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Fixed and Coincidence Best Proximity Points in Sets With Certain Distance Structures

Thesis Info

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External Link

Author

Hussain, Azhar

Program

PhD

Institute

University of Sargodha

City

Sargodha

Province

Punjab

Country

Pakistan

Thesis Completing Year

2017

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12657/1/Azhar%20Hussain_Maths_2017_UoSargodha.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726211514

Similar


Fixed point theory provides a useful tool for solving problems arising in various branches of mathematical analysis and its related disciplines. This theory deals with the investigation of less restrictive conditions on mappings that guarantee the of existence of their fixed point and study of conditions which assure uniqueness of fixed point. On the other hand, though best approximation theorems ensure the existence of approximation solutions, such results need not yield optimal solutions. But, best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal as well. In this thesis, we are mainly concerned convex type contractions, α-η-Weakly Zamfirescu contraction and α-η-Ciri´c strong al- ´ most contraction. We obtained related fixed point and approximate fixed theorems in the framework of complete G-metric space which are more general than ordinary metric spaces. We also obtained coincidence best proximity point results of Fg-weak contraction mapping and (ϕ, θ, α, g)-contraction mapping in partially ordered metric spaces and in G-metric spaces, respectively. Best proximity points for proximal cyclic contraction of Perov type, quasi contraction, Perov-Ciric quasi contraction and Perov-Fisher quasi contraction in the setup of complete cone metric space are also obtained.
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مولوی محمد یحییٰ تنہاؔ

مولوی محمد یحییٰ تنہاؔ
افسوس ہے کہ گزشتہ رمضان میں مولوی محمد یحییٰ تنہا مرحوم نے کراچی میں انتقال کیا، مرحوم اردو زبان کے مشہور اہل قلم، اس کے پرانے خدمت گزار اور متعدد کتابوں کے مصنف و مترجم تھے، ان کی سب سے مشہور و مفید کتاب سیر المصنفین اور مراۃ الشعراء ہے، دونوں دو جلدوں میں ہیں، سیرالمصنفین میں شروع سے لے کر موجودہ دور تک کے مصنفین کے حالات اور ان کی تصانیف پر تبصرہ ہے، اس میں اردو کی پوری تاریخ آگئی ہے، مراۃ الشعراء اردو کے شعراء کا تذکرہ ہے، اﷲ تعالیٰ ان کی مغفرت فرمائے۔
(شاہ معین الدین ندوی، فروری ۱۹۶۷ء)

 

Financial Misgivings of Married Working Women in Lahore

In Pakistan, the financial issues of married working women are rarely discussed. There is an absence of literature on the subject. Nevertheless, the social sciences literature has been debating financial aspect of gender, either from the perspective of employer or employee; ignoring the working women’s reservations regarding their financial contribution in their marital life, particularly with reference to spouse and in-laws expectations. This issue becomes more acute when it comes to patriarchal conservative developing societies. This study is a delicate attempt to understand the magnitude of financial support of Pakistani married working women for spouse and in-laws, in the city of Lahore. In routine, the husband as well as the in-laws expect that a working wife should surrender her income, in entirety or partially towards the household budget, thus taking the financial responsibility of her spouse along with his extended family. Such financial misgivings create tensions and pressure for the already burdened woman having a disadvantaged status. In certain cases, the contribution is by free will of the wife, as well. This research will make an attempt through a survey with 50 married working women of different social classes, residing in Lahore. The purpose is to understand the rationale of approval or disapproval of this practice. Refusal for cooperation often create problems for the earning woman, even at the risk of separation or divorce, or at the least, tensions in the married life. At the end, the study will debate over possible adjustments and compromises, which could reduce tense situation for the married working woman, and at the same time maintaining her financial independence.

سورة الانفال: دراسۃ تحلیلیۃ تحویۃ و بلاغیۃ

(Analytical, Grammatical and Rhetorical study of Surah Al-Anfal) We have created man from Nutfah Amshaj, in order to try him, so, We made him hearer and seer. We guided him to the way, whether he be grateful or ungrateful. Man, the vicegerent of Allah, envy and adore of Angels, captor of heavens and earths, titled best of creatures, clothed in His qualities, blessed with divine command, created in preeminent form, dressed in the physical and spiritual nobility, fashioned in the finest of images, wrapped in freedom, equality and accountability, adorned with will, crowned with knowledge, is the nucleus of this universe. Down from centuries Allah sent His Apostles, Commands, Scripture and Books to illuminate the path that leads to Him and shown guidance to those who accept His Divinity and yield to Him. These conjunctions were commensurate with mental, social and civil progress of man in the stream of time and were always lucid, simple, practicable and all embracing. As the time pass by, the caliber of man grew to accept and understand last message of his Creator that invites him to ponder over the reasons of his own and universe creation, the prime objective of his coming into being and fellow communities under, on and above the earth. The message urges him to bring into use his inherit abilities to explore the abundance secrets apparent and hidden for the benefit of his fraternity and follow the path of those who have been graced by Allah. The message is delivered through His last Messenger Hazrat Muhammad (PBUH) and contained in Quran; and the way of Allah of showering His blessings is "And whomsoever Allah& wills to guide, He opens his breast to Islam". The subject matter of the Quran is essentially man: man in relation to his Lord and Creator, Allah; man in relation to himself; and man in relation to the rest of creation. The gist of the revelations is to invite man to the right way of dealing with his Lord, with himself, and with rest of the creations. The Quran is the main miracle given to the Prophet Muhammad (peace be upon him) to prove that he was a true prophet of Allah and not an imposter. The Quran has been preserved in both the oral as well as written form in a way no other religious book in the history. Undoubtedly, it is the Quran's preservation which has kept the religion Islam in its original purity. Humanity can always revert to the sources of Islam no matter what people may have added or forgotten in time to reinforce the essential principles of Islam. Consequently, the preservation of the Quran meant the preservation of Islam in its final form, unique in the way it presents its subject matter; it is not .a book in the usual sense of the word wherein there is an introduction, explanation of the subject, followed by a conclusion, neither is it restricted to only a presentation of historical events, problems of philosophy, facts of science or social laws, though all may be found woven together in it without any apparent connection and links. The fluency of the words explaining an event, multitude of expression describing the same event, and stating varied tales in similar phrases establishes its truth of Divine Words. The eloquence, commentary, rhetoric surpasses the established standards that could be compiled by humanity so far. That is why Allah challenges to groups of mankind and jinn in various parts of the Qur'an for brining in a book, a chapter or even an ayat parallel to it having same level of clarity, lucidity and rhetoric (Say: "If mankind and the Jinn were together to produce the like of this Qur'an, they could not produce the like thereof; even if they helped one another. (And if you (Arab pagans, Jews, and Christians) are in doubt concerning that which We have sent down (i.e. the Qur'an) to Our servant (Muhammad), then produce a Surah (chapter) of the like thereof and call your witnesses (supporters and helpers) besides Allah&, if you are truthful). But if you do it not, and you can never do it, then fear the Fire (Hell) whose .fuel is men and stones, prepared for the disbelievers. (Say (to them, 0 Muhammad): "Then bring a Book front Allah4, which is a better guide than these two (the Tawrah (Torah) and the Qur'an), that may follow it, if you are truthful"). (Or they say, "He (Prophet Muhammad) forged it (the Qur'an)." Say: "Bring you then ten forged Surahs (chapters) like it, and call whomsoever you can, other than Allah& (to your help), if you speak the truth!"), (And this Qur'an is not such as could ever be produced by other than Allah& (Lord of the heavens and the earth), but it is a confirmation of (the revelation) which was before it (i.e. the Tawrah, and the Injil), and a full explanation of the Book (i.e. Laws decreed for mankind) . wherein there is no doubt from the Lord of all that exists.) (Or do they say: "He (Muhammad) has forged it" Say: "Bring then a Sunlit (chapter) like it, and call upon whomsoever you can besides Allah, if you are truthfitl!"). All of these Ayat were revealed in Makkah. Whoever reads through the Qur'an will realize that it contains various levels of superiority through both the apparent and hidden meanings that it mentions. Allah said, (Alif Lam Ra. (This is) a Book, the verses whereof are perfect (in every sphere of knowledge, etc.), and then explained in detail from One (Allah&), Who is Wise and well-acquainted (with all things. So the expressions in the Qur'an are perfect and its meanings are explained. Furthermore, no one can deny the authenticity and veracity of the events and several accounts that are narrated in the Holy Book for they occurred in the same manner. The Qur'an also mentioned the stories of the people of the past; and these accounts and stories occurred exactly as the Qur'an stated. And the Word of your Lord has been fulfilled in truth and in justice). The Qur'an is true, just and full of guidance; it is entirely eloquent in the most perfect manner, while reading through the stories in the Qur'an, one will find them fruitful, whether they are in extended or short forms, repeated or not. When the Qur'an mentions the subject of warning and promises, it presents truths that would make solid, firm mountains shake, so what about the comprehending, understanding hearts. When the Qur'an promises, it opens the hearts and the ears, making them eager to attain the abode of peace - Paradise -and to be the neighbors of the Throne of the Most Beneficent while graphically sketching the Resurrection and the horrors that will occur on that Day, and Paradise and the Fire and the joys and safe refuge that Allah prepared for His loyal friends, or torment and Hell for His enemies, one can easily visualize the whole scene whether an abode of eternal bliss. I have chosen the topic Analytical, Grammatical and Rhetorical study of Surah Al-Anfal to discover the beauty of Quran, the style it adopts to narrate an event, fashion it chooses to reveal the Divine instructions and mode select to deduce lessons; understand, solidify and compile underlying principles of writings; explore, coagulate and identify instances of rhythms in phrases. The paper touches upon topics in different parts and throws light on angles directing to discourse, essay writing and elucidation. It is a vast topic and demands yeoman effort to enlighten these aspects by going through the works of our legendry scholars, philosophers and historians yet with the possibility of leaving great part undiscovered. The thesis consists of five chapters having subsections dealing with details and combining relevant material of the topic in that part. The division has been made for topical and chronological study of the topic for sequential clarity. The first chapter covers all headings under "Analytical study of Surah Al-Anfal" with three sections each synthesizing lexical, idiomatical /phraseological meaning of Anfal and its name, revelation and importance as well as reason and circumstances leading to the revelation of different ayals-verses. The second chapter deals with commands mentioned about Jihad in the Surah whereas situations stemming out and/or matters incidental to Jihad have been elaborated in sections. These ranging from non exposure/sharing of secrets of Unirnah, urging continuous preparations for war to strike fear in the hearts of the enemies of Allah, prohibition on fleeing from battle and its punishment, manner of war and its prime objective-repelling of aggression and transgression, respect for covenants and treaties and allowance of asylum whosoever seeks and ruling on the spoils of war- booty. The third chapter deliberates upon other important subjects discussed in the Surah vis-a-vis the command to answer and obey Allah, response to His Messenger's call, fear of Allah, states and affairs of Idolaters and qualities/characteristics of believers. The chapter four constitutes grammatical study of the Surah and contains sections on types of informative and compositional sentences, auxiliaries, objects, specifications, circumstantial phrases and genitives. The fifth chapter which is the last of this paper attempts to explore rhetorical shades of the Surah and sheds light on rhetoric and associated themes, brings forth examples of eloquence, oratory, commentary, meaningfulness, expressiveness and novelty craft of an expression found in the Surah. I expounded Ilm-ul-Ma'ani, the science of meanings', Semantics that deals with phrase constructions and how meanings change with changes in phraseology. In this area we look into the characteristics of speech composition by virtue of which they conform to the requirements of the occasion. This is considered the most important science of Arabic balaaghah (rhetoric). llm-ul-Bayan a science of language that discusses figures of speech such as similes and metaphors convey shades of meanings i.e., different ways of expressing the various shades of a single word. Ilm-ul-Badi knowledge of rhetoric or sciences of rhetorical figures that reveal the beauty of language and its implications and tend expose the artistic embellishment of speech. In the outset, I thought it to be small topic with limited scope and exposure, but when I started my study and gathered subject material of the paper it happened to be a cosmic subject deeper than oceans teemed with multifold commands, instructions, guidance, varied colours of meanings and linguistics. In the preparation of my thesis, I have gone through the works of classical and modern scholars and concluded my own premises. It rendered me an opportunity to get myself acquainted with beauty of Quran and explore the mystical and epical majesty of its style. I tried to add a line in the vast ocean of writings on the topic, a baton in simile, for the coming generation and clue to new horizons of prognosis. It was a wonderful journey more than Alice in wonder land that not only added to my mental, intellectual and religious knowledge but also triggered a persuasion to expand my voyage beyond the boundaries of thesis.