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 |
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