Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported

The aim of the present paper is to consider a fully elastic beam equation with left-end fixed and right-end simply supported, i.e., u 4 t = f t , u t , u ′ t , u ″ t , u ‴ t , t ∈ 0,1 u 0 = u ′ 0 = u 1 = u ″ 1 = 0 , where f : 0,1 × ℝ 4 ⟶ ℝ is a continuous function. By applying Leray–Schauder fixed point theorem of the completely continuous operator, the existence and uniqueness of solutions are obtained under the conditions that the nonlinear function satisfies the linear growth and superlinear growth. For the case of superlinear growth, a Nagumo-type condition is introduced to limit that f t , x 0 , x 1 , x 2 , x 3 is quadratical growth on x 3 at most.

The Existence, Uniqueness, and Stability Analysis of the Discrete Fractional Three-Point Boundary Value Problem for the Elastic Beam Equation

An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The findings are obtained by employing properties of discrete fractional equations, Banach contraction, and Brouwer fixed-point theorems. Further, we discuss our problem’s results concerning Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam–Rassias (HUR), and generalized Hyers–Ulam–Rassias (GHUR) stability. Specific examples with graphs and numerical experiment are presented to demonstrate the effectiveness of our results.