Abstract : In this thesis we introduce the notion of modified w-distance (mw-distance) on a quasi-metric space which generalizes the concept of quasi-metric. We obtain a fixed point theorem for generalized contractions with respect to mw-distances on complete quasi-metric spaces

 ‎This thesis is concerned with maximization and minimization problems of the energy‎
‎integral associated to p-Laplace equations depending on functions that belong to a class of rearrangements‎.
‎Let‎
‎$ \Omega$‎
‎be a bounded smooth domain in‎
‎$ \mathbb{R}^{N} $‎
‎If‎
‎$E\subset \mathbb{R}^{N} $‎
‎is a measurable set‎, ‎we denote by‎
‎$ |E| $‎
‎its Lebesgue measure‎. ‎We say‎
‎that two measurable functions‎
‎$ f(x) $‎
‎and‎
‎$ g(x) $‎
‎defined in‎
‎$ \Omega $‎
‎have the same rearrangement if‎
‎\begin{eqnarray*}‎
‎\big\vert\lbrace x\in \Omega‎ :‎f(x)\geq \beta \rbrace\big\vert = \big\vert\lbrace x\in \Omega‎ :‎g(x)\geq \beta \rbrace\big\vert &&~~~\forall\beta\in\mathbb{R}‎.
‎\end{eqnarray*}‎

 The aim of this thesis, is to study the optimization in problems involving the p-laplacian. Also we proof the existence and uniqueness solution of this problems. Morever, we study an eigenvalue problem and investigate its maximum solution.

 

Let X be a Banach space and let K be an Hausdorff compact topological space. We provide that every nonempty weak open subset relative to the unit ball of C(K,X), WC(K,X)and W*C(K,X*) has diameter 2. Whenever K be infinit.C(K,X) the banach space of all continuous functions on K into X with the norm topology, WC(K,X) the banach space of all continuous functions K into X with the weak topology and W*C(K,X*) the banach space of all continuous functions on K into X with the weak-* topology. also, we characterize when every nonempty weak open subset relative to the unit ball has diameter 2 for the operators space L(L₁(µ),X), Where µis a positive measure
 

In this thesis, we defined hyperbolic metric space uniformly convex space
and 2-uniformuly convex metric space. eke, defind asymptotic pointwise nonexpansive
mappings. Moreover,we discuss the behavior of the modified mann iteration
process associated with asymptptic pointwise mappings and we prove that modified
mann iteration process coverges to a fixed point of T.Moreover,we discuss the existence
of a fixed point of a single for asymptotic pointwise nonexpansive mappings in
uniformly convex hyperbolic metric spaces

This thesis explores the questions raised by Eldred and Veeramani about the existence of Best proximity point for cyclic contraction maps in a reflexive Banach space. Moreover, we introduce cyclicφ -contraction maps and Consider the existence of Best proximity point for this mapping