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Leggett--Williams theorems for coincidences of multivalued operators. (English) Zbl 1152.47041
Let $X,Y$ be Banach spaces, $C$ be a cone in $X$ and let $\Omega_1,\Omega_2$ be open bounded subsets of $X$ with $\overline{\Omega}_1\subset\Omega_2$. For a pair $(L,N)$ consisting of a linear Fredholm operator $L:\text{dom}\,L\subset X\to Y$ of zero index and an upper semicontinuous (compact convex)-valued multimap $N:X\multimap Y$, the authors study the existence of a coincidence point $(Lx \in Nx)$ in $C\cap(\overline{\Omega}_2\setminus\Omega_1)$. They use topological degree methods under some compactness or condensivity conditions. As application, the existence of positive periodic solutions for a differential inclusion is considered.

MSC:
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34A60Differential inclusions
34C25Periodic solutions of ODE
47H04Set-valued operators
47H11Degree theory (nonlinear operators)
54H25Fixed-point and coincidence theorems in topological spaces
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References:
[1] Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[2] Deimling, K.: Multivalued differential equations, (1992) · Zbl 0760.34002
[3] Cremins, C. T.: Existence theorems for semilinear equations in cones, J. math. Anal. appl. 265, 447-457 (2002) · Zbl 1015.47042 · doi:10.1006/jmaa.2001.7746
[4] Cremins, C. T.: Existence theorems for weakly inward semilinear operators, Discrete contin. Dyn. syst. (Suppl.), 200-205 (2003) · Zbl 1073.47054
[5] Gaines, R. E.; Santanilla, J.: A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky mountain J. Math. 12, 669-678 (1982) · Zbl 0508.34030 · doi:10.1216/RMJ-1982-12-4-669
[6] Leggett, R. W.; Williams, L. R.: A fixed point theorem with application to an infectious disease model, J. math. Anal. appl. 76, 91-97 (1980) · Zbl 0448.47044 · doi:10.1016/0022-247X(80)90062-1
[7] Ma, T. W.: Topological degrees of set-valued compact fields in locally convex spaces, Dissertationes math. (Rozprawy mat.) 92 (1972) · Zbl 0211.25903
[8] Mawhin, J.: Equivalence theorems for nonlinear operator equations and coincidence degree theory for mappings in locally convex topological vector spaces, J. differential equations 12, 610-636 (1972) · Zbl 0244.47049 · doi:10.1016/0022-0396(72)90028-9
[9] Müller, V.: Spectral theory of linear operators, (2003)
[10] O’regan, D.; Zima, M.: Leggett--Williams norm-type theorems for coincidences, Arch. math. 87, 233-244 (2006) · Zbl 1109.47051 · doi:10.1007/s00013-006-1661-6
[11] Petryshyn, W. V.: On the solvability of x$\inTx+{\lambda}$Fx in quasinormal cones with T and fk-set contractive, Nonlinear anal. 5, 585-591 (1981) · Zbl 0474.47028 · doi:10.1016/0362-546X(81)90105-X
[12] Petryshyn, W. V.: Existence of fixed points of positive k-set-contractive maps as consequences of suitable boundary conditions, J. London math. Soc. (2) 38, 503-512 (1988) · Zbl 0681.47024
[13] Santanilla, J.: Some coincidence theorems in wedges, cones, and convex sets, J. math. Anal. appl. 105, 357-371 (1985) · Zbl 0576.34018 · doi:10.1016/0022-247X(85)90053-8
[14] Tarafdar, E.; Teo, S. K.: On the existence of solutions of the equation lx$\inNx $and a coincidence degree theory, J. austral. Math. soc. Ser. A 28, 139-173 (1979) · Zbl 0431.47038
[15] Zima, M.: Leggett--Williams norm-type fixed point theorems, Dynam. systems appl. 14, 551-560 (2005) · Zbl 1098.47048