On fixed point theorems for mixed monotone ope rators without normality of cone

In this paper, we establish existence results of fixed point for mixed monotone operators in a real Hausdorff locally convex topological vector space without normality of cone. Our result is an extension of Z. Zhang- K. Wang.

http://jst.tnu.edu.vn 88 Email: jst@tnu.edu.vn

ON FIXED POINT THEOREMS FOR MIXED MONOTONE OPE RATORS WITHOUT NORMALITY OF CONE

Trinh Van Ha *

TNU - University of Information and Communication Technology

Received: 04/4/2022 Mixed monotone operators were introduced by Dajun Guo and

V Lakshmikantham in 1987 Thereafter many authors have investigated these kinds of operators in Banach spaces In 2009,

Z Zhang- K Wang proved the fixed point theorems for mixed monotone operators in Banach spaces under the normal cone assumption In this paper, we establish existence results of fixed point for mixed monotone operators in a real Hausdorff locally convex topological vector space without normality of cone Our result is an extension of Z Zhang- K Wang.

Revised: 12/5/2022

Published: 16/5/2022

KEYWORDS

Cone

Normal cone

Neighborhood properties

Fixed point

Mixed monotone operators

VỀ ĐỊNH LÝ ĐIỂM BẤT ĐỘNG ĐỐI VỚI TOÁN TỬ ĐƠN ĐIỆU HỖN

HỢP MẶC DÙ KHÔNG CẦN TÍNH CHUẨN TẮC CỦA NÓN

Trịnh Văn Hà

Trường Đại học Công nghệ thông tin và Truyền thông- ĐH Thái Nguyên

THÔNG TIN BÀI BÁO TÓM TẮT

Ngày nhận bài: 04/4/2022 Toán tử đơn điệu hỗn hợp được giới thiệu bởi Dajun Guo và

V Lakshmikantham năm 1987 Sau đó nhiều tác giả đã nghiên cứu các loại toán tử này trong không gian Banach Năm 2009, Z Zhang- K Wang chứng minh định lý điểm bất động đối với toán

tử đơn điệu hỗn hợp trong không gian Banach dưới giả thiết nón chuẩn tắc Trong bài báo này, chúng tôi chứng minh định lý điểm bất động đối với toán tử đơn điệu hỗn hợp trong không gian lồi địa phương Hausdorff không có tính chuẩn tắc của nón Kết quả của chúng tôi là mở rộng kết quả của Z Zhang- K Wang.

Ngày hoàn thiện: 12/5/2022

Ngày đăng: 16/5/2022

TỪ KHÓA

Nón

Nón chuẩn tắc

Tính chất lân cận

Điểm bất động

Toán tử đơn điệu hỗn hợp

DOI: https://doi.org/10.34238/tnu-jst.5686

Email: tvha@ictu.edu.vn

1 Introduction and preliminaries

Mixed monotone operators were introduced by Dajun Guo and V Lakshmikantham in [1] in

1987 Thereafter many authors have investigated these kinds of operators in Banach spaces and obtained a lot of interesting and important results (see [2]-[6]) In this paper, we establish existence results of fixed point for mixed monotone operators in a real Hausdorff locally convex topological vector space without normality of cone

Let E always be a real Hausdorff locally convex topological vector spaces with its zero vector θ and P is subset of E We say that P is a cone in E if

(i) P is closed, nonempty and P ̸= {θ},

(ii) ax + by ∈ P for all x, y ∈ P and non-negative real numbers a, b,

(iii) P ∩ (−P ) = {θ}

For a given cone P in E, we can define a partial ordering ⪯ with respect to P by x ⪯ y if and only if y − x ∈ P , while x ≪ y will stand for y − x ∈ int P , where int P denotes the interior

of P , x ≺ y if and only if x ⪯ y and x ̸= y In this paper, we always suppose E be a real Hausdorff locally convex topological vector spaces, P be a cone in E with int P ̸= ∅ and ⪯ is

Definition 1.1 Let P be a cone in E We say that P satisfies the neighborhood property if for any neighborhood U of θ in E, there is neighborhood V of θ in E such that

(V + P ) ∩ (V − P ) ⊂ U

Remark 1.2 If P has a closed convex bounded base then P satisfies the neighborhood property (see, Proposition 1.8 in [4])

property

lim

n,m→∞(xn− xm) = θ

(iii) E is complete if every Cauchy sequence is converges

Lemma 1.5 Suppose that P satisfies the neighborhood property in a real Hausdorff locally

for all n ≥ 1 and lim

n→∞un= θ

Proof Let U be an arbitrary neighborhood of θ in E Since P satisfies the neighborhood prop-erty, there is neighborhood V of θ in E such that

(V + P ) ∩ (V − P ) ⊂ U

Since lim

n→∞un= θ Definition 1.6 (See [5]) Let P be a cone in normed space Ẹ We say that P is normal if there

is a number M > 0 such that for all x, y ∈ E,

θ ⪯ x ⪯ y implies ∥x∥ ≤ M ∥y∥

Proposition 1.7 Let P be a normal cone in normed space Ẹ Then P satisfies the neighborhood propertỵ

Proof Assume that P does not satisfy the neighborhood propertỵ Then there exists ϵ > 0 such that for any n ≥ 1, we have

1

n)

n→∞un= θ Hence θ ̸∈ B(θ, ϵ) This is a contradiction

Definition 2.1 Let P be a cone in Ẹ We say that A : P ×P → P is a mixed monotone operator

Theorem 2.2 Suppose that the cone P satisfy the neighborhood property in a complete real Hausdorff locally convex topological vector space E and A : P × P → P be a mixed monotone operator satisfying the following conditions:

(i) for each v ∈ P , Ặ, v) : P → P is concave;

(ii) for any u ∈ P , there exists N > 0 such that

Proof We prove the theorem in three steps:

Because A is mixed monotone, B(u, v) is increasing in u and v Moreover, we have B(u, v) = v

This implies that

Thus, we obtain

N + 1

Therefore

N + 1

By Lemma 1.5, we get

lim

On the other hand, for m > n we have

Since lim

lim

m,n→∞(um− un) = θ

lim

n→∞vn For each n ≥ 0, we have

Letting k → ∞, we have

Since B is increasing, we have

By Lemma 1.5, we get

lim

Step 2 We show that

(1) T (.) is increasing;

(2) [(1 − δ)t + δ]T (u) ⪯ T (tu) for all t ∈ [0, 1]

in both variables, we have

Thus, T is increasing On the other hand, for t ∈ [0, 1], by A is concave in the first variable and

A is mixed monotone, we have

[(1 − δ)t + δ]T (u) = tT (u) + (1 − t)δT (u)

⪯ tA(u, T (u)) + (1 − t)A(θ, T (tu))

⪯ tA(u, T (tu)) + (1 − t)A(θ, T (tu))

⪯ A(tu, T (tu))

= T (tu)

Thus,

Hence, lim

lim

we have

lim

n→∞vn= u∗

Remark When E is a Banach space and P is a normal cone in E then Theorem 2.2 becomes Theorem 2.1 in [7]

3 Conclusion

In this paper, we establish existence results of fixed point for mixed monotone operators in a real Hausdorff locally convex topological vector space without normality of cone Our result is

an extension of Z Zhang- K Wang [7]

References

[6] Z.Zhao,"Existence anduniquenessoffixedpointsforsomemixed monotoneoperators",