Extremal solutions for a class of unilateral problems. (English) Zbl 1044.47041
Fixed point theorems for monotone increasing operators in ordered Banach spaces are an important tool in the study of elliptic and parabolic differential equations. The existence and approximation of solutions for both continuous and discontinuous equations can be investigated with this tool [S. Heikikilä and V. Lakshmikantham, “Monotone iterative techniques for discontinuous nonlinear differential equations” (1994; Zbl 0804.34001)].
In this paper, the authors prove the existence of extremal solutions for a class of differential variational inequalities using a fixed point theorem for monotone increasing operators in ordered spaces. Under certain assumptions, the variational inequality under consideration is reduced to the problem of finding fixed points for monotone increasing operators in ordered Banach spaces.
The results of this paper are related with the recent papers of V. K. Le [J. Math. Anal. Appl. 252, 65–90 (2000; Zbl 0980.49011)] and P. Drábek and J. Hernández [Nonlinear Anal., Theory Methods. Appl. 44A, 189–204 (2001; Zbl 0991.35035)].
In this paper, the authors prove the existence of extremal solutions for a class of differential variational inequalities using a fixed point theorem for monotone increasing operators in ordered spaces. Under certain assumptions, the variational inequality under consideration is reduced to the problem of finding fixed points for monotone increasing operators in ordered Banach spaces.
The results of this paper are related with the recent papers of V. K. Le [J. Math. Anal. Appl. 252, 65–90 (2000; Zbl 0980.49011)] and P. Drábek and J. Hernández [Nonlinear Anal., Theory Methods. Appl. 44A, 189–204 (2001; Zbl 0991.35035)].
Reviewer: R. K. Bose (New Delhi)
MSC:
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
References:
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| [2] | Boccardo, L., Giachetti, D. and F. Murat: A generalization of a theorem of H. Brezis and F. Browder and applications to some unilateral problems. Publ. du Lab. d’Anal. Num., Univ. Pierre et Marie Curie R89014, 1989. |
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| [8] | Vy Khoi Le: Existence of positive solutions of variational inequalities by a subsolution-supersolution approach. J. Math. Anal. Appl. 252 (2000), 65 - 90. · Zbl 0980.49011 · doi:10.1006/jmaa.2000.6907 |
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