Positive Solutions for a System of Neumann Boundary Value Problems of Second-Order Difference Equations Involving Sign-Changing Nonlinearities
In this paper, we study the existence of positive solutions for the system of second-order difference equations involving Neumann boundary conditions: , , , , , , where is a given positive integer, , and . Under some appropriate conditions for our sign-changing nonlinearities, we use the fixed point index to establish our main results.
For with , let . Consider the system of second-order difference equations involving Neumann boundary conditions:where () are two continuous functions and there exist with on () and a positive number such that
(H0) , , i=1,2.
As known to all, semipositone problems arise in bulking of mechanical systems, design of suspension bridges, chemical reactions, astrophysics, combustion, and management of natural resources; for example, see [1–4]. We note that studying positive solutions for semipositone problems is more difficult than that for positive problems. There are many methods to deal with semipositone (positive) problems, with the usual approaches being variational methods, fixed point theory, subsuper solutions methods, and degree theory; for example, see [3–32] and references therein.
In , the author used the Guo-Krasnosel’skii fixed point theorem to study the existence of at least one positive solution for the discrete fractional equation:where is a parameter; the semipositone nonlinear term satisfies the condition
For semipositone systems, the authors [9, 17, 18] used the similar conditions of (3) to obtain some results for boundary value problems of differential (difference) equations.
However, we note that systems of boundary value problems for difference problems have seldom been considered in the literature; we refer to only [8, 9, 33–37] and references therein.
In , the authors used the Krasnosel’skii-Zabreiko fixed point theorem to investigate the existence and multiplicity of positive solutions for the system of second-order discrete boundary value problems:where and are nonnegative continuous functions on .
Inspired by the works aforementioned, in this paper we use the fixed point index to consider the existence of positive solutions for (1). The novelty is threefold: The nonlinearities may be either bounded or unbounded below; ultimately nonpositive or nonnegative or oscillating, see [6, Page 2]; this improves some conditions for the nonlinearities in [36, 37]. Some appropriate nonnegative concave and convex functions are employed to depict the coupling behaviors of nonlinearities. Our conditions are better than (3). Our a priori estimates for positive solutions are derived by unknown functions ,; see Section 3. This is different from [36, 37].
For convenience, letThenis the Green’s function associated with the linear Neumann boundary value problemswhich is equivalent to
Let , . Then , for . Consequently, from  we obtain that the Green’s function has the following properties.
Lemma 1. (i) for all .
(ii) , for all .
(iii) If , for , then we haveThese involve direct computations, and so we omit their proofs. For convenience, we denoteDefine and consider the following modified discrete Neumann boundary value problems
Lemma 2 (see ). is a positive solution of (1) if and only if is a solution of (12) with for .
Let be the collection of all maps from to equipped with the max norm, . Then is a Banach space. Define a set by Then is a cone on . Moreover, is a Banach space with the norm , and is a cone on .
Note that (12) can be expressed in the formAs a result, for , and , we define the operatorsandThen we use the Arzelà-Ascoli theorem in a standard way to establish that is a completely continuous operator. It is clear that is a positive solution for (12) if and only if is a fixed point of .
On the other hand, let . Then from Lemma 1(ii) we haveTherefore, if we seek a fixed point of with (i.e., is a positive solution for (1)). Then if we havewhere . As a result, implies that for .
Lemma 3 (see ). Let be a real Banach space and a cone on . Suppose that is a bounded open set and that is a continuous compact operator. If there exists such thatthen , where denotes the fixed point index on .
Lemma 4 (see ). Let be a real Banach space and a cone on . Suppose that is a bounded open set with and that is a continuous compact operator. Ifthen .
3. Main Results
For convenience, we use to denote distinct positive constants. Let for . Now, we list our assumptions on .
(H1) There exist such that
(i) is concave and strictly increasing on ,
(ii) there exist such that
(iii) there is a such thatfor and .
(H2) Let . Then for any , we suppose that
(H3) There exist such that
(i) is convex and strictly increasing on ,
(ii) there exists such that
(iii) there are and such thatfor and .
(H4) For any , we suppose that
We now present a list of remarks and examples in which we discuss how our hypotheses and assumptions are better (weaker) hypotheses and assumptions in some of the closely related papers cited in the reference.
Remark 5. We first provide the growth conditions for the nonlinearities of [36, (H3)(ii) on page 4] given bywhere are nonnegative real numbers. However, note that our condition (H3)(i), (ii) of this paper is given byand obviously, this includes (26) as a special case.
On the other hand, we note that our growth conditions for nonlinearity depends on two variables ; however, in [37, (H2)(ii) and (H4)(i)], the corresponding conditions only involve one variable. Finally, our nonlinearities here are allowed to be unbounded from below, which are better than the nonlinearities in [36, 37], which are bounded below due to being semipositone.
Remark 6. Note that (3) is the superlinear condition; i.e., the degree is ; however, for our conditions, the corresponding degree can be any arbitrary positive number. For example, if we take with for , we see that this function does not satisfy the condition (3) if .
Remark 7. In this paper, we use the functions (see (H1) and (H3)) to act on and then estimate the norms of ; however, in [36, 37] the corresponding parts only involve . Moreover, when the nonlinearities in  grow sublinearly at , nonnegative matrices are used to depict the coupling behaviors, yet this is not used in our paper.
Example 8. Let , , . Then , and satisfy (H1). Moreover, we take as follows:where , . We next show that satisfy (H1)(ii) and (H2). Suppose that ; we haveOn the other hand, we also haveandAnd so, (H1)(ii) and (H2) hold.
It follows, from (11), thatand this function may be unbounded from below if and are large enough. So, this function is not applicable in [36, 37]. Moreover, we also note that is a linear function about , and it does not satisfy the condition (3).
Example 9. Let , , and . Then , and satisfy (H3). Moreover, we chose as follows:where , . In what follows, we prove that satisfy (H3)(ii) and (H4). Indeed, if , we haveandOn the other hand, we obtainandConsequently, (H3)(ii) and (H4) hold.
Next, from (11) we see thatand this function may be unbounded from below if and are large enough. So, this function is not applicable in [36, 37]. Moreover, we also note that is a sublinear function about , and it does not satisfy the condition (3).
Theorem 10. Suppose that (H0)-(H2) hold. Then (1) has at least one positive solution.
Proof. There exists a sufficiently large , for which we will prove thatwhere are two given functions. Indeed, if not, there exist and such that , and thenThis implies , and for . Note that with ; this implies for . From (H1) we haveandAs a result, for , we haveThis meansTherefore,Multiply both sides of the above inequality by and sum from to . Then together with Lemma 1(iii) we obtainHence, we haveFrom (16), (40), and we have . This impliesNote that from (16), (40), and , we find . From the definition of , we know , and this implies . Moreover, we may assume , for . Then and . Thus, from the concavity of , we haveThis implies thatOn the other hand, from (41) and Lemma 1(ii) we obtainCombining the above two inequalities, we getNote that triangular inequality and from (H1), , and thus there exists such that .
Consequently, we obtain that and . As a result, we can choose and thus (39) holds true. Consequently, Lemma 3 indicates thatThen we show that