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In this thesis, it has been introduced Hardy inequality and it’s extends. The fractional Hardy inequality for Ω ⊆ Rn and f ∈ C ∞ (Ω) is:1f (x) − f ( y)2∫α +n2dx dy ≥ kn,α ∫f (x)dx( )αΩ×Ωx − yΩ M α x In this thesis we are going to introduce the fractional and derivative forms of Hardy inequality then the Hardy inequality will be proved to fractional form on Euclidean and Hyperbolic domains, and finally we will get right into Hardy inequality with remainder.
 

In this thesis, we present some extension of Burton and Kirk fixed point theorem for the sum of two nonlinear operators, which one of them is compact and the other one is strict contraction; and we investigate the existence of fixed point where these two operators don’t need to be weakly continuous. Next, we will check the necessity of some conditions, like being strict contraction or compactness, in some specific Banach spaces, such as reflective spaces or Banach spaces equipped with uniformly convex norm.Finally, we analysis the existence of the solution for integral equations in L1 space by using the expressed theorems  

The so called T(b) theorem for boundedness of generalized CalderonZygmund singular integrals is investigated. This theorem provides a criterion, in which boundedness of the operator is equivalent to a special condition for the image of some function under action of that operator.Then we discuss some applications of the theorem in analysis and partial differential equations  

For a Banach space E and a compact metric space (X, d), a function F : X → E is a Lipschitz function if there exists k > 0 such that ∥F (x) − F (y)∥ ≤ kd(x, y) for all x, y ∈ X.The smallest such k is called the Lipschitz constant L(F ) for F . The space Lip(X, E) of all Lipschitz functions from X to E is a Banach space under the norm defined by ∥F ∥ = max{L(F ), ∥F ∥∞}, where ∥F ∥∞ = sup{∥F (x)∥ : x ∈ X}.Recent results characterizing isometries on these vector-valued Lipschitz spaces require the Banach space E to be strictly convex. We inves- tigate the nature of the extreme points of the dual ball for Lip(X, E) and use the information to describe the surjective isometries on Lip(X, E) under...