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Metric Fixed Point Theorems for Single-Valued and Multi-Valued Mappings

Thesis Info

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Author

Jamshaid Ahmad

Supervisor

Akbar Azam

Program

PhD

Institute

COMSATS University Islamabad

City

Islamabad

Province

Islamabad.

Country

Pakistan

Thesis Completing Year

2015

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/785

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726636638

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Metric Fixed Point Theorems for Single-valued and Multi-valued Mappings The main goal of the present research is to establish existence and uniqueness theorems regarding fixed points, coincidence points and common fixed points of single-valued and multi-valued mappings in connection with contractive type inequalities in complete metric spaces and generalized metric spaces. Some notions namely generalized α*- Mizoguchi-Takahashi type contraction, generalized Mizoguchi-Takahashi G contraction and α-admissible mappings for cone metric space are also obtained. We also include an application in which we prove the existence and uniqueness of a solution for a general class of Fredholm integral equation of 2nd kind. To enhance the validity of this work some interesting examples are also furnished.
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البــاب الثالث: دراسة الأدب القراني الكريم في آثار مصطفی صادق الرافعي

الفصــل الأول: اعجاز القرآن وبلاغة النبوية

أن الألفاظ القران منزلة بحروفها ونسقها والابطل الاعجاز لأن الاعجاز لا يكون انسانيا وقد كان الوحي ينزل علي النبي صلي الله عليه وسلم فتعتريه حالة روحية وردت صفتها في البخاري و غيره.

قال سعد باشا زغلول في رسالته الذي كتب إلی الرافعي عن تقريظ كتاب اعجاز القران:

تحدي القران أهل البيان في عبارت قارعة محرجة ، ولهجة واخزة مرغمة أن يأتو بمثله أو سورة منه ، فما فعلوا ، ولو قدروا ماتأخروا لشدةحرصهم علي تكذيبهم و معارضته بكل ما ملكت أيمانهم واتسع له امكانهم ،هذا العجز الوضيع بعد ذاك التحدي الصارخ هو أثر تلك القدرة الفائقة ، وهذا السكون الذليل بعد ذاك الاستفزاز الشامخ هو أثر ذلك الكلام العزيز فجاءكتابكم ’’اعجاز القران ‘‘مصدقا لاياتها مكذبا لانكارهم وأيد بلاغة القران واعجازها بأدلة مشتقة من أسرارها في بيان مستمد من روحها ’’كانه تنزيل من التنزيل أو قبس من نور الذكر الحكيم‘‘([1])

وقال أيضاً عن تقريظ اعجاز القران والبلاغة النبوية شبَّه فيه أسلوب المؤلف بالتنزيل الحكيم، وهو تشبيه يصور حقيقة كبيرة، فإن الرافعي يتأثر في نثره العبارة القرآنية في بلاغتها وسموها.([2]) أن القرآن معجزة هذه في بلاغة نظمه واتساق أوضاعه وأسراره، فمن ثم كانت مادة الاتصال في نسق التإلیف بين هذا الجزء والذي قبله. القرآن الكريم مما يتعلق بلغته ويتصل ببلاغته ويكشف عن أوجه الإعجاز في ذلك، لا ننفذ في غير سبب لما نحن بسبيله، ولا نذهب في الكلام عن نتيجة من نتائجه. ولقد أراد الله أن لا تضعف قوة هذا الكتاب، وأن لا يكون في أمره على تقادم الزمن خَضغ أو تَطامن، فجاءت هذه القوة فيه بأسبابها المختلفة على مقدار ما أراد، وهي قوة الخلودالأرضي التي خرج بها القرآن مخرج الشذوذ الطبيعي، فلا سبيل عليه ليد الزمن وحوادثه مما تُبليه أو تستجدُّه، إنما هو روحٌ من أمر الله تعالى هو...

عصرحاضر کے تقاضوں کے تناظر میں جامعات دینیہ کا قضیہ: عملی تجاویز

The Muslim religious seminaries (Jameaat-i-diniyya/dini madaris) have become a theme of the global academic agenda, particularly in the wake of the rise of political Islam and the Afghan resistance against Soviet invasion. The theme continuously looms large and has attracted reputable scholars to address the issue in a critical manner. In the present article the author describes various aspects of madrassa education and suggests a number of workable solutions including a new curriculum under the auspices of the Madrassa Education Board and the Higher Education Commission (HEC) in consultation with the traditional madaris and university scholars.

Biomass Derived Adsorbents for Chromium Vi Treatment from Industrial Wastewater

Environmental concerns are being aggravated with the industrial progress of the country. Untreated wastewater disposal into the aquatic bodies pose a serious environmental threat through heavy metals‘ contamination of food chain, especially the chromium (Cr(VI)). This metal is well known for carcinogenicity and mutagenicity in humans. Inexpensive wastewater treatment technologies with local input of raw materials can be an optimal choice for large scale application for the industries of the developing countries, like Pakistan. Therefore, the major objective of the proposed research was to explore and develop the efficient adsorbents from the locally available biomass resources for Cr(VI) removal from aqueous solution. For this purpose, plenty of waste biomasses were collected, processed and subjected to batch screening for the respective Cr(VI) treatment. All the screened adsorbents, i.e., Eucalyptus camaldulensis sawdust (ECS), Eucalyptus bark, white cedar sawdust and tobacco stem showed maximum Cr(VI) removal efficiency at pH 2. The potential and an abundantly available precursor waste adsorbent, i.e., ECS, was converted to highly efficient (≥ 80% removal) activated carbon (AC-ECS) through two stage phosphoric acid activation (method V). A mesoporous activated carbon, i.e., AC-ECS was developed successfully and it displayed highly efficient Cr(VI) removal of 87% at pH 3. The morphological changes were detected through scanning electron microscopy. An increase in surface area of AC-ECS was noticed i.e., 217 m2 g-1 than its precursor biomass of ECS, i.e., 40 m2 g-1 using Brunauer–Emmett–Teller analysis. The evidence of Cr(VI) ions adsorption was confirmed through shift of wave number of the various surface functional groups and the appearance of crystalline peaks using fourier transform infrared spectroscopy, energy dispersive X-ray spectroscopy and X-ray diffraction analyses respectively. Cr(VI) adsorption data of AC-ECS exhibited the best fit to Langmuir isotherm (R2 = 0.999) revealing monolayer adsorption than its precursor biomass, i.e., ECS, where Cr(VI) adsorption data displayed best fit to Freundlich isotherm (R2 = 0.992) showing multilayer adsorption. Kinetic studies demonstrated the chemisorption as the rate limiting step. Thermodynamic studies exposed the non-spontaneous nature of reaction in non-activated adsorbents, while only AC-ECS displayed spontaneous reaction on the basis of Gibbs free energy. Fixed bed column studies of both AC-ECS and its precursor ECS showed that the exhaust time reduced with an increase of solution flow rate and contaminant concentration, whereas an inverse trend was observed with increasing bed height. ECS and AC-ECS packed columns indicated the increased breakthrough time from 670-1270 min and 5595-12270 min with respect to bed height (5 to 15 cm), respectively. However the increasing flow rate (5 to 15 mL min-1) and contaminant concentration (50-90 mg L-1) resulted the decrease in breakthrough time from 670-20 min and from 670-90 min for ECS packed column, respectively and from 5595-1330 min and 5595-800 min in case of AC-ECS packed column, respectively. Column study confirmed that AC-ECS packed columns were found more effective for Cr(VI) removal than ECS, as the AC-ECS exhibited the high exhaust and breakthrough times than ECS. Economic feasibility showed that total cost for the preparation of 1 kg non-activated ECS was approximately 4.4 PKR, while the cost of phosphoric acid activated carbon, i.e., AC-ECS prepared in the lab was found to be 4493 PKR. 1 kg of AC-ECS can treat approximately 1562 L of wastewater containing Cr(VI), which is larger than ECS (1388 L). It can be concluded that Eucalyptus camaldulensis sawdust is a good adsorbent for Cr(VI) adsorption among other tested non-activated forms of biomasses, and this is a good lignocellulosic material for the development of highly efficient activated carbon, i.e., phosphoric acid activation. The study recommends to assess the Cr(VI) adsorption performance of both ECS and AC-ECS in multiple solutes mixtures and to uncover the interference of other salts or metal ions through further batch and column studies at pilot scale and ultimately for the launch of these adsorbents for field scale treatment of wastewater.