## 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.

### MSC:

 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
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### References:

 [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
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