Fixed-point theorems for discontinuous multivalued operators on ordered spaces with applications. (English) Zbl 1110.47043
In the paper, some fixed point theorems for discontinuous multivalued maps on ordered Banach spaces are proved. The continuity assumption is replaced by strict monotonicity of multivalued maps. The author generalizes his previous results obtained for complete lattices. The abstract fixed point results are applied to first-order discontinuous differential inclusions, namely proving the existence of extremal solutions to a BVP with \(x(0)=x(T)\) in \(\mathbb R\), and under suitable monotonicity assumptions on the right-hand side of the inclusion.
Reviewer: Grzegorz Gabor (Toruń)
MSC:
| 47H10 | Fixed-point theorems |
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 47H04 | Set-valued operators |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34A60 | Ordinary differential inclusions |
| 34K30 | Functional-differential equations in abstract spaces |
| 65J15 | Numerical solutions to equations with nonlinear operators |
References:
| [1] | Dhage, B. C., A lattice fixed-point theorem for multivalued mappings and applications, Chinese Journal of Math., 19, 11-22 (1991) · Zbl 0765.47015 |
| [2] | Dhage, B. C.; O’Regan, D., A lattice fixed-point theorem and multivalued differential equations, Functional Dif. Equations, 9, 109-115 (2002) · Zbl 1095.34510 |
| [3] | Amann, H., Order structures and fixed points, ATTI 20, Sem. Anal. Punz. Appl. (1977) |
| [4] | Heikkila, S.; Lakshmikantham, V., Monotone Iterative Technique for Nonlinear Discontinues Differential Equations (1994), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0882.34063 |
| [5] | Agarwal, R. P.; Dhage, B. C.; O’Regan, D., The method of upper and lower solution for differential inclusions via a lattice fixed-point theorem, Dynamic Systems and Appl., 12, 1-7 (2003) · Zbl 1054.34020 |
| [6] | Heikkila, S.; Hu, S., On fixed points of multi-functions in ordered spaces, Appl. Anal., 53, 115-127 (1993) · Zbl 0839.54033 |
| [7] | Dhage, B. C., Some algebraic fixed-point theorems for multivalued operators with applications, Diss. Math. Differential Inclusions, Control & Optim., 26 (2006) · Zbl 1110.47043 |
| [8] | Guo, D.; Lakshmikanthm, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045 |
| [9] | Dhage, B. C., A fixed-point theorem for multivalued mappings in ordered Banach spaces with applications, PanAmer. Math. Journal, 1, 15-34 (2005) · Zbl 1215.47039 |
| [10] | Dhage, B. C., A functional integral inclusion involving discontinuities, Fixed Point Theory, 5, 1, 53-64 (2004) · Zbl 1057.47063 |
| [11] | Dhage, B. C., On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations, J. Math. Phy. Sci., 25, 603-611 (1988) · Zbl 0673.47043 |
| [12] | Dhage, B. C., Fixed point theorems in ordered Banach algebras and applications, PanAmer. Math. Journal, 9, 4, 93-102 (1999) · Zbl 0964.47026 |
| [13] | Dhage, B. C., Multi-valued operators and fixed-point theorems in Banach algebras I, Taiwanese Journal Math, 6 (2006) · Zbl 1144.47321 |
| [14] | Halidias, N.; Papageorgiou, N., Second order multivalued boundary value problems, Arch. Math. (Brno), 34, 267-284 (1988) · Zbl 0915.34021 |
| [15] | Benchohra, M.; Ntouyas, S. K., On second order differential inclusions with periodic boundary conditions, Acta Mathematica Univ. Comenianea, LXIX, 173-181 (2000) · Zbl 0971.34003 |
| [16] | Nieto, J. J., Nonlinear second order periodic boundary value problems, J. Math. Anal. Appl., 130, 22-29 (1988) · Zbl 0678.34022 |
| [17] | Lasota, A.; Opial, Z., An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phy., 13, 781-786 (1965) · Zbl 0151.10703 |
| [18] | Aubin, J.; Cellina, A., Differential Inclusions (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0538.34007 |
| [19] | Deimling, K., Multi-valued Differential Equations (1998), De Gruyter: De Gruyter New York |
| [20] | Hu, S.; Papageorgiou, N. S., (Handbook of Multivalued Analysis, Volume I: Theory (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Berlin) · Zbl 0887.47001 |
| [21] | Krasnoselskii, M. A., Topological Methods in The Theory of Nonlinear Integral Equations (1964), Pergamon Press: Pergamon Press Dordrechet · Zbl 0111.30303 |
| [22] | Dhage, B. C., Monotone method for discontinuous differential inclusions, Math. Sci. Res. Journal, 8, 3, 104-113 (2004) · Zbl 1067.47069 |
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