In this paper, we introduce quasi $M$-metric spaces as a generalization of $M$-metric spaces. We establish some fixed point results along with the examples and application of our results to integral equations and system of linear equations.

In this manuscript, we introduce the notion of ℜ $\alpha$- $\theta$-contractions and prove some fixed-point theorems in the sense of ℜ-complete metric spaces. These results generalize existing ones in the literature. Also, we provide some illustrative non-trivial examples and applications to a non-linear fractional differential equation.

In this paper, we introduce the concept of elliptic-valued b-metric spaces, extending the notions of elliptic-valued metric spaces and complex-valued metric spaces. We present several fixed-point results that involve rational and product terms within this novel space framework. To support our main findings, we offer numerical examples. Additionally, we demonstrate an application of Urysohn integral equations.

Salem's equivalence of the Riemann hypothesis asserts that the hypothesis is true if and only if $g(t)=0$ is the only nontrivial bounded measurable solution (for fixed $1/2<r<1$) of the following integral equation:

$${\int }_0^{\mathrm{\infty }}\frac{t^{r-1}g(t)}{e^{xt}+1}dt=0,x>0.$$

Recent works have investigated this integral equation and shown that various classes of bounded measurable functions, subject to suitable growth assumptions, cannot furnish counterexamples to Salem's criterion. These approaches typically employ Mellin inversion, Widder–Lambert–type transforms, or distribution-theoretic methods. The aim of this paper is to study the associated nonhomogeneous Salem-type equation

$$f(x)=g(x)+\lambda {\int }_0^{\mathrm{\infty }}\frac{t^{r-1}f(t)}{1+e^{xt}}dt,x>0,$$

where $g$ is a given measurable function, $r\in (1/2,1)$, and $\lambda \in \mathbb{R}$, from the viewpoint of fixed–point theory. We show that the corresponding integral operator does not map the space of bounded measurable functions into itself on $(0,\mathrm{\infty })$. We therefore introduce a suitable weighted complete function space in order to apply Banach's contraction principle. The nonhomogeneous equation is important because it is a more general form of Salem's equation. As we shall see, its analysis allows one to identify precisely why Banach's contraction principle succeeds for the nonhomogeneous equation in an appropriate weighted space, yet does not directly settle Salem's original integral equation. This paper suggests that possible future directions may include the use of generalized metric spaces and generalized contraction mappings to study Salem's integral equation.

In this manuscript, utilizing interpolative contractions with fractional forms, some unique fixed-point results were studied in the context of extended $b$-metric spaces. For the validity of the presented results some examples are given. In the last section an existence theorem is provided to study the existence of a solution for the Fredholm integral equation.