Constrained topological degree and positive solutions of fully nonlinear boundary value problems. (English) Zbl 1188.35088

This article consists of two parts: in the first one the authors define a “constrained degree”: Let \(E\) be a Banach space and let \(M\subset E\) (the “set of constraints”) be an \(\mathcal{L}\)-retract, i.e., there is an \(\eta>0\) and an \(L\geq1\) and a retraction \(r:B:=\{x\in E\mid d(x,M)<\eta\}\to M\) such that \(\|r(x)-x\|\leq Ld(x,M)\) whenever \(x\in B\). Let then \(U\subset M\) be open, let \(A:D(A)\to E\) be a densely defined \(m\)-accretive operator and \(F:U\to E\) be a continuous map such that (1) there is a \(\lambda_0>0\) such that \((I+\lambda A)^{-1}(M)\subset M\) whenever \(0<\lambda\leq\lambda_0\), (2) for positive \(\lambda\) the resolvent \((I+\lambda A)^{-1}\) is completely continuous when restricted to \(M\), (3) \(F(x)\in \{u\in E\mid \liminf_{h\to0+}\frac1h d(x+hu,M)=0\}\) for \(x\in U\), (4) the zero set \(\{x\in U\cap D(A)\mid 0\in -A(x)+F(x)\}\) is compact. In this situation, the authors define a degree, \(\deg_M((A,F),U)\), that besides the existence, additivity, and homotopy property satisfies the following normalization property: \(\deg_M((A,F),M)=\chi(M)\) provided \(M\) and \(F(M)\) are bounded.
In the second part, the authors apply their degree theory to problems which can be cast in the form \(0\in -Ax+F(x)\), \(x\in M\). In this context they deal with stationary reaction-diffusion problems, one-dimensional non-homogeneous \(p\)-Laplacian problems, and controlled Neumann-like problems.


35J66 Nonlinear boundary value problems for nonlinear elliptic equations
47H04 Set-valued operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H11 Degree theory for nonlinear operators
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35A16 Topological and monotonicity methods applied to PDEs
Full Text: DOI


[1] Aubin, J.-P.; Frankowska, H., Set-valued analysis, (1991), Birkhäuser
[2] Bader, R.; Kryszewski, W., On the solutions of differential inclusions and the periodic problem in Banach spaces, Nonlinear anal., 54, 707-754, (2003) · Zbl 1034.34072
[3] Barbu, V., Nonlinear and differential equations in Banach spaces, (1979), Nordhoff · Zbl 0462.45022
[4] Ben-El-Mechaiekh, H.; Kryszewski, W., Equilibria of set-valued maps on nonconvex domains, Trans. amer. math. soc., 349, 4159-4179, (1997) · Zbl 0887.47040
[5] Ph. Benilan, Equations d’evolution dans un space de Banach quelconque et applications, These d’Etat, Paris, 1972
[6] Bothe, D., Periodic solutions of a nonlinear evolution problem from heterogeneous catalysis, Differential integral equations, 6, 6, 641-670, (2001) · Zbl 1032.34061
[7] Brezis, H., Analyse fonctionelle. théorie et applications, (1983), Masson Paris
[8] Brown, R., The Lefschetz fixed point theorem, (1971), Scott-Foresman Glenview, IL, London · Zbl 0216.19601
[9] Chen, Y.Z., The generalized degree for compact perturbations of m-accretive operators and applications, Nonlinear anal., 13, 393-403, (1989) · Zbl 0692.47044
[10] Ćwiszewski, A., Topological degree methods for perturbations of operators generating compact \(C_0\) semigroups, J. differential equations, 220, 2, 434-477, (2006) · Zbl 1086.47030
[11] Ćwiszewski, A., Degree theory for perturbations of m-accretive operators generating compact semigroups with constraints, J. evol. equ., 7, 1-33, (2007) · Zbl 1124.47039
[12] A. Ćwiszewski, PhD thesis, Nicolaus Copernicus University, 2003
[13] Ćwiszewski, A.; Kryszewski, W., Homotopy invariants for tangent vector fields on closed sets, Nonlinear anal., 65, 175-209, (2006) · Zbl 1107.54019
[14] Ćwiszewski, A.; Kryszewski, W., The constrained degree and fixed-point index theory for set-valued maps, Nonlinear anal., 64, 2643-2664, (2006) · Zbl 1108.54014
[15] Dancer, E.N., On the indices of fixed points of mappings in cones and applications, J. math. anal. appl., 91, 131-151, (1983) · Zbl 0512.47045
[16] Drábek, P., Solvability and bifurcations of nonlinear equations, Pitman res. notes math. ser., vol. 264, (1992), Longman Scientific & Technical · Zbl 0753.34002
[17] Evans, L.C., Partial differential equations, Grad. stud. math., vol. 19, (1997), Amer. Math. Soc.
[18] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1977), Springer-Verlag · Zbl 0691.35001
[19] Granas, A., The leray – schauder index and the fixed point theory for arbitrary anrs, Bull. soc. math. France, 100, 209-228, (1972) · Zbl 0236.55004
[20] Granas, A.; Dugundji, J., Fixed point theory, (2004), Springer-Verlag
[21] Guan, Z.G.; Kartsatos, A.G., A degree for maximal monotone operators, () · Zbl 0862.47040
[22] Hu, S.; Papageorgiou, N., Handbook of multivalued analysis. vol. I: theory, (1997), Kluwer · Zbl 0887.47001
[23] Kartsatos, A.G., Recent results involving compact perturbations and compact resolvents of accretive operators in Banach spaces, () · Zbl 0849.47027
[24] 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, (2002) · Zbl 1005.47058
[25] 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
[26] Kobayashi, J.; Ôtani, M., An index formula for the degree of \((S_+)\) mappings associated with one-dimensional p-Laplacian, Abstr. appl. anal., 2004, 11, 981-995, (2004) · Zbl 1078.47021
[27] del Pino, M.; Elgueta, M.; Manásevich, R., A homotopic deformation along p of a leray – schauder degree result and existence for (u′|u′|p−2)′+f(x,u)=0, u(0)=u(T)=0, p>1, J. differential equations, 80, 1, 1-13, (1989) · Zbl 0708.34019
[28] del Pino, M.; Manásevich, R., Multiple solutions for the p-Laplacian under global resonance, Proc. amer. math. soc., 112, 1, 131-138, (1991) · Zbl 0725.34021
[29] Ôtani, M., A remark on certain nonlinear elliptic equations, Proc. fac. sci. tokai univ., 19, 23-28, (1984) · Zbl 0559.35027
[30] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag · Zbl 0516.47023
[31] Sato, K., On the generators of non-negative contraction semi-groups in Banach lattices, J. math. soc. Japan, 20, 423-436, (1968) · Zbl 0167.13601
[32] Showalter, R.E., Monotone operators in Banach space and nonlinear partial differential equations, Math. surveys monogr., vol. 49, (1997), Amer. Math. Soc. · Zbl 0870.35004
[33] Vrabie, I.I., Compactness methods for nonlinear evolutions, (1985), Longman · Zbl 0842.47040
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.