European Journal of Molecular & Clinical Medicine ISSN 2515-8260 Volume 07, Issue 09, 2020 3154 Some Common Fixed Point Theorems for Hardy Roger Type Contraction on Partial b-Metric Spaces Dr. A. Mary Priya Dharsini 1 , Dr.A. Leema Maria Prakasam 2 , Dr. A. Jennie Sebasty Pritha 3 1,2,3 - Assistant Professor, PG and Research Department of Mathematics, Holy Cross College (Autonomous), Tiruchirappalli – 620002. Mail id : priyairudayam@gmail.com Abstract We prove some fixed point theorems using Hardy Roger type contraction in the setting of b-metric as well as partial b-metric spaces in order to find the existence and uniqueness of the common fixed point. We also provide examples to illustrate the existence of fixed point and its uniqueness. Keywords: Common fixed point, Hardy roger type contraction, Partial b-metric Mathematics Subject Classification: Primary 47H10; Secondary 54H25 1. Introduction Fixed point theory is the most important and unique instrument in the field of Science, Engineering and Technological World. The method of fixed point theory is used in analysis from 20 th Century onwards. It was introduced by Joseph Liouville in 1837 and by Charles Emile Picard in 1890 based on the method of successive approximations and it is relevant in finding the existence of solutions in differential equations. The pioneering work of Classical Theory was given by Stephan Banach which was established in 1922. In point of the historical view, there are some Mathematicians who completed the results in Fixed Point Theory, they are L.E.T. Brower, W.A. Kirk, Silms, Granas and Dugundiji. The concept of b – metric space was introduced by Bakhtin in 1989. Further it was worked out and expanded by Czerwik in 1993. Making use of their results as better tools, many scholars derived some renowned Banach fixed point theorems in the b - metric spaces and partial b-metric spaces. The partial b-metric was introduced by O’Neill and it is also known as dualistic partial metric space.