The double-controlled metric-type space $(X,\mathcal{D})$ is a metric space in which the triangle inequality has the form $\mathcal{D}(\eta ,\mu )\le {\zeta }_1(\eta ,\theta )\mathcal{D}(\eta ,\theta )+{\zeta }_2(\theta ,\mu )\mathcal{D}(\theta ,\mu )$ for all $\eta ,\theta ,\mu \in X$. The maps ${\zeta }_1,{\zeta }_2:X\times X\to [1,\infty )$ are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form $\mathcal{D}(\eta ,\mu )\le \alpha \left(\mathcal{D}(\eta ,\theta )\right)+\beta \left(\mathcal{D}(\theta ,\mu )\right)$ for all $\eta ,\theta ,\mu \in X$. In our new space, the control functions $\alpha ,\beta :[0,\infty )\to [0,\infty )$ are composed of the metric $\mathcal{D}$ in the triangle inequality, where the control functions ${\zeta }_1,{\zeta }_2:X\times X\to [1,\infty )$ in a double-controlled metric-type space are multiplied with the metric $\mathcal{D}$. We establish some fixed-point theorems along with the examples and applications.

Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to ${\mathcal{G}}_b$-metric spaces. Stimulated by the notion of the $\mathcal{F}$-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with $\mathcal{F}$-contractions in ${\mathcal{G}}_b$-metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research.

One of the most intensively studied and generalized results in metric fixed point theory is Branciari’s fixed point theorem, which asserts that in a

One of the well-studied generalizations of a metric space is known as a partial metric space. The partial metric space was further generalized to the so-called M-metric space. In this paper, we introduce the Double-Controlled Quasi M-metric space as a new generalization of the M-metric space. In our new generalization of the M-metric space, the symmetry condition is not necessarily satisfied and the triangle inequality is controlled by two binary functions. We establish some fixed point results, along with the examples and applications to illustrate our results.

We study the following three definite integrals, previously posed as open problems by another researcher: \( I(\alpha) = \int_0^{\infty} x^{-1/2} \ln(1 + x^{-\alpha})\,dx \), \( I_n(\alpha) = \int_0^{\infty} \frac{1}{\sqrt{x}(x^2 + 4\alpha^2)^n}\,dx \), and \( I(\beta)=\int_0^{\infty} x^{-3/2} \big[f(2\beta/x) - f(2/x)\big]\,dx \). We establish sufficient conditions for the convergence of these integrals and evaluate them in closed form using special functions. In particular, the third integral $I(\beta)$ turns out to be similar to Frullani integral, and we obtain two interesting formulas for this integral. These types ofintegrals have been used to establish logarithmic Hardy-Hilbert-type inequalities.