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Best Proximity Point Results in Fuzzy Metric Spaces

Thesis Info

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External Link

Author

Saleem, Naeem

Program

PhD

Institute

National University of Computer and Emerging Sciences

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2017

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/13693/1/naeem.saleem.phd.thesis.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725572523

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The aim of this research is to investigate some contractions and establish the existence results of single and multivalued mappings in frame work of some abstract space. Banach (1922) presented a fixed point theorem, namely Banach contraction theorem, which states that, a contraction mapping has a unique fixed point in a complete metric space. This theorem is widely applicable in proving the existence of solutions of functional equations under certain conditions. This theorem is based upon the iterative process, so it can be easily implemented on computer. This theorem was a revolution in fixed point theory. Later on , several generalizations of this theorem have been obtained. Fixed point results are widely studied in the framework of topological, metric and ordered oriented spaces. It is interesting to study that a nonlinear functional equation T(x) = x (where T is some linear or nonlinear operator defined on metric or fuzzy metric space) has either a solution or possess no solution. If the nonlinear functional equation has any solution, then it is interesting to find some algorithms which lead to an appropriate solution, but if nonlinear functional equation has no solution, then to try and find some approximate solution. The approximation, optimization and fixed point theory incorporate the main theme of nonlinear analysis. The best approximation theory is applicable in variety of problems arising in nonlinear functional analysis. Banach (1922) fixed point theorem is vastly applicable in proving the existence of solutions of functional equations under certain conditions. Many authors generalized Banach contraction principle in various directions. To address the question, if a nonlinear functional equation has no solution, it is recommended to approximate the solution which solves the optimization problem such that d(x, T x) is minimum. Ky Fan’s (Fan (1969)) best approximation result has been used when some nonlinear functional equations have no solution. In this thesis, best proximity point results in non-Archimedean fuzzy metric space are proved and some optimal best proximity point results are also obtained. These results provides the existence of optimal approximate solutions to some equations which have no solution. Further, fuzzy optimal coincidence point results for different proximal contractions in the framework of complete non-Archimedean fuzzy metric space are obtained. These results also holds in fuzzy metric spaces when some mild assumptions are added in the domain of involved mappings. In the next part of thesis, best proximity point results in a complete non-Archimedean fuzzy metric space with ordered structure have been discussed. Further, a class of multivalued mappings is introduced which satisfies Suzuki type generalized contractive condition in the framework of fuzzy metric spaces and some fixed point results are obtained for such kind of mappings. In next part of this thesis, Suzuki type contraction conditions are further generalized as Suzuki type F−contraction fuzzy mapping in ordered metric spaces. As an application, a common fixed point result for hybrid pair of single and multivalued mappings, the existence and uniqueness of common bounded solution of functional equations arising in dynamic programming are obtained. The obtained results generalize and extend various results in the existing literature. In each chapter, some examples are provided, which shows the validity of the results along with couple of remarks about the comparison of obtained results with the existing ones in the literature.
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المبحث الرابع: شعرها

المبحث الرابع: شعرها

 حصلت الشاعرۃ علی فن الشعر منذ صغرھا، وخاصۃً من أجدادها لأمها، وحسن عسکري لہ الدور الکبیر في تربیۃ بروین شاکر وتھذیبھا من الناحیۃ الأدبیۃ کانت الشاعرۃ تحب ان تقرأ من الشعر أو ما شابہ ذلک من الجرائد والمجلات وکانت تحب ان تسمع القصائد والأغاني في الرادیو وکانت تحب ان تُردد خلف الأغاني والاشعار۔ وعندما وصلت مرحلۃ الثانویۃ تقابلت مع الشاعرۃ المعروفۃ عرفانہ عزیز، فقامت عرفانہ بتربیۃ بروین من الناحیۃ الأدبیۃ والشعریۃ وتھذیب ألفاظھا، ومنحتھا ببعض النصائح والإرشادات التي أفادتھا في مستقبلھا۔ وأول عمل شعري لبروین شاکر کان في 6 - سبتمبر وکانت قصیدتھا الأولیٰ المعنونۃ ب(صبحِ وطن) ۔ (صباح الوطن) وکانت حول 6- سبتمبر، ویقول الدکتور ناظم جعفري عن بروین بأنھا تربت وترعرعت تحت رعایۃ حسن عسکري[1].

 تعرفت الشاعرۃ علی أحمد ندیم قاسمي عن طریق منتج رادیو پاکستان یاور مھدي ویقول أحمد ندیم قاسمي وصلتني رسالۃ من شاعر شاب عبیداﷲ علیم في 1971م وفیہ بعض أشعار لبروین شاکر ویقول ھذا الشاب عن بروین شاکر أنھا شابۃ بارعۃ ولھا قصائد رائعۃ، فیقول أحمد ندیم عندما وصتلني أشعارھا وبعد قراءتھا صححتُ بعض الأخطاء البسیطۃ ثم تم نشرها في 1971م[2]، وساعدھا الأستاذ القاسمي في المھارۃ الفنیۃ والشعریۃ وأعطاھا الکثیر من النصائح والإرشادات المھمۃ التي ساعدتھا في براعتھا وتھذیب لُغۃ الشعر وتطور أسالبیھا الشعریۃ۔

 



[1] الدکتور ناظم جعفري، خوشبو کی ہمسفر، ص78ـ79

[2] أحمد ندیم قاسمي، خوشبو کی ہمسفر، ص17۔

عظیم شخصیات کے اوصاف وخصائص: سیرت طیبہ کی روشنی میں

Great lineage, good manners, trustworthiness, sincerity, courage and bravery, obedience, generosity, modesty, humility, thankfulness, absolute trust in Allah, education and upbringing, affection and compassion towards the elderly, the youth, slaves and animals, the regard for truth, the endurance and patience etc. Are such features that at them neither was the match for the Holy Master (saw) born, nor will any such person be ever born. In addition, let alone friends, his enemies were helpless to hold him as-Sadiq (the Truthful) and al-Ameen (the Trustworthy) due to these and many other great attributes; and they would say: Muhammad is not a liar at all, but we can’t believe in him as a prophet sent by Allah. Moreover, even a non-Muslim or polytheist has never criticized his character to this day, nor will anybody do till the judgment day. Whenever the greatest personalities of the world were listed or somebody committed to writing on the said topic, the Holy Prophet (saw) remained at the top of the list. Drawing of the swords of father and son against each other, the change of the standard of friendship and enmity, fighting of master and his slave against each other, chanting the slogan of Ahad-Ahad (the only One, the only One) on the blazing sand, giving priority to martyrdom over the utterance of a word of unbelief, taking a husband away from his wife and separating a mother from her infant, deforming the condition after a blasphemous blow on the blessed face, forced migration to Habsha (Abyssinia) at times, and laying siege to Shb-e-Abi Talib (the valley of Abi Talib) at times.

Synthesis and Bioactivity Studies of Various 2, 5-Disubstituted-1, 3-4-Oxadiazole and 3, 4, 5-Trisubstituted-1, 2, 4-Triazole Molecules Having Azinane Core

The understanding of universe has also been decorated efficiently by the chemistry like the other sciences. The organic chemistry has launched the tentative challenges in the broad spectrum to understand the chemistry of life. A chemist facilitates the humanity in all the disciplines of life especially in the field of health and care based on the pharmacological efficiencies. Here, we do not aim at discussing the skeleton of chemical sciences but actually we want to equip the thinking to realize the demands of organic chemistry. An organic chemist is always in attempts to design unique synthetic molecules or to extract natural products to quench his thirst for the study of interaction of these molecules with life. Even from a common observer it is evident that the running medicines have entered in the inefficient process from therapeutics point of view because of development of resistance and tolerance by the threatening agents. The current time extremely claims the synthetic chemists to design, discover and invent more potent therapeutic compounds to ensure the well-being, health, care and happiness of humanity like the other advancements on this sphere. The literature survey of synthetic chemistry is witness for the need of more potent and biologically active compounds. This is the motivational force which has compelled us to design heterocyclic compounds having 1,3,4-oxadiazole, 1,2,4-triazole and azinane with minimum cost, better yield and active pharmacological applications. Based on the applicability of these compounds, placement of wide variety of substituents has been designed to evaluate them for their pharmacological profile against different enzymes (acetyl cholinesterase, α-glucosidase and urease), various bacterial strains (S. typhi (-), E. coli (-), P. aeruginosa (-), S. aureus (+) and B. subtilis (+)) supported by the molecular docking to understand their active sites responsible for their pharmacological profile. BSA binding studies were also in progress parallel to the other investigations to check the binding constant which in turn justifies the pharmacodynamics and efficiency of designed drugs. The current research was organized in twelve schemes to design unique, biologically active compounds. The first scheme was furnished with the synthesis 5-(1-(4- chlorophenylsulfonyl)piperidin-4-yl)-1,3,4-oxadiazole-2-thiol (5a) and 5-(1-(4- nitrophenylsulfonyl)piperidin-4-yl)-1,3,4-oxadiazole-2-thiol (5b) by the moieties of 4- chlorophenylsulfonyl chloride (1a) and 4-nitrophenylsulfonyl chloride (1b) treated with ethyl piperidine-4-carboxylate (2) to generate ethyl 1-(4-(chloro/nitro)phenyl xiii sulfonyl)piperidine-4-carboxylate (3a-b). Ethyl 1-(4-(chloro/nitro)phenylsulfonyl) piperidine-4-carboxylate (3a-b) was treated with hydrazine monohydrate to produce 1-(4-(chloro/nitro)phenylsulfonyl)piperidine-4-carbohydrazide (4a-b) respectively. Carbohydrazides were finally converted into their respective 1,3,4-oxadiazoles. A series of 27 N-substituted-2-bromoacetamides (10a-z, 10aa) (scheme 3) and a series of 17 N-substituted-2-bromopropanamide (15b, 15c, 15e-g, 15j, 15m, 15o-t, 15v-x, 15aa) (scheme 7) were synthesized in the aqueous medium by the reaction of 2- bromoacetyl bromide (9), 2-brompropionyl bromide (14) and different substituted/ unsubstituted alkyl/aralkyl/phenyl/aryl amines. Both 5-(1-(4-chlorophenylsulfonyl) piperidin-4-yl)-1,3,4-oxadiazole-2-thiol (5a) and 5-(1-(4-nitrophenylsulfonyl) piperidin-4-yl)-1,3,4-oxadiazole-2-thiol (5b) were treated with alkyl/aryl/aralkyl halides (6a-z) to synthesize twenty six 2-(alkyl/arylthio)-5-(1-(4-chlorophenyl sulfonyl)piperidin-4-yl)-1,3,4-oxadiazole (7a-z) (scheme 2) and fifteen 2-(alkyl/aryl thio)-5-(1-(4-nitrophenylsulfonyl)piperidin-4-yl)-1,3,4-oxadiazole (12b-g, 12i, 12k-n, 12p, 12r, 12y, 12aa) (scheme 5) respectively. Twenty six 2-(5-(1-(4-chlorophenyl sulfonyl)piperidin-4-yl)-1,3,4-oxadiazol-2-ylthio)-N-(substituted) acetamide (11a-z) (scheme 4) and sixteen 2-(5-(1-(4-nitrophenylsulfonyl)piperidin-4-yl)-1,3,4- oxadiazol-2-ylthio)-N-(substituted)acetamides (13b, 13e-g, 13j, 13m, 13o, 13r-v, 13x-z, 13aa) (sheme 6) were synthesized by treatment of different N-substituted-2- bromoacetamides (10a-z, 10aa) (scheme 3) in the presence of DMF with 5-(1-(4- chlorophenylsulfonyl)piperidin-4-yl)-1,3,4-oxadiazole-2-thiol (5a) and 5-(1-(4-nitro phenylsulfonyl)piperidin-4-yl)-1,3,4-oxadiazole-2-thiol (5b) respectively. By the reaction of N-substituted-2-bromopropanamide (15b, 15c, 15e-g, 15j, 15m, 15o-t, 15v-x, 15aa) (scheme 7) and 5-(1-(4-nitrophenylsulfonyl)piperidin-4-yl)-1,3,4- oxadiazole-2-thiol (5b), twelve 2-(5-(1-(4-nitrophenylsulfonyl)piperidin-4-yl)-1,3,4- oxadiazol-2-ylthio)-N-(substituted)propanamides (16b, 16e-g, 16j, 16m, 16q, 16s-t, 16v, 16w, 16aa) (scheme 8) were synthesized. Scheme 9 was based on the synthesis of 5-(1-(4-nitrophenylsulfonyl)piperidin-4-yl)-4-phenyl-4H-1,2,4-triazole-3-thiol (19) through the reaction of 1-(4-nitrophenylsulfonyl)piperidine-4-carbohydrazide (4b) and phenylisothiocyanate (17) in the presence of ethanol through the formation of an intermediate 2-(1-(4-nitrophenylsulfonyl)piperidine-4-carbonyl)-N-phenylhydrazine carbothioamide (18) product which was cyclized into aimed product 5-(1-(4- nitrophenylsulfonyl)piperidin-4-yl)-4-phenyl-4H-1,2,4-triazole-3-thiol (19) of scheme 9. 5-(1-(4-Nitrophenylsulfonyl)piperidin-4-yl)-4-phenyl-4H-1,2,4-triazole-3-thiol (19) xiv was reacted at room temperature with equimolar quantities of alkyl/aryl/aralkyl halides (6b-e, 6g-j, 6l, 6o-p, 6r, 6t, 6x, 6z, 6aa, 6bb), N-substituted-2-bromo acetamides (10a, 10c-g, 10j, 10m, 10o-p, 10r-t, 10v, 10x-z, 10aa) (scheme 3) and Nsubstituted- 2-bromopropanamide (15c, 15f-g, 15j, 15m, 15o-s, 15v-x) (scheme 7) to synthesize seventeen 4-(5-(substituted)thio)-4-phenyl-4H-1,2,4-triazol-3-yl)-1-(4- nitrophenylsulfonyl)piperidine (20b-e, 20g-j, 20l, 20o-p, 20r, 20t, 20x, 20z, 20aa, 20bb) (scheme 10), eighteen N-(substituted)-2-(5-(1-(4-nitrophenylsulfonyl)piperidin- 4-yl)-4-phenyl-4H-1,2,4-triazol-3-ylthio)acetamides (21a, 21c-g, 21j, 21m, 21o-p, 21r-t, 21v, 21x-z, 21aa) (scheme 11) and fourteen N-(substituted)-2-(5-(1-(4- nitrophenylsulfonyl)piperidin-4-yl)-4-phenyl-4H-1,2,4-triazol-3-ylthio)propionamides (22c, 22f-g, 22j, 22m, 22o-s, 22v-x) (scheme 12) respectively. The whole library of synthesized compounds was spectroscopically characterized by using IR, 1H-NMR, 13C-NMR and EIMS spectral information to justify the available main functional groups, hydrogen atoms, carbon atoms and the fragmentation pattern of the structures of synthesized compounds. All the synthesized compounds were screened against five different bacterial strains in order to judge their antibacterial potential and almost half were found active. Compounds of current research were also subjected to check their anti-enzymatic potential against AChE, α-glucosidase and urease enzyme. Almost all the compounds were found to be excellent active agents against these enzymes. Anticancer and antiinflammatory activities of all the synthesized molecules were also tested in search of some unique drug candidates but unluckily no compound was found active against these activities. The chemistry of active sites and different functionalities responsible for the best pharmacological potential of all the synthesized compounds was verified through docking studies. In addition to it, the evaluation of protein drug interaction assisted us in understanding the various binding sites and binding constant to justify the stay of the drugs in the body, their circulation, metabolism, elimination and pharmacodynamics. The sketched library of the compounds in the twelve various schemes were synthesized efficiently with high yield and purity through environment friendly protocol with minimum cost and time. The following synthetic as well as biological screening studies resulted in the identification of a list of compounds (54) with broad spectrum of biological and pharmacological applications. These compounds may be admitted by the pharmacological world as new unique cost effective and human friendly therapeutic agents for the betterment of humanity." xml:lang="en_US