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This article introduces the concept of partially controlled J metric spaces; in particular, the J metric space with self-distance is not necessarily zero, which is important in computer science. We prove the existence of a unique fixed point for linear and nonlinear contractions, provide some examples to prove the existence of this metric space, and present some important applications in fractional differential equations, i.e., "Riemann–Liouville derivatives". [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]