Equations in Banach Spaces

In this chapter we present general existence principles for continuous and discrete problems on the infinite interval. Two continuous problems, namely (6.1.1) $$x(t) = h(t) + \int_0^t {g(t,s)f(s,x(s))ds,t \in [0,\infty )}$$ and (6.1.2) $$ x(t) = h(t) + \int_0^\infty {g(t,s)f(s,x(s))ds,t \in [0,\infty )}$$ are discussed. Also we examine the discrete problem (6.1.3) $$ x(k) = h(k) + \sum\limits_{i = 0}^\infty {G(k,i)f(i,x(i)),k \in \mathbb{N}.}$$ 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.