Positive Solutions of Fourth-Order Problem Subject to Nonlocal Boundary Conditions
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Positive Solutions of Fourth-Order Problem Subject to Nonlocal Boundary Conditions Lin Han1 · Guowei Zhang1
· Hongyu Li2
Received: 13 May 2019 / Revised: 13 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. 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. Keywords Positive solution · Fixed point index · Cone · Spectral radius Mathematics Subject Classification 34B18 · 34B10 · 34B15
1 Introduction In this paper, we study the existence of positive solutions for fourth-order boundary value problem (BVP) with dependence on the second derivative in nonlinearity under Stieltjes integral boundary value conditions
Communicated by Shangjiang Guo. Zhang is supported by the National Natural Science Foundation of China (Grant No. 61473065). Li is supported by the National Natural Science Foundation of China (Grant No. 11801322).
B
Guowei Zhang [email protected] ; [email protected]
1
Department of Mathematics, Northeastern University, Shenyang 110819, China
2
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
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L. Han et al.
u (4) (t) = f (t, u(t), u (t)), t ∈ [0, 1], u(0) = u(1) = β1 [u], u (0) + β2 [u] = 0, u (1) + β2 [u] = 0,
(1.1)
1 where βi [u] denotes linear functional given by βi [u] = 0 u(t)dBi (t) involving Stieltjes integral with suitable function Bi of bounded variation (i = 1, 2). This equation models the stationary state of the deflection of elastic beam, and the second derivative stands for the bending moment stiffness. Alves et al. [1] considered the existence of positive solutions for the beam equation u (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, where g is a continuous function. Using the monotonically iterative technique, Yao [18] dealt with the positive solution for fourth-order two-point boundary value problem
u (4) (t) = f (t, u(t), u (t)), t ∈ (0, 1), u(0) = u (0) = u (1) = u (1) = 0.
Li [10] and Ma [12] discussed the existence of positive solutions for the fourth-order boundary value problem
u (4) (t) = f (t, u(t), u (t)), t ∈ (0, 1), u(0) = u (0) = u(1) = u (1) = 0.
Their methods are, respectively, based upon fixed point index theory on cones and global bifurcation techniques. Bai [2] and Guo et al. [5] explored the existence of positive solu
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