Nonlinear Analysis: Theory, Methods & Applications
Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions
Introduction
Consider the th-order singular nonlocal boundary value problem (BVP) where is a parameter, , may be singular at and/or may also have singularity at . denotes the Riemann–Stieltjes integral with a signed measure, that is, has bounded variation.
The nonlocal BVPs have been studied extensively. The methods used therein mainly depend on the fixed-point theorems, degree theory, upper and lower techniques, and monotone iteration. The existence results are available in the literature [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Recently, Webb [5] studied the th-order nonlocal BVP where have singularity, and the nonlinearity satisfies Caratheodory conditions. Under weak assumptions, Webb obtained sharp results on the existence of one positive solution and multiple positive solutions under suitable conditions on .
In BVP (1.1), (1.2), denotes the Riemann–Stieltjes integral, where is of bounded variation, that is can be a signed measure. This includes as special cases the multi-point problems and integral problems. Particularly, when , where and , the authors of papers [6], [7], [8], [9] established the existence and multiplicity of positive solutions for the th-order three-point BVP (1.2) by applying the fixed point theorems on cones. When , where with , the th-order -point BVP (1.2) has been studied in [10], [11], [12]. For the nonsingular case, the existence and multiplicity results of positive solutions were shown by using various fixed point theorems and fixed point index theory. Few papers have considered the existence of positive solutions for higher order nonlocal BVP (1.1) when has singularity at .
In this paper, we will consider the higher order singular nonlocal BVP (1.1). We obtain the existence of positive solutions by means of the fixed point index theory under some conditions on concerning the first eigenvalue corresponding to the relevant linear operator. We extend one result of [5] to the case where is singular when . The nonlocal condition is given by a Riemann–Stieltjes integral with a signed measure. It is worth mentioning that the idea using a Riemann–Stieltjes integral with a signed measure is due to Webb and Infante in [2], [3]. The papers [1], [2], [3] contain several new ideas, and give a unified approach to many BVPs. Some ideas of this paper are also from [20], [21], [22], [23], [24].
Let be a cone in Banach space . For any , let and . A function is said to be a solution of the BVP (1.1) if and exist, and satisfies the BVP (1.1). A function is said to be a positive solution of the BVP (1.1) if is a solution of BVP (1.1) and for .
The following lemmas are needed in our argument.
Lemma 1.1 [25] Let be a cone in Banach space . Suppose that is a completely continuous operator. If there exists such that for any and , then .
Lemma 1.2 [25] Let be a cone in Banach space . Suppose that is a completely continuous operator. If for any and , then .
Section snippets
Preliminaries and lemmas
Let be the Green’s function for then As in [5], let . Defining , it is shown in [5] that the Green’s function for a nonlocal BVP (1.1) is given by where .
Lemma 2.1 Let and for , the Green’s function defined by (2.1) satisfies: is continuous for all ;[5]
Main results
Theorem 3.1 Assume that conditions (H1)–(H4) hold. Further assume that the following conditions hold:Then for anyBVP (1.1) has at least one positive solution, where is the first eigenvalue of defined by (2.2).
Proof First, by Lemma 2.2, Lemma 2.3, we know that is completely continuous. Thus for any , from the extension theorem of a completely continuous operator (see Proposition 8.3 on page 56 of [26]), there
Examples
We give two explicit examples to illustrate our results in Section 3.
Example 4.1 A 3-Point BVP We consider the BVP (1.1) with , and then , and . BVP (1.1) becomes the fourth order three-point BVP Obviously, is singular at . The Green’s function of the BVP (4.1) is from (2.1), in this special case see also [6], [7], [8], [9], [10], [11], [12], by
Acknowledgements
The authors thank the referee for helpful comments and suggestions, which lead to an improvement of the paper. The first and second authors were supported financially by the National Natural Science Foundation of China (10771117) and the Natural Science Foundation of Shandong Province of China (Y2007A23, Y2008A24). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
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