A composite coincidence degree with applications to boundary value problems of neutral equations. (English) Zbl 0770.34053
The authors extend the notion of essential map and generalize the topological transversality theorem of Granas to the nonlinear problem \(L(I-B)(x)=G(x)\), where \(L\) is an unbounded Fredholm operator of index zero, \(B\) is condensing and \(G\) is \(L\)-compact. They also develop a topological degree theory to detect essential maps. Their topological degree called composite coincidence degree includes Leray-Schauder degree, Nussbaum degree, Sadovskii degree, Mawhin degree and Hetzer degree as special cases. The theory is applied to extend various existence results of boundary value problems from retarded equations to neutral equations.
Reviewer: R.Precup (Cluj-Napoca)
MSC:
| 34K40 | Neutral functional-differential equations |
| 34K10 | Boundary value problems for functional-differential equations |
| 47H10 | Fixed-point theorems |
| 47H11 | Degree theory for nonlinear operators |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47J05 | Equations involving nonlinear operators (general) |
| 47J25 | Iterative procedures involving nonlinear operators |
Keywords:
essential map; topological transversality theorem of Granas; unbounded Fredholm operator of index zero; topological degree theory; composite coincidence degree; boundary value problems; retarded equations; neutral equationsReferences:
| [1] | 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). · Zbl 0309.34052 |
| [2] | L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1911), no. 1, 97 – 115 (German). · JFM 42.0417.01 · doi:10.1007/BF01456931 |
| [3] | Jane Cronin, Periodic solutions of nonautonomous equations, Boll. Un. Mat. Ital. (4) 6 (1972), 45 – 54 (English, with Italian summary). · Zbl 0269.34062 |
| [4] | Jane Cronin-Scanlon, Equations with bounded nonlinearities, J. Differential Equations 14 (1973), 581 – 596. · Zbl 0251.34049 · doi:10.1016/0022-0396(73)90069-7 |
| [5] | 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. · Zbl 0729.47058 · doi:10.1016/0362-546X(90)90091-T |
| [6] | 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. · Zbl 0599.34089 |
| [7] | 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. · Zbl 0339.47031 |
| [8] | A. Granas, The theory of compact vector fields and some of its applications to topology of functional spaces. I, Rozprawy Mat. 30 (1962), 93. · Zbl 0111.11001 |
| [9] | G. B. Gustafson and Klaus Schmitt, Periodic solutions of hereditary differential systems, J. Differential Equations 13 (1973), 567 – 587. · Zbl 0254.34070 · doi:10.1016/0022-0396(73)90012-0 |
| [10] | G. B. Gustafson and K. Schmitt, A note on periodic solutions for delay-differential systems, Proc. Amer. Math. Soc. 42 (1974), 161 – 166. · Zbl 0273.34054 |
| [11] | Jack K. Hale and J. Mawhin, Coincidence degree and periodic solutions of neutral equations, J. Differential Equations 15 (1974), 295 – 307. · Zbl 0274.34070 · doi:10.1016/0022-0396(74)90081-3 |
| [12] | 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. · Zbl 0316.47041 |
| [13] | 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. · Zbl 0298.47034 |
| [14] | M. A. Krasnosel\(^{\prime}\)skiĭ, The operator of translation along the trajectories of differential equations, Translations of Mathematical Monographs, Vol. 19. Translated from the Russian by Scripta Technica, American Mathematical Society, Providence, R.I., 1968. |
| [15] | M. A. Krasnosel’skii and E. A. Lifschits, Some criteria for the existence of periodic oscillation in nonlinear systems, Automat. Remote Control 9 (1973), 1374-1377. · Zbl 0296.93024 |
| [16] | Wiesław Krawcewicz, Contribution à la théorie des équations non linéaires dans les espaces de Banach, Dissertationes Math. (Rozprawy Mat.) 273 (1988), 83 (French). · Zbl 0677.47038 |
| [17] | L. Leray and J. Schauder, Topologie et équations functionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45-78. · JFM 60.0322.02 |
| [18] | J. Mawhin, Periodic solutions of nonlinear functional differential equations, J. Differential Equations 10 (1971), 240 – 261. · Zbl 0223.34055 · doi:10.1016/0022-0396(71)90049-0 |
| [19] | 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. · Zbl 0244.47049 · doi:10.1016/0022-0396(72)90028-9 |
| [20] | 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. · Zbl 0414.34025 |
| [21] | -, Compacité monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires, Séminaire d’Analyse Moderne 19, Université de Sherbrooke, 1981. · Zbl 0497.47033 |
| [22] | 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). · Zbl 0255.47004 |
| [23] | Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217 – 258. · Zbl 0226.47031 · doi:10.1007/BF02414948 |
| [24] | Roger D. Nussbaum, The fixed point index and asymptotic fixed point theorems for \?-set-contractions, Bull. Amer. Math. Soc. 75 (1969), 490 – 495. · Zbl 0174.45402 |
| [25] | Roger D. Nussbaum, Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972), 741 – 766. · Zbl 0232.47062 · doi:10.1016/0022-247X(72)90253-3 |
| [26] | 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). |
| [27] | K. Schmidt, Equations différentielles et fonctionnelles non linéaires, Hermann, Paris, 1973, pp. 65-78. |
| [28] | Peter Volkmann, Démonstration d’un théoreme de coincidence par la méthode de Granas, Bull. Soc. Math. Belg. Sér. B 36 (1984), 235 – 242 (French). · Zbl 0576.47033 |
| [29] | 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. · Zbl 0263.34072 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
