On Existence and Uniqueness Solution of Heat Equations in Quasi-Metric Spaces
Ahmad Hisbu Zakiyudin, Kistosil Fahim (*), Mahmud Yunus, Sunarsini, I Gst Ngr Rai Usadha, Sadjidon

Department of Mathematics,
Institut Teknologi Sepuluh Nopember,
Kampus ITS Sukolilo, Surabaya 60111, Indonesia

Abstract

In this paper, we prove the existence and uniqueness of solutions of heat equations in quasi-metric spaces by applying the \(\phi G-\)contraction in the setting of quasi-metric spaces. This type of contraction is analogous to \(\psi F\)-contraction which is introduced by Secelean et al. in 2019. In the \(\psi F\)-contraction, we have \(F:\mathbb{R}^+\rightarrow \mathbb{R}\) is an increasing mapping and \(\psi:(-\infty,\mu)\rightarrow \mathbb{R}\) for some \(\mu\) in \(\mathbb{R}^+\cup\{\infty\}\) is an increasing and continuous function such that \(\psi(t)<t\) for every \(t\) in \((-\infty, \mu)\). Meanwhile, in the \(\phi G\)-contraction, we have \(G\) is a strictly increasing mapping from \(\mathbb{R}^+\cup \{0\}\) to \(\mathbb{R}^+\cup \{0\}\). Also \(\phi:(-\infty,\mu)\rightarrow \mathbb{R}^+\cup\{0\}\) is a strictly increasing and continuous mapping such that \(\phi(t)<t\) for all \(t\) and \(\phi(0)=0\).

Keywords: Fixed point theories- Heat equations- Quasi-metric spaces

Topic: Functional Analysis