Some Results on G-Normed Linear Spaces

(1)

第 55 卷第 3 期

2020 年 6 月

JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY

Vol. 55 No. 3 June 2020

ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.3.1 Research article

Mathematics

S

OME

R

ESULTS ON

G-N

ORMED

L

INEAR

S

PACES

G 范数线性空间的一些结果

Maiada Nazar Mohammedali, Raghad Ibraham Sabri, Mohammed Rasheed, Suha Shihab

Applied Science Department, University of Technology

Baghdad, Iraq, [email protected], [email protected],[email protected],

[email protected], [email protected], [email protected], [email protected], [email protected]

Received: February 1, 2020 ▪ Review: February 28, 2020 ▪ Accepted: April 20, 2020 This article is an open access article distributed under the terms and conditions of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/4.0)

Abstract

In the present work, our goal is to define the Cartesian product of two generalized normed spaces depending on the notion of generalized normed space. It is a background to state and prove that the Cartesian product of two complete generalized normed spaces is also a complete generalized normed space. Furthermore, the definition of the pseudo-generalized normed space is introduced and essential concepts related to this space are discussed and proved.

Keywords:Pseudo-Generalized Normed Space, Generalized Normed Space, Cartesian Product

摘要在当前工作中，我们的目标是回顾广义范数空间的概念，以定义两个广义范数空间的笛卡尔积作为陈述的背景，并证明两个完全广义范数空间的笛卡尔积也是一个完全广义范数空间。此外，介绍了伪广义赋范空间的定义，并讨论和证明了与该空间有关的基本概念。

关键词: 伪广义范数空间，广义范数空间，笛卡尔积

I. INTRODUCTION

The solution space of a mathematical problem is a necessary condition for existence. The solution space is the base of establishing the quantitative property of mathematical structure. Hence, the concepts of metric and normed spaces play an increasingly important role in

mathematics and applied sciences. Because of the strong link between the theoretical frameworks of such two spaces were introduced and studied in [1] using different generalizations notations and the theory was subsequently extended to n-normed spaces in [2]. Numerous studies were devoted to these concepts [3], [4], [5], [6], [7],

(2)

[8], [9], [10]. The widely known G-metric space is a generalized metric space first defined by Mustafa and Sims [11], [12], [13], [14], [15] as an alternative to D-metric spaces [16]. On the other hand, to generalize the normed linear space concept, the author proposed the concept of generalized norm (G-norm) [17]. To further develop the theory of G-normed spaces, a study of product spaces in a probabilistic framework is presented in [18]. The first main purpose of the present paper is the definition of the G-normed space used to introduce the Cartesian product of G-normed spaces. Next, we prove that the Cartesian product of two complete G-normed spaces is also a complete G-normed linear space. The second purpose is to introduce the notion of a Pseudo G-normed space and prove a series of important concepts related to it. Other applications of these spaces are illustrated in [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].

Several other fundamental results and useful concepts are discussed in this paper, as follows. In Section Three, the Cartesian product of generalized normed spaces is defined, while in Section four, we introduce the notion of Pseudo G-normed space. In addition, different important properties are discussed and proved; for example, the space ( ) is G-normed when the space ( is Pseudo G-normed.

II. SOME PRELIMINARY RESULTS

This section presents a few necessary definitions and properties regarding G-normed spaces that are used in this work.

Definition 2.1: A real valued function

Ɽ is said to be G-normed on , where is a real linear space, if satisfies the following properties:

(Ɲ1) and if and only if

(Ɲ2) is invariant under permutations of all variables (Ɲ3) for all and Ɽ (Ɲ4) for all (Ɲ5) for all

Then, the space is called a G-normed space.

Definition 2.2: Let be a

G-normed space. A sequence is said to be

convergent if an element _in for a given , such that .

Definition 2.3: Let be a

G-normed space. A sequence is said to be a Cauchy sequence if given , such that

Definition 2.4: The space is

called a complete G-normed space if each Cauchy sequence in converges in

III. COMPLETENESS

OF THE

SPACE

This section is devoted to presenting the representation of the Cartesian product of two G-normed spaces and , followed by proving some fundamental results.

First, we introduce the definition of the Cartesian product of two G-normed spaces, which is the key of our result in this section.

Definition 3.1: The Cartesian product of two

G-normed spaces and

is the product space , where is the Cartesian product of sets and and is a mapping

Ɽ defined by:

for all .

We now state and prove that the Cartesian product of two normed spaces is also a G-normed space.

Theorem 3.2: If and

are two G-normed spaces, then is a G-normed space.

Proof: Suppose that

and . (Ɲ1)

.

Since and , then

and and . (Ɲ2) = Hence,

(3)

for all . (Ɲ3) (Ɲ4) for all (Ɲ5) Therefore, is a G-normed space. Proposition 3.3: If is a sequence in a G-normed space that converges to in and is a sequence in a G-normed space that converges to in , then is a sequence in that converges to in .

Proof: Theorem 3.2 gives us that

is a G-normed space. Now, since converges to , for a given

, such that

.

Since converges to , for a given , such that

. Now for

. Therefore,

. This means that converges to .

Proposition 3.4: Let be a Cauchy

sequence in a G-normed space and a Cauchy sequence in a G-normed space . Then, is a Cauchy sequence in .

Proof: By Theorem 3.2, is

G-normed. Since and are Cauchy sequences, thus, if given , such that

and . Now, for

This means that for

Therefore, is a Cauchy sequence in .

According to the previous propositions, we are now ready to establish the complete G-normed space theorem.

Theorem 3.5: If and

are two complete G-normed spaces, then is also a complete G-normed space.

Proof: The product space

represents a G-normed space (Theorem 3.2). Let be a Cauchy sequence in — that is, given , such that

, which

,

(4)

such that

and .

As a result, is a Cauchy sequence in and is a Cauchy sequence in . Because and are G-normed spaces, in and in converge to and converge to that is,

and . Now

Thus, converges to in —that is, completes the G-normed space.

Theorem 3.6: If is a

G-normed space, then and are G-normed spaces by defining

and for all and .

Proof: (Ɲ1)

, which implies that and

if and only if and if and only if .

(Ɲ2)

Thus,

for all . (Ɲ3)

= for all and

(Ɲ4)

for all

(Ɲ5)

for all .

Therefore, is a G-normed space. In the same way, we can prove that

is a G-normed space.

Theorem 3.7: If is a

complete G-normed space, then

and are complete G-normed spaces.

Proof: We have shown and

are G-normed spaces by way of Theorem 3.6. Suppose that is a Cauchy sequence in —that is, for , , we have . Now for given , such that

, this implies that is Cauchy sequence in . But we have is complete G-normed space hence

converges to in , that is

. Hence

, that is converges to in . Therefore, is complete G-normed space. Similarly we can prove that

is complete G-normed space.

IV. PSEUDO G-NORMED SPACE

In this section, the definition of the concept of pseudo G-normed space are introduced and the main properties pointed out. They are then studied and proved. First, the key results are introduced as follows:

Definition 4.1: A pair is said to be a pseudo G-normed space, where is a linear over the field of real numbers space, a function from to , satisfying the following axioms:

( Ɲ1) for each ( Ɲ2) If, then, ( Ɲ3)

That is, invariant under all variables ( Ɲ4)

for all and Ɽ

( Ɲ5)

(5)

( Ɲ6) for all

Proposition 4.2: Let, be a pseudo

G-normed space. Define the relation, on by and if, and only if, = 0. Then the relation on is a G-equivalence

Proof:

1. The relation satisfies the reflexive property since = 0.

2. The relation satisfies the symmetric property because if we have , then there exists since

.

3. The relation satisfies the transitive property. Suppose that , then ; and . Hence, .

Now,

It follows that , and therefore, .

The following notation gives the space in terms of the space and the relation

Notation 4.3: Let be the real linear space,

and _{be the relation on . Then, the} element in is [ = { .

Lemma 4.4: Let be a pseudo

G-normed space. Then the map : , defined by = , does not depend on the representatives .

Proof: Let , and , then

, hence . This implies that

The axioms of G-normed space is proved with respect to the space ( ) in case a pseudo G-normed space is available for the space ( in the next theorem.

Theorem 4.5: Let ( be a pseudo G-normed space, then ( ) is a G-normed space where .

Proof: We have ( ) as a pseudo G-normed space, so (Ɲ2), (Ɲ3), (Ɲ4), and (Ɲ5) are satisfied to prove that ( ) is a G-normed space. Now we verify the second part of condition (Ɲ1), that is if , then and that is

, , , it means .Therefore ( ) G-normed space.

Proposition 4.6: Let ( is a Pseudo G-normed space, ( ) is a G-normed space. If is an operator defined by

then is a G-isometry.

Proof: We want prove that is well defined

by the following two cases

a- If and then

b- If and then

Based on (a) and (b), and and and

can be derived. From this, it follows that or

= = since

. Therefore, is a G-isometry

To demonstrate that a Pseudo G-normed space ( is G-equivalent to another Pseudo G-normed space ( , the following definition is introduced:

Definition 4.7: A Pseudo G-normed space

( is said to be G-equivalent to a Pseudo G-normed space ( if

a sequence and there exists a point in then if and only if

Theorem 4.8: ( is a Pseudo

G-normed space. If the operator is continuous, then a Pseudo G-normed space ( is present, such that and

are G-equivalent.

Proof: Define

+ for in

Firstly, is a Pseudo G-normed space, based on which is demonstrated

( Ɲ1) since and

( Ɲ2) If then =0 and thus, we get

( Ɲ3) let Ɽ, then

( Ɲ4) it is now shown that, for all

(6)

=

( Ɲ5)

Therefore, is a Pseudo G-normed space

Finally, to demonstrate that ( is G-equivalent to ( .

is a sequence in and a point in , assuming that = 0. Then,

as is continuous. So,

. This means that

Conversely, assuming that , this implies that . It follows that . The calculation is complete.

The result demonstrates the relationship between the finite set of Pseudo G-normed spaces and Pseudo G-normed product space.

Theorem 4.9: Let

be Pseudo G-normed spaces. If , and

, arbitrary elements in the product set The function is subsequently defined by , which is the product Pseudo G-normed on and ( is Pseudo G-normed product space.

Proof: ( Ɲ1) since for all

, and for every , so for all , and for every , hence for

.

Ɲ2) if , this implies that , for every . Subsequently, for every .

Ɲ3) since =

for every which is invariant under all . Therefore,

Ɲ4) let , Since for . Therefore = for all ( Ɲ4) let , , and then Ɲ5)

(7)

The calculation is subsequently complete.

REFERENCES

[1] SAADATI, R. and PARK, C. (2010) Non-Archimedean L-fuzzy normed spaces and stability of functional equations.

Computers & Mathematics with

Applications, 60 (8), pp. 2488-2496.

[2] ANAND, R. and RAJ, K. (2017) Complete paranormed Orlicz Lorentz sequence spaces over n-normed spaces.

Journal of the Egyptian Mathematical Society, 25 (2), pp. 151-154.

[3] CHEN, X.Y. and SONG, M.M. (2010) Characterizations on isometries in linear n-normed spaces. Nonlinear Analysis: Theory,

Methods & Applications, 72 (3-41), pp.

1895-1901.

[4] SHARMA, S.K. and ESI, A. (2013) Some I-convergent sequence spaces defined by using sequence of moduli and n-normed space. Journal of the Egyptian Mathematical

Society, 21, pp. 103-107.

[5] HAZARIKA, B. (2016) Lacunary ideal convergence of multiple sequences in probabilistic normed spaces. Applied Mathematics and Computation, 279, pp.

139-153.

[6] TRIPATHY, B.C., SEN, M., and NATH, S. (2015) Lacunary I-convergence in probabilistic n-normed space. Journal of the

Egyptian Mathematical Society, 23, pp.

90-94.

[7] WANG, D., LIU, Y., and SONG, M. (2012) The Aleksandrov problem on non-Archimedean normed space. Arab Journal of

Mathematical Sciences, 18 (2), pp. 135-140.

[8] SAHINER, A., GURDALl, M., and YIGIT, T. (2011) Ideal convergence characterization of the completion of linear n-normed spaces. Computers & Mathematics

with Applications, 61 (3), pp. 683-689.

[9] DUTTA, H. (2011) On some n-normed linear space valued difference sequences.

Journal of the Franklin Institute, 348 (10),

pp. 2876-2883.

[10] SAHINER, A. and AKIN, H.S. (2011) New sequence spaces in multiple normed

spaces. Applied Mathematics and Computation, 218 (31), pp. 1030-1035.

[11] KUNDU, A., BAG, T., and NAZMUL, S.K. (2019) A new generalization of normed linear space. Topology and its Applications, 256, pp. 159-176.

[12] LIANG, M., ZHU, C., CHEN, C., and WU, Z. (2019) Some new theorems for cyclic contractions in Gb-metric spaces and some applications. Applied Mathematics and

Computation, 346 (C), pp. 545-558.

[13] GABA, Y.U. (2017) Fixed point theorems in G-metric spaces. Journal of

Mathematical Analysis and Applications, 455

(11), pp. 528-537.

[14] SHOAIB, A., ARSHAD, M., RASHAM, T., and ABBAS, M. (2017) Unique fixed point results on closed ball for dislocated quasi G-metric spaces.

Transactions of A. Razmadze Mathematical Institute, 171 (2), pp. 221-230.

[15] AYDI, H., BILGILI, N., and KARAPINAR, E. (2015) Common fixed point results from quasi-metric spaces to G-metric spaces. Journal of the Egyptian

Mathematical Society, 23 (2), pp. 356-361.

[16] PASICKI, L. (2017) Meir and Keeler were right. Topology and its Applications, 228, pp. 382-390.

[17] WU, X., ZHENG, J., WU, C., and CAI, Y. (2013) Variational structure–texture image decomposition on manifolds. Signal

Processing, 93 (7), pp. 1773-1784.

[18] LIU, J., LUO, Y., WANG, L., YANG, X., and GUO, X. (2019) A probabilistic framework for stability assessment of existing spatial structures. Journal of

Constructional Steel Research, 156, pp.

96-104.

[19] DKHILALI, F., MEGDICHE, B.S., RASHEED, M., BARILLE, R., SHIHAB, S., GUIDARA, K., and MEGDICHE, M. (2018) Characterizations and morphology of sodium tungstate particles. Royal Society Open

Science, 5 (18), pp. 1-16.

[20] RASHEED, M. and BARILLE, R. (2017) Optical constants of DC sputtering derived ITO, TiO2 and TiO2: Nb thin films

characterized by spectrophotometry and spectroscopic ellipsometry for optoelectronic devices. Journal of Non-Crystalline Solids, 476, pp. 1-14.

(8)

[21] RASHEED, M. and BARILLE, R. (2017) Room temperature deposition of ZnO and Al: ZnO ultrathin films on glass and PET substrates by DC sputtering technique.

Optical and Quantum Electronics, 49 (5), pp.

1-14.

[22] RASHEED, M. and BARILLE, R. (2017) Comparison the optical properties for Bi2O3 and NiO ultrathin films deposited on

different substrates by DC sputtering technique for transparent electronics. Journal

of Alloys and Compounds, 728, pp.

1186-1198.

[23] AUKŠTUOLIS, A., GIRTAN, M., MOUSDIS, G.A., MALLET, R., SOCOL, M., RASHEED, M., and STANCULESCU, A. (2017) Measurement of charge carrier mobility in perovskite nanowire films by photo-CELIV method. Proceedings of the

Romanian Academy - Series A: Mathematics, Physics, Technical Sciences, Information Science, 18 (1), pp. 34-41.

[24] BOURAS, D., MECIF, A., BARILLE, R., HARABI, A., RASHEED, M., MAHDJOUB, A., and ZAABAT, M. (2018) Cu: ZnO deposited on porous ceramic substrates by a simple thermal method for photocatalytic application. Ceramics International, 44 (17), pp. 21546-21555.

[25] DKILALLI, F., MEGDICHE, S., GUIDARA, K., RASHEED, M., BARILLE, R., and MEGDICHE, M. (2018) AC conductivity evolution in bulk and grain boundary response of sodium tungstate Na2WO4. Ionics, 24 (1), pp. 169-180.

[26] SAIDANI, T., ZAABAT, M., AIDA, M.S., BARILLE, R., RASHEED, M., and ALMOHAMMED, Y. (2017) Influence of precursor source on sol–gel deposited ZnO thin films properties. Journal of Materials

Science: Materials in Electronics, 28 (13),

pp. 9252-9257.

[27] BOUMEZOUED, A., GUERGOURI, K., BARILLE, R., RECHEM, D., ZAABAT, M., and RASHEED, M. (2019) ZnO nanopowders doped with bismuth oxide, from synthesis to electrical application.

Journal of Alloys and Compounds, 791, pp.

550-558.

[28] SAIDAI, W., HFAIDH, N., RASHEED, M., GIRTAN, M., MEGRICHE, A., and MAAOUI, M.E. (2016) Effect of B2O3

addition on optical and structural properties of TiO2 as a new blocking layer for multiple

dye sensitive solar cell application (DSSC).

RSC Advances, 6 (73), pp. 68819-68826.

[29] DKHILALLI, F., BORCHANI, S.M., RASHEED, M., BARILLE, R., GUIDARA, K., and MEGDICHE, M. (2018) Structural, dielectric, and optical properties of the zinc tungstate ZnWO4 compound. Journal of Materials Science: Materials in Electronics,

29 (8), pp. 6297-6307.

[30] ENNEFFATI, M., LOUATI, B., GUIDARA, K., RASHEED, M., and BARILLE, R. (2018) Crystal structure characterization and AC electrical conduction behavior of sodium cadmium orthophosphate. Journal of Materials Science: Materials in Electronics, 29 (1), pp.

171-179.

[31] KADRI, E., KRICHEN, M., MOHAMMED, R., ZOUARI, A., and KHIROUNI, K. (2016) Electrical transport mechanisms in amorphous silicon/crystalline silicon germanium heterojunction solar cell: impact of passivation layer in conversion efficiency. Optical and Quantum Electronics, 48 (12), 546.

[32] KADRI, E., MESSAOUDI, O., KRICHEN, M., DHAHRI, K., RASHEED, M., DHAHRI, E., ZOUARI, A., KHIROUNI, K., and BARILLE, R. (2017) Optical and electrical properties of SiGe/Si solar cell heterostructures: Ellipsometric study. Journal of Alloys and Compounds, 721, pp. 779-783.

[33] AZAZA, N.B., ELLEUCH, S., RASHEED, M., GINDRE, D., ABID, S., BARILLE, R., ABID, Y., and AMMAR, H. (2019) 3-(p-nitrophenyl) Coumarin derivatives: Synthesis, linear and nonlinear optical properties. Optical Materials, 96, 109328.

[34] ENNEFFATI, M., RASHEED, M., LOUATI, B., GUIDARA, K., and BARILLE, R. (2019) Morphology, UV– visible and ellipsometric studies of sodium lithium orthovanadate. Optical and Quantum

Electronics, 51 (9), 299.

[35] KADRI, E., DHAHRI, K., ZAAFOURI, A., KRICHEN, M., RASHEED, M., KHIROUNI, K., and BARILLE, R. (2017) Ac conductivity and dielectric behavior of

(9)

a−Si:H/c−Si1−yGey/p−Si thin films synthesized by molecular beam epitaxial method. Journal of Alloys and Compounds, 705, pp. 708-713.

[36] OUDA, E.H., SHIHAB, S., and RASHEED, M. (2020) Boubaker Wavelet Functions for Solving Higher Order Integro-Differential Equations. Journal of Southwest

Jiaotong University, 55 (2). Available from

http://jsju.org/index.php/journal/article/view/ 525.

[37] SARHAN, A.M., SHIHAB, S., and RASHEED, M. (2020) A New Boubaker Wavelets Operational Matrix of Integration.

Journal of Southwest Jiaotong University, 55

(2). Available from

[38] SARHAN, A.M., SHIHAB, S., and RASHEED, M. (2020) On the Properties of Two Dimensional Normalized Boubaker Polynomials. Journal of Southwest Jiaotong

University, 55 (3).

[39] ASMAA, A.A. SHIHAB, S., and RASHEED, M. (2020) Discrete Chebyshev Wavelet Transformation with Image Processing. Journal of Southwest Jiaotong

University, 55 (2). Available from

[40] ASMAA, A.A. SHIHAB, S., and RASHEED, M. (2020) Discrete Hermite Wavelet Filters with Prove Mathematical Aspects. Journal of Southwest Jiaotong

University, 55 (2). Available from

[41] ABBAS, M.M. and RASHEED, M. (2020) Solid State Reaction Synthesis and Characterization of Aluminum Doped Titanium Dioxide Nanomaterials. Journal of

Southwest Jiaotong University, 55 (2).

Available from

[42] SARHAN, A.M., SHIHAB, S., and RASHEED, M. (2020) On the Properties of Two Dimensional Normalized Boubaker Polynomials. Journal of Southwest Jiaotong

[43] AZIZ, S.H., RASHEED, M., and SHIHAB, S. (2020) New Properties of

Modified Second Kind Chebyshev Polynomials. Journal of Southwest Jiaotong

[44] MITILIF, R.J., RASHEED, M., and SHIHAB, S. (2020) An Optimal Algorithm for a Fuzzy Transportation Problem. Journal

of Southwest Jiaotong University, 55 (3).

[45] KASHEM, B.E., OUDA, E.H., AZIZ, S.H., RASHEED, M., and SHIHAB, S. (2020) Some Results for Orthonormal Boubaker Polynomials with Application.

(3).

[46] SABRI, R.I., MOHAMMEDALI, M.N., RASHEED, M., and SHIHAB, S. (2020) Compactness of Soft Fuzzy Metric Space.

(4).

[47] SHUKUR, A.M., ALABDALI, O., RASHEED, M., and SHIHAB, S. (2020) Decomposing Method for Space-Time Fractional Order Partial Differential Equations. Journal of Southwest Jiaotong

参考文:

[1] SAADATI，R. 和 PARK，C.（2010）非阿基米德大号-模糊赋范空间和泛函方程的稳定性。计算机与数学与应用，60 （8），第 2488-2496 页。 [2] ANAND，R. 和 RAJ，K.（2017）在 n 赋范空间上完成超范奥利兹·洛伦兹序列空间。埃及数学学会杂志，25（2），第 151-154 页。 [3] 陈晓燕和 SONG，M.M.（2010）在线性 n 赋范空间中的等距特征。非线性分析：理论，方法与应用，72（3-41），第 1895-1901 页。 [4] SHARMA，S.K. 和 ESI，A.（2013）一些一世收敛序列空间是通过使用模和 n 范空间序列来定义的。埃及数学学会杂志， 21，第 103-107 页。 [5] HAZARIKA，B.（2016）概率范空间中多个序列的理想理想收敛。应用数学和计算，279，第 139-153 页。 [6] TRIPATHY ， B.C. ， SEN ， M., 和 NATH，S.（2015）概率 n 赋范空间中的

(10)

腔一世-收敛。埃及数学学会杂志，23，第 90-94 页。 [7] 王丹，刘永和，宋 M.（2012）非阿希米德规范空间上的亚历山德罗夫问题。阿拉伯数学科学杂志，18（2），第 135-140 页。 [8] SAHINER ， A. ， GURDALl ， M., 和 YIGIT，T.（2011）线性 n 赋范空间的完成的理想收敛性。计算机与数学与应用， 61（3），第 683-689 页。 [9] DUTTA，H.（2011）在一些 n 范线性空间值差分序列上。富兰克林学院学报， 348（10），第 2876-2883 页。 [10] A. SAHINER 和 H.S. AKIN 。（2011）多个规范空间中的新序列空间。应用数学与计算，218（31），第 1030-1035 页。 [11] A. KUNDU ， T 。 BAG, 和 S.K. NAZMUL。（2019）规范线性空间的新概括。拓扑及其应用，256，第 159-176 页。 [12] LIANG, M.，ZHU, C.，CHEN, C., 和 WU，Z.（2019）千兆位-度量空间中循环收缩的一些新定理及其应用。应用数学与计算, 346（C），第 545-558 页。 [13] GABA，Y.U。（2017）G-度量空间中的不动点定理。数学分析与应用学报， 455（11），第 528-537 页。 [14] SHOAIB ， A. ， ARSHAD ， M. ， RASHAM，T., 和 ABBAS，M.（2017）错位准 G-度量空间在闭合球上的唯一不动点结果。拉兹马兹数学学院学报， 171 （2），第 221-230 页。 [15] H. AYDI ， N 。 BILGILI, 和 E. KARAPINAR（2015）从准度量空间到 G 度量空间的公共不动点结果。埃及数学学会杂志，23（2），第 356-361 页。 [16] PASICKI，L.（2017）梅尔和基勒是正确的。拓扑及其应用，228，第 382-390 页。 [17] WU，X.，ZHENG，J.，WU，C.，和 CAI，Y.（2013）变分结构—流形上的纹理图像分解。信号处理， 93 （ 7 ），第 1773-1784 页。

[18] LIU, J. ， LUO ， Y. ， WANG ， L. ， YANG，X.，和 GUO，X.（2019）现有空间结构稳定性评估的概率框架。建筑钢研究杂志，156，第 96-104 页。 [19] DKHILALI， F.， MEGDICHE，B.S.， RASHEED ， M. ， BARILLE ， R. ， SHIHAB ， S. ， GUIDARA ， K., 和 MEGDICHE，M.（2018）钨酸钠颗粒的表征和形态。皇家学会开放科学， 5 （18），第 1-16 页。 [20] RASHEED ， M. 和 BARILLE ， R. （2017）直流溅射衍生的 ITO，二氧化钛和二氧化钛的光学常数：铌薄膜，其特征是用于光电子设备的分光光度法和椭圆偏振光度法。非晶体固体杂志，476，第 1-14 页。 [21] RASHEED ， M. 和 BARILLE ， R. （2017）室温沉积氧化锌和铝：通过直流电溅射技术在玻璃和宠物基板上沉积氧化锌超薄膜。光学与量子电子学, 49（5），第 1-14 页。 [22] RASHEED ， M. 和 BARILLE ， R. （2017）比较透明电子学中通过直流电溅射技术沉积在不同基板上的氧化铋和氧化镍超薄膜的光学性质。合金与化合物学报， 728，第 1186-1198 页。 [23] AUKŠTUOLIS ， A 。 GIRTAN ， MOUSDIS ， G.A. ， MALLET ， R. ， SOCOL ， M. ， RASHEED ， M 。和 STANCULESCU，A。（2017）钙钛矿纳米线薄膜中载流子迁移率的测量相片塞利夫方法。罗马尼亚科学院院刊 - 一个系列：数学，物理学，技术科学，信息科学， 18（1），第 34-41 页。 [24] BOURAS ， D. ， MECIF ， A. ， BARILLE ， R. ， HARABI ， A. ， RASHEED ， M. ， MAHDJOUB ， A., 和 ZAABAT，M.（2018）铜：氧化锌通过以下方法沉积在多孔陶瓷基板上用于光催化应用的简单热方法。陶瓷国际，44（17），第 21546-21555 页。 [25] DKILALLI， F.， MEGDICHE，S.， GUIDARA ， K. ， RASHEED ， M. ， BARILLE ， R., 和 MEGDICHE ， M. （2018）钨酸钠硫酸钠的体积和晶界响应中的交流电导率演变。离子学，24（1），第 169-180 页。

(11)

[26] SAIDANI ， T. ， ZAABAT ， M. ， AIDA，M.S.，BARILLE，R.，RASHEED， M., 和 ALMOHAMMED，Y.（2017）前驱物源对溶胶-凝胶沉积氧化锌薄膜性能的影响。材料科学杂志：电子材料，28 （13），第 9252-9257 页。 [27] BOUMEZOUED，A.，GUERGOURI， K. ， BARILLE ， R 。 RECHEM ， D. ， ZAABAT ， M 。和 RASHEED ， M 。（2019）从合成到电气应用，均掺有氧化铋的氧化锌纳米粉。合金与化合物学报， 791，第 550-558 页。 [28] SAIDAI ， W. ， HFAIDH ， N. ， RASHEED ， M. ， GIRTAN ， M. ， MEGRICHE ， A., 和 MAAOUI ， M.E. （2016）飞溅的添加对二氧化钛光学和结构性质的影响作为一种新的阻挡多层染料敏感太阳能电池应用（DSSC）的涂层。 RSC 进展，6（73），第 68819-68826 页。 [29] DKHILALLI，F.，BORCHANI，S.M.， RASHEED ， M. ， BILLILLE ， R. ， GUIDARA ， K., 和 MEGDICHE ， M. （2018）钨酸锌 ZnWO4 化合物的结构，介电和光学性质。材料科学学报：电子材料，29（8），第 6297-6307 页。 [30] ENNEFFATI ， M. ， LOUATI ， B. ， GUIDARA ， K. ， RASHEED ， M., 和 BARILLE，R.（2018）正磷酸钠镉的晶体结构表征和交流电传导行为。材料科学杂志：电子材料，29（1），第 171-179 页。 [31] KADRI ， E. ， KRICHEN ， M. ， MOHAMMED ， R. ， ZOUARI ， A., 和 KHIROUNI，K.（2016）非晶硅/晶体硅锗异质结太阳能电池中的电传输机制：钝化层对硅的影响转换效率。光学与量子电子学，48（12），546。 [32] KADRI ， E. ， MESSAOUDI ， O. ， KRICHEN ， M. ， DHAHRI ， K. ， RASHEED，M.，DHAHRI，E.，ZOUARI， A. ， KHIROUNI ， K., 和 BARILLE ， R. （2017）硅锗/硅太阳能电池异质结构的光学和电学性质：椭偏研究。合金与化合物， 721，第 779-783 页。 [33] AZAZA ， N.B. ， ELLEUCH ， S. ， RASHEED，M.，GINDRE，D.，ABID，

S. ， BARILLE ， R. ， ABID ， Y., 和 AMMAR，H.（2019）3-（p-硝基苯）香豆素衍生物：合成，线性和非线性光学性质。光学材料，96，109328。 [34] ENNEFFATI，M.，RASHEED，M.， LOUATI ， B. ， GUIDARA ， K., 和 BARILLE，R.（2019）原钒酸锂钠的形态，紫外可见光和椭偏研究。光学与量子电子学，51（9），299。 [35] E. KADRI ， DHAHRI ， K. ZAAFOURI ， A. KRICHEN ， M. ， RASHEED ， M. ， KHIROUNI ， K., 和 BARILLE，R.（2017）非晶硅的交流电导率和介电行为通过分子束外延法合成的： H/c-硅 1-yGey/硅薄膜。合金与化合物， 705，第 708-713 页。

[36] OUDA ， E.H. ， SHIHAB ， S., 和 RASHEED，M.（2020）布贝克小波函数，用于求解高阶积分微分方程。西南交通大学学报， 55 （ 2 ）。可从 http://jsju.org/index.php/journal/article/view/ 525 获得。

[37] SARHAN ， A.M. ， SHIHAB ， S., 和 RASHEED，M.（2020）的新布贝克小波积分运算矩阵。西南交通大学学报，55

（ 2 ）。可从

http://jsju.org/index.php/journal/article/view/ 524 获得。

[38] SARHAN ， A.M. ， SHIHAB ， S., 和 RASHEED，M.（2020）关于二维标准化布贝克多项式的性质。西南交通大学学报， 55（3）。

[39] ASMAA ， A.A. SHIHAB ， S., 和 RASHEED，M.（2020）离散切比雪夫小波变换与图像处理。西南交通大学学报，

55 （ 2 ）。可从

[40] ASMAA ， A.A. SHIHAB ， S., 和 RASHEED，M.（2020）具有证明数学意义的离散赫米特人小波滤波器。西南交通大学学报， 55 （ 2 ）。可从 http://jsju.org/index.php/journal/article/view/ 559 获得。 [41] ABBAS，M.M。和 RASHEED，M. （2020）铝掺杂的二氧化钛纳米材料的固

(12)

相反应合成与表征。西南交通大学学报，

55 （ 2 ）。可从

[42] SARHAN ， A.M. ， SHIHAB ， S., 和 RASHEED，M.（2020）关于二维规范化布贝克多项式的性质。西南交通大学学报， 55（3）。 [43] AZIZ ， S.H. ， RASHEED ， M., 和 SHIHAB，S.（2020）修改的第二类切比雪夫多项式的新性质。西南交通大学学报， 55（3）。 [44] MITILIF，R.J.，RASHEED，M。和 SHIHAB，S。（2020）提出的一种模糊运输问题的最优算法。西南交通大学学报， 55（3）。

[45] KASHEM ， B.E. ， OUDA ， E.H. ， AZIZ，S.H.，RASHEED，M., 和 SHIHAB， S.（2020）正交布贝克多项式的一些结果及其应用。西南交通大学学报，55（3）。 [46] SABRI ， R.I. ， MOHAMMEDALI ， M.N. ， RASHEED ， M., 和 SHIHAB ， S. （2020）软模糊度量空间的紧凑性。西南交通大学学报，55（4）。 [47] SHUKUR，A.M.，ALABDALI，O.， RASHEED，M., 和 SHIHAB，S.（2020）时空分数阶偏微分方程的分解方法。西南交通大学学报，55（4）。