Topological degree theories and nonlinear operator equations in Banach spaces. (English) Zbl 1142.47346

Let \(X\) be a real Banach space, \(G_1\), \(G_2\) open and bounded such that \(0\in G_2\subset\bar{G}_2\subset G_1\). Let \(T:D(T)\to X\) be accretive such that \(0\in D(T)\) and \(T(0)=0\). Let \(C:D(C)\to X\) be compact or continuous and bounded with the resolvents of \(T\) compact. The authors use various degree theories to find zeros of \(T+C\) in \(D(T+C)\cap(G_1\setminus G_2)\). As a matter of fact, the article contains much more results: the range space may be \(X^*\) instead of \(X\) and \(C\) may belong to more complicated classes of operators. There is a short application to partial differential equations.


47H11 Degree theory for nonlinear operators
47B44 Linear accretive operators, dissipative operators, etc.
47H05 Monotone operators and generalizations
Full Text: DOI


[1] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1975), Noordhoff Int. Publ. Leyden, The Netherlands
[2] Brézis, H.; Crandall, M.G.; Pazy, A., Perturbations of nonlinear maximal monotone sets in Banach spaces, Comm. pure appl. math., 23, 123-144, (1970) · Zbl 0182.47501
[3] F. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, in: Proc. Symp. Pure Appl. Math., vol. 18, Part 2, Providence, 1976 · Zbl 0327.47022
[4] Browder, F.E., Fixed point theory and nonlinear problems, Bull. amer. math. soc., 9, 1-39, (1983) · Zbl 0533.47053
[5] Brown, R.F., A topological introduction to nonlinear analysis, (1993), Birkhäuser Boston · Zbl 0794.47034
[6] Cac, N.P.; Gatica, J.A., Fixed point theorems for mappings in ordered Banach spaces, J. math. anal. appl., 71, 547-557, (1979) · Zbl 0448.47035
[7] Cioranescu, I., Geometry of Banach spaces, ()
[8] Ding, Z.; Kartsatos, A.G., Nonzero solutions of nonlinear equations involving compact perturbations of accretive operators in Banach spaces, Nonlinear anal., 25, 1333-1342, (1995) · Zbl 0869.47033
[9] Guo, D., Nonlinear analysis, (1985), Shandong Press, Inc. Shandong, China
[10] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press, Inc. New York · Zbl 0661.47045
[11] Hamilton, J.D., Noncompact mappings and cones in Banach spaces, Arch. ration mech. anal., 48, 153-162, (1972) · Zbl 0246.47063
[12] Kartsatos, A.G., Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, (), 2197-2222 · Zbl 0849.47027
[13] Kartsatos, A.G.; Skrypnik, I.V., Topological degree theories for densely defined mappings involving operators of type \((S_+)\), Adv. differential equations, 4, 413-456, (1999) · Zbl 0959.47037
[14] Kartsatos, A.G.; Skrypnik, I.V., Normalized eigenvectors for nonlinear abstract and elliptic operators, J. differential equations, 155, 443-475, (1999) · Zbl 0931.47050
[15] Kartsatos, A.G.; Skrypnik, I.V., The index of a critical point for nonlinear elliptic operators with strong coefficient growth, J. math. soc. Japan, 52, 109-137, (2000) · Zbl 0953.47042
[16] Kartsatos, A.G.; Skrypnik, I.V., The index of a critical point for densely defined operators of type \((S_+)_L\) in Banach spaces, Trans. amer. math. soc., 354, 1601-1630, (2001) · Zbl 1005.47058
[17] A.G. Kartsatos, I.V. Skrypnik, Invariance of domain for perturbations of maximal monotone operators in Banach spaces, Adv. Differential Equations (in press) · Zbl 1160.47044
[18] Kartsatos, A.G.; Skrypnik, I.V., On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces, Trans. amer. math. soc., 358, 3851-3881, (2006) · Zbl 1100.47050
[19] Kartsatos, A.G.; Skrypnik, I.V.; Shramenko, V., The index of an isolated critical point for a class of non-differentiable elliptic operators in reflexive Banach spaces, J. differential equations, 214, 189-231, (2005) · Zbl 1086.47032
[20] Kittilä, A., On the topological degree for a class of mappings of monotone type and applications to strongly nonlinear elliptic problems, Ann. acad. sci. fenn. ser. A I math. diss., 91, (1994), 48pp · Zbl 0816.47066
[21] Krasnoselskii, M.A., Positive solutions of operator equations, (1964), Noordhoff and Sijthoff Groningen
[22] Lakshmikantham, V.; Leela, S., Nonlinear differential equations in abstract spaces, (1981), Pergamon Press Oxford · Zbl 0456.34002
[23] Nagumo, M., Degree of mapping in convex linear topological spaces, Amer. J. math., 73, 497-511, (1951) · Zbl 0043.17801
[24] Pascali, D.; Sburlan, S., Nonlinear mappings of monotone type, (1978), Sijthoff and Noordhoof Bucharest · Zbl 0392.47026
[25] Rothe, E.H., ()
[26] Simons, S., ()
[27] Skrypnik, I.V., Nonlinear higher order elliptic equations, (1973), Nauka Dumka Kiev, (in Russian) · Zbl 0296.35032
[28] Skrypnik, I.V., ()
[29] Yu, Q., Condensing mappings and positive fixed points in semiordered Banach spaces, J. Lanzhou univ., 2, 23-32, (1979)
[30] Zeidler, E., Nonlinear functional analysis and its applications, vol. II/B, (1990), Springer-Verlag New York
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.