Continuous Systems

Abstract

Consider the differential system

$$ x' = A(t)x + f(t,x),t \in [0,\infty )$$

((3.1.1))

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

$$L[x] = \ell \in \mathbb{R}^n .$$

((3.1.2))

In Section 3.2 we consider the system (3.1.1) with f(t,x) = b(t) i.e. the linear system

$$ x' = A(t)x + b(t),t \in [0,\infty )$$

((3.1.3))

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

$$ \begin{gathered}x' = A(t)x + g(t,x,\lambda ),t \in [0,\infty ) \hfill \\x(0) = \xi \hfill \\\end{gathered}$$

((3.1.4))

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.

$$ x' = f(t,x)t \in [0,\infty )$$

((3.1.5))

together with the boundary conditions

$$N[x] = 0,$$

((3.1.6))

where N is a nonlinear operator mapping B[0, ∞) (or a subspace of B[0, ∞)) into ℝⁿ.