Equations in Banach Spaces

摘要: In this chapter we present general existence principles for continuous and discrete problems on the infinite interval. Two problems, namely $$x(t) = h(t) + \int_0^t {g(t,s)f(s,x(s))ds,t \in [0,\infty )}$$ (6.1.1) and $$ x(t) \int_0^\infty )}$$ (6.1.2) are discussed. Also examine problem x(k) h(k) \sum\limits_{i 0}^\infty {G(k,i)f(i,x(i)),k \mathbb{N}.}$$ (6.1.3) In all of these values solution lie in some real Banach space E (here (E, ‖ · ‖) is not necessarily finite dimensional). Section 6.2 establish (6.1.1) (6.1.2). Here are interested solutions BC([0, ∞), E), where E) denotes bounded functions u : [0, ∞) → with norm |u|0 sup t∈[0, ‖u(t)‖. 6.3 concerns (6.1.3). We look BC(ℒ, E). maps w ℒ (discrete topology) ‖w‖0 k∈ℒ‖w(k)‖. Our main result here immediately yields an interesting exis tence criterion intervals.