Positive solutions of beam equations under nonlocal boundary value conditions

Abstract

In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions2, 3). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in C 2 [0, 1]. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel. MSC: Primary 34B18; 34B10; secondary 34B15 Keywords: Positive solution; Fixed point index; Cone where β i [u] = 1 0 u(t) dB i (t) is Stieltjes integral with B i of bounded variation (i = 1, 2, 3). This equation describes the deflection of an elastic beam. Alves et al. [1] established the existence of positive solutions for the beam equationu (4) (t) = f t, u(t), u (t) under boundary conditions u(0) = u (0) = 0, u (1) = g u(1) , u (1) = 0 or u (1) = 0,

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

“…Among them, the solvability of two-point ones has received much attention in the literature. Such problems for equations of the form x (4) = f (t, x), t ∈ (0, 1), were studied by A. Cabada et al [1], J. Caballero et al [2], J. Cid et al [3], G. Han and Z. Xu [4], J. Harjani et al [5], G. Infante and P. Pietramala [6], J. Li [7] (here f (t, x) may be singular at t = 0, 1 and x = 0), B. Yang [8] and C. Zhai and C. Jiang [9]. In [1,[3][4][5] the BCs are…”

“…homogeneous (12) in [8], and those in [6,9] include homogeneous ones (6). J. Liu and W. Xu [10], D. O'Regan [11] and Q. Yao [12] studied BVPs for…”

“…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,…”

“…λ with one of the boundary conditions (k), k = 2, 13. Section 3 is devoted to the solvability of BVPs (1), (2), ..., (1), (8) and (1), (9). In Section 4 we study (1), (10) and (1), (11), and in Section 5 give existence results for (1), (12) and (1), (13).…”