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Variational Inclusions Involving Difference of Operators

Thesis Info

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Author

Kamal, Rabia

Program

PhD

Institute

COMSATS University Islamabad

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/123456789/13622/1/Rabia_kamal_Maths_2017_CIIT_HSR_14.04.2017.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727657740

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A new class of general variational inclusions involving difference of operators has been investigated. Using resolvent operator and resolvent equations technique, it is shown that general variational inclusions and fixed point problems are equivalent. Using this equivalent formulations, we prove the existence of a solution of general variational inclusion. Iterative schemes and its convergence is also discussed. We study some dynamical systems associated with variational inclusion. The dynamical system is used to discuss the existence of the variational inclusion using Lipschitz continuity only. Some Merit functions are constructed for the variational inclusions. Using these merit functions, some error estimates are obtained under suitable assumptions. Sensitivity analysis is analyzed using the resolvent equations. Several classes of variational inclusion and optimization problem are discussed which can be obtained as special cases from our results.
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Zulfikar Ali Bhutto: The Psychodynamics of his Rise and Fall by Shamim Ahmad

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Management of Growth, Productivity and Quality of Kinnow Mandarin Citrus Nobilis Lour. X Citrus Deliciosa Tenora. Through Exogenous Application of Moringa Moringa Oleifera L. Leaf Extract

„Kinnow‟ mandarin is the leading citrus commercial cultivar of Pakistan; however, it bears 2-3 times lower yield than its potential. Poor tree health and nutrient deficiency are considered to be major causes of its low yield. Moringa leaf extract (MLE) being rich in nutrients, antioxidants (such as ascorbic acid), phenolics and growth hormone proved to be potential plant growth enhancer that improves growth and fruit quality. Thus, the aim of the the present study was to a) optimize dose and time for foliar application of MLE, b) optimize dose of ascorbic acid, c) comparison of MLE with growth promoters [benzylaminopurine (BAP) and ascorbic acid] and nutrients [potassium (K) and zinc (Zn)] that affect growth, production and fruit quality of „Kinnow‟ mandarin trees. To find out optimize dose of MLE various concentrations (2%, 3%, 4% and 5%) were sprayed after fruit harvesting. To find out the best suitable time for foliar application of MLE different growth stages selected, were before flowering, flowering, fruit set stage (pea stage), before flowering + fruit set stages, premature stage and flowering + fruit set + premature stages; to find out optimize dose of ascorbic acid different concentration of ascorbic acid (100 mg L-1, 200 mg L-1, 300 mg L-1 and 400 mg L-1) were sprayed after fruit harvesting. In first experiment leaf K, phosphorus (P), Zn, manganese (Mn), chlorophyll b and ascorbic acid contents were increased by 1.5-fold, 1.18-fold, 1.1-fold, 1.18-fold, 1.75-fold and 2.3-fold with 3% MLE treatment compared to control, respectively. In second experiment highest leaf K, calcium (Ca) and Zn contents were found in trees treated with 3% MLE at premature stage. However, maximum leaf P, Mn, chlorophyll a and b contents were exhibited by trees treated with 3% MLE at before flowering + fruit set stages. Foliar application of 3% MLE increased trunk width and leaf age as compared to untreated control trees. Yield and number of marketable fruits were increased in trees treated with 3% MLE at premature stage. Maximum fruit weight, fruit size, juice weight, ascorbic acid, total sugars and non-reducing sugars were observed in trees treated with 3% MLE applied before flowering + fruit set stages. Total phenolic contents were highest in trees treated with 3% MLE at premature stage. Therefore, it can be concluded that 3% MLE at premature stage increased leaf nutrient and yield, whereas, 3% MLE at before flowering + fruit set stages improved fruit quality of „Kinnow‟ mandarin. In third experiment leaf K, P, Zn, Fe and chlorophyll a concentration were significantly increased after foliar application of 300 mg L-1 of ascorbic acid as compared to untreated trees. Highest Abstract 2 chlorophyll b contents were determined with 200 mg L-1 of ascorbic acid treatment and leaf ascorbic acid contents with 400 mg L-1 ascorbic acid treatment. Hence, it may be concluded that 300 mg L-1 of ascorbic acid is best suited dose for foliar application to „Kinnow‟ mandarin trees. A fourth experiment was carried out to check the effect of MLE (3% MLE at before flowering + fruit set stages and 3% MLE at premature stage), ascorbic acid (300 mg L-1) and BAP (800 mg L-1) on „Kinnow‟ mandarin growth, yield and fruit quality. Results revealed that highest leaf nitrogen (N), K, P and Mn contents were found with application of 3% MLE at premature stage. Highest increase in tree height, trunk width and tree diameter were found with foliar application of 800 mg L-1 BAP at fruit set stage. Maximum fruit set, yield, fruit weight, fruit size, juice weight, soluble solids content (SSC), total sugars and non-reducing sugars were recorded in 3% MLE at premature stage treated trees. Highest total number of fruits and number of marketable fruits were found with 3% MLE at before flowering + fruit set stages treatment. Maximum ascorbic acid and total phenolic contents in fruit juice were found with foliar application of 300 mg L-1 ascorbic acid. Hence, it can be concluded that foliar application of 3% MLE at premature stage increased nutrient content, yield and fruit quality of „Kinnow‟ mandarin trees. In fifth experiment; trees were sprayed with 3% MLE at before flowering + fruit set stage, 3% MLE at premature stage and 0.6% Zn or 0.25% K alone and in combination with MLE at fruit set stage, respectively. Leaf N, P, K, Ca, Mn and Zn were significantly increased with all treatments. Less fruit drop, maximum fruit set, yield, fruit weight, juice weight, SSC, vitamin C, sugars, total antioxidants and total phenolics were recorded with combined application of 3% MLE, 0.25% K and 0.6% Zn at fruit set stage. Activities of superoxide dismutase (SOD) and catalase (CAT) enzymes in fruit juice were significantly increased with 3% MLE application. Hence, it can be concluded that foliar application of 3% MLE+ 0.25% K + 0.6% Zn at fruit set stage can effectively improve the growth, productivity and fruit quality of „Kinnow‟ mandarin. In the sixth experiment, the best treatments from the above experiments were selected i.e., 3% MLE at pre-mature stage, 300 mg L-1 ascorbic acid at fruit set stage and 3% MLE + 0.25% K + 0.6% Zn at fruit set stages. Results showed that leaf nutrient level (N, P, K, Zn, Mn and Fe), ascorbic acid contents, fruit set, yield and number of fruits were significantly increased with combined application 3% MLE, 0.25% K and 0.6% Zn. Highest fruit size and fruit weight was obtained with 3% MLE. Maximum fruit juice, SSC, pH, ascorbic acid and total phenolic contents were Abstract 3 obtained with combined application of 3% MLE, 0.25% K and 0.6% Zn at fruit set stage. Highest total antioxidants, SOD enzyme and CAT enzyme were determined with 3% MLE at premature stage. Maximum peroxidase (POD) activity was obtained with ascorbic acid treatment. Conclusively, combined foliar application of 3% MLE, 0.25% potassium sulphate and 0.6% zinc sulphate at fruit set stage can be used effectively to improve the leaf nutrient status, fruit yield and quality of „Kinnow‟ mandarin.