Journal of Function Spaces
Volume 2020 (2020), Article ID 3527430, 7 pages
https://doi.org/10.1155/2020/3527430

Research Article

A Fixed-Point Theorem for Ordered Contraction-Type Decreasing Operators in Banach Space with Lattice Structure

Chenguang Wang,1,2 Jinxiu Mao,³ and Zengqin Zhao³

¹School of Mathematics, Shandong University, Jinan, Shandong 272100, China
²Department of Mathematics, Jining University, Qufu, Shandong 273155, China
³School of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Jinxiu Mao; maojinxiu1982@163.com

Received 31 March 2020; Accepted 1 May 2020; Published 5 June 2020

Guest Editor: Chuanjun Chen

Copyright © 2020 Chenguang Wang 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.

Abstract.

In this work, we mainly improve the results in Amini-Harandi and Emami (2010). By introducing a new kind of ordered contraction-type decreasing operator in Banach space, we obtain a unique fixed point by using the iterative algorithm. An example is also presented to illustrate the theorem.

1. Introduction

In this work, we obtain a unique fixed point for a kind of ordered contraction-type decreasing operator in Banach space by using the iterative algorithm. The fixed-point study is mainly focused on two aspects. On the one hand, it is about the research of contraction-type mapping, for example, in [1–8]. On the other hand, it is about the study of monotone operators with concavity and convexity, for example, in [9–15]. There is little research on operators that only satisfy the partial-order constrictions. Applications of operator theory in fractional differential equations can be seen in [16–43].

The following generalization of Banach’s contraction principle is due to Geraghty [44].

Lemma 1. Let be a complete metric space and let be a map. Suppose there exists such that for each ,

where denotes the class of those functions which satisfy the condition . Then has a unique fixed-point , and converges to , for each .

Very recently, Amini-Harandi and Emami [6] proved the following existence theorem which is a version of Lemma 1 in the context of partially ordered complete metric spaces:

Lemma 2. Let () be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be an increasing map such that there exists an element with . Suppose that there exists such that

Assume that either is continuous or is such that if an increasing sequence in , then .
Besides, if for each , there exists , which is comparable to and .
Then has a unique fixed point.

We found that in (2), the contraction is concerning a metric. But in fact, the relation of partial order does not play any role in (2). That is to say, in (1), the constriction is not effective because . A question appears naturally in the authors’ minds: “Can the contraction condition be merely about partial order so that can be effective?” The authors have been haunted by this question since it was found. Driven by this idea, we introduce a new kind of ordered contraction-type decreasing operator in Banach space with lattice structure and obtain a unique fixed point of the operator. Our results are helpful and meaningful for studies of fixed point. Comparing to [6], our improvements are in three aspects.

First, the contraction is merely about partial order, and the relation of partial-order does play an important role in the contraction condition. This has never been seen. Second, we consider the situation when the operator is decreasing. Third, we only use the iterative algorithm, and we can start the iterative process with any initial point, i.e., we do not need any assumptions of the existence of upper or lower solutions. An example is also presented to illustrate the theorem.

The outline of this paper is as follows. In the remainder of this section, we will give some preliminaries. In Section 2 of this paper, we present the existence and uniqueness theorem. In Section 3, an example is illustrated.

Definition 3 (see [45]). Let be a real Banach space. A nonempty convex closed set is called a cone if

(i)(ii); is the zero element in

In the case that is a given cone in a real Banach space , a partial order “” can be induced on by . The cone is called normal if there exists a constant , such that for all implies that . Details about cones and fixed point of operators can be found in [45, 46].

Definition 4 (see [47, 48]). We call a set a lattice under the partial ordering , if sup and inf exist for arbitrary .

Lemma 5 (see [45]). A cone is normal if and only if there exists a norm in which is equivalent to such that for any is monotone. The equivalence of means that there exist such that .

Lemma 6 (see [45]). Let be a normal cone in a real Banach space . Suppose that is a monotone sequence which has a subsequence converging to , then also converges to . Moreover, if is an increasing sequence, then ; if is a decreasing sequence, then .

2. The Main Results

We suppose that is a partially ordered Banach space. is a normal cone. The partial-order “” on is induced by the cone .

Let denote the class of those functionals which satisfy the condition

Theorem 7 (main theorem). Suppose that is a closed subset, . is a lattice. is a decreasing operator and satisfies the following ordered contraction condition:
(H) Suppose that there exists such that

Then has unique fixed-point . Moreover, constructing successively sequence

for any initial , we have

Remark 8. Here, we study the decreasing operator while most of the contractions are about increasing operators. And the contraction condition (4) is merely about the partial order, while most of the contractions are about metric.

Remark 9. Two elements and in an ordered set are said to be comparable if either , and we denote it as .

Proof. Let , we have . So we have the following two cases.

Case 1. When is comparable with . Firstly, without loss of generality, we suppose that

If , then the proof is finished. Suppose that . Since is decreasing, we obtain , and it is easy to prove that is increasing. Using the contractive condition (4), we have

hence,

From (4), we have

Let

From (9) and (10), we have the following two conclusions:

(a)There exists a functional such that for with

(b)There exists such that

We assert that the operator has unique fixed point in . And the unique fixed point of is also the unique fixed point of . In order to be clear, we divide the process of proof into three steps.

Step 1. We will use the method of iteration to construct a fixed point of . In fact, consider the iterative sequence

Since and the operator is increasing, we have

This means that is an increasing sequence. It follows from (12) that

Since is normal, from the equivalence of in Lemma 5 we have

Then, is a decreasing sequence and bounded as follows. So

Assume . Then, from (16), we have

The above inequality yields

And implies . Then,

and .

Now we show that is a Cauchy sequence. On the contrary, assume that

For any fixed natural number . From (16), by the triangle inequality

Hence, we have .

Since and , then

from which we obtain

But since , we get

This contradicts (21) and shows that is a Cauchy sequence in .

Since is closed, we can suppose that there exists a such that

Since is normal, (14) together with Lemma 5 implies that

(26), together with (12) and the equivalence of and , implies that

So

Let , we obtain . So

This proves that is a fixed point of in and

Step 2. We will obtain the uniqueness of the fixed point of . On the contrary, if is another fixed point of , we will get .

In fact, the first case, whenis comparable with, without loss of generality, we suppose that. Since is increasing,

Moreover, by (12)

and so

Consequently, the sequence

is nonnegative and decreasing, and so . Now we show that

On the contrary, assume that . By passing to subsequences, if necessary, we may assume that exists. From (31), we obtain , and so . Since , then , and .

This contradiction proves .

It can be obtained that

The second case, when cannot compare with . Fromwhich is a lattice, we obtain

satisfying

i.e., is comparable with and is comparable with . Since is increasing, we know

Moreover, by (12)

So we have

Similar to the process of (31), (32), and (33),

From (36), we have

So from (39) and (40), we get

(33) together with (41) implies that is unique fixed point of .

Step 3. We will point that the unique fixed point of is also the unique fixed point of . Since

Thus,

i.e., . From the uniqueness of the fixed point of , we know

So is the unique fixed point of in .

Case 2. Another case, when is not comparable to . From which is a lattice, we know there exists such that . That is, . Since is a decreasing operator, we have

This shows that

For any initial , constructing successively sequence

From , we can get

From (28), we know

Since and from the arbitrary of in (28), we obtain

(49) and (50) imply that

holds. Similarly to the proof of Step 2 and Step 3 in case 1, we get that is the unique fixed point of .

3. An Example

Let , equipped with usual normal and usual partial order . . Then, . is a normal cone in . is a partially ordered Banach space. And is a lattice under the partial order induced by the cone .

Then, satisfying the assumptions of Theorem 7. Define the mapping by where is a fixed real number. Then, is nonincreasing. Define by , then . Now, for all with , we have

so that and satisfy the assumption of Theorem 7. Observing that all the other conditions of Theorem 7 are also satisfied, has a unique fixed-point . Moreover, constructing successively sequence

for any initial , we have

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Supported by the Natural Science Foundation of China (11571197), Shandong Province Higher Educational Science and Technology Program (J17KB143), Shandong Education Science 13th Five-Year Plan Project (BYK2017003), and Jining University Youth Research Foundation (2015QNK102).

References

J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, English Series, vol. 23, no. 12, article 769, pp. 2205–2212, 2007.

J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2518, 2007.

D. O'Regan and A. Petruşel, “Fixed point theorems for generalized contractions in ordered metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1241–1252, 2008.

M. Borkowski, D. Bugajewski, and M. Zima, “On some fixed-point theorems for generalized contractions and their perturbations,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 464–475, 2010.

J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis, vol. 71, no. 7-8, pp. 3403–3410, 2009.

A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2238–2242, 2010.

B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for α–ψ-contractive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012.

Z. Kadelburg, M. Pavlovic, and S. Radenovic, “Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3148–3159, 2010.

C. Zhai and L. Wang, “φ − (h,e)-concave operators and applications,” Mathematical Analysis and Applications, vol. 454, no. 2, pp. 571–584, 2017.

Z. Q. Zhao, “Fixed points of τ–φ-convex operators and applications,” Applied Mathematics Letters, vol. 23, no. 5, pp. 561–566, 2010.

X. Li and Z. Zhao, “On a fixed point theorem of mixed monotone operators and applications,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 94, no. 94, pp. 1–7, 2011.

Z. Zhao, “Existence and uniqueness of fixed points for some mixed monotone operators,” Nonlinear Analysis, vol. 73, no. 6, pp. 1481–1490, 2010.

L. Wang, B. Liu, and R. Bai, “Stability of a mixed type functional equation on multi-Banach spaces: a fixed point approach,” Fixed Point Theory and Applications, vol. 2010, no. 1, Article ID 283827, 9 pages, 2010.

L. Wang, “Intuitionistic fuzzy stability of a quadratic functional equation,” Fixed Point Theory and Applications, vol. 2010, no. 1, Article ID 107182, 7 pages, 2010.

T. Wang and Z. Hao, “New results of mixed monotone operator equations,” Topol. Methods Nonlinear Anal, vol. 53, p. 1, 2019.

D. Kong, L. Liu, and Y. Wu, “Isotonicity of the metric projection with applications to variational inequalities and fixed point theory in Banach spaces,” Journal of Fixed Point Theory and Applications, vol. 19, no. 3, pp. 1889–1903, 2017.

L. Liu, F. Sun, X. Zhang, and Y. Wu, “Bifurcation analysis for a singular differential system with two parameters via to topological degree theory,” Nonlinear Analysis, vol. 2017, no. 1, pp. 31–50, 2017.

X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4680–4691, 2013.

X. Zhang, L. Liu, Y. Wu, and B. Wiwatanapataphee, “The spectral analysis for a singular fractional differential equation with a signed measure,” Applied Mathematics and Computation, vol. 257, pp. 252–263, 2015.

J. Mao, Z. Zhao, and C. Wang, “The exact iterative solution of fractional differential equation with nonlocal boundary value conditions,” Journal of Function Spaces, vol. 2018, Article ID 8346398, 6 pages, 2018.

J. Mao, Z. Zhao, and C. Wang, “The unique positive solution for singular Hadamard fractional boundary value problems,” Journal of Function Spaces, vol. 2019, Article ID 5923490, 6 pages, 2019.

J. Mao, Z. Zhao, and C. Wang, “The unique iterative positive solution of fractional boundary value problem with q-difference,” Applied Mathematics Letters, vol. 100, article 106002, 2020.

C. Chen, X. Zhang, G. Zhang, and Y. Zhang, “A two-grid finite element method for nonlinear parabolic integro-differential equations,” International Journal of Computer Mathematics, vol. 96, no. 10, pp. 2010–2023, 2018.

C. Chen, W. Liu, and C. Bi, “A two-grid characteristic finite volume element method for semilinear advection-dominated diffusion equations,” Numerical Methods for Partial Differential Equations, vol. 29, no. 5, pp. 1543–1562, 2013.

C. Chen and X. Zhao, “A posteriori error estimate for finite volume element method of the parabolic equations,” Numerical Methods for Partial Differential Equations, vol. 33, no. 1, article 259C275, pp. 259–275, 2017.

C. Chen, K. Li, Y. Chen, and Y. Huang, “Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations,” Advances in Computational Mathematics, vol. 45, no. 2, pp. 611–630, 2019.

C. Chen, H. Liu, X. Zheng, and H. Wang, “A two-grid MMOC finite element method for nonlinear variable-order time-fractional mobile/immobile advection–diffusion equations,” Computers & Mathematcs with Applications, vol. 79, no. 9, pp. 2771–2783, 2020.

X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.

X. Zhang, Y. Wu, and L. Caccetta, “Nonlocal fractional order differential equations with changing-sign singular perturbation,” Applied Mathematical Modelling, vol. 39, no. 21, pp. 6543–6552, 2015.

X. Zhang, L. Liu, Y. Wu, and Y. Cui, “A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator,” Nonlinear Analysis: Modelling and Control, vol. 25, no. 1, pp. 126–143, 2020.

X. Zhang, J. Xu, J. Jiang, Y. Wu, and Y. Cui, “The convergence analysis and uniqueness of blow-up solutions for a Dirichlet problem of the general k-Hessian equations,” Applied Mathematics Letters, vol. 102, article 106124, 2020.

X. Zhang, J. Jiang, Y. Wu, and Y. Cui, “The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach,” Applied Mathematics Letters, vol. 100, article 106018, 2020.

J. He, X. Zhang, L. Liu, Y. Wu, and Y. Cui, “A singular fractional Kelvin–Voigt model involving a nonlinear operator and their convergence properties,” Boundary Value Problems, vol. 2019, no. 1, 2019.

X. Zhang, J. Jiang, Y. Wu, and Y. Cui, “Existence and asymptotic properties of solutions for a nonlinear Schrödinger elliptic equation from geophysical fluid flows,” Applied Mathematics Letters, vol. 90, pp. 229–237, 2019.

X. Zhang, L. Liu, Y. Wu, and Y. Cui, “The existence and nonexistence of entire large solutions for a quasilinear Schrödinger elliptic system by dual approach,” Journal of Mathematical Analysis and Applications, vol. 464, no. 2, pp. 1089–1106, 2018.

X. Zhang, L. Liu, Y. Wu, and Y. Cui, “Existence of infinitely solutions for a modified nonlinear Schrödinger equation via dual approach,” Electronic Journal of Differential Equations, vol. 2018, no. 147, pp. 1–15, 2018.

J. He, X. Zhang, L. Liu, and Y. Wu, “Existence and nonexistence of radial solutions of the Dirichlet problem for a class of general k-Hessian equations,” Nonlinear Analysis: Modelling and Control, vol. 23, no. 4, pp. 475–492, 2018.

X. Zhang, Y. Wu, and Y. Cui, “Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator,” Applied Mathematics Letters, vol. 82, pp. 85–91, 2018.

Q. Han, “Compact Sobolev embeddings and positive solutions to a quasilinear equation with mixed nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 481, no. 2, article 123150, 2020.

Q. Han, “Elliptic variational problems with mixed nonlinearities,” Mathematical Methods in the Applied Sciences, vol. 43, pp. 1675–1684, 2019.

Q. Han, “Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions,” Monatshefte für Mathematik, vol. 176, no. 1, pp. 107–141, 2015.

Q. Han, “Compact embedding results of Sobolev spaces and positive solutions to an elliptic equation,” Proceedings of the Royal Society of Edinburgh, vol. 146, no. 4, pp. 693–721, 2016.

Q. Han, “Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations,” Bulletin des Sciences Mathematiques, vol. 141, no. 1, pp. 46–71, 2017.

M. A. Geraghty, “On contractive mappings,” Proceedings of the American Mathematical Society, vol. 40, no. 2, pp. 604–604, 1973.

D. Guo, Y. J. Cho, and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publ, New York, NY, USA, 2004.

M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, The Netherlands, 1964.

J. Sun and X. Liu, “Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 348, no. 2, pp. 927–937, 2008.

X. Lin and Z. Zhao, “Sign-changing solution for a third-order boundary value problem in ordered Banach space with lattice structure,” Boundary Value Problems, vol. 2014, no. 1, 2014.