Abstract
In this chapter we present general existence principles for continuous and discrete problems on the infinite interval. Two continuous problems, namely
and
are discussed. Also we examine the discrete problem
In all of these problems values of the solution lie in some real Banach space E (here (E, ‖ · ‖) is not necessarily finite dimensional). In Section 6.2 we establish existence principles for (6.1.1) and (6.1.2). Here we are interested in solutions in the space BC([0, ∞), E), where BC([0, ∞), E) denotes the Banach space of all bounded and continuous functions u : [0, ∞) → E with norm |u|0 = sup t∈[0, ∞) ‖u(t)‖. Section 6.3 concerns with the existence principles for the discrete problem (6.1.3). We look for solutions in BC(ℒ, E). Here BC(ℒ, E) denotes the Banach space of maps w continuous and bounded on ℒ (discrete topology) with norm ‖w‖0 = sup k∈ℒ‖w(k)‖. Our main result here immediately yields an interesting exis tence criterion for the discrete problems on finite intervals.
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6.6. References
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Agarwal, R.P., O’Regan, D. (2001). Equations in Banach Spaces. In: Infinite Interval Problems for Differential, Difference and Integral Equations. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0718-4_6
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DOI: https://doi.org/10.1007/978-94-010-0718-4_6
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