Systems of Inclusions Involving Fredholm Operators and Noncompact Maps
Gabor, Dorota
Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 119-158 / Harvested from Biblioteca Digitale Italiana di Matematica
Access to full text
Full (PDF)
Full (DjVu)

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.

In questa nota si studia l'esistenza di soluzioni per un sistema di due inclusioni con operatori di Fredholm aventi indice non negativo e multifunzioni «fondamentalmente restringibili» e a valori non necessariamente convessi. Si applica la tecnica della mappa soluzione e, poiché le ipotesi consentono un «difetto di dimensione», l'indice di coincidenza, cioé un invariante omotopico basato sulla teoria della co-omotopia. Si forniscono poi due applicazioni ai problemi ai limiti.

Publié le : 2007-02-01
@article{BUMI_2007_8_10B_1_119_0,
     author = {Dorota Gabor},
     title = {Systems of Inclusions Involving Fredholm Operators and Noncompact Maps},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {10-A},
     year = {2007},
     pages = {119-158},
     zbl = {1129.34006},
     mrnumber = {2310962},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2007_8_10B_1_119_0}
}

Gabor, Dorota. Systems of Inclusions Involving Fredholm Operators and Noncompact Maps. Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007) pp. 119-158. http://gdmltest.u-ga.fr/item/BUMI_2007_8_10B_1_119_0/

[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-Berlin 1992. | MR 1153247 | Zbl 0748.47045

[2] YBorisovich, U. G., Modern approach to the theory of topological characteristics of nonlinear operators, Lecture Notes in Math. 1334, Springer, Berlin, New York 1988. | MR 964702 | Zbl 0666.47037

[3] Borsuk, K., Theory of retracts, PWN, Warszawa1967. | MR 216473 | Zbl 0153.52905

[4] Brezis, H., Analyse Fonctionelle, Masson, Paris1983. | MR 697382

[5] Bryszewski, J. - Górniewicz, L., Multivalued maps of subsets of Euclidean spaces, Fund. Math. 90 (1976), 233-251. | MR 402726 | Zbl 0355.55013

[6] Bryszewski, J., On a class of multi-valued vector fields in Banach spacesFund. Math.97 (1977), 79-94. | MR 515321 | Zbl 0369.47034

[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. Surveys 35 (1980), 65-143. | MR 565568

[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. | MR 1163216 | Zbl 0754.47039

[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. | MR 1169080 | Zbl 0770.34053

[10] Gabor, D., The coincidence index for fundamentally contractible multivalued maps with nonconvex values, Ann. Polon. Math.75 (2), (2000), 143-166. | MR 1821162 | Zbl 0969.47041

[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 Systems 6 (1998), 377-403. | MR 1790183 | Zbl 0998.47034

[13] Gabor, D. - Kryszewski, W., A coincidence theory involving Fredholm operators of nonnegative index, Topol. Methods Nonlinear Anal. 15 (2000), 43-59. | MR 1786250 | Zbl 0971.47046

[14] Gabor, D. - Kryszewski, W., Systems of nonconvex inclusions involving Fredholm operators of nonnegative index, Set-Valued Anal. 13 (2005), 337-379. | MR 2187347 | Zbl 1100.47052

[15] Geba, K., Fredholm s-proper maps of Banach spaces, Fund. Math.64 (1969), 341- 373. | MR 250341 | Zbl 0191.21802

[16] Goldberg, S., Unbounded linear operators. Theory and applications, McGraw-Hill Book Co., 1966. | MR 200692 | Zbl 0148.12501

[17] Górniewicz, L., Topological fixed point theory of multivalued mappings, Kluwer Acad. Publ., Dordrecht, Boston, London1999. | MR 1748378

[18] Górniewicz, L., Homological methods in fixed-point theory of multivalued maps, Dissertationes Math.129 (1976), 1-66. | MR 394637

[19] Hu, S. T., Homotopy theory, Academic Press, New York1959. | MR 106454

[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. | MR 1307460

[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. | MR 1313856 | Zbl 0849.55010

[24] Kryszewski, W., Remarks to the Vietoris Theorem, Topol. Methods Nonlinear Anal.8 (1996), 371-382. | MR 1483636

[25] Kryszewski, W., Homotopy properties of set-valued mappings, Wyd. Uniwersytetu Mikolaja Kopernika, Toruń1997. | Zbl 1250.54022

[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. | MR 643012

[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. | MR 500350

[28] Mawhin, J., Topological degree methods in nonlinear boundary value problems. CBMS Regional Conference Series in Mathematics40, Amer. Math. Soc., Providence, R.I. 1979. | MR 525202 | Zbl 0414.34025

[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 York 1993, 74-142. | MR 1226930 | Zbl 0798.34025

[30] Nistri, P., On a general notion of controllability for nonlinear systems Boll. Un. Mat. Ital., vol. V-C (1986), 383-403. | MR 897207 | Zbl 0636.93005

[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. | MR 1232189 | Zbl 0793.47050

[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. | MR 1920145 | Zbl 1038.47042

[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. | MR 2478129 | Zbl 0297.47049

[34] Spanier, E., Algebraic Topology, McGraw-Hill Book Co., New York 1966. | MR 210112

[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. | MR 385836 | Zbl 0305.55001

[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).