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

Thesis Info

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Author

Zahid Akhtar

Program

PhD

Institute

The Islamia University of Bahawalpur

City

Bahawalpur

Province

Punjab

Country

Pakistan

Thesis Completing Year

2016

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/13389/1/Zahid_Akhtar_Maths_2016_HSR_IUB_Bahawalpur_28.08.2017.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726093149

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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.
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دولتِ دردِ یار مل جائے

دولتِ دردِ یار مل جائے
زندگی کو قرار مل جائے

عشق پابندِ رنگ و ذات نہیں
دل کا جس سے بھی تار مل جائے

وقتِ رخصت ہے اب خدا کے لیے
ہم سے وہ ایک بار مل جائے

اُس کا ملنا خزاں رسیدہ کو
جیسے فصلِ بہار مل جائے

درد مندوں کو رشک ہو تائبؔ
درد یوں بے شمار مل جائے

جنین کا عصری اور شرعی تناظر میں تحقیقی جائزہ

Allah (SWT) dignified human beings among all the creatures and made these (creatures) all subjugated to men. Allah says in the Holy Qur’ān: (He is such a Lord who has created all the things in the universe for you). The beginning of human generation is traced back to the miraculous birth of Adam (A.S), and Allah entrusted their survival in their physical bodies in the process of fusing the male and female gametes in the womb of mother. And new offspring are born undergoing many developmental stages. The Holy Qur’ān has mentioned all the stages of the human being like: its creation started from dust, then from seamen, then a clot, then from a flesh (Al- Qur’ān, 22: 5). The same are described by modern medical sciences. The medical sciences also describes it in details that how a “fetus” is created and the process which it passes through. This article discusses how the “Fetus” is described in Islamic Sources, modern medical sciences and what are the similarities between these two?

Effect of Bacillus Subtilis and Sodium Butyrate on the Morphometry of the Small Intestine and Immune System in Healthy and Salmonella-Challenged Broiler Chickens

Supplementation of Bacillus subtilis and microencapsulated sodium butyrate in the feed is being practiced as a substitute for antibiotics growth promoters. An expansive range of encouraging health-related properties exhibited by B. subtilis and SB has been published, but their exact effect on gut and immune system is not completely understood. Consequently, the evaluation of B. subtilis and SB as feed supplements is desired. To achieve this goal, the present study was aimed to investigate the effects of B. subtilis and SB on performance, immune system, gut and lymphoid organs microarchitecture in healthy and Salmonella-challenged broiler chickens. In the first experiment the research was targeted to investigate the effects of B. subtilis on performance, immune system, gut and lymphoid organ microarchitecture in broilers. A total of 120 d-old broiler chicks were randomly distributed into four groups, each group with three replicates containing 10 birds per replicate. The birds were fed a corn-soy-based basal diet (BD, control) or BD supplemented with 10% zinc bacitracin (ZnB), and 0.05g/kg or 0.1g/kg of B. subtilis, respectively. On d 21 and 35, six birds from each group were killed to collect blood and visceral organs (thymus, spleen, bursa of Fabricius, liver and small intestine). Parameters evaluated included growth performance, immune responses, relative organ weights, lymphoid organs and gut mucosal morphometry, intraepithelial lymphocytes (IEL) count and goblet cell histochemistry in mucosa. Results showed that the group fed 0.1g/kg of B. subtilis had superior (P < 0.05) mean body weight and weight gain, and lower FCR compared to the nonsupplemented or ZnB-fed groups. The BS-0.1 group revealed higher antibody titer against Newcastle disease (ND) virus and the supplemented groups against sheep RBCs (SRBCs) on d 35. Cell-mediated immune response post-phytohemagglutinin-P injection was attained (P < 0.05) by birds in the BS-0.1 group at 24 h, and by both the BS-0.1 and BS-0.05 groups at 48 and 72 h compared to the ZnB and control groups. The BS-0.1 group gained higher (P < 0.05) relative bursal weight on d 21 compared to the other groups. Compared to the control group, the liver, spleen and thymus weighed more (P < 0.05) in the experimental groups on d 35. The histomorphological study revealed increased (P < 0.05) thymus cortical width, and cortex/medulla ratio in the BS-0.1 group compared to the control. The area of the bursal follicles and germinal centers of the spleen also improved (P < 0.05) in the BS-0.1 group compared to the control. Compared to the ZnB and control, higher (P < 0.05) villus height, villus surface area and villus crypt ratio of the duodenum and jejunum were recorded on d 21, and higher (P < 0.05) villus height of the duodenum and ileum was noted on d 35 in the BS-0.1 and BS-0.05 groups. The number of goblet cells having acid mucin was significantly higher in the ileal mucosae of the BS-0.1 group chickens compared to the ZnB and control. In conclusion, B. subtilis type probiotics effectuated better growth performance, improved immune system and modulated morphology of lymphoid organs and gut mucosa in broilers. The second experiment was carried out to evaluate the effects of sodium butyrate on growth performance, immune status, organ weights and the microarchitecture of lymphoid organs and the small intestine compared to the effects brought about by an antibiotic. The cell-mediated immune response at 48 h post-phytohemagglutinin-P injection, and antibody titer against NDV and sheep RBCs on d 35 was higher (P < 0.05) in SB-1 chicks compared to those in the ZnB and control groups. Higher (P < 0.05) weight gain, and lower (P < 0.05) FCR were attained by the supplemented groups compared to the control. The thymus and spleen weighed more (P < 0.05) in the SB-1 group and bursa registered more (P < 0.05) weight in both SB groups compared to the control. On d 21, areas of the thymus medulla and the spleen germinal centers were larger (P < 0.05) in SB-1 chicks compared to ZnB and control chicks. The VH and VSA increased (P < 0.05) in the duodenum and jejunum in both SB groups on d 21, and in SB-1 on d 35 compared to the ZnB and control groups. The villus to crypt ratio was higher (P < 0.05) in the duodenum in SB-1 chicks compared to ZnB and control chicks. On d 35, VH in all segments and VSA in the duodenum and jejunum increased (P < 0.05) in SB-1 chicks compared to ZnB and control chicks. Statistically, IEL count was not significant among supplemented groups. On d 21, the number of goblet cells containing acidic mucin increased (P < 0.05) in all the segments of the small intestines in the SB-1 group compared to the control group and on d 35 in the ileum compared to the other groups. In conclusion sodium butyrate elicited better growth performance, improved immune system and modulated the morphology of lymphoid organs and the gut mucosa in broiler chickens. The third experiment was focused to assess the effect of B. subtilis and SB on gut development, growth performance and immune system in broilers challenged with S. Gallinarum. Better growth performance was reported in the supplemented groups compared to the NC-S group due to better feed efficiency. The B. subtilis-supplemented group exhibited higher (P < 0.05) cellular immunity and antibody titer against NDV compared to the PC-S and NC-S groups. Furthermore, B. subtilis- and SB-supplemented groups reflected higher (P < 0.05) relative thymus and bursa weights, and improved microarchitecture of the lymphoid organs compared to the NC-S group. On d 21, villus surface area in the jejunum and ileum increased (P < 0.05) in sodium butyratetreated birds. The crypt depth of the jejunum decreased (P < 0.05) in B. subtilis and sodium butyrate groups compared to NC-S and PC-S groups. On d 35, the villus height, villus surface area and VH:CD ratio of the duodenum increased (P < 0.05) in the supplemented groups compared to the NC-S group. The FCR, Salmonella population in ceca and mortality were higher (P < 0.05) in the NC-S group. In conclusion, the prophylactic use of the B. subtilis probiotic and SB alleviated stress associated with Salmonella Gallinarum infection and improved performance, immune function, lymphoid organs and gut mucosal development in infected broilers. Further analyses are needed to reveal the mechanism(s) by which B. subtilis and sodium butyrate produce such effects.