical Systems6 (1998), 377-403. Zbl0998.47034MR1790183
  13. GABOR, D. - KRYSZEWSKI, W., A coincidence theory involving Fredholm operators of nonnegative index, Topol. Methods Nonlinear Anal.15 (2000), 43-59. Zbl0971.47046MR1786250DOI10.12775/TMNA.2000.004
  14. GABOR, D. - KRYSZEWSKI, W., Systems of nonconvex inclusions involving Fredholm operators of nonnegative index, Set-Valued Anal.13 (2005), 337-379. Zbl1100.47052MR2187347DOI10.1007/s11228-004-6344-5
  15. GEBA, K., Fredholm s-proper maps of Banach spaces, Fund. Math.64 (1969), 341- 373. Zbl0191.21802MR250341DOI10.4064/fm-64-3-341-373
  16. GOLDBERG, S., Unbounded linear operators. Theory and applications, McGraw-Hill Book Co., 1966. Zbl0148.12501MR200692
  17. GÓRNIEWICZ, L., Topological fixed point theory of multivalued mappings, Kluwer Acad. Publ., Dordrecht, Boston, London1999. MR1748378DOI10.1007/978-94-015-9195-9
  18. GÓRNIEWICZ, L., Homological methods in fixed-point theory of multivalued maps, Dissertationes Math.129 (1976), 1-66. MR394637
  19. HU, S. T., Homotopy theory, Academic Press, New York1959. MR106454
  20. KACZYŃSKI, T., Topological transversality of set-valued condensing maps, Doctoral Diss., McGill Univ., Montreal1986. 
  21. KACZYŃSKI, T. and KRAWCEWICZ, W., Fixed point and coincidence theory for condensing maps, Preprint, 1984. 
  22. KRYSZEWSKI, W., Topological and approximation methods in the degree theory of setvalued maps, Dissertationes Math.336 (1994), 1-102. MR1307460
  23. KRYSZEWSKI, W., Some homotopy classification and extension theorems for the class of compositions of acyclic set-valued maps, Bull. Sci. Math.119 (1995), 21-48. Zbl0849.55010MR1313856
  24. KRYSZEWSKI, W., Remarks to the Vietoris Theorem, Topol. Methods Nonlinear Anal.8 (1996), 371-382. MR1483636DOI10.12775/TMNA.1996.041
  25. KRYSZEWSKI, W., Homotopy properties of set-valued mappings, Wyd. Uniwersytetu Mikolaja Kopernika, Toruń1997. Zbl1250.54022
  26. MASSABO, I. - NISTRI, P. - PEJSACHOWICZ, J., On the solvability of nonlinear equations in Banach spaces, Fixed Point Theory, (Proc. Sherbrooke, Quebec 1980), (E. Fadell and G. Fournier, eds.) Lecture Notes in Math.886, Springer-Verlag, 1980, 270-289. MR643012
  27. MAWHIN, J., Nonlinear boundary value problems for ordinary differential equa- tions: from Schauder theory to stable homotopy, Collection Nonlinear analysis, Academic Press, New York1978, 145-160. MR500350
  28. MAWHIN, J., Topological degree methods in nonlinear boundary value problems. CBMS Regional Conference Series in Mathematics40, Amer. Math. Soc., Providence, R.I. 1979. Zbl0414.34025MR525202
  29. MAWHIN, J., Topological degree and boundary value problems for nonlinear differential equations, in M. Furi, P. Zecca eds., Topological Methods for ordinary differential equations. Lecture Notes in Math.1537, Springer, Berlin, New York1993, 74-142. Zbl0798.34025MR1226930DOI10.1007/BFb0085076
  30. NISTRI, P., On a general notion of controllability for nonlinear systems Boll. Un. Mat. Ital., vol. V-C (1986), 383-403. Zbl0636.93005MR897207
  31. NISTRI, P. - OBUKHOVSKII, V. V. - ZECCA, P., On the solvability of systems of inclusions involving noncompact operators, Trans. Amer. Math. Soc.342 (1994), 543-562. Zbl0793.47050MR1232189DOI10.2307/2154640
  32. OBUKHOVSKII, V. - ZECCA, P. - ZVYAGIN, V., On the coincidence index for multivalued perturbations of nonlinear Fredholm and some applications, Abstract Appl. Anal.7 (2002), 295-322. Zbl1038.47042MR1920145DOI10.1155/S1085337502002853
  33. PETRYSHYN, W. V. - FITZPATRICK, M., A degree theory, fixed points theorems and mappings theorems for multivalued noncompact mappings, Trans. Amer. Math. Soc.194 (1974), 1-25. Zbl0297.47049MR2478129DOI10.2307/1996791
  34. SPANIER, E., Algebraic Topology, McGraw-Hill Book Co., New York1966. MR210112
  35. SVARC, A. S., The homotopic topology of Banach spaces, Soviet Math. Dokl.5 (1964), 57-59. 
  36. SWITZER, R. M., Algebraic topology - homotopy and homology, Springer-Verlag, Berlin1975. Zbl0305.55001MR385836
  37. ZABREJKO, P. P. - KOSHELEW, A. I. - KRASNOSELSKIJ, M. A. - MIKHLIN, S. G. - RAKOVSHCHIK, L. S. - STETSENKO, V. YA., Integral equations, Leyden, The Netherlands: Noordhoff International Publishing. XIX, (1975). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.