In this article, we introduce a new extension of b-metric spaces, called controlled metric type spaces, by employing a control function $\alpha (x,y)$ of the right-hand side of the b-triangle inequality. Namely, the triangle inequality in the new defined extension will have the form, $d(x,y)\le \alpha (x,z)d(x,z)+\alpha (z,y)d(z,y),\mathrm{for}\mathrm{all}x,y,z\in X.$ Examples of controlled metric type spaces that are not extended b-metric spaces in the sense of Kamran et al. are given to show that our extension is different. A Banach contraction principle on controlled metric type spaces and an example are given to illustrate the usefulness of the structure of the new extension.

In this article, in the sequel of extending b-metric spaces, we modify controlled metric type spaces via two control functions $\alpha (x,y)$ and $\mu (x,y)$ on the right-hand side of the $b-$ triangle inequality, that is, $$d(x,y)\le \alpha (x,z)d(x,z)+\mu (z,y)d(z,y),\mathrm{for}\mathrm{all}x,y,z\in X.$$ Some examples of a double controlled metric type space by two incomparable functions, which is not a controlled metric type by one of the given functions, are presented. We also provide some fixed point results involving Banach type, Kannan type and $\varphi$ -nonlinear type contractions in the setting of double controlled metric type spaces.

The Klein–Fock–Gordon equation (KFGE), defined as the equation of relativistic wave related to NLEEs, has numerous implications for energy particle physics and is useful as a model for several types of matter, with deviation in the basic stuffs of particles and in crystals. In this work, the Sardar subequation method (SSM) is used for finding the solution of this KFGE. The advantage of SSM is that it provides many different kinds of solitons, such as dark, bright, singular, periodic singular, combined dark–singular and combined dark–bright solitons. The results show that the SSM is very reliable, simple and can be functionalized to other nonlinear equations. It is verified that all the attained solutions are stable by modulation instability process. To enhance the physical description of solutions, some 3D, contour and 2D graphs are plotted by taking precise values of parameters using Maple 18.

In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of $F_c$ -contractions such as Ćirić type, Hardy-Rogers type, Reich type and Chatterjea type. Two illustrative examples are presented to show the effectiveness of our results. Also, it is given an application of a Ćirić type $F_c$ -contraction to discontinuous self-mappings which have fixed circles.

In this manuscript, some fixed point results for generalized contractive type mappings under mild conditions in the setting of double controlled metric spaces (in short, $\eta _{\gimel }^{\nu }$η ℷ ν -metric spaces) are obtained. Moreover, some related consequences dealing with a common fixed point concept and nontrivial examples to support our results are presented. Ultimately, we use the theoretical results to discuss the existence and uniqueness of solutions of 2D Volterra integral equations, Riemann–Liouville integrals and Atangana–Baleanu integral operators are given.

In this paper, we introduce a new extension of the double controlled metric-type spaces, called double controlled metric-like spaces, by assuming that the “self-distance” may not be zero On the other hand, if the value of the metric is zero, then it has to be a “self-distance” (i.e., we replace $[\varsigma(g,h)=0 \Leftrightarrow g=h]$[ ς ( g , h ) = 0 ⇔ g = h ] by $[\varsigma(g,h)=0 \Rightarrow g=h]$[ ς ( g , h ) = 0 ⇒ g = h ] ). Using this new type of metric spaces, we generalize many results in the literature. We prove fixed point results along with examples illustrating our theorems. Also, we present double controlled metric-like spaces endowed with a graph along with an open question.