JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

Itô-Henstock integration of the fuzzy stochastic process

GONG Zeng-tai, SU Ai

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 1-13. doi:10.6040/j.issn.1671-9352.0.2019.235

Abstract ( 1418 )

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The fuzzy Itô-Henstock integral and the fuzzy Itô-McShane integrals for adapted fuzzy stochastic process with respect to a Brownian motion are defined and their properties are discussed. In addition, the characterization theorems are given and their interrelation of between the fuzzy Itô-Henstock integral and the fuzzy Itô-McShane integral is investigated. The result shows that the fuzzy Itô-Henstock integral is equivalent to the fuzzy Itô-McShane integral when its primitive of fuzzy Itô-Henstock integral is Itô absolutely continuous.

Strong hyper deductive systems in hyper equality algebras

CHENG Xiao-yun, XIN Xiao-long

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 14-19. doi:10.6040/j.issn.1671-9352.0.2018.587

Abstract ( 1200 )

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Some types of strong hyper deductive systems in hyper equality algebras are introduced and investigated. Some representations of generated strong hyper deductive systems are given. Then, Two types of important strong hyper deductive systems, namely, implicative and positive implicative strong hyper deductive systems are studied. In particular, characterizations of them are given and the relations between them are discussed.

Characterization of fuzzy implications satisfying the law of importation with respect to conjunctive 2-uninorms

CHENG Ya-fei, ZHAO Bin

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 20-32. doi:10.6040/j.issn.1671-9352.0.2018.429

Abstract ( 1031 )

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The notion of fuzzy negations satisfying(COMP_U²)with respect to 2-uninorms is introduced, and the situations that continuous fuzzy negations satisfying(COMP_U²)with respect to four types of conjunctive 2-uninorms are discussed respectively. The definition and related properties of fuzzy implications satisfying the law of importation with conjunctive 2-uninorms are given. Finally, using(COMP_U²), a characterization that fuzzy implications satisfy respectively the law of importation with respect to four types of conjunctive 2-uninorms is provided.

Positive periodic solutions for second-order singular differential equations with damping terms

CHEN Rui-peng, LI Xiao-ya

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 33-41. doi:10.6040/j.issn.1671-9352.0.2018.707

Abstract ( 1237 )

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This paper studies the existence of positive periodic solutions of u″+p(t)u'+q(t)u=f(t,u)+c(t), where p,q,c∈L1(R/TZ;R), f is a Carathéodory function and is singular when u=0. By means of the fixed point theory, several existence theorems are established for the above equation, and some recent results in the literature are generalized and improved.

Non-uniform dependence on initial data for the periodic two-coupled Camassa-Holm system

WANG Hai-quan

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 42-49. doi:10.6040/j.issn.1671-9352.0.2019.125

Abstract ( 1213 )

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Considered herein is the initial value problem for the periodic two-coupled Camassa-Holm system. It is shown that the solution map of this problem is not uniformly continuous in Besov spaces B^3/2_2,1(T)×B^3/2_2,1(T). Based on the well-posedness result and the lifespan for this problem, the method of approximate solutions is utilized. The same approach can be used to discuss this property of the solutions for the other nonlinear partial differential equations.

Inverse spectral results for AKNS systems with partial information on the potentials

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 50-54. doi:10.6040/j.issn.1671-9352.0.2018.334

Abstract ( 1059 )

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We consider the AKNS systems on the interval［0,1］ and prove that two potentials already known on ［a,1］(a∈(0,1/2］)and their difference in L^p are equal if the number of their common eigenvalues is sufficiently large. The result here is to write down explicitly this number in terms of p(and a)showing the role of p.

Property(ω₁)and the single-valued extension property

DAI Lei, HUANG Xiao-jing, GUO Qi

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 55-61. doi:10.6040/j.issn.1671-9352.0.2019.099

Abstract ( 3880 )

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A bounded linear operator T satisfies property(ω₁), if the complement in the approximate point spectrum σa(T)of the upper semi-Weyl spectrum σ_ea(T)is contained in the set of all isolated points of the spectrum σ(T)which are finite eigenvalues. In this paper, by means of the new spectrum defined in view of the single-valued extension property, the sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space satisfying the property(ω₁)are established. As an application, the property(ω₁)for hypercyclic(or supercyclic)operators are characterised.

Existence of ground states for linear coupled systems of lower critical Choquard type

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 62-67. doi:10.6040/j.issn.1671-9352.0.2018.670

Abstract ( 1118 )

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To the best of our knowledge, there is few result about linearly coupled systems of Choquard type with the lower critical up to now. The existence of ground state solutions for a class of Choquard-type linear coupled systems with lower critical exponents is studied by variational methods. It is a promotion and supplement to the previous research results.

Weighted estimates of fractional maximal operator and its commutator on weighted λ-central Morrey spaces

TAO Shuang-ping, YANG Yu-he

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 68-75. doi:10.6040/j.issn.1671-9352.0.2018.718

Abstract ( 1244 )

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By applying the weighted inequalities and the real variable methods, the boundedness of the fractional maximal operator with rough kernel is obtained in the weighted λ-central Morrey spaces. And also, the boundedness of its commutator generated by a λ-central mean oscillation function is proved.

On decompositions of continuous generalized frames in Hilbert spaces

ZHANG Wei, FU Yan-ling

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 76-80. doi:10.6040/j.issn.1671-9352.0.2018.724

Abstract ( 1051 )

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This paper establishes the characterization of continuous generalized frames,Parseval continuous g-frames,continuous generalized Riesz bases and continuous generalized orthonormal bases in term of the continuous generalized preframe operator. Using the established characterization results and decompositions of bounded operators, the representation of continuous generalized frames in term of linear combinations of simpler ones such as continuous generalized orthonormal bases, continuous generalized Riesz bases and Parseval continuous generalized frames is studied.

Meromorphic solutions of a class of auxiliary differential equation and its applications

GU Yong-yi, KONG Yin-ying

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 81-89. doi:10.6040/j.issn.1671-9352.0.2018.706

Abstract ( 959 )

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This paper introduce a method to find exact solutions of nonlinear partial differential equations—complex method, and derive meromorphic solutions for a class of algebraic differential equation by the mentioned method. The results are used to seek exact solutions of nonlinear differential equations. Exact solutions of the Vakhnenko-Parkes equation and Dodd-Bullough-Mikhailov equation are obtained.

Uniqueness of differential polynomials of meromorphic functions IM sharing a value

ZHANG Wei-jie, WANG Xin-li, WANG Han-jie

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 90-96. doi:10.6040/j.issn.1671-9352.0.2018.720

Abstract ( 1042 )

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This paper study uniqueness of differential polynomials of meromorphic functions sharing 1 IM. Let f(z)and g(z)be two non-constant meromorphic functions, whose zeros and poles are of multiplicities at least s, where s is a positive integer. Let n≥2 be an integer satisfying(n+1)s≥24. If f ⁿf ' and gⁿg' share the value 1 IM, then either f(z)=tg(z)for some(n+1)-th root of unity t or g(z)=c1e^cz, f(z)=c2e^-cz, where c1,c2, c are constants satisfying(c1c2)ⁿ⁺1c2=-1.

Convergence and contractivity of boundary value methods for nonlinear delay differential equations

ZHANG Ru, HAN Xu, LIU Xiao-gang

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 97-101. doi:10.6040/j.issn.1671-9352.0.2018.758

Abstract ( 1072 )

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The boundary value methods are applied to the nonlinear delay differential equations. Under the assumptions of Lipschitz conditions, the convergence and the global contractivity, the weakly global contractivity of the boundary value methods are analyzed. Finally, some numerical experiments are carried out to illustrate the theoretical results.

Strongly Gorenstein injective modules with respect to a cotorsion pair

WANG Zhan-ping, YUAN Kai-ying

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 102-107. doi:10.6040/j.issn.1671-9352.0.2018.517

Abstract ( 1058 )

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As a generalization of strongly Gorenstein injective modules, this paper introduces and studys strongly Gorenstein injective modules with respect to a complete and hereditary cotorsion pair(X,Y), that is, strongly Gorenstein(X∩Y,Y)-injective modules, and give some properties and equivalent characterizations. Moreover, the stability of strongly Gorenstein(X∩Y,Y)-injective modules is studied, and the relation between it and Gorenstein(X∩Y,Y)-injective modules is discussed.

Counting solutions of a binary quadratic congruence equation

DUAN Ran

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 108-120. doi:10.6040/j.issn.1671-9352.0.2018.407

Abstract ( 1346 )

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Let n be a positive integer. Denote by Z_n the ring of residue classes mod n, and by Z^*_n the group of units in Z_n, i.e. Z^*_n={s:1≤s≤n and gcd(s,n)=1}. The main purpose of this paper is using congruence conclusion and some results of exponential sums to study the number of elements of the set T(a,b,c,n)={(x,y)∈(Z^*_n)2:ax2+by2+c≡0 mod n} and give an exact computational formula for the number of elements of T(a,b,c,n).

The inclusion graph of S-acts

SUN Shuang, LIU Hong-xing

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2019, 54(8): 121-126. doi:10.6040/j.issn.1671-9352.0.2018.504

Abstract ( 1287 )

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Let S be a semigroup and M be an S-act. The inclusion graph of M, denoted by G(M), is the undirected simple graph whose vertices are all non-trivial subact of M and defining two distinct vertices I and J to be adjacent if and only if I⊆J or J⊆I. Some results on completeness, connectivity, diameter, girth, the clique number and the chromatic number of G(M) are presented.

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