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Fixed Point Theorems for Various Spaces and Their Applications

Thesis Info

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Author

Shah, Rahim

Program

PhD

Institute

University of Peshawar

City

Peshawar

Province

KPK

Country

Pakistan

Thesis Completing Year

2018

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12788/1/Rahim%20Shah_Math_2018_UoPsw_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726213558

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In this thesis, we prove some fixed point theorems in various spaces with applications. We discuss the uniqueness and existence of a fixed point in a dislocated metric and dislocated quasi metric spaces. Several results of fixed point under different kind of contractive assumptions are introduced that generalize and unify a number of related results in the existing literature. Some important examples are presented to support our main results. We also establish some fixed point theorems in ordered cone b–metric spaces. These new results generalize and unify numerous popular theorems in literature to ordered cone b–metric spaces. Several auxiliary examples are also provided. Some new theorems of fixed point are established in G–metric spaces by using integral type contraction with supporting examples and applications. We also analyze the Ulam type stability to some families of function equations via fixed point theory. In particular, of certain delay Volterra integro–differential equations are established by using a fixed point method in a generalized complete metric spaces. We provide some examples of our established results. After applying the idea of fixed point technique, we examine the Hyers–Ulam stability and the Hyers–Ulam–Rassias stability of impulsive Volterra integral equations in complete generalized metric spaces.
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دکھاں دی پنڈ ہوئی تیار

کھول کے ویکھیں اس نوں یار
اوہناں نوں پنجاب دیاں رسماں دے گھٹ جاون دا وی بڑا دکھ اے۔ لکھدے نے
حقے وی نے گھٹ دے جاندے

ایڈیاں کج ادائیاں ہویاں
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عصر حاضر كی اجتہادی كاوشوں میں مقاصد شریعت كی رعایت

Ijtiha'd is to make utmost endeavor in the light of the basic sources of Islam, the Qur'an and Sunnah to develop the most viable rulings on the new challenges of the time. Muslim scholars have always risen to new challenges. In the contemporary period many Muslim scholars attempted to resolve new problems keeping in view the very objectives of Islamic Law (mtufd al-shari’ah) , which is totally different from literal understanding and application of the rulings of the Qur'an and Sunnah. This article seeks to highlight the contribution of some Muslim scholars in the contemporary period to the application of matpsid al-shari'ah as a principle of ijtihud

Preparation and Evaluation of Legume Based Weaning Foods

Protein energy malnutrition (PEM) is the leading nutritional problem worldwide especially in infants. It refers to the macronutrient deficiency resulting from an inadequate intake of protein and energy. Legumes are locally available, cheaper source of protein and can complement cereals to prepare weaning foods to combat PEM. Among the legumes, cowpea and mungbean have good protein and amino acid profile, but may contain certain anti-nutritional components. Roasting was employed to increase the nutritional profile of the product and to reduce anti-nutritional factors in the product. In the present study, roasted cowpea and mungbean flours were analyzed for anti-nutritional factors (hemagglutinin & trypsin inhibitor activities and phytate content) and were mixed with wheat flour to make composite flour. The proximate analysis of mungbean depicted higher moisture and ash contents (5.93 and 4.02%) as compared to cowpea (3.37 and 3.35%), whereas crude protein, fat & fiber contents in cowpea were higher (23.10, 2.05 & 3.57%) than mungbean (22.45, 1.34 & 3.07%), respectively. The mineral profile explicated that potassium, calcium, sodium, iron and zinc were 28.19, 387.51, 61.20, 78.33 and 45.75 mg/100g in cowpea while 10.46, 7.55, 5.31, 1.28 and 2.17 mg/100g in mungbean respectively. After roasting, phytate contents presented decrement in cowpea and mungbean up to 67.03 and 71.12% as compared to raw counterparts, respectively. Likewise, the percent decrease in hemagglutinin-lectin and trypsin inhibitor activities were 68.85 & 78.47% and 79.81 & 77.00% in roasted cowpea & mungbean as compared to raw versions, accordingly. Amongst composite flours, T0 (100% Wheat flour) had highest moisture content 12.22%, T3 (30% cowpea) showed highest protein content 13.14% while crude fat, crude fiber and ash contents were maximum in T9 (15% cowpea & 15% mungbean) as 1.69, 1.12 and 1.29%, correspondingly. Furthermore, maximum nitrogen free extract (NFE) value was observed in T5 (20% mungbean) as 76.32% and highest water absorption & oil absorption capacity was noticed in T3 1.44 mL/g & T8 (10% cowpea & 10% mungbean) 1.16 mL/g, respectively. Additionally, the emulsion capacity was highest in T8 as 75.75% while emulsion stability was maximum in treatment; T2 (20% cowpea) i.e. 53.33%. Highest foaming stability was observed in T8 as 68.67%, whereas highest foaming capacity (44.74%) was reported in T3. In the product development phase, nine weaning foods were prepared using different ratios of cowpea and mungbean. Three most acceptable weaning food prototypes; W2= Roasted cowpea (20%)+wheat flour (30%)+rice flour (13%)+sugar (10%)+vegetable oil (10%)+milk powder (16%)+vitamin-mineral mix (1%), W5= Roasted mungbean (20%)+wheat flour (30%)+rice flour (13%)+sugar (10%)+vegetable oil (10%)+milk powder (16%)+vitamin-mineral mix (1%) and W8= Roasted cowpea & mungbean (10:10%)+wheat flour (30%)+rice flour (13%)+sugar (10%)+vegetable oil (10%)+milk powder (16%)+vitamin-mineral mix (1%) were selected on the basis of sensory response with overall acceptability scored as 7.67, 7.66 and 7.92 and were renamed as W1S, W2S and W3S, respectively however, Wc= soybean (20%)+wheat flour (30%)+rice flour (13%)+sugar (10%)+vegetable oil (10%)+milk powder (16%)+vitamin-mineral mix (1%) was run as control. Amongst the selected weaning foods, W2S had more moisture content (4.58%) while higher crude protein was found in W3S (17.01%). On the other hand, fat, fiber and ash were highest in WC as 11.11, 4.57 and 2.67% respectively. W3S showed higher loose and packed bulk density values as well as reconstitution index, viscosity and gross calorific value. Regarding growth study parameters, protein efficiency ratio (PER), net protein ratio (NPR) and relative net protein ratio (RNPR) were viewed maximum in W3 as 2.65, 3.34 and 53.18 among the prepared treatments and found relatively comparable with Wc (2.84, 6.28 and 100), correspondingly. In nitrogen balance study, true digestibility (TD), biological value (BV) and net protein utilization (NPU) were highest in W3S as 80.31, 85.09 and 82.67% i.e. quite equivalent to Wc values (91.25, 91.84 & 89.37%), respectively. The acceptability appraisal of prepared infant formulations was conducted through short term infant feeding trails with the help of nursing mothers. The acceptability scores were ranged from 3.18 to 4.07 (out of 5) from day 1 to day 15. Conclusively, the selected legume based weaning foods presented comparable characteristics to that of existing soy based equivalents in terms of protein content, energy value and acceptability test as well as its impact on growth & nitrogen balance aspects.