The Unique Solution to the Differential Equations of the Fourth Order with Non-Homogeneous Boundary Conditions

Abstract

This research paper aims to establish the uniqueness of the solution to fourth-order nonlinear differential equations v(4)(x) + f (x,v(x)) = 0, x ε [a,b], with non-homogeneous boundary conditions where 0 ≤ a < ζ < b, the constants α, ???? are real numbers and f : [a,b] x R →R is a continuous function with f (x, 0] ≠ 0. Using the sharper bounds on the integral of the kernel, the uniqueness of the solution to the problem is established based on Banach and Rus fixed point theorems on metric spaces. AMS Subject Classification: 34B15, 34B10.

Existence Results for Differential Equations of Fourth Order with Non-Homogeneous Boundary Conditions

“…Thus, for any given ϵ > 0, we can take such that d(v, w) < δ, which implies that d(Fv, Fw) < ϵ. Hence the operator F is continuous on D with respect to the metric d given in (14). Furthermore, we prove that the inequality (13) of Theorem 2.7 is fulfilled.…”

“…Due to their enormous importance in theory and applications, researchers have studied the existence and uniqueness of solutions to the differential equations of various orders under certain conditions by different methods. We mention a few papers on the existence and uniqueness of solutions to the boundary value problems of third order [5][6][7][8][9] and fourth order [10][11][12][13][14] for readers reference.…”