Nonlinear Functional Analysis and Applications Vol. 25, No. 4 (2020), pp. 769-797 ISSN: 1229-1595(print), 2466-0973(online) https://doi.org/10.22771/nfaa.2020.25.04.10 http://nfaa.kyungnam.ac.kr/journal-nfaa Copyright c  2020 Kyungnam University Press KUP ress NEW GERAGHTY TYPE CONDENSING OPERATORS AND SOLVABILITY OF NONLINEAR QUADRATIC VOLTERRA-STIELTJES INTEGRAL EQUATION Vishal Nikam 1 , Dhananjay Gopal 1,2 , Habib ur Rehman 3 and Thanatporn Bantaojai 4 1 S.V. National Institute of Technology, Surat- 395007 Gujarat, India e-mail: nikamvishal832@gmail.com 2 Department of Pure & Applied Mathematics Guru Ghasidas Vishwavidyalaya (A Central University) Bilaspur (C.G.) India e-mail: gopal.dhananjay@rediffmail.com 3 Department of Mathematics, Faculty of Science King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand e-mail: hrehman.hed@gmail.com 4 Mathematics English Program, Faculty of Education Valaya Alongkorn Rajabhat University under the Royal Patronage (VRU), Pathumtani 13180, Thailand e-mail: thanatporn.ban@vru.ac.th Abstract. The true motivation of this article is to provide sufficient conditions with the aid of Geraghty type condensing operators that guarantee the existence of a solution of non- linear quadratic Volterra-Stieltjes integral equation. We also address several new fixed point theorems that ensure the existence of a fixed point for Geraghty type condensing operators in real Banach spaces. An example and numerical approximations are presented to justify the basis of our results. 0 Received June 17, 2020. Revised September 13, 2020. Accepted September 17, 2020. 0 2010 Mathematics Subject Classification: 47H09, 47H10, 37C25. 0 Keywords: Monotone operator, ξμ-admissible, ξμ-β condensing operators, measure of noncompactness, non-linear quadratic Volterra-Stieltjes integral equation. 0 Corresponding author: T. Bantaojai(thanatporn.ban@vru.ac.th).