Abstract
Consider the differential system
where the n × n matrix A is defined and continuous on [0, ∞), and f is a n-vector defined and continuous on [0, ∞) × ℝn. Let B[0,∞) be the space of all bounded, continuous n-vector valued functions and let L be a bounded linear operator mapping B[0, ∞) (or a subspace of B[0, ∞)) into ℝn. In this chapter we mainly study the differential system (3.1.1) subject to the boundary conditions
In Section 3.2 we consider the system (3.1.1) with f(t,x) = b(t) i.e. the linear system
together with (3.1.2). Here we provide necessary and sufficient conditions for the existence of solutions. In Section 3.3 we apply various fixed point theorems to establish the existence of solutions to the nonlinear problem (3.1.1), (3.1.2). Then in Section 3.4 we offer sufficient conditions for the existence of at least one value of the IRn-valued parameter λ so that the system
has a solution satisfying (3.1.2). Finally, in Section 3.5 we establish existence theory for the system (3.1.1) with A ≡ 0 i.e.
together with the boundary conditions
where N is a nonlinear operator mapping B[0, ∞) (or a subspace of B[0, ∞)) into ℝn.
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Agarwal, R.P., O’Regan, D. (2001). Continuous Systems. In: Infinite Interval Problems for Differential, Difference and Integral Equations. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0718-4_3
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DOI: https://doi.org/10.1007/978-94-010-0718-4_3
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