كمال فلاحي

In this thesis, using weakly null sequences (weakly p.summable sequences) and different classes of sets (such as Dunford-Pettis sets (resp. almost Dunford -Pettis sets)) in Banach spaces and especially Banach Lattices, Various properties related to the fixed point property will be investigated and studied. Just as a weaker property than the fixed point has been introduced with the norm topology, called the weak fixed point property, we will also use the Right topology, wich, is stronger than the weak topology and weaker than the norm topoloy , and we will study the Right fixed point property of order p and some properties related to it. in this regard, the consept of R-orthogonality of order p, R- WORTH of order p, R-non-strict opial of order p and stronger version of these properties will be of most use in this thesis. We will also find sufficient conditions for a closed subspace of M ⊂ K(X, F) (we means the K(X,F), is the class of all compact oprators from Banach space X to a Banach lattice F) to have the property of a Right fixed point property of order p.

In this thesis, we generalize well­known results on pairwise fixed points and best approximate pairwise points. We generalize the symbols of ordered pairs of contractile cyclic maps and obtain sufficient conditions for the existence and uniqueness of best approximate points. We also obtain a prior and a posterior estimate for the pairwise fixed points and provide conditions for the best pairwise approximate points under which a Banach space with a convex power­type modulus at the best approximate points can be obtained using a sequence of successive iterations. We illustrate the main result by providing an example. We also generalize the most famous results on paired fixed points and best approximation points of circular contractions of ordered maps. We prove the uniqueness of the best paired approximation points for circular contractions of paired maps in a uniformly convex Banach space. We also obtain a prior error and a posterior error for the best paired approximation points using sequences of successive iterations when the Banach space has modules of the sequence of successive iterations. With a reduced condition, we obtain the existence and uniqueness of pairwisefixed points of distance contractions of ordered coupled mappings in a complete metric space, and also obtain a prior error, a posterior error, and a convergence rate for pairwise fixed points for repeated sequences. We apply these results to solve systems of integral equations, systems of linear and nonlinear equations.

In this thesis, we will first define various cyclic G-(φ − ψ)-weak contractive mappings and then we will investigate the existence of fixed points and best proximity point for such mappings in metric spaces equipped with graphs regarding various properties. Various results will also be derived from the main theorems by considering certain cases of graphs that include partially ordered, comparable and ε-closed elements. Also, as an application, the integral type of these contractions will be studied. In addition, different examples will be presented throughout the text of the thesis to show the effciency of the previous results and new obtained theorems. The comparisons made between the previous theorems and the new obtained results will show that the existing results on fixed point and best proximity point are well generalized in this thesis.

According to the three concepts, weak orthogonality, WORT H property and nonstrictly Opial condition in Banach spaces, the disjoint (positive) version of them in Banach lattices is introduced. Also, Banach lattices in which these three properties are equivalent to the order continuity of the norm are characterized. As an application, some conditions for a Banach lattice under which any of these three properties implies the weak fixed point property are provided. Next, the disjoint (positive) weak fixed point property for some operator spaces is studied. In particular, it is established that for each Banach space X and a suitable Banach lattice F, a Banach lattice M ⊂ K(X, F) has the weak fixed point property (resp. disjoint (positive) weak fixed point property) if and only if each evaluation operator ψy∗ on M is completely continuous (resp. almost Dunford-Pettis) operator, where ψy∗ : M → X∗ is defined by ψy∗ = T∗y∗ for y∗ ∈ Y ∗ and T ∈ M.

The purpose of this thesis is to define mappings and integral contractions, and obtain the approximate fixed points of these contractions in metric spaces equipped with graphs. Subsequently, we generalize the best approximate points for integral contractions and derive several results. Furthermore, we present several well-known fixed point results and provide interesting examples to illustrate these results.

In this thesis, after defining the weak fixed point property in Banach spaces, we study some conditions for a Banach space under which any of these properties implies the weak fixed point property is provided. For example, if the Banach space X has a weak normal structure, then it has a weak fixed point property. Also, if X has Schur property, Opial property, or K property, then it has weak normal structure. In this regard, we define some of the properties such as weak orthogonality in Banach lattices. The disjoint (positive) version of them in Banach lattices is introduced. Also, Banach lattices in which these three properties are equivalent to the order continuity of the norm are characterized. As an application, some conditions for a Banach lattice under which any of these three properties implies the weak fixed point property is provided. Next, the disjoint (positive) weak fixed point property for some operator spaces is studied. Then we will introduce the new concept of weak fixed point property of order p and we will study the previous conditions in Banach spaces and Banach lattices with order p. In final, using the Right topology three new properties the so called Right orthogonality, Right WORTH property, and non-strictly Right Opial condition (and also the disjoint version of them) in Banach lattices are introduced. Moreover, Banach lattices in which these three properties coincide with order continuity of the norm are characterized. As an application, we give some suffcient conditions under which a Banach lattice has the Right fixed point property (or, disjoint Right fixed point property).

In this thesis,By using of the concept the reproducing kernl Hilbert space and Absolutely continuous, we study existence and solution of the differential equations. Also, we Consider examples of reproducing kernels and the role of Sobolev spaces in reproducing kernels Hilbert spaces. For this, we represent some of initial definitions and needed theorems. In the following, we bring Sobolev spaces and Also define the reproducing kernels Hilbert spaces on the two-dimensional rectangle. The reproducing kernel Hilbert space is a useful framework to construct approximate solutions of (PDEs) partial differential equations.Many of numerical methods considered to solution linear and nonlinear PDEs. But we didnt find the method By using reproducing kernels. For this aim, we focus on the exact and approximate solutions of PDEs with linear boundary initial conditions. In the following, we consider the singularly perturbed second order initial boundary value problem in the reproducing kernel space. Finale chapters includes definitions and properties of Chebyshev polynomials in the one-dimensional spaces. In this section, we discuss reproducing kernel on two set of nodes in two dimensions and we obtain reproducing kernel space by re-defining the inner product of Chebyshev polynomials. In the end, we consider description of the method mentioned equations with initial-boundary conditions in the reproducing kernel space Pn2(Ω)

The aim of this research is to define ⊥ −proximally increasing mapping and obtain several best proximity point results concerning this mapping in the framework of new spaces, which is called orthogonal bmetric spaces. Also, several well-known fixed point results in such spaces are established. All main results and new definitions are supported by some illustrative and interesting examples.

In this thesis, we introduce some new concepts such as n-cyclic Fisher quasi-contraction mappings, full-n-noncyclic and regular-n-noncyclic Fisher quasi-contraction mappings in metric spaces and we generalize the results of Safari-Hafshejan, Amini-Harandi and Fakhar [Numer.Func. Anal. Optim. 40 (5) (2019), 603619]. Meanwhile, we answer the question under what conditions does a mapping full-n-noncyclic Fisher quasicontraction have n(n−1) 2 unique optimal pair of fixed points?”. Further, to support the main results, we highlight all of the new concepts by non-trivial examples. Next, we introduce a new type of graph contraction using a special class of functions and give a best proximity point theorem for this contraction in complete b-metric spaces endowed with a graph. Finally, we support our main theorem by a non-trivial example and give some consequences of it for usual graphs.

In this thesis we present the concepts of proximal contraction and proximal nonexpansive mappings on star-shaped sets in probabilistic Banach (Menger) spaces and investigates properties of convergence of distances of p-cyclic contractions on the union of the p subsets of an abstract set X defining probabilistic metric spaces and Menger probabilistic metric spaces as well as the characterization of Cauchy sequences which converge to the best proximity points. The purpose of this paper is to present some definitions and basic concepts of best proximity point in a new class of probabilistic metric spaces and to prove the best proximity point theorems for the contractive mappings and weak contractive mappings. At the end In this paper, we define the concepts of commute proximally, dominate proximally, weakly dominate proximally, proximal generalized φ-contraction and common best proximity point in probabilistic Menger space.