Existence and uniqueness theorems for some fourth‐order nonlinear boundary value problems

Abstract

Abstract. Let G and f : [0, 1] × R 4 → R be two functions satisfying Caratheodory conditions. This paper is concerned with the problems of existence and uniqueness of solutions for the nonlinear fourth-order ordinary differential equation G x, y, y ,y ,y = f x, y, y ,y ,y with one of a particular set of boundary conditions.

On the Solvability of Fourth-Order Two-Point Boundary Value Problems

“…Homogeneous BCs (4), (6), (12), (14) and (15) are considered in [20], in [16] they are again (17), in [21] are (14), and (4) in [22]. BVPs for equations of the form (1) were considered by R. Agarwal [23], Z. Bai [24], C. De Coster et al [25], J. Ehme et al [26], D. Franco et al [27], A. Granas et al [28], Y. Li and Q. Liang [29], Y. Liu and W. Ge [30], R. Ma [31], F. Minhós et al [32], B. Rynne [33], F. Sadyrbaev [34] and Q. Yao [35]. The BCs in those works are as follows: [23,34], homogeneous (16) in [24], homogeneous (4) in [24,31], (14) in [29,32,33], periodic in [25], (15) in [28,33],…”

“…In the works mentioned above, the main nonlinearity is a Carathéodory function on unbounded set, see [6,31,35], or is defined and continuous on a set such that each dependent variable changes in a left-and/or right-unbounded set, see [1][2][3][4][5]7,8,10,11,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][32][33][34]36,37]; an exception is [9] where it is supposed that f (t, x) is continuous on a set of the form (0, 1) × [a, b]. Several results guaranteeing a unique solution, at least one or multiple solutions are obtained by using various techniques, conditions and tools as: the classical lower and upper solutions technique [1,7,13,16,17,24,26,27,32,34], requirements to the main nonlinearity of the equation to be positive or non-negative...…”

Existence of a solution for a general order boundary value problem using the Leray-Schauder fixed point theorem

“…This kind of phenomena could be observed in convection patterns in several situations, for instance, in the atmosphere, in the oceans, when considering the coupling with a strong electromagnetic field, or on the Sun's surface (Chandrasekhar, 1961). This work will extend the results of (Fabiano et al, 2020), (Ahmad & Ntouyas, 2012) and (Ma, 2000) to an equation of a generic order 2n.…”

Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray–Schauder Alternative Fixed Point Theorem

“…Recently Zouaoui Bekri [21] studied sixth-order nonlinear boundary value problem using the Leray-Schauder alternative theorem. Ma [22] has given the existence and uniqueness theorems based on the Leray-Schauder fixed point theorem for some fourth-order nonlinear boundary value problems. Zvyagin and Baranovskii [23] have constructed a topological characteristic to investigate a class of controllable systems.…”