Perceived Online Social Support, Internet Addiction, and Psychological Wellbeing among University students

This study aims to explore the relationship between perceived online social support, internet addiction, and psychological well-being among university students. A sample of 250 university students from different universities in Lahore was included in this study. A purposive sampling strategy was used to collect survey data using internationally standardized tools. Pearson Product Moment Correlation and Multiple Hierarchical Linear Regression results showed that two subscales of perceived online social support, friends, and family, had a significant positive relationship with psychological well-being among university students. Among predictors, the current level of the semester and one subscale of perceived online social support- family, emerged as a significant positive predictor of psychological well-being among university students. The study concludes that family and friends are integral to the psychological well-being of university students and interventions must be planned to ensure that university students living far from home and those with excessive workloads have sufficient online and offline time with family and friends for mental health and well-being.

Fixed Points Results for Various Contractions in Different Distance Spaces

In this thesis, we present our contribution to develop Fixed Point Theory. The main purpose is to introduce different notions of contractive inequalities in the frame work of different distance spaces and to obtain fixed point, common fixed point, best proximity point, common best proximity point results for such inequalities by adding and relaxing some conditions and generalizing the existing results. This thesis is comprised on six chapters. Chapter 1 recapitulates some basic definitions and existing results related to fixed point, common fixed point and best proximity points. Chapter 2 consists of eight sections. Section 2.1 covers the introduction of the chapter and in Section 2.2 we obtain fixed point results for α-η-GF-contractions in the setting of modular metric spaces. In Section 2.3, we derive new results in partially ordered metric spaces from previous section. In Section 2.4, 2.5 and 2.6, as an application of our results proved in last sections, we deduce, respectively, SuzukiWardowski type fixed point theorems, fixed point results for orbitally continuous mappings and more general fixed point theorems for integral type GF-contractions. In Section 2.7, we introduce the concept of ω-weak compatibility and prove the existence and uniqueness of common fixed point results for ω-weak contractive inequalities of integral type in modular metric spaces. The presented results in this section elongate and generalize the Theorems 2.2-4.3 of [29], Theorem 2.1 of [44], Theorems 2.1 and 2.4 of [40], Theorems 2.1-2.4 of [124], Theorem 2.1 and 3.1 of [129], Theorem 2 of [150] and Theorems 3.1 and 3.4 of [122] in the set-up of modular metric space. In Section 2.8, we deduce fixed point results and common fixed point results in a triangular fuzzy metric spaces. The results of Sections 2.2, 2.3, 2.4, 2.5, 2.6 and Sections 2.7, 2.8 have appeared, respectively in [85] and [84]. Chapter 3 consists of six sections. Section 3.1 covers the introduction of the chapter, in Section 3.2 we introduce the concept of α-η lower semi-continuous multivalued mappings, α-η upper semi-continuous multivalued mappings and prove some fundamental lemmas related to these concepts. In Section 3.3, we develop fixed point results for modified α-η-GF-contractions with the help of the newly introduced concept in previous section. The obtained results generalize the Theorem 2.5 and Theorem 2.6 of [126]. In Section 3.4 we prove fixed point results for F-contraction of Hardy-Rogers type. These results extend and generalize Theorem 10 and Theorem 11 of [17]. In Section 3.5, we find common fixed point and fixed point results for multivalued α∗η∗ manageable contractions. The obtained results generalize Theorems 3.2 and 3.3 of [12], Theorem 9 of [14], Theorem 4.1 of [91] and Theorem 5 of [127] and [133]. Lastly, in Section 3.6, as an application of our results, we derive fixed Point Results in Partially Ordered Metric Space and establish the existence of solution of Volterra integral equation of the second kind. The results of Sections 3.2, 3.3, 3.4 and Sections 3.5, 3.6 have appeared, respectively, in [95] and [80]. Chapter 4 consists of five sections. Section 4.1 covers the introduction of the chapter and in Section 4.2, we define multivalued α-orbital admissible mappings and prove some supplementary results, which will be used in further sections. In Section 4.3, we proved the existence of fixed points for multivalued α-type F-contractions in complete metric spaces. It is also worth mentioning that, to prove these results we only use two conditions from already defined F-contraction by Wardowski. In Section 4.4, we derive the best proximity results for multivalued cyclic α-F contraction with proximally complete property. The obtained results generalize Theorem 2.2 and Theorem 2.5 of [16]. In Section 4.5, as an application of previous section, we obtain best proximity point results and fixed point results for single-valued mappings. As a special case of our results, we obtain Theorem 3.4 of [66], Theorems 2.1 and 2.2 of [135] and Theorem 2.1 of [167], we also present an example which illustrates thesolvability of our results but Theorems 2.1 and 2.2 of [135], Theorems 2 and 5 of [88] are not applicable for this example. The results of Sections 4.2, 4.4 and 4.5 have appeared in [79]. Chapter 5 consists of six sections. Section 5.1 covers the introduction of the chapter and in Section 5.2, we define T-orbitally continuous, T-orbitally lower semicontinuous, T-orbitally upper semi-continuous mappings and prove some related lemmas. In Section 5.3, we prove Variational Principle and as a consequence we obtain Ekeland’s-Variational Principle in the setting of T-orbitally complete metric spaces. The obtained results generalize Theorem 1.1 of [64] and Theorem 1 of [170]. In Section 5.4, we derive some fixed point results from the results proved in previous section. These fixed point results extend and generalize Theorem 2 of [161], Theorem 1 of [38] and [55] and main results of [104], [49] and [149]. In Section 5.5, as a consequence, we obtain minimax theorems in incomplete metric spaces without assumption of convexity and also obtain the existence of a solution of equilibrium problem in incomplete metric spaces. We also present an example here, which satisfies our obtained equilibrium formulation but the equilibrium formulations of Ekeland’s variational principles given in [11, 41, 46, 116, 136, 140] can not be applied for this example. In Section 5.6, we define α-orbital admissible mapping with respect to η and utilize this concept to obtain the extension of Theorem 6 of [54], Theorem 10 and 11 of [17] in the frame work of T-orbitally complete metric spaces. The obtained results also generalize Theorem 5.1 and Theorem 5.3 of [95]. Chapter 6 consists of seven sections. Section 6.1 covers the introduction of the chapter and in Section 6.2, we introduce the concept of cyclic orbital simulative contractions and explore the existence of best proximity points for these type of mappings via enriched class of simulation functions. For this purpose, we adopt only one condition from the concept of simulation functions and show that other conditions are superfluous. In Section 6.3, we deduce some fixed point results from previous section. The presented results generalize Theorem 2.8 of [111]. In Section 6.4, we deduce some new and existing best proximity points results and fixed point results in the Literature from previous sections. As a consequence, we obtain Theorem 4 of [15], Theorem 2.4 of [63], Theorem 3.4 of [66], Theorem 2.2 of [106], Theorem 2.1 of [139], and Theorem 1.8 of [142]. In Section 6.5, we refine Theorem 1 and Theorem 2 of [4], Theorem 2.1 and Theorem 2.2 of [154], Theorem 3.1 and Theorem 3.2 of [163]. In Section 6.6, We give an application to the variational inequalities and provide the solvability theorems of an optimization problem. We also explore the solution for an elliptic boundary value problem in Hilbert spaces. Finally, in Section 6.7, we introduce the notion of α∗-proximal contractions for multivalued mappings and obtain the existence of common best proximity points for both multivalued mappings and single valued mappings. Here we get the generalizations of Theorem 13 in [13], Theorem 3.3 in [12] and Theorem 2.1 in [155]. We also give a generalization to the concepts of compatibility and weak compatibility due to Jungck ([101] and [100]).