Some Existence Results of Positive Solution to Second-Order Boundary Value Problems

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 516452, 7 pages http://dx.doi.org/10.1155/2014/516452 Research Article Some Existence Results of Positive Solution to Second-Order Boundary Value Problems Shuhong Li,1 Xiaoping Zhang,2 and Yongping Sun2 1 2 School of Medicine and School of Science, Lishui University, Lishui, Zhejiang 323000, China School of Electron and Information, Zhejiang University of Media and Communications, Hangzhou, Zhejiang 310018, China Correspondence should be addressed to Shuhong Li; Received 4 February 2014; Accepted 16 March 2014; Published 20 May 2014 Academic Editor: Xinan Hao Copyright © 2014 Shuhong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the existence of positive and monotone solution to the boundary value problem 𝑢󸀠󸀠 (𝑡) + 𝑓(𝑡, 𝑢(𝑡)) = 0, 0 ⩽ 𝑡 ⩽ 1, 𝑢(0) = 𝜉𝑢(1), 𝑢󸀠 (1) = 𝜂𝑢󸀠 (0), where 𝜉, 𝜂 ∈ (0, 1) ∪ (1, ∞). The main tool is the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O’Regan. Finally, four examples are provided to demonstrate the availability of our main results. 1. Introduction Boundary value problems for ordinary differential equations play a very important role in both theory and applications. They are used to describe a large number of physical, biological, and chemical phenomena. In recent years many papers have been devoted to second-order two-point boundary value problem. For a small sample of such work, we refer the reader to the monographs of Agarwal [1], Agarwal et al. [2], and Guo and Lakshmikantham [3], the papers of Avery et al. [4] and Henderson and Thompson [5], and references therein along this line. In the literature, many attempts have been made by researchers to develop criteria which guarantee the existence and uniqueness of positive solutions to ordinary differential equations; this subject has attracted a lot of interests; see, for example, Cid et al. [6], Ehme [7], Ehme and Lanz [8], Ibrahim and Momani [9], Kong [10], Ma and An [11], Zhang and Liu [12], Zhang et al. [13], and Zhong and Zhang [14]. In this paper, we study the existence of positive and monotone solution for the second-order two-point boundary value problem 𝑢󸀠󸀠 (𝑡) + 𝑓 (𝑡, 𝑢 (𝑡)) = 0, 𝑢 (0) = 𝜉𝑢 (1) , 0 ⩽ 𝑡 ⩽ 1, 𝑢󸀠 (1) = 𝜂𝑢󸀠 (0) , (1) where 𝑓 : [0, 1] × [0, ∞) → [0, ∞) is continuous function and 𝜉, 𝜂 ∈ (0, 1) ∪ (1, ∞) are two constants. The boundary conditions in problem (1) are closely related to some other boundary conditions. If 𝜉 = 𝜂 = 1, the boundary conditions in problem (1) reduce to periodic boundary conditions. If 𝜉 = 𝜂 = −1, the boundary conditions in problem (1) reduce to antiperiodic boundary conditions. If 𝜉 = 𝜂 = 0, problem (1) reduces to second-order right focal boundary value problem. In a recent paper [15], by applying a fixed point theorem by Avery et al. [16], Sun studied the existence of monotone positive solutions to problem (1). In this paper we will prove some new existence results for problem (1) by using the new fixed point theorem of cone expansion and compression of functional type by Avery et al. [17]. This paper is organized as follows. In Section 2 we present some notations, definitions, and lemmas. In Section 3 we establish some sufficient conditions which guarantee the existence of positive solutions to problem (1). In Section 4 we give four examples to illustrate the effectiveness and applications of the results presented in Section 3. 2. Preliminary Results For the convenience of the reader, we present here the necessary definitions and background results. We also state 2 Abstract and Applied Analysis the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O’Regan. To prove our results, we will need the following fixed point theorem, which is presented by Avery et al. [17]. Definition 1. Let 𝐸 be a real Banach space. A nonempty closed convex set 𝑃 ⊂ 𝐸 is called a 𝑐𝑜𝑛𝑒 of 𝐸 if it satisfies the following two conditions: Theorem 4. Let Ω1 and Ω2 be two bounded open sets in a Banach space 𝐸 such that 0 ∈ Ω1 and Ω1 ⊆ Ω2 and 𝑃 is a cone in 𝐸. Suppose that 𝑇 : 𝑃 ∩ (Ω2 \ Ω1 ) → 𝑃 is a completely continuous operator, 𝛼 and 𝛾 are nonnegative continuous functional on 𝑃, and one of the two conditions (1) 𝑢 ∈ 𝑃, 𝜆 ⩾ 0, implies 𝜆𝑢 ∈ 𝑃; (2) 𝑢 ∈ 𝑃, −𝑢 ∈ 𝑃, implies 𝑢 = 0. Every cone 𝑃 ⊂ 𝐸 induces an ordering in 𝐸 given by 𝑢 ⩽ V if and only if V − 𝑢 ∈ 𝑃. Definition 2. Let 𝐸 be a real Banach space. An operator 𝑇 : 𝐸 → 𝐸 is said to be completely continuous if it is continuous and maps bounded sets into precompact sets. Definition 3. A map 𝛼 is said to be a nonnegative continuous concave functional on a cone 𝑃 of a real Banach space 𝐸 if 𝛼 : 𝑃 → [0, +∞) is continuous and 𝛼 (𝜆𝑢 + (1 − 𝜆) V) ⩾ 𝜆𝛼 (𝑢) + (1 − 𝜆) 𝛼 (V) , 𝑢, V ∈ 𝑃, 0 ⩽ 𝜆 ⩽ 1. (2) Similarly we said the map 𝛽 is a nonnegative continuous convex functional on a cone 𝑃 of a real Banach space 𝐸 if 𝛽 : 𝑃 → [0, +∞) is continuous and 𝛽 (𝜆𝑢 + (1 − 𝜆) V) ⩽ 𝜆𝛽 (𝑢) + (1 − 𝜆) 𝛽 (V) , 𝑢, V ∈ 𝑃, 0 ⩽ 𝜆 ⩽ 1. 𝑢 ∈ 𝑃, 0 ⩽ 𝜆 ⩽ 1. (K2) 𝛾 satisfies Property A2 with 𝛾(𝑇𝑢) ⩽ 𝛾(𝑢), for all 𝑢 ∈ 𝑃 ∩ 𝜕Ω1 , and 𝛼 satisfies Property A1 with 𝛼(𝑇𝑢) ⩾ 𝛼(𝑢), for all 𝑢 ∈ 𝑃 ∩ 𝜕Ω2 is satisfied. Then 𝑇 has at least one fixed point in 𝑃 ∩ (Ω2 \ Ω1 ). To study problem (1), we need the following lemmas (see [15]). Lemma 5. Green’s function 𝐺 : [0, 1] × [0, 1] → [0, ∞) for the BVP (1.1) is given by 𝐺 (𝑡, 𝑠) = 1 (1 − 𝜉) (1 − 𝜂) ×{ 𝑠 + 𝜂 (𝑡 − 𝑠) + 𝜉𝜂 (1 − 𝑡) , 0 ⩽ 𝑠 ⩽ 𝑡 ⩽ 1, 𝑡 + 𝜉 (𝑠 − 𝑡) + 𝜉𝜂 (1 − 𝑠) , 0 ⩽ 𝑡 ⩽ 𝑠 ⩽ 1. (5) (3) Lemma 6. Suppose that 𝜉, 𝜂, 𝛿 ∈ (0, 1). Then Green’s function 𝐺(𝑡, 𝑠) defined by (5) has the following properties: We say that the map 𝛾 is sublinear functional if 𝛾 (𝜆𝑢) ⩽ 𝜆𝛾 (𝑢) , (K1) 𝛼 satisfies Property A1 with 𝛼(𝑇𝑢) ⩾ 𝛼(𝑢), for all 𝑢 ∈ 𝑃 ∩ 𝜕Ω1 , and 𝛾 satisfies Property A2 with 𝛾(𝑇𝑢) ⩽ 𝛾(𝑢), for all 𝑢 ∈ 𝑃 ∩ 𝜕Ω2 ; or (4) All the concepts discussed above can be found in [3]. (a) 𝐺(𝑡, 𝑠) ⩾ 0, 𝜕𝐺(𝑡, 𝑠)/𝜕𝑡 ⩾ 0, ∀𝑡, 𝑠 ∈ [0, 1]; (b) 𝑡𝐺(1, 𝑠) ⩽ 𝐺(𝑡, 𝑠) ⩽ 𝐺(1, 𝑠), ∀𝑡, 𝑠 ∈ [0, 1]; 1 1 1 1 (c) max0⩽𝑡⩽1 ∫0 𝐺(𝑡, 𝑠)𝑑𝑠 = ∫0 𝐺(1, 𝑠)𝑑𝑠 = (1 + 𝜂)/2(1 − 𝜉)(1 − 𝜂); Property A1. Let 𝑃 be a cone in a real Banach space 𝐸 and Ω a bounded open subset of 𝐸 with 0 ∈ Ω. Then a continuous functional 𝛽 : 𝑃 → [0, ∞) is said to satisfy Property A1 if one of the following conditions holds: (d) min𝛿⩽𝑡⩽1 ∫𝛿 𝐺(𝑡, 𝑠)𝑑𝑠 = ∫𝛿 𝐺(𝛿, 𝑠)𝑑𝑠 = (1 − 𝛿)𝛿/(1 − 𝜂) + (1 − 𝛿)(1 + 𝛿 + 𝜂 − 𝜂𝛿)𝜉/2(1 − 𝜉)(1 − 𝜂) ⩾ (1 − 𝛿)𝛿/(1 − 𝜂). (a) 𝛽 is convex, 𝛽(0) = 0, and 𝛽(𝑢) ≠ 0 if 𝑢 ≠ 0 and inf 𝑢∈𝑃∩𝜕Ω 𝛽(𝑢) > 0, Lemma 7. Suppose that 𝜉, 𝜂 ∈ (1, ∞), 𝛿 ∈ (0, 1). Then Green’s function 𝐺(𝑡, 𝑠) defined by (5) has the following properties: (b) 𝛽 is sublinear, 𝛽(0) = 0, and 𝛽(𝑢) ≠ 0 if 𝑢 ≠ 0 and inf 𝑢∈𝑃∩𝜕Ω 𝛽(𝑢) > 0, (c) 𝛽 is concave and unbounded. Property A2. Let 𝑃 be a cone in a real (...truncated)