Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions

Abstract

We study nth order boundary value problems with a nonlinear term f (t, x) subject to nonhomogeneous multi-point boundary conditions. Criteria for the existence of positive solutions of such problems are established. Conditions are determined by the relationship between the behavior of f (t, x)/x near 0 and ∞ when compared with the smallest positive characteristic value of an associated linear integral operator. This work improves and extends some recent results in the literature for the second order problems. The results are illustrated with examples.

On positive solutions for nonhomogeneous m-point boundary value problems with two parameters

“…China. 2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China.…”

Uniqueness and Parameter Dependence of Positive Solution of Fourth-Order Nonhomogeneous BVPs

“…The authors in 12 studied the multiplicity of positive solutions for some fourth-order two-point nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. For more recent results on higher-order BVPs with nonhomogeneous boundary conditions, one can see [13][14][15][16] . Inspired greatly by the above-mentioned excellent works, in this paper we are concerned with the following Sturm-Liouville BVP consisting of the fourth-order differential equation:…”

On positive solutions of an -point nonhomogeneous singular boundary value problem

“…Since this paper was submitted some other relevant papers have been published, see, for example, [14][15][16][17][18][19]. In [14], Kwong and Wong studied the second order multi-point BVPs with a nonhomogeneous BC at the right endpoint.…”

“…In [16], by employing the Krasnosel'skii-Guo fixed point theorem and Schauder's fixed point theorem, Sun studied the existence and nonexistence of positive solutions to the third order three-point nonhomogeneous BVP. The higher order multi-point BVPs with nonhomogeneous BCs were considered in [17,18]. The 2nth order BVP was studied by Kong and Kong [17] who showed the existence, nonexistence and multiplicity of positive solutions by using the fixed point index theory, the Schauder fixed point theorem, and the lower and upper solutions method.…”

“…The 2nth order BVP was studied by Kong and Kong [17] who showed the existence, nonexistence and multiplicity of positive solutions by using the fixed point index theory, the Schauder fixed point theorem, and the lower and upper solutions method. In [18], criteria for the existence of positive solutions of nth order BVPs are established via the Krein-Rutman theorem and fixed point index theory. The existence of multiple positive solutions for more general nonlocal BVPs of arbitrary order was studied by Webb and Infante in [19], and nonhomogeneous nonlocal BVPs also have been studied.…”