Fixed Point Theorems of Superlinear Operators with Applications

Xu, Shaoyuan; Han, Yan

doi:https://doi.org/10.1155/2022/2965300

Journal of Function Spaces

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Abstract Introduction Preliminaries Applications Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Unique and Non-Unique Fixed Points and their Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 2965300 | https://doi.org/10.1155/2022/2965300

Fixed Point Theorems of Superlinear Operators with Applications

Shaoyuan Xu¹and Yan Han²

Academic Editor: Santosh Kumar

Received16 Mar 2022

Accepted22 Apr 2022

Published18 May 2022

Abstract

In this paper, by using the partial order method and monotone iterative techniques, the existence and uniqueness of fixed points for a class of superlinear operators are studied, without requiring any compactness or continuity. As corollaries, the new fixed point theorems for -convex operators , -convex operators, positive homogeneous operator , generalized -convex operator, and convex operators are obtained. The results are applied to nonlinear integral equations and partial differential equations.

1. Introduction

Linear operators are a kind of operators with good properties and rich theoretical results, which have formed a classical branch in functional analysis. However, in order to solve the fixed point problems involving operators or equations in practical applications, we need a large number of nonlinear operators, including two classes of significant operators, namely, superlinear operators and sublinear operators. Since some of these operators have concavity or convexity, they bring convenience to the related research. The concepts of concave operators and convex operators were proposed in 1960s, which attracted people’s great interest. Many authors obtained a lot of meaningful results, see [1–27]. Among them, -convex operators [12, 17], -convex operators [13], and generalized -convex operators [16] are a very important class of convex operators. It has important applications in many fields. However, it was difficult to study the -convex operators (including positive -homogeneous operators) and -convex operators because they had strong superlinear properties [13] and described nonlinear problems [12]. Until now, the results are still very few and not very ideal (see [7], P457). Therefore, under what conditions, these operators have a unique fixed point remains a very important and meaningful problem.

In [7], a fixed point theorem for a class of superlinear operators was obtained by topological degree method under the condition that there are inverse upward and downward solutions. In [17], using some results of -concave operator, the author transformed the positive -homogeneous superlinear operator into -concave operator and studied the existence and uniqueness of the solutions of positive -homogeneous superlinear operator equations. In [13], the existence of fixed points was investigated when the -convex operators was a strict set contraction. In [16], Zhao and Du obtained the existence of fixed points of generalized -concave operators and generalized -convex operators. As an application, the singular boundary value problems for second order differential equations were discussed. In [10], according to the properties of totally ordered sets, the existence and uniqueness of new positive fixed points for a class of superlinear homogeneous operators were studied in abstract spaces. The results were applied to a class of superlinear Hammerstein-type integral equations.

In this paper, we study a class of superlinear operators without requiring any compactness or continuity and obtain some new fixed point theorems for superlinear operators by using the partial order and the monotone iteration which are different from those mentioned above in the literature. As corollaries, new fixed point theorems for -convex operators , -convex operators, positive homogeneous operator , generalized -convex operator, and convex operators are obtained. The results are applied to nonlinear integral equations and partial differential equations.

2. Preliminaries

Let be a real Banach space and be a subset of , denotes the zero element of and int denotes the interior of . The subset is called a cone if: (i) and , then (ii) and , then .

Given a cone , we define a partial ordering with respect to by if and only if . We shall write if and , while will stand for . A cone is called normal if there is a number such that for all ,

The least positive number satisfying the above inequality is called the normal constant of .

Let , be an operator. If there exists a point such that , then is called a fixed point of in . Let , and , then

is said to be an ordering interval. The operator is said to be increasing; if for any , implies .

Throughout this paper, we always assume that is a real Banach space and is a partial ordering with respect to ; denotes the null element of .

Definition 1 (see [19]). Let . is called a star-shaped subset of the real Banach space ; if for any and , it holds that .
Note that a convex set in the real Banach space with the null element is a star-shaped subset of . Especially, any cone in the real Banach space is a star-shaped subset of .

Definition 2 (see [7]). Let be a star-shaped subset of the real Banach space and be an operator, then (1) is said to be sublinear, if for all and , ;(2) is said to be superlinear, if for all and , .

Definition 3 (see [4, 7]). Let . is called an -concave operator, if
(i) is -positive, that is, , where (ii) For all and , there exists such that where is called the characteristic function of .
Similarly, if in the above definition, (ii) is replaced by the following (ii):
(ii) For all and , there exists such that where is called the characteristic function of ; then, is called an -convex operator.

Definition 4 (see [16]). Let . is called a generalized -concave operator, if
(i) , where (ii) For all and , there exists such that where is called the characteristic function of .
Similarly, if in the above definition, we replace (ii) by the following (ii):
(ii) For all and , there exists such that where is called the characteristic function of ; then, is called a generalized -convex operator.

Definition 5 (see [4, 17]). Let be an operator, . (1) is said to be an -concave operator, if for any and , (2) is said to be an -convex operator, if for any and , (3) is said to be a positive -homogeneous operator, if for any and , .

Remark 6 (see [9]). Any -convex operator must be an -convex operator, where the characteristic function .

Remark 7. Clearly, any -convex operator must be a superlinear operator. Thus, -convex operators and -convex operators are special superlinear operators.

Remark 8. Any generalized -convex operator must be a superlinear operator if for any and where is the characteristic function of . Thus, generalized -convex operators are special superlinear operators under suitable conditions.

Remark 9. Noting is called a convex operator if for all and ; we can easily see that any convex operator satisfying must be a superlinear operator.

3. Main Results

In [18], the author proved that there was no operator which was decreasing and -convex, where . Now, we give some important theorems of increasing superlinear operators, which generalize increasing -convex operators.

Theorem 10. Let be a normal cone in and be an increasing superlinear operator. If there exist and , such that , then the operator has a unique fixed point in . For any and iterated sequence , we have .

Proof. We firstly prove the existence of the fixed point. Let . Since is increasing, we have Take then Equation (10) can be proved by iteration. Indeed, for , we get which means equation (10) holds when . Suppose that equation (10) holds for , that is By the fact that is increasing, we obtain , then which implies . Thus, equation (10) holds for all . Now, we prove that equation (11) is also true. Indeed, if , then that is, (11) holds when . Suppose (11) holds for , i.e., It follows that since is an increasing superlinear operator. Hence, we see that which gives . So, equation (11) holds for all .

Combining equations (9), (10), and (11), for any , we know

By equations (18) and (19) and the normality of , we can check that , which implies that {} and {} are Cauchy sequences in . Then, there exist , such that , , and . Denote . We have by (9). Therefore,

Let in (13), then . This gives ; that is, the operator has a fixed point in .

Next, we prove the uniqueness of the fixed point. If there exists [, ] such that , then . By the monotonicity of , we see , i.e., . It is easy to deduce that , for any 1. So as .

At last, for any [, ], the sequence satisfies

by iteration. Letting , we know ().

Similarly, if the superlinear operator has an upward solution, we have the following result.

Theorem 11. Let be a normal cone in and be an increasing superlinear operator. If there exist and , such that , then the equation has a unique fixed point in . For any and the iterated sequence , we have .

Proof. Let , then For any and , we obtain

Thus, is a superlinear operator which satisfies all conditions of Theorem 10. The conclusions are true by Theorem 10.

Similar to Theorem 10, we immediately get the following result.

Theorem 12. Let be a normal cone in and be an increasing superlinear operator. If there exists such that , , then the operator has a unique fixed point in . For any and iterated sequence , we have .

Proof. We use Theorem 10 to give the proof of Theorem 12. Set . Then, . Since the operator is increasing and , we have . Obviously, we have (otherwise if , then , which implies that , so . This is a contradiction since .
Now letting , we see that So, all conditions of Theorem 10 are satisfied. By Theorem 10, we know that the conclusions of Theorem 12 hold true.

Remark 13. Compared with ([7], Theorem 3.1), in order to obtain the existence and uniqueness of positive fixed points, the superlinear operator in Theorem 10 and Theorem 11 does not need any compactness or continuity. It is quite different from [7] (Theorem 3.1), which required that is a condensing operator.

Remark 14. Since superlinear operators include three classes of operators: generalized -convex operators, -convex operators, and -convex operators, Theorem 10 and Theorem 11 improve or generalize lots of famous results in [5, 7, 9, 12–17].

Corollary 15. Let be a normal cone in and be an increasing -convex operator. If there exist and , such that , then the operator has a unique fixed point in . For any and iterated sequence , we have .

Corollary 16. Let be a normal cone in and be an increasing -convex operator. If there exist and , such that , then the equation has a unique fixed point in . For any and the iterated sequence , we have .

Corollary 17. Let be a normal cone in and be an increasing -convex () operator. If there exist and , such that , then the operator has a unique fixed point in . For any and the iterated sequence , we have .

Corollary 18. Let be a normal cone in and be an increasing -convex () operator. If there exist and , such that , then the equation has a unique fixed point in . For any and the iterated sequence , we have .

Corollary 19. Let be a normal cone in and be an increasing positive homogeneous operator. If there exist and , such that , then the operator has a unique fixed point in . For any and the iterated sequence , we have .

Corollary 20. Let be a normal cone in and be an increasing positive homogeneous operator. If there exist and , such that , then the equation has a unique fixed point in . For any and the iterated sequence , we have .

Corollary 21. Let be a normal cone in and be an increasing generalized -convex operator satisfying for any and where is the characteristic function of . If there exist and , such that , then the operator has a unique fixed point in . For any and iterated sequence , we have .

Corollary 22. Let be a normal cone in and be an increasing generalized -convex operator satisfying for any and where is the characteristic function of . If there exist and , such that , then the equation has a unique fixed point in . For any and the iterated sequence , we have .

Corollary 23. Let be a normal cone in and be an increasing convex operator satisfying . If there exist and , such that , then the operator has a unique fixed point in . For any and iterated sequence , we have .

Corollary 24. Let be a normal cone in and be an increasing convex operator satisfying . If there exist and , such that , then the equation has a unique fixed point in . For any and the iterated sequence , we have .

Remark 25. In Corollary 15 and Corollary 16, the existence and uniqueness of positive fixed points are proved, without appealing to the monotonicity or any compactness and continuity of the -convex operator . This is very different from [9] (Theorem 9), which required that there existed homogeneous increasing functional . In addition, Corollary 15 and Corollary 16 in the paper are quite different from [14] (Corollary 2.4), which only obtained the existence of positive fixed points while the condition required the strong condition of that there existed such that with .

Remark 26. In Corollary 17 and Corollary 18, the existence and uniqueness of positive fixed points are proved, without appealing to the monotonicity of -convex operator or any compactness and continuity of the operator . This is very different from [12] (Theorem 9), [8] (Theorem 2), and [15] (Theorem 1.3), which required that there existed a linear operator which satisfied certain conditions, and the increasing -convex operator () was completely continuous, respectively.

Remark 27. In Corollary 19 and Corollary 20, the positive -homogeneous operator does not need to have any compactness or continuity, but Theorem 1 in [17] requested that the -homogeneous operator can be decomposed into , where was an increasing positive functional and was an increasing operator in . Therefore, the methods and techniques of Corollary 19 and Corollary 20 are different from those of [17] (Theorem 1).

Remark 28. In this paper, we use the partial order and the monotone iteration to study the fixed point theorems of superlinear operators in Banach spaces. The methods and techniques are different from those used in the literature [7–10, 12, 14, 15, 17], but the existence and uniqueness of the fixed points and the convergence of the iterative sequences of superlinear operators are obtained.

4. Applications

Now, we give some examples to show the applications of our main results in nonlinear integral equations and partial differential equations.

Example 1. Let . Consider Hammerstein integral equation

Conclusion 29. Let be a nonnegative continuous function. If there exists a constant and two continuous functions satisfying , , and Then, equation (26) has a unique solution satisfying . For any which satisfies , the iterated sequence uniformly converges to in .

Proof. Let be a bounded continuous function space in . Define , then is a Banach space. Let denote all nonnegative continuous functions in , then is a normal cone in . We claim that is a homogeneous operator. In fact, by equation (26), we have which means is a homogeneous operator. It is clear that satisfies all conditions of Theorem 10. The conclusion is true.

Similarly, we also have the following.

Example 2. Let . Consider Hammerstein integral equation (see the equation (9) in [10])

Conclusion 30. Let be a nonnegative continuous function. If there exists a constant and two continuous functions satisfying , , and Then, equation (30) has a unique solution satisfying . For any which satisfies , the iterated sequence uniformly converges to in .

Remark 31. In Example 2, we obtain the existence of positive solutions of the integral equation (30), without requiring that the integral kernel can be decomposed into (see condition C1 in [10]). The methods and techniques used in this paper are different from those in [10].

Example 3. Let be a bounded convex domain in whose boundary belongs to for some . Consider the Dirichlet problem where is nonnegative and continuous on and and i.e., there exists a positive constant such that Here, all functions , and belong to (see [3]).
Finding the solution of the above problem is equivalent to finding the fixed point of the integral operator : where is the corresponding Green function, which satisfies Hence (see Guo and Lakshmikantham [4]), the linear integral operator is a completely continuous operator from into , and therefore, operator maps into and is completely continuous, where is a normal cone of space .

Conclusion 32. Let the function be increasing and satisfy If there exist and , such that for some , then the Dirichlet problem has a unique fixed point in .

Proof. Firstly, we prove that the operator is -convex, where Here, we need to use a conclusion about integral operator (17), which can be found in Amann [2]: linear integral operator (17) is -positive, i.e., for any , there exist and such that i.e., Now, let . Then, there exists an such that , and it follows from (39) that Consequently, , where denotes the Nemitskyi operator: Thus, from (41), we know that there exist and such that i.e., satisfies condition (i) of Definition 4.
Next, suppose satisfying and . Since for any , we have by (39) and hence, by (41), there exists such that On the other hand, it is clear that where It follows therefore from (46) and (47) that i.e., , where . Thus, the operator satisfies condition (ii) of Definition 4, and therefore, is -convex.
Take , then and . Therefore, all conditions of Corollary 15 are satisfied. By Corollary 15, we see that the conclusion is true.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is partially supported by the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities’ Association (No. 202101BA070001-045).

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Copyright © 2022 Shaoyuan Xu and Yan Han. 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.

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