A composite coincidence degree with applications to boundary value problems of neutral equations

Erbe, L. H.; Krawcewicz, W.; Wu, J. H.

doi:10.1090/S0002-9947-1993-1169080-3

A composite coincidence degree with applications to boundary value problems of neutral equations
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by L. H. Erbe, W. Krawcewicz and J. H. Wu PDF

Trans. Amer. Math. Soc. 335 (1993), 459-478 Request permission

Abstract:

We present a topological degree theory for the nonlinear problem $L(I - B)(x) = G(x)$ with applications to a class of boundary value problems of neutral equations, where $L$ is an unbounded Fredholm operator of index zero, $B$ is condensing and $G$ is $L$-compact.

References

Stephen R. Bernfeld, Gangaram Ladde, and V. Lakshmikantham, Nonlinear boundary value problems and several Lyapunov functions for functional differential equations, Boll. Un. Mat. Ital. (4) 10 (1974), 602–613 (English, with Italian summary). MR 0369857
L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1911), no. 1, 97–115 (German). MR 1511644, DOI 10.1007/BF01456931
Jane Cronin, Periodic solutions of nonautonomous equations, Boll. Un. Mat. Ital. (4) 6 (1972), 45–54 (English, with Italian summary). MR 0330703
Jane Cronin-Scanlon, Equations with bounded nonlinearities, J. Differential Equations 14 (1973), 581–596. MR 344956, DOI 10.1016/0022-0396(73)90069-7
L. H. Erbe, W. Krawcewicz, and J. Wu, Leray-Schauder degree for semilinear Fredholm maps and periodic boundary value problems of neutral equations, Nonlinear Anal. 15 (1990), no. 8, 747–764. MR 1074953, DOI 10.1016/0362-546X(90)90091-T
Mohd. Faheem and Rama Mohana Rao, A boundary value problem for functional-differential equations of neutral type, J. Math. Phys. Sci. 18 (1984), no. 4, 381–404. MR 803965
Robert E. Gaines and Jean L. Mawhin, Coincidence degree, and nonlinear differential equations, Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin-New York, 1977. MR 0637067
A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962), 93. MR 149253
G. B. Gustafson and Klaus Schmitt, Periodic solutions of hereditary differential systems, J. Differential Equations 13 (1973), 567–587. MR 342775, DOI 10.1016/0022-0396(73)90012-0
G. B. Gustafson and K. Schmitt, A note on periodic solutions for delay-differential systems, Proc. Amer. Math. Soc. 42 (1974), 161–166. MR 326109, DOI 10.1090/S0002-9939-1974-0326109-3
Jack K. Hale and J. Mawhin, Coincidence degree and periodic solutions of neutral equations, J. Differential Equations 15 (1974), 295–307. MR 336004, DOI 10.1016/0022-0396(74)90081-3
G. Hetzer, Some remarks on $\Phi _{+}$-operators and on the coincidence degree for a Fredholm equation with noncompact nonlinear perturbations, Ann. Soc. Sci. Bruxelles Sér. I 89 (1975), no. 4, 497–508. MR 385653
Georg Hetzer, Some applications of the coincidence degree for set-contractions to functional differential equations of neutral type, Comment. Math. Univ. Carolinae 16 (1975), 121–138. MR 364814
M. A. Krasnosel′skiĭ, The operator of translation along the trajectories of differential equations, Translations of Mathematical Monographs, Vol. 19, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by Scripta Technica. MR 0223640

Some criteria for the existence of periodic oscillation in nonlinear systems

9

Wiesław Krawcewicz, Contribution à la théorie des équations non linéaires dans les espaces de Banach, Dissertationes Math. (Rozprawy Mat.) 273 (1988), 83 (French). MR 989165

Topologie et équations functionnelles

51

J. Mawhin, Periodic solutions of nonlinear functional differential equations, J. Differential Equations 10 (1971), 240–261. MR 294823, DOI 10.1016/0022-0396(71)90049-0
J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636. MR 328703, DOI 10.1016/0022-0396(72)90028-9
J. Mawhin, Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 40, American Mathematical Society, Providence, R.I., 1979. Expository lectures from the CBMS Regional Conference held at Harvey Mudd College, Claremont, Calif., June 9–15, 1977. MR 525202

Compacité monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires

È. Muhamadiev and B. N. Sadovskiĭ, Estimation of the spectral radius of a certain operator that is connected with equations of neutral type, Mat. Zametki 13 (1973), 67–78 (Russian). MR 320482
Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
Roger D. Nussbaum, The fixed point index and asymptotic fixed point theorems for $k$-set-contractions, Bull. Amer. Math. Soc. 75 (1969), 490–495. MR 246285, DOI 10.1090/S0002-9904-1969-12213-5
Roger D. Nussbaum, Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972), 741–766. MR 306986, DOI 10.1016/0022-247X(72)90253-3
B. N. Sadovskiĭ, Application of topological methods to the theory of periodic solutions of nonlinear differential-operator equations of neutral type, Dokl. Akad. Nauk SSSR 200 (1971), 1037–1040 (Russian). MR 0310721

Equations différentielles et fonctionnelles non linéaires

Peter Volkmann, Démonstration d’un théoreme de coincidence par la méthode de Granas, Bull. Soc. Math. Belg. Sér. B 36 (1984), 235–242 (French). MR 777199
Paul Waltman and James S. W. Wong, Two point boundary value problems for nonlinear functional differential equations, Trans. Amer. Math. Soc. 164 (1972), 39–54. MR 287126, DOI 10.1090/S0002-9947-1972-0287126-8