Mathematical Analysis & Convex Optimization Vol. 1 (2020), No. 2, 109-122 https:\\ maco.lu.ac.ir DOI: 10.29252/maco.1.2.11 Research Paper FIXED POINT RESULTS IN COMPLEX VALUED RECTANGULAR EXTENDED b-METRIC SPACES WITH APPLICATIONS NAIMAT ULLAH AND MOHAMMED SHEHU SHAGARI * Abstract. In this article, two new fxed point results in the framework of complex-valued rectangular extended b-metric space are established. Our results include as special cases, some well-known results in the cor- responding literature. We provide nontrivial examples and an existence theorem of a Fredholm type integral equation to support our assertions and to indicate a usability of the results presented herein. MSC(2010): 46S40, 47H10, 54H25. Keywords: Complex valued metric; Complex Valued Rectangular Ex- tended b-metric; Fixed point; Integral equation. 1. Introduction and Background The Banach contraction principle [9] is the frst most well-known, simple and versatile classical result in fxed point theory with metric space struc- ture. More than a handful of literature embrace applications and general- izations of this principle in diferent directions, for example, by weakening the hypotheses, employing diferent mappings and various forms of metric spaces. In this context, the work of Taskovic [21] is handy for recollecting various modifcations of Banach type contractive defnitions. The study of new spaces and their properties have been an interesting topic among the mathematical research community. In this direction, the notion of b-metric spaces is presently fourishing. The idea commenced with the work of Bakhtin [8] and Bourbaki [10]. Later on, Czerwik [11] gave a postulate which is weaker than the classical triangle inequality and formally established a b-metric space with a view of improving the Banach fxed point theorem. Meanwhile, the notion of b-metric spaces has gained enormous generalizations, see, for example, [15, 19, 20]. For a recent short survey on basic concepts and results in fxed point theory in the framework of b-metric spaces, we refer the interested reader to Karapinar [17]. Along the line, Branciari [24] invented the concept of rectangular metric space by changing Date: Received: November 10, 2020 , Accepted: December 27, 2020. * Corresponding author. 109