×

Multivalued differential inequalities. (English) Zbl 0706.34013

Given a cone \(K\subset {\mathbb{R}}^ n\) and a point \(x_ 0\in {\mathbb{R}}^ n\), the authors study the problem of existence of solutions of the differential inclusion \(x'\in F(t,x)\) in an order interval \([u,v]\subset {\mathbb{R}}^ n\). Here F takes compact convex values in \({\mathbb{R}}^ n\), F(\(\cdot,x)\) is measurable for all \(x\in {\mathbb{R}}^ n\), and F(T,\(\cdot)\) is upper semicontinuous for all \(t\in [0,a]\). Some interesting results and illuminating examples are given.
Reviewer: J.Appell

MSC:

34A60 Ordinary differential inclusions
34A40 Differential inequalities involving functions of a single real variable
54C60 Set-valued maps in general topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aubin, J. P.; Cellina, A., Differential Inclusions (1984), Springer: Springer Berlin
[2] Deimling, K., Ordinary differential equations in Banach spaces, (Lecture Notes in Mathematics, 596 (1977), Springer: Springer Berlin) · Zbl 0555.60036
[3] Deimling, K., Multivalued differential equations on closed sets, Diff. Integral Eqns, 1, 23-30 (1988) · Zbl 0715.34114
[4] Deimling, K., Extremal solutions of multivalued differential equations, Results Math., 14, 38-47 (1988) · Zbl 0655.34014
[5] Deimling, K., Extremal solutions of multivalued differential equations II, Results Math., 15, 197-201 (1989) · Zbl 0776.34012
[8] Deimling, K.; Lakshmikantham, V., On existence of extremal solutions of differential equations in Banach spaces, Nonlinear Analysis, 3, 563-568 (1979) · Zbl 0418.34061
[9] Deimling, K.; Lakshmikantham, V., Existence and comparison theorems for differential equations in Banach spaces, Nonlinear Analysis, 3, 569-575 (1979) · Zbl 0418.34062
[10] Deimling, K.; Rao, M. R.M., On solution sets of multivalued differential equations, Appl. Analysis, 30, 129-135 (1988) · Zbl 0635.34014
[11] Lakshmikantham, V.; Leela, S., Differential and Integral Inequalities II (1969), Academic Press: Academic Press New York · Zbl 0177.12403
[12] Szarski, J., Differential inequalities, (Monografie Mat., 43 (1967), Polish Sci. Publishers) · Zbl 0135.25804
[13] Walter, W., Differential and Integral Inequalities (1970), Springer: Springer Berlin
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.