Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

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

“…x, x , x ), t ∈ (0, 1), was studied by M. Elgindi and Z. Guan [20], A. El-Haffaf [16], T. Ma [21] and Q. Yao [22] (here f (t, x, p, q) may be singular at t = 0, 1, x = 0, p = 0 and q = 0). 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].…”

“…x, x , x ), t ∈ (0, 1), was studied by M. Elgindi and Z. Guan [20], A. El-Haffaf [16], T. Ma [21] and Q. Yao [22] (here f (t, x, p, q) may be singular at t = 0, 1, x = 0, p = 0 and q = 0). 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].…”

“…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...…”

“…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 [2,3,[5][6][7][8]10,14,18,19,21,22], Nagumo-type growth conditions [24,26,27,32,34], nonresonance conditions [15,25], monotone conditions [7,24], maximum principles [1,17], various applications of Greens function [2,3,6,8,10,14,16,29,[36][37][38]. Because of the nature of processes leading to fourth-order BVPs, a variety of authors study the existence of positive solutions, see [2,3,…”

Positive solutions and eigenvalue intervals for a second order p-Laplacian discrete system

“…We prove that system (2) has at least one positive solution for each λ in an explicit eigenvalue interval. Recently, several eigenvalue characterizations for different kinds of boundary value problems have appeared, and we refer the reader to [5,12,14,16,18]. The rest of this paper is organized as follows.…”