Nonlinear Sum Operator Equations with a Parameter and Application to Second-Order Three-Point BVPs

Abstract

A class of nonlinear sum operator equations with a parameter on order Banach spaces were considered. The existence and uniqueness of positive solutions for this kind of operator equations and the dependence of solutions on the parameter have been obtained by using the properties of cone and nonlinear analysis methods. The critical value of the parameter was estimated. Further, the application to some nonlinear three-point boundary value problems was given to show the significance of the discussion.

1. Introduction and Preliminaries

The aim of this paper is to investigate the existence and uniqueness of positive solution for the following operator equations: where is an operator with concavity, is a pseudo subhomogeneous operator, and is a parameter. In addition, by applying our results to the second-order three-point boundary value problem (BVP), where , and ; we obtain that there exists a , such that BVP(2) has a unique positive solution and no positive solution for and , respectively. In particular, such a positive solution of BVP(2) is increasing and continuous in for and . Further, we estimate the critical value .

In recent years, many authors focus on multipoint boundary value problems for differential equations, since these problems arise in a variety of different areas of applied mathematics and physics (see [1]). For example, by using degree-theoretic arguments, Gupta [2] obtained the existence and uniqueness theorems for the following three-point boundary value problem: Since then, the existence of solutions for nonlinear multipoint boundary value problems has been studied by many authors (see [3–8] and their references). The cases with special boundary value conditions in BVP(2) were discussed by [2–7], where and or and . However, to the best of our knowledge, little has been done for the multipoint boundary value problem (2) with parameter, perturbed loading force, and nonlinear boundary conditions, especially on the existence and uniqueness of positive solution and the dependence of solutions on the parameter .

It is well known that fixed point theory is an effective tool in the treatment of existence results of boundary value problems for nonlinear differential equations. Many researchers were concerned with the existence and uniqueness of fixed point for concave operators and sum operators. For example, [9–15] investigated eigenvalue problems of concave operators, the existence and uniqueness of positive fixed point for concave operators, and the existence and uniqueness of positive fixed point for sum operator, respectively. However, to our knowledge, few of the results in literature can be applied to BVP(2) successfully. The above reasons stimulate us to do this work.

First, we consider the existence and property of positive solutions for nonlinear operator equations (1) on order Banach space .

To be clear, some definitions, notations, and lemmas are presented as follows.

Let be a real Banach space which is partially ordered by a cone of , that is, if and only if , and let be the zero element of . If and , then we denote or . A nonempty closed convex set is a cone if it satisfies (i) ; (ii) . A cone is said to be normal if there exists a positive number , called the normal constant of , such that, for all implies .

Let . An operator is said to be increasing if, for . An element is called a fixed point of if . For details on cone theory, see [16, 17].

Given , let Then and

Lemma 1. Let be a normal cone in and an increasing operator. Suppose that(H1);(H2)for any and there exists such that Then has a unique fixed point in if and only if there exist such that . Moreover, for any initial value and a sequence , one has .

The proof of Lemma 1 is standard; we omit it here.

Definition 2 (see [14]). An operator is said to be generalized -concave if it satisfies (H1) and the following condition:(H3)there exists an such that
From Lemma 1 it is easy to show that the following lemma holds.

Lemma 3 (see [15]). Let be a normal cone of and increasing generalized -concave. Then has a unique fixed point in . Moreover, for any initial value and a sequence , one has .

In what follows, we introduce definitions of pseudo subhomogeneous operator and pseudo generalized -concave operator.

Definition 4. An operator is said to be pseudo subhomogeneous if it satisfies(H4);(H5).

Definition 5. An operator is said to be pseudo generalized -concave if it satisfies (H3) and (H4).

Remark 6. An increasing pseudo subhomogeneous operator and an increasing pseudo generalized -concave operator may have no fixed point in . For example, Let , and . Obviously, is increasing pseudo subhomogeneous, but has no fixed point in . Let . Clearly, is increasing pseudo generalized -concave. Since for , then for all . Therefore, is not generalized -concave and has no fixed point in .

Remark 7. From Definitions 4 and 5, it is clear that a pseudo generalized -concave operator is a pseudo subhomogeneous operator.

Remark 8. It is easy to show that (8) is equivalent to and (H5) is equivalent to

2. Positive Solutions of Operator Equation

In this section, we assume that is a real Banach space, is a normal cone in with the normal constant , , and are increasing operators.

Theorem 9. Assume that is a generalized -concave operator and is a pseudo subhomogeneous operator. Then the following four results are true.(i)There exists a such that (1) has a unique solution in for . For any initial value , set ; then .(ii)Equation (1) has no solution in for .(iii) is increasing in for .(iv) is continuous with respect to for .

Proof. By Lemma 3, has a unique fixed point and
If for any . Set ; it is obvious that conclusions (i)–(iv) hold.
If , there exists such that . By (6), for any , there exists such that . Moreover, Definition 4 shows that there exists such that The facts that and imply that
Next, we prove all statements by five steps.
Step  1. Existence of the critical value . Set and . Now we show that .
Define a mapping by By (13) it is obvious that . In addition, for any , , we have , which implies that . That is, is increasing in for .
For a given , set . Then and . It follows from Remark 8 that This means that We assert that . If , from (15) it is obvious that . Suppose that and . Then again by (15) there exists with such that . Set for a given . Then and . Note that ; we can take a number sufficiently small such that From (16)–(19) and the fact that , we obtain which means that . This contradicts the definition of . Therefore, we conclude that
Step  2. Conclusion (i) holds. For given , consider (1). Since and are increasing, is increasing in . Moreover, combining (12) and (15) gives Besides, by (H2) and (H4) we obtain where For any with , (H1) implies that . Further, there exists such that . Hence, Evidently, Therefore, let ; then it follows from (23) and (26) that The application of Lemma 1 concludes the proof of (i).
Step  3. Conclusion (ii) holds. Suppose that there exists such that has a solution in . Since , by (6) there exists such that . Set . Then which means that . Equation (21) implies that , which is a contradiction to the hypothesis .
Step  4. Conclusion (iii) holds. Let with . Then , and, further, . By the proof of conclusion (i), has a unique solution in , which implies that . Thus, .
Step  5. Conclusion (iv) holds. Let . By conclusion (iii) we have Let Then is nondecreasing in for , and We assert that if, otherwise, there exists a sequence such that By (23), (24), and (30), we obtain Therefore, That is, Taking the limit , we get which is a contradiction. So, (32) holds.
From (29), (31), and (32), we have which implies that .
A similar argument shows that, for any as . Thus, conclusion (iv) holds. The proof of Theorem 9 is complete.

Noting (18) and (21), we can easily obtain the following result.

Theorem 10. Assume that the hypotheses in Theorem 9 hold. If , then in Theorem 9 satisfies where are defined by (16).

Corollary 11. Assume that is a generalized -concave operator and is a pseudo generalized -concave operator. Then(i)(1) has a unique solution for . Moreover, for any , set ; then ;(ii) is increasing in for ;(iii) is continuous with respect to for ;(iv)either (the unique fixed point of in ) or .

Proof. From Definitions 2 and 4, it is clear that . Since and are increasing, is increasing in for any . Let ; then Thus, conclusion (i) follows from Lemma 3. Similar to the proofs of Theorem 9, the proofs of (ii) and (iii) can be completed.
Note that and, therefore, normality of implies that conclusion (iv) holds. This ends the proof.

Theorem 12. Assume that is a pseudo generalized -concave operator and is a generalized -concave operator. Then,(i)(1) has a unique solution for . Moreover, for any , set ; then ;(ii) is increasing with respect to for ;(iii) is continuous with respect to for ;(iv).

Proof. Similar to the proof of Corollary 11, the proofs of (i), (ii), and (iii) can be completed. To prove (iv), let be the unique solution of (1) with . From conclusion (ii) of this theorem, we obtain Therefore, which implies that and . This ends the proof.

Next, we discuss the case of (1) with ; that is, which can be widely applied to various problems for differential equations.

Theorem 13. Assume that and is a pseudo subhomogeneous operator. Then(i)there exists a such that (45) has a unique solution for . Moreover, for any and a sequence , one has ;(ii)(45) has no solution in for ;(iii) is increasing in for ;(iv) is continuous with respect to for ; moreover, ;(v)further, if for and is completely continuous, then .
Here .

Proof. Define an operator by for . Then is increasing and is the unique fixed point of in .
Conclusions (i)–(iv) can be proved similarly to the proof of Theorem 9. We only prove conclusion (v) by considering the following two cases.
Case  1  . Note that for ; it is clear that .
Case  2  . In this case, suppose, to the contrary, that . Then, there exists a nondecreasing sequence and a constant such that and .
Since is completely continuous, there exist and subsequence such that . Taking the limitation to both sides of we have . Equation (46) implies that The relation shows that . Further, . This means that . Therefore, (47) gives rise to the contradiction . This finishes the proof.

Corollary 14. Assume that and is pseudo generalized -concave. Then,(i)(45) has a unique solution for . Moreover, for any , set ; then ;(ii) is increasing with respect to for ;(iii) is continuous with respect to for ;(iv)either or .

Proof. It is obvious that is increasing in and for any . In addition, This means that is generalized -concave. Thus, conclusion (i) follows from Lemma 3. Similarly to the proof of Corollary 11, the proofs of (ii), (iii), and (iv) can be completed.

From the proofs of Theorem 12, Theorem 13, and Corollary 14, we easily prove the following results.

Theorem 15. Assume that and is a generalized -concave operator. Then,(i)(45) has a unique solution for . Moreover, for any , set ; then ;(ii) is increasing in and for ;(iii) is continuous with respect to for ;(iv) and .

Remark 16. Different from Theorem 9 and Corollary 11, even if (1) satisfies the conditions in Theorem 12, (1) may not have a fixed point in when .

3. Three-Point Nonlinear Boundary Value Problem

In this section, based on the discussion of the previous section, we study the existence and uniqueness of positive solutions for the three-point BVP(2) and the dependence of solutions on the parameter .

In what follows, set , the Banach space of continuous functions on with the norm . Consider that . It is clear that is a normal cone with the normality constant 1; is given as in (5) with . Let It is easy to prove that Define two operators and by It is clear that a positive solution of BVP(2) is equivalent to nontrivial solution of (1) in .

The following lemma can be proved easily by the Ascoli-Arzela theorem.

Lemma 17. is completely continuous.

The following hypotheses are needed in this section.(L1) is continuous.(L2) is increasing in for fixed and (L3) is continuous and .(L4) is increasing in and there exists a function such that (L5) is continuous.(L6).

Lemma 18. Suppose that (L1) and (L2) hold. Then is increasing and satisfies(i);(ii).

Proof. From (50), (52), and (L1), it is clear that . By (L2) we obtain that is an increasing operator and, for any , where , and The proof is complete.

Theorem 19. Suppose that (L1)–(L5) hold. Then there exists such that BVP(2) has a unique positive solution in for and has no solution in for . Furthermore such a solution satisfies the following properties:(i)for any , set and then uniformly converges to on ;(ii) is nondecreasing in for ;(iii) is continuous with respect to for .

Proof. Consider and defined by (51) and (52). By (L3)–(L5), is increasing. For any , from (50) we have which means that . In addition, for any and , by (L4) we obtain So, the operator is generalized -concave. Lemma 18 implies that is pseudo subhomogeneous. The conclusion follows from Theorem 9. The proof is complete.

In the following, we consider three special cases of BVP(2).

Case 1. BVP(2) has no perturbation; that is, .
From the proof of Theorem 19, we have the following result.

Theorem 20. Suppose that (L1)–(L4) hold. Then there exists such that have a unique positive solution for and no solution in for . Furthermore such a solution satisfies the following properties:(i)for any , set and then uniformly converges to on ;(ii) is nondecreasing in for ;(iii) is continuous with respect to for .

Case 2. The nonlinear boundary value control function in BVP(2) reduces to the linear one .

Theorem 21. Suppose that (L1), (L2), and (L5) hold. Then, there exists such that have a unique positive solution for and no solution in for ; furthermore such a solution satisfies the following properties:(i)for any , set and then uniformly converges to on ;(ii) is nondecreasing in for ;(iii) is continuous with respect to for ;(iv).

Proof. Define operator as (52) and let Then . By Lemmas 17 and 18, is completely continuous and is pseudo subhomogeneous, and for . The application of Theorem 13 completes the proof.

Case 3. The nonlinear boundary value control function in BVP(2) vanishes; that is, for .

Theorem 22. Suppose that (L1), (L2), (L5), and (L6) hold. Then there exists such that have a unique positive solution for and no solution in for ; furthermore such a solution satisfies the following properties:(i)for any , set and then uniformly converges to on ;(ii) is nondecreasing in for ;(iii) is continuous with respect to for ;(iv).

Proof. Define operator as (52) and let Note the monotonicity of in ; it follows that which together with (50), (L5), and (L6) leads to . By Lemma 18, is pseudo subhomogeneous and for . The application of Theorem 13 finishes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Sincere thanks are due to the referees for their comments and suggestions. This research was supported by the NNSF of China (11361047, 10961020), the Science Foundation of Qinghai Province of China (2012-Z-910), Research Project Supported by Shanxi Scholarship Council of China (2013-102), and the University Natural Science Research Development Foundation of Shanxi Province of China (20111021, 2013156).