<span>Several studies have been carried out to investigate the effect of Lanthanum (La³⁺) ion substitution on the structural and magnetic properties of manganese-chromium (Mn-Cr) ferrite of chemical formula Mn La_xCr Fe₂O₄(x=0.0, 0.25 and 0.5). Such studies have made efforts to improve the magnetic and structural properties of manganese-chromium (Mn-Cr) ferrite by using lanthanum substituted nano ferrites and then synthesized using the sol-gel method and annealed at a temperature of 700^oC. The changes that occurred in the structure of the nano ferrites as a result of lanthanum substitution were identified using X-ray diffraction (XRD). Based on Debye-Scherrer equation, the XRD data were used in measuring the particle sizes of different diffraction and average crystallite size by means of Fourier Transform infrared spectroscopy (FTIR). In analyzing the morphology of the nano ferrites, scanning electron microscopy (SEM) was used, elemental compassion was studied using energy dispersive X-ray spectroscopy (EDAX), and the average particle diameter was determined using Transmission electron microscopy (TEM) studies. FTIR spectral analysis of the prepared samples under investigations revealed the formation of a single phase spherical particles. Two important absorption bands were observed; one (<em>ν₁</em>) around 556 cm^-1, which is attributed to the intrinsic vibrations of tetrahedral complexes, while the other low frequency band (<em>ν₂</em>) was around 430 cm^-1, and attributed to octahedral complexes.</span>

<span class="s19">In this paper, An important definitions are to be used to prove the existence of a common fixed point theorem for four mappings incomplete,  partial – b – metric spaces, as well as prove a unique common fixed point by assuming another point and getting that, these points are finally equal. We presented an example thus enhancing us the outcome.</span>