Existence of Three Positive Solutions for Boundary Value Problem of Fourth Order with Sign-Changing Green's Function

In this paper, we examine a fourth-order equation that has parameter dependency and boundary conditions in three different places. We prove some of the features of the relevant asymmetric Green's function and infer its exact form. The resulting solutions are still positive and decreasing functions on the entire interval of the Green's function definition, and they are concave in a specific subinterval, despite the fact that the function's sign changes on the square of its definition. The fixed point theorem of Krasnoselskii is the foundation of the existence arguments. Next, using the Leggett-Williams fixed point theorem, it is concluded that there are at least three positive solutions. Lastly, an example is provided, to highlight the primary findings of the manuscript.