Analytical Investigation of a Fourth-order Boundary Value Problem in Deformation of Beams and Plate Deflection Theory

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

“…It is enough to prove the operator F has a unique fixed point ϑ * ∈ D with Fϑ * = ϑ * . According to (17), every such fixed point will also lie in C (4) ([a, b]) as can be directly shown by differentiating (10).…”

“…□ Lemma 2. 4 The kernel (ζ, ) given in (6) satisfies the following integral inequality Proof. We can consider, then 5 The kernel (x, ) in (4) satisfies the the following integral inequality ) 1 ( , ) .…”

“…The notion that the laws concerning these situations can be expressed as differential equations of various orders with certain initial or boundary conditions. Especially, the fourth order differential equations appear in the modeling of inelastic flows, viscoelastic, theory of plate deflection, bending of beams and various areas of applied mathematics as well as engineering [1][2][3][4]. 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.…”

“…In 1988, Gupta [15] demonstrated the existence results for the deformation of stretchy beam with completely supported edges and is characterized by a boundary problem of fourth order of the form In 2019, Li and Gao [16] discussed the solvability of fourth order boundary value problem (4)…”

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

“…The laws concerning these situations can be expressed as differential equations of various orders satisfying certain conditions. In particular, the fourth-order differential equations arise in the modeling of the concepts in the inelastic flows, viscoelastic, electric circuits, theory of plate deflection, bending of beams, and various areas of applied mathematics, as well as the concepts in engineering [1][2][3][4][5]. Due to their immense importance in theory and applications, researchers have shown an interest in studying the existence of solutions to differential equations of different orders under certain conditions.…”