# Characteristic Inequalities in Banach Spaces and Applications

Thesis

The contribution of this project falls within the general area of nonlinear functional analysis and applications. We focus on an important topic within this area: Inequalities in Banach spaces and applications. As is well known, among all infinite dimensional Banach spaces, Hilbert spaces generally have simple geometric structures. This makes problems posed in them easier to handle, this is as a result of the existence of inner product, the proximity map, and the two characteristic identities which we state below. kx + yk 2 = kxk 2 + 2hy, xi + kyk 2 , (1) kλx + (1 − λ)yk 2 = λkxk 2 + (1 − λ)kyk 2 − λ(1 − λ)kx − yk 2 , (2) for any x, y ∈ H, and λ ∈ (0, 1) where H is a real Hilbert space. These are some of the properties which characterize inner product space and also make certain problems posed in Hilbert spaces more manageable to handle than those in the general Banach spaces. Another important tool which characterizes Hilbert spaces is the fact that the proximity map P K from a Hilbert space H onto a nonempty, closed, convex subset K of H is non-expansive, i.e., if P K : H → K is defined by P K x = z, where kx − zk = inf kx − uk, u∈K then kP K u − P K vk ≤ ku − vk ∀u, v ∈ H. This property of P K which is central in solving numerous problems in Hilbert spaces, does not hold in all Banach spaces more general than Hilbert spaces. However, in applications, many problems do not naturally live in Hilbert spaces, therefore to extend some of the Hilbert space techniques to more general Banach spaces, analogue of the inner product, the proximity map, and the two identities (1) and (2) have to be developed. For this development, the duality mapping J : X → 2 X ∗ of a Banach space X defined by J(x) = {x ∗ ∈ X ∗ : hx, x ∗ i = kxk 2 , kxk = kx ∗ k}, which we shall see in the next chapter, has become one of the most important tool in non-linear functional analysis. It serves as the replacement for inner product in a Banach space more general than Hilbert space. Sunny nonexpansive retractions in Banach spaces, when they exist, generalize the so called proximity map which exists in Hilbert spaces as we shall discuss in chapter three of this work. We study inequalities obtained in various Banach spaces more general than Hilbert spaces as analogues of (1) and (2) and their applications. Lastly, we constructed explicitly the sunny nonexpansive retraction in certain Banach space. As application of sunny nonexpansive retraction, we approximate a fixed point of nonself nonexpansive map.