Symmetric positive solutions to singular system with multi-point coupled boundary conditions
Introduction
Coupled boundary conditions (BCs) arise in the study of reaction–diffusion equations, Sturm–Liouville problems, mathematical biology and so on, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and [18, Chapter 13]. In [4], Cardanobile and Mugnolo studied the following parabolic systemwith a general class of coupled BCs of the formwhere is a closed subspace of , the unknown function u takes values in a separable Hilbert space W and is an elliptic operator with operator-valued symbol.
Delgado et al. [5] investigated the following system with coupled BCs of the typewhere denote nonnegative constants. The authors proved the existence of a unique positive global in time classical solution and analyzed the associated stationary problem.
Leung [7] studied the following reaction–diffusion system for prey-predator interactionsubject to the coupled BCswhere the functions respectively represent the density of prey and predator at time and at position . Similar coupled BCs are also studied in [2] for biochemical system.
At the same time, the theory of multi-point boundary value problems (BVPs) for ordinary differential equations arises in different areas of applied mathematics and physics. For example, the vibrations of a guy wire of uniform cross-section and composed of N parts of different densities can be set up as a multi-point BVPs; many problems in the theory of elastic stability can be handled as multi-point BVPs too. Recently, the existence and multiplicity of positive solutions for nonlinear ordinary differential equations have received a great deal of attention. To identify a few, we refer the readers to [19], [18], [19], [20], [21], [22], [23], [24], [25], [26] and the reference therein for three-point BVPs and [27], [28], [29], [30], [31] for multi-point BVPs.
Very recently, Asif and Khan [32] studied the following a coupled singular system subject to four-point coupled BCs of the typewhere the parameters satisfy , are continuous and singular at . The authors obtained at least one positive solution to the system (1.1) by using Guo–Krasnosel’skii fixed-point theorem.
We notice that a type of symmetric problem has received much attention, for instance, [19], [27], [28], [29], [31], [32], [33], [34], [36], and the references therein. In [33], Sun discussed the existence and multiplicity of the BVPsby applying the Krasnosel’skii fixed-point theorem. Sun [34] also considered the optimal existence criteria for symmetric positive solutions to the Eq. (1.2) with the following BCs: . Recently, Wang and Sun [35] studied the existence of symmetric positive solutions to Eq. (1.2) with the following BCs: . Inspired by the above mentioned work and wide applications of coupled BCs in various fields of sciences and engineering, we study the existence of symmetric positive solutions to a singular systemwhere is an integer, the parameters , and . We assume are continuous and are symmetric on for all are symmetric on , and may be singular at and . By a symmetric positive solution of the system (1.4), we mean that satisfies (1.4), are symmetric and for all . To the best knowledge of the author, there is no earlier literature studying this problem. This paper attempts to fill part of this gap in the literatures.
The rest of the paper is organized as follows. In Section 2, we present a positive cone, a fixed point theorem which will be used to prove existence of symmetric positive solutions, Green’s function for the system of BVPs (1.4) and some related lemmas. In Section 3, we present main results of the paper and finally an example to illustrate the application of our main results.
Section snippets
Preliminaries and lemmas
The basic space used in this paper is . Obviously, the space E is a Banach space if it is endowed with the norm as follows:for any . Denote . LetwhereClearly . It is easy to see that K is a cone of E.
Main results
In this section, we apply Lemma 2.1 to establish the existence of positive solutions for the system (1.4). Writewhere denotes 0 or , , , and Theorem 3.1 Assume that hold. In addition, suppose that and or
Acknowledgements
The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11201260), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) , the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province and the Natural Science Foundation of Shandong Province of China (ZR2010AM017, ZR2011AQ008). The third author was supported financially by the Australia
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