سیرت نبوی ﷺ: ایک تحقیقی جائزہ

Prophet Hood consists of guidance from Allah to humankind. It is a Allah given blessing and a favor that is bestowed on an individual chosen be Him to convey His message, which cannot be acquired or earned otherwise. There has never been a human being so well-respected, loved and followed as Muhammad (SAW), the final messenger of Allah. There has never been a person who has changed world history so dramatically as Muhammad (SAW) and his message. The Prophet (SAW) was the single most important person in the history of the world. Knowledge of the Prophetic Biography is necessary for every Muslim and sharing it with everyone is a responsibility. The importance of a complete biography of the Messenger as available to us cannot be under estimated in this troubled time since both Muslims as well as Non-Muslims have serious knowledge gap when it comes to even approaching the nature of the Final Prophet and the Ultimate Messenger of God sent to all of humanity, who came to restore the primordial religion of Man, the submission to Allah and His Commands. Muhammad (SAW) serves as: - Allah’s messenger and prophet to all mankind as an example of human behavior and noble character Therefore, in studying his life-story we should derive lessons and morals that can help us in our lives today.

Estimation of Fixed Point of Certain Non-Linear Maps in Convex Metric Spaces

The Banach contraction principle states that a contraction on a complete metric space has a unique fixed point and its proof hinges on "Picard iterations". This principle is applicable to a variety of subjects such as integral equations, partial differential equations and engineering of image processing. This principle fails for nonexpansive mappings on a Banach space. Mann [Proc. Amer. Math. Soc. 4(1953), 506-510] introduced an iterative scheme to approximate fixed points of a nonexpansive mapping on a Banach space. Mann scheme is inadequate for the approximation of fixed points of pseudocontractive mappings even on a Hilbert space. Consequently, Ishikawa [Proc. Amer. Math. Soc., 44 (1974), 147-150] upgraded Mann iterative scheme which is extensively used to approximate common fixed points of nonlinear mappings including nonexpansive mappings. The purpose of this dissertation is two-fold: (i) To prove the existence of fixed point for the classes of nonlinear mappings, namely generalized nonexpansive mappings and generalized quasi- contractive mappings in the setting of uniformly convex metric spaces. (ii) To establish approximate fixed point property for the classes of nonlinear mappings, namely generalized nonexpansive mappings, asymptotically nonexpansive mappings and generalized quasi-contractive mappings in the setting of (uniformly / strictly) convex metric spaces and cone metric type spaces. The approximation of fixed points is obtained by using appropriate iterative schemes; for example, the averaged iteration scheme, Ishikawa iteration scheme, multi-step iterative scheme, three-step explicit iterative scheme and Jungck three-step implicit iterative scheme. The strong convergence analysis of different iterative schemes contribute significantly in metric fixed point theory of nonlinear mappings. Most of the results presented here are new in the setting of a metric space and are: (a) established with the limited set of conditions on the control parameters (b) supported by real world applications, such as the existence of a solution for the first order periodic boundary value problem and the existence of a solution for an implicit integral equation.