A unified approach to coincidence and common fixed point results with application to Euler-Bernoulli beam equation

Abstract In this manuscript, we introduce a new class of proximal contractions, termed a modified proximal contraction, to establish coincidence and best proximity point results supported by illustrative examples. These results are then specialized to recover existing best proximity point theorems and further reduced to well-known common fixed point and fixed point results, thereby highlighting the unifying power and flexibility of the proximal contraction framework. In particular, we show how classical results of Banach and Reich arise naturally as special cases, demonstrating that common fixed point theorems can be reduced to classical fixed point theorems under suitable conditions. As an application, we investigate the fourth-order Euler-Bernoulli beam equation using Green’s function techniques. Several graphical representations of the derived Green’s function, including surface plots, heatmaps, and slice profiles, are provided to visualize symmetry, boundary effects, and maximum midspan deflection, thereby connecting the abstract theory to the physical behavior of beams under point loads. Finally, the existence and uniqueness of solutions to the Euler-Bernoulli boundary value problem are established through the developed fixed-point results.