3D analysis of modified F-contractions in convex b-metric spaces with application to Fredholm integral equations

1 Department of Math & Stats, International Islamic University Islamabad, Pakistan 2 Department of Mathematics, Gomal University, D. I. Khan, 29050, KPK, Pakistan 3 Department of Mathematics and General Sciences, Prince Sultan University, P.O Box 66833, Riyadh 4 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 5 Department of Computer Science and Information Engineering, Asia University, Taichung 40402, Taiwan 6 Department of Mathematics, Usak University, Turkey


Introduction and Preliminaries
Czerwik [1] introduced b-metric and proved fixed point theorems in it. It was further extended to partial b-metric and dislocated b-metric spaces in the past years. Chen et al. [2] introduced convex b-metric space and established various fixed point theorems. On the other hand, various authors generalized the metric space into many other spaces (see [3][4][5][6][7][8][9][10][11]).
Wardowski [12] introduced the idea of -contraction which was later followed by many authors who delivered interesting results of -contraction. One of them was presented by Cosentino et al. [13] who expanded -contraction in -contraction of Hardy Roger's type. For more generalization of -contraction, we refer the readers to see ( [14][15][16]).
In this article we discuss -contraction in the frame of convex b-metric space using generalized Mann's iteration algorithm. However, we have modified definition of -contraction of Nadler type by eliminating two of its conditions (F3) and (F4). Cosentino et al. [17] have proved the results for multivalued simple F-contraction of Banach type, while our results have been proved for F-Reich type contraction for single valued mappings. As the conditions (F3) and (F4) of the mappings belonging to the set ℱ have been removed, thus, our results are more generalized than the results presented by Cosentino for each and .

Definition 1.6 ([2]):
Assume that is a convex structure on a b-metric space . Then is known as convex b-metric space. Suppose that is convex b-metric space with a self mapping . Then for and a generalize Mann's iteration sequence is defined as;

Fixed point results of -Kannan contraction in convex b-metric spaces
This section evaluates -Kannan contraction for the existence of unique fixed point results. . This shows that .

Fixed point results of -Reich contraction
This section examines -Reich contraction for the possible existence of a unique fixed point. Also, an example is given to explain the proved theorem. which is a contradiction. Therefore, is the only fixed point of . Figure 1. The higher the graph is at z-axis, the greater the value of the function is.
In Figure  1, the graph in blue and purple colour represent while that in red and green shows . The first one clearly dominates the later one, hence confirming the contractive inequality. Figure 2. The higher the graph is at z-axis, the greater the value of the function is.
In Figure 2, the graph in red colour represent while that in blue shows . Note that the higher is the graph at z-axis, the bigger is the value of the mapping . Hence, both the figures clearly demonstrate that the inequality (3.2) holds true.
In the next section, we discuss the application of our results to Fredholm integral equation of the second type.  for all . Then the integral equation (4.1) has a unique solution.

Conclusion
This paper has modified the definition of generalized -contractions by eliminating the conditions (F3) and (F4) and thus proved some principal fixed point results in the setting of convex b-metric spaces. Throughout this research, it was observed that the elements are taken from the convex structure . Thus, investigated fixed point for -Reich contractions and -Kannan contractions followed by the verification of our results with the help of example and graphs. Further, an application of our results in finding a unique solution to the Fredholm integral equation is