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This paper is concerned with a system governed by nonsingular delay differential equations. We study the β-Ulam-type stability of the mentioned system. The investigations are carried out over compact and unbounded intervals. Before proceeding to the main results, we convert the system into an equivalent integral equation and then establish an existence theorem for the addressed system. To justify the application of the reported results, an example along with graphical representation is illustrated at the end of the paper.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

In the present paper, we derive some subordination and superordination results for p-valent functions in the open unit disk by using certain fractional derivative operator. Relevant connections of the results, which are presented in the paper, with various known results are also considered.

The purpose of the paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial. The results of the paper improve and generalize the recent results due to X. B. Zhang and J. F. Xu [19]. We also solve an open problem as posed in the last section of [19].

A class of (right) Rickart rings, called strong, is isolated. In particular, every Rickart *-ring is strong. It is shown in the paper that every strong Rickart ring R admits a binary operation which turns R into a right normal band having an upper bound property with respect to its natural order ≤; such bands are known as right normal skew nearlattices. The poset (R, ≤) is relatively orthocomplemented; in particular, every initial segment in it is orthomodular.The order ≤ is actually a version of the so called right-star order. The one-sided star orders are well-investigated for matrices and recently have been generalized to bounded linear Hilbert space operators and to abstract Rickart *-rings. The paper demonstrates that they can successfully be treated also in Rickart rings without involution.

In this paper, we study the asymptotic behavior of the solutions of the one-dimensional Cauchy problem in Timoshenko system with thermal effect. The heat conduction is given by the type III theory of Green and Naghdi. We prove that the dissipation induced by the heat conduction alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow and, in the case of nonequal wave speeds, are of regularity-loss type. This paper solves the open problem stated in [10] and shows that the stability of the solution holds without any additional mechanical damping term.

The applications of q-calculus in the approximation theory is a very interesting area of research in the recent years, several new q-operators were introduced and their behaviour were discussed by many researchers. This paper is the extension of the paper [15], in which Durrmeyer type generalization of q-Baskakov-Stancu type operators were discussed by using the concept of q-integral operators. Here, we propose to study the Stancu variant of q-Baskakov-Stancu type operators. We establish an estimate for the rate of convergence in terms of modulus of continuity and weighted approximation properties of these operators.

On account of analytical importance or application in certain problems in radiation physics and nuclear technology, several interesting families of elliptic-type integrals were recently studied by many authors. The aim and objective of present paper is to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. In addition, to deriving known and various new elliptic-type integrals and their generalizations, these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.

In [8], Csaszar introduced some generalized open sets in generalized topological spaces, namely, -semi-open, -preopen, -open, and -open. The primary purpose of this paper is to study some new properties and characterizations of these concepts as we also introduce and study a new generalized open set in generalized topological spaces, namely, -b-open. We also investigate some relationships between these concepts. 1. Introduction and Preliminaries A generalized topology (brei‡y GT) [7] on a nonempty set X is a collection of subsets of X such that ; 2 and is closed under arbitrary unions. Elements of will be called -open sets, and a subset A of (X; ) will be called -closed if XnA is -open. Clearly, a subset A of (X; ) is -open if and only if for each x 2 A, there exists Ux 2 such that x 2 Ux A, or equivalently, A is the union of -open sets. The pair (X; ) will be called generalized topological space (brei‡y GTS). By a space X or (X; ), we will always mean a GTS. A space (X; ) is called a -structure [16] if X 2 . (X; ) is called a quasi-topological space [9] if is closed under …nite intersections. Clearly, every topological space is a quasi-topological space, every quasi-topological space is a GTS, and a space (X; ) is a topological space if and only if (X; ) is both -structure and quasi-topological space. 2000 Mathematics Subject Classi…cation. Primary 54A05, 54A10. Key words and phrases. -open, -closed, -regular open, -regular closed, -semi-open, -dense, -preopen, -open, -open, -b-open, generalized topology. 1 2 MOHAMMAD. S. SARSAK If A is a subset of a space (X; ), then the -closure of A [8], c (A), is the intersection of all -closed sets containing A and the -interior of A [8], i (A), is the union of all -open sets contained in A. It was pointed out in [8] that each of the operators c and i are monotonic [10], i.e. if A B X, then c (A) c (B) and i (A) i (B), idempotent [10], i.e. if A X, then c (c (A)) = c (A) and i (i (A)) = i (A), c is enlarging [10], i.e. if A X, then c (A) A, i is restricting [10], i.e. if A X, then i (A) A, A is -open if and only if A = i (A), and c (A) = Xni (XnA). Clearly, A is -closed if and only if A = c (A), c (A) is the smallest -closed set containing A, i (A) is the largest -open set contained in A, x 2 c (A) if and only if any -open set containing x intersects A, and x 2 i (A) if and only if there exists a -open set U such that x 2 U A. A subset A of a topological space X is called regular open if A =IntA, and regular closed if XnA is regular open, or equivalently, if A = IntA. A is called semi-open [13] (resp. preopen [14] where called locally dense in [6], semi-preopen [3] where called -open in [1], b-open [4], -open [15]) if A IntA (resp. A IntA, A IntA, A IntA[IntA, A IntIntA). A is called semi-regular [11] if it is both semi-open and semi-closed, or equivalently, if there exists a regular open set U such that U A U , where A and IntA denote respectively, the closure of A in X and the interior of A in X. In [5], the term regular semi-open were used for a semi-regular set. For the concepts and terminology not de…ned here, we refer the reader to [12]. In concluding this section, we recall the following de…nitions and facts for their importance in the material of our paper. De…nition 1.1. [8] Let A be a subset of a space (X; ). Then A is called (i) -semi-open if A c (i (A)). (ii) -preopen if A i (c (A)). ON SOME PROPERTIES OF GENERALIZED OPEN SETS... 3 (iii) -open if A c (i (c (A))). (iv) -open if A i (c (i (A))). As in [8], for a space (X; ), we will denote the class of -semi-open sets by , the class of -preopen sets by , the class of -open sets by , and the class of -open sets by . Proposition 1.2. [8] Let (X; ) be a space. Then (i) . (ii) . (iii) = \ . Proposition 1.3. [8] Let (X; ) be a space. Then each of the families , , , and is a generalized topological space. De…nition 1.4. [18] Let A be a subset of a space (X; ). Then A is called (i) -regular closed if A = c (i (A)). (ii) -regular open if XnA is -regular closed. (iii) -semi-closed if XnA is -semi-open. (iv) -preclosed if XnA is -preopen. (iii) -closed if XnA is -open. (iv) -closed if XnA is -open. We will denote the collection of all -regular open sets by r. Proposition 1.5. [18] Let A be a subset of a space (X; ). Then (i) A is -semi-closed if and only if i (c (A)) A. (ii) A is -preclosed if and only if c (i (A)) A. (iii) A is -regular open if and only if A = i (c (A)). (iv) A is -closed if and only if i (c (i (A))) A. (v) A is -closed if and only if c (i (c (A))) A. (vi) A is -regular open if and only if A = i (B) for some -closed set B. (vii) A is -regular closed if and only if A = c (B) for some -open set B. 4 MOHAMMAD. S. SARSAK Proposition 1.6. [18] For a subset A of a space (X; ), the following are equivalent: (i) A is -regular open, (ii) A is -open and -semi-closed, (iii) A is -open and -closed, (iv) A is -open and -closed, (v) A is -open and -semi-closed, (vi) A is -preopen and -semi-closed. Corollary 1.7. [18] For a subset A of a space (X; ), the following are equivalent: (i) A is -regular closed, (ii) A is -closed and -semi-open, (iii) A is -closed and -open, (iv) A is -closed and -open, (v) A is -closed and -semi-open, (vi) A is -preclosed and -semi-open. 2. -b-open sets De…nition 2.1. Let A be a subset of a space (X; ). Then A is called -b-open if A c (i (A))[i (c (A)). A is called if XnA is -b-open, or equivalently, if c (i (A))\i (c (A)) A. We will denote the class of all -b-open subsets of (X; ) by b. Remark 2.2. It is easy to see that for a space (X; ), [ b . It was shown in [4] that in case of a topological space, the above inclusions are not reversible. Remark 2.3. It is also easy to see using the monotonicity of c and i , that for a space (X; ), (X; b) is a generalized topological space. The following proposition is an immediate consequence of Remarks 2.2 and 2.3. ON SOME PROPERTIES OF GENERALIZED OPEN SETS... 5 Proposition 2.4. Let A be a subset of a space (X; ). If A = B [ C, where A is -semi-open and C is -preopen, then A is -b-open. In [17], it is pointed out that a subset A of a topological space X is b-open if and only if A = B [ C, where B is semi-open and C is preopen. However, this is not true for generalized topological spaces since the converse of Proposition 2.4 need not be true as following example shows. Example 2.5. Let B = ff4g ; f1; 2; 3g ; f3; 4; 5g ; f5; 6; 7gg and let = f;; all possible unions of members of Bg Then (X; ) is a generalized topological space, whereX = f1; 2; 3; 4; 5; 6; 7g. If A = f1; 2; 4g. Then A is -b-open because c (A) = f1; 2; 3; 4g , and thus i (c (A)) = f1; 2; 3g and i (A) = f4g , and thus c (i (A)) = f4g We will show now that there is no -semi-open set B and -preopen set C such that A = B [ C. Observe …rst that the only -semi-open subsets of A are ; and f4g, that is because c (i (A)) = f4g. We discuss the following cases: (i) If B = ;, then C = f1; 2; 4g which is not -preopen because c (U) = f1; 2; 3; 4g and i (c (C)) = f1; 2; 3g. (ii) If B = f4g and C = f1; 2; 4g, then C is not -preopen. (iii) If B = f4g and C = f1; 2g, then C is not -preopen because c (C) = f1; 2g and i (c (C)) = ;. The following result is an immediate consequence of Proposition 1.6 and Remark 2.2. Corollary 2.6. For a subset A of a space (X; ), the following are equivalent: 6 MOHAMMAD. S. SARSAK (i) A is -regular open, (ii) A is -open and -b-closed, (iii) A is -open and -b-closed. Lemma 2.7. Let A be a subset a space (X; ). If A is both -semiclosed and -open, then A is -semi-open. Proof. Since A is -semi-closed, it follows from Proposition 1.5 (i) that

The purpose of this paper is to introduce an implicit iterative process with errors for approximating common fixed point of two finite families of asymptotically nonexpansive mappings in the framework of Banach space. The results presented in this paper extend and generalize the corresponding results of Qin et al. [Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings, Appl. Math. Comp. 210 (2009), 542–550], Thakur [Weak and strong convergence of composite implicit iteration process, Appl. Math. Comp. 190 (2007), 965–973] and some others.

The following is an errata to the paper “On the coset structure of a skew lattice” published in Demonstratio Mathematica as [4]. All the corrections are relevant to the clear understanding of the paper by the reader. The author thanks to K. Cvetko-Vah and J. Leech for discussions that lead to these corrections. This survey paper revisits some of the known decompositions in the theory of skew lattices (non-commutative generalizations of lattices) including both Leech’s decomposition theorems, and some of the base knowledge regarding the coset decomposition. In the introduction it states that skew lattices “consist(s) of the left version of Slavik’s algebras” which, in fact, should state that “Slavik’s algebras lead to a left version of Leech’s skew lattices”. This statement is referred by J. Leech in his review paper [2] and it is proved in Proposition 6 of the present paper, followed by the corresponding discussion. At a late stage of preparation of the paper an error occured into the statement of Proposition 5 on p. 678. The Proposition is assigned to the preprint [3] by J. Leech and M. Kinyon (which remains today to be a preprint, soon to be submitted according to the authors). However, a confusion arises due to the definition of ∧-distributive and ∨distributive skew lattices. By the way they are defined in this paper (using axioms S21, S22 and S23, S24, respectively notice the existing typo) Proposition 5 is not as in the cited reference, and moreover it fails to be true. Therefore, Proposition 5 as stated in the paper is neither a citation of their result nor true. To overcome this confusion, the identities S21 and S22 that define ∧-distributive skew lattices in this paper should be replaced by the identity: x ∧ (y ∨ z) ∧ x ≈ (x ∧ y ∧ x) ∨ (x ∧ z ∧ x)

In this paper, we present conditions which are equivalent to the Darboux property for non-constant polynomials in Golomb's and Kirch's topologies on the set of positive integers. 1. Preliminaries The letters Z, N and N0 denote the sets of integers, positive integers, and non-negative integers, respectively. The symbol ( a) denotes the set of all prime divisors of a 2 N. For all a;b 2 N, we use the symbol (a;b) to denote the greatest common divisor of a and b. Moreover, for all a;b2 N, the symbolsfan+bg andfang stand for the infinite arithmetic progressions: fan + bg := a N0 + b and fang := a N: Let X and Y be the topological spaces. We say that a function f : X!Y has the Darboux property, if the set f(E) is connected in Y for every connected set E X. Assume that k2 N. In this paper, we will examine polynomials f : N!N of the form (1) f(x) = k X i=0 aix i ; where ak 1;:::;a02 N0, and ak2 N.

The current paper represents a suplement for papers [

Let V be an open convex subset of a nontrivial real normed space X. In the paper we give a partial generalization of Bernstein-Doetsch Theorem. We prove that if there exist a base B of X and a point x ∈ V such that a midconvex function f : X → R is locally bounded above on b-ray at x for each b ∈ B, then f is convex. Moreover, we show that under the above assumption, f is also continuous in case X = RN , but not in general. Let X be a real normed space and V be a convex subset of X. A function f : V → R is called convex if f(tx + (1− t)y) ≤ tf(x) + (1− t)f(y) for x, y ∈ V, t ∈ [0, 1]. If the above inequality holds for t = 1 2 , then f is said to be midconvex (or Jensen convex). F. Bernstein and G. Doetsch [1] proved that every midconvex function f : (a, b) → R locally bounded above at a point is continuous (clearly a continuous midconvex function must be convex). The above statement has many generalizations (see e. g. [2]–[10]). Nowadays by Bernstein-Doetsch Theorem we usually mean the following one. Bernstein-Doetsch Theorem Let V be an open convex subset of X and let f : V → R be midconvex. If f is locally bounded above at a point, then f is continuous and convex. In the paper we give a generalization of Bernstein-Doetsch Theorem. To formulate this generalization, we introduce the following definition. Mathematics Subject Classification (2000): 26B25

The aim of this paper is to review and clarify some facts concerning the uniform convergence of statistics like Xn and random variables like √ n(Xn − μ(θ))/σ(θ). We consider convergence in distribution or in probability, uniform with respect to a family of probability distributions. It seems that these concepts are appropriate tools for asymptotic theory of mathematical statistics, but in reality they are rather rarely used or even mentioned. Little in this paper is new, we focus on relations between known results. We examine a few rather paradoxical examples which hopefully shed some light on the subtleties of the underlying definitions and the role of asymptotic approximations in statistics. 1 Definitions Consider a statistical space (Ω,F , {Pθ : θ ∈ Θ}). Let us say that a random variable is a function Z : Θ × Ω → R such that for every θ ∈ Θ the mapping Z(θ) : ω 7→ Z(θ, ω) is (F ,B(R))-measurable. As usual, the argument ω will most often be supressed, while the argument θ will be explicitly written to avoid misunderstanding. Thus we write e.g. Pθ(Z(θ) ∈ B) = Pθ {ω : Z(θ, ω) ∈ B} for B ∈ B(R). A random variable T which does not

The main purpose of this paper is to extend our previous construction of T -Fock spaces from a given Yang-Baxter operator satisfying the inequalities −1 ≤ T ≤ 1 to the constructions of T -symmetric Fock spaces related to the class of Yang-Baxter-Hecke operators meeting a weaker condition that T ≥ −1. The new representation of the monotone Fock space of N. Muraki will be given. The main idea of this paper is the new class of generalized Gaussian random variables acting on suitable T -symmetric Fock spaces. Relations with the row and column operator space will be also given. Introduction In this note we will present the following subjects: 1. Generalized symmetric Fock spaces of Yang-Baxter-Hecke type. 2. (a) Hecke operators and (b) positivity of T -symmetrizators. 3. Connections of Pusz-Woronowicz operators T μ with monotone Fock space of Muraki-Lu (μ = 0) and with Bose monotone Fock space and also with mixed Bose-Fermi Fock spaces related to μ = −1. The work was partially supported by Grant No. N N 201 364436 of Polish Ministry of Science.

We prove the result announced by the title as well as some of its consequences. It is well known that the variety of Lukasiewicz algebras is generated by its finite members (see e.g. [18]). W. Blok and I. Ferreirin [3] proved that the variety of all Lukasiewicz algebras is generated by its finite members as a quasi-variety. In this paper we show that it is so also the case for the variety of all commutative BCK-algebras as well as some of its subvarieties. It is worth to note that unlike the case of subvarieties of the variety of all Lukasiewicz algebras there are subvarieties of the variety of all commutative BCK-algebras which are not generated by their finite members [27, 36]. The main result of the paper was obtained in early ’90s. Since then there was many papers on BCK-algebras. BCK-algebras with condition (S) (pocrims) and bounded commutative BCK-algebras (Wajsberg algebras, MV-algebras). Let me list some important papers suggested by an anonymous refferee. • A good study of the varieties of BCK-algebras (including some aspects of commutative BCK-algebras) was done by W. Blok and J. Raftery in [5]. The references given in this paper provide a fairly comprehensive vision of the status of BCK-algebras until 1995. The implicative presentation of BCK-algebras tends to be used today, because they are the implicational subreduct of commutative integral residuated lattices. • BCK-algebras with condition S are actually known as pocrims: partial ordered, commutative, resituated, integral monoids. The varieties of pocrims were also studied by W. Blok and J. Raftery in [6]. This paper contains a complete bibliography on pocrims and BCK-algebras.

In this paper we prove the existence of solutions and Carathéodory’s type solutions of the dynamic Cauchy problemThe Sadovskii fixed point theorem and Ambrosetti’s lemma are used to prove the main result.As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.

In the first part of this paper, we considered the prob- lem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes. We provided a general construction of such semirings, along with basic examples and some general proper- ties. In the second part of the paper we discussed some selected varieties of modes, in particular, varieties of affine spaces, vari- eties of barycentric algebras and varieties of semilattice modes, and described the semirings determining their semi-linearizations, the varieties of semimodules having these algebras as idempotent subreducts. The third part is devoted to varieties of differential groupoids and more general differential modes, and provides the semirings of the semi-linearizations of these varieties. This paper is a direct continuation of the first and second parts appearing with the same title (4) and (5). In the first part, we considered the problem of constructing a (commutative unital) semiring defining the variety of semimodules whose idempotent subreducts lie in a given variety of modes, and such that each semimodule-embeddable member of this mode variety embeds into a semimodule over the constructed semiring. We described a general construction of such semirings, with basic examples and some general properties. In the second part, we investigated selected varieties of modes, and described the semirings determining varieties of semimodules having algebras of these classes as subreducts, and discussed properties of the corresponding semi-affine spaces. In particular, we investigated varieties of affine spaces, varieties of barycentric algebras and varieties of semilattice modes. The third part is devoted to varieties of differential groupoids and more general

The purpose of this paper is to prove a common fixed point theorem for a set of four mappings on a complete metric space, using weak compatibility and a general implicit relation without appeal to continuity. Our results improve and generalize all the results obtained by A. Djoudi in a paper published in 2003.

This paper deals with the solvability of systems of linear operator equations in a linear space. Namely, the paper provides necessary and sufficient conditions for the operators under which certain kinds of systems of operator equations are solvable.

This paper is a survey on the recent study of homotopy theory of generalized phrases in Turaev’s theory of words and phrases. In this paper, we introduce the homotopy classification of generalized phrases with some conditions on numbers of letters. The theory of topology of words and phrases are closely related to the theory of surface-curves. We also introduce applications of topology of words to the topology of surface-curves.

In this paper, we de…ne and study a new concept of rdominated multilinear operators in the category of operator spaces, which we call lr-dominated m-linear operators. We give some characterizations of this concept such as the theorem of factorization. 0. Introduction The concept of r-dominated multilinear operators was mainly introduced at the beginning of the 1980s by Pietsch [18] where the idea of generalizing the theory of ideals of linear operators to the multilinear setting appears. Motivated by the importance of this theory, several authors have developed and studied many concepts relating to summability (we mention for example [13, 14, 16, 17] among so many authors). Regarding this, it is natural to try to develop analogous in the non commutative case. Hence, this paper deals with the equivalent of this concept in the theory of operator spaces; which will be called the “lr-dominated m-linear operators”. Firstly, we will introduce the notion of lr-dominated m-linear operators and we prove an analogue to the Pietsch domination theorem. After that, we shall give the factorization theorem of such operators. We …nish this paper by giving the …nite dimensional version of our result. This paper is organized as follows. In the …rst section, we recall some basic de…nitions and properties concerning the theory of operator spaces. In the second section of the present paper, we introduce and study the notion of lr-dominated multilinear operators. We give the Pietsch domination theorem and related properties. 2000 Mathematics Subject Classi…cation. 46B28, 47H60, 46G25, 46B25.

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

Sarsak [On μ \mu -compact sets in μ \mu -spaces, Questions Answers Gen. Topology 31 (2013), no. 1, 49–57] introduced and studied the class of μ \mu -Lindelöf sets in μ \mu -spaces. Mustafa [ μ \mu -semi compactness and μ \mu -semi Lindelöfness in generalized topological spaces, Int. J. Pure Appl. Math. 78 (2012), no. 4, 535–541] introduced and studied the class of μ \mu -semi-Lindelöf sets in generalized topological spaces (GTSs); the primary purpose of this paper is to investigate more properties and mapping properties of μ \mu -semi-Lindelöf sets in μ \mu -spaces. The class of μ \mu -semi-Lindelöf sets in μ \mu -spaces is a proper subclass of the class of μ \mu -Lindelöf sets in μ \mu -spaces. It is shown that the property of being μ \mu -semi-Lindelöf is not monotonic, that is, if ( X , μ ) \left(X,\mu ) is a μ \mu -space and A ⊂ B ⊂ X A\subset B\subset X , where A A is μ B {\mu }_{B} -semi-Lindelöf, then A A need not be μ \mu -semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called ω μ {\omega }_{\mu } -semi-open sets, and investigate them to obtain new properties and characterizations of μ \mu -semi-Lindelöf sets in μ \mu -spaces.

In this paper, we introduce a shrinking projection method of an inertial type with self-adaptive step size for finding a common element of the set of solutions of a split generalized equilibrium problem and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step size incorporated helps to overcome the difficulty of having to compute the operator norm, while the inertial term accelerates the rate of convergence of the proposed algorithm. Under standard and mild conditions, we prove a strong convergence theorem for the problems under consideration and obtain some consequent results. Finally, we apply our result to solve split mixed variational inequality and split minimization problems, and we present numerical examples to illustrate the efficiency of our algorithm in comparison with other existing algorithms. Our results complement and generalize several other results in this direction in the current literature.

In this paper, we introduced the Lyapunov exponents (LEs) as a significant tool that is used to study the numerical solution behavior of the dynamical systems. Moreover, Hide’s coupled dynamo model presents a valuable dynamical study. We simulate the convergence of the LEs of the model in three cases by means of periodic flow, regular flow, and chaos flow. In addition, we compared these cases in logic connections and proved them in a mathematical way.

In this study, a new iterative method for third-order boundary value problems based on embedding Green’s function is introduced. The existence and uniqueness theorems are established, and necessary conditions are derived for convergence. The accuracy, efficiency and applicability of the results are demonstrated by comparing with the exact results and existing methods. The results of this paper extend and generalize the corresponding results in the literature.

In this paper, we show how to approximate the solution to the generalized time-fractional Huxley-Burgers’ equation by a numerical method based on the cubic B-spline collocation method and the mean value theorem for integrals. We use the mean value theorem for integrals to replace the time-fractional derivative with a suitable approximation. The approximate solution is constructed by the cubic B-spline. The stability of the proposed method is discussed by applying the von Neumann technique. The proposed method is shown to be conditionally stable. Several numerical examples are introduced to show the efficiency and accuracy of the method.

In this paper, we prove some common fixed point theorems on bicomplex metric space. Our results generalize and expand some of the literature’s well-known results. We also explore some of the applications of our key results.

In this paper, we study the problem of finding a common solution of split generalized vector mixed equlibrium problem (SGVMEP), fixed point problem (FPP) and variational inequality problem (VIP). We propose an inertial-type iterative algorithm, which uses a projection onto a feasible set and a linesearch, which can be easily calculated. We prove a strong convergence of the sequence generated by the proposed algorithm to a common solution of SGVMEP, fixed point of a quasi- ϕ \phi -nonexpansive mapping and VIP for a general class of monotone mapping in 2-uniformly convex and uniformly smooth Banach space E 1 {E}_{1} and a smooth, strictly convex and reflexive Banach space E 2 {E}_{2} . Some numerical examples are presented to illustrate the performance of our method. Our result improves some existing results in the literature.

LetGbe an abelian group with identityee. LetRbe aG-graded commutative ring with identity 1, andMMbe a gradedR-module. In this paper, we introduce the concept of gradedJgr{J}_{gr}-classical 2-absorbing submodule as a generalization of a graded classical 2-absorbing submodule. We give some results concerning of these classes of graded submodules. A proper graded submoduleCCofMMis called a gradedJgr{J}_{gr}-classical 2-absorbing submodule ofMM, if wheneverrg,sh,ti∈h(R){r}_{g},{s}_{h},{t}_{i}\in h\left(R)andxj∈h(M){x}_{j}\in h\left(M)withrgshtixj∈C{r}_{g}{s}_{h}{t}_{i}{x}_{j}\in C, then eitherrgshxj∈C+Jgr(M){r}_{g}{s}_{h}{x}_{j}\in C+{J}_{gr}\left(M)orrgtixj∈C+Jgr(M){r}_{g}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M)orshtixj∈C+Jgr(M),{s}_{h}{t}_{i}{x}_{j}\in C+{J}_{gr}\left(M),whereJgr(M){J}_{gr}\left(M)is the graded Jacobson radical.

The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

Operators preserving singularity and nonsingularity of matrices were studied in paper of P. Botta under the assumption that operators are linear. In the present paper the linearity of operators is not assumed: we only assume that operators are of the form T = (fi,j), where f i j : K —• K and K is a field, i,j € {1,2, . . . , n } . If n > 3, then in the matrix space Mn(K) operators preserving singularity and nonsingularity of matrices must be as in paper of P. Botta. If n < 2, operators may be nonlinear. In this case the forms of the operators are presented. Let R, N denote the set of real numbers or positive integer numbers, respectively. Let Mn(K) be the set of n x n matrices over a field K, i.e. Mn{K) e Knxn, where n e N. We denote by Ej^ the matrix whose j,k entry is 1 and the remaining entries of which are 0. First of all let us introduce.

In this paper, we study multi-step random iteration scheme with errors for a common random fixed point of a finite family of nonself asymptotically nonexpansive random mappings in real uniformly convex separable Banach spaces. The results presented in this paper extend the recent ones announced by Zhou and Wang [

In this paper, we show that the paper mentioned in the title includes some wrong results. We also provide a counter example.

In this paper we investigate some properties of quotient structured spaces. The notion of a structured space was originally considered by Mostow [In the beginning we present some basic notions and definitions from structured space theory. Then we discuss some properties of quotient structured spaces. In the third part we present a space-time as a quotient space. At the end of this paper we consider F-quasiregular equivalence relation and the structured space with malicious singularity.

In this paper we study divided difference operators of any order acting in function algebras. In the definition of difference quotient operators we use a tension structure defined on the set of points on which depend the functions of the algebras considered. In the paper we mention the oportunity for partial difference quotient operators as well as for some purely algebraic definition of divided difference operators in terms of the suitable Leibniz product rules.

In this paper, the notion of an operator \gamma on a supra topological space (X,\mu ) is studied and then utilized to analyze supra \gamma -open sets. The notions of {\mu }_{\gamma }-g.closed sets on the subspace are introduced and investigated. Furthermore, some new {\mu }_{\gamma }-separation axioms are formulated and the relationships between them are shown. Moreover, some characterizations of the new functions via operator \gamma on \mu are presented and investigated. Finally, we give some properties of {S}_{(\gamma ,\beta )}-closed graph and strongly {S}_{(\gamma ,\beta )}-closed graph.

Properties of linear regression of order statistics and their functions are usually utilized for the characterization of distributions. In this paper, based on such statistics, the concept of Pearson covariance and the pseudo-covariance measure of dependence is used to characterize the exponential, Pearson and Pareto distributions.

In this paper, we study the Hyers-Ulam stability of a nonautonomous semilinear reaction-diffusion equation. More precisely, we consider a nonautonomous parabolic equation with a diffusion given by the fractional Laplacian. We see that such a stability is a consequence of a Gronwall-type inequality.

The combined systems of integral equations have become of great importance in various fields of sciences such as electromagnetic and nuclear physics. New classes of the merged type of Urysohn Volterra-Chandrasekhar quadratic integral equations are proposed in this paper. This proposed system involves fractional Urysohn Volterra kernels and also Chandrasekhar kernels. The solvability of a coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type is studied. To realize the existence of a solution of those mixed systems, we use the Perov’s fixed point combined with the Leray-Schauder fixed point approach in generalized Banach algebra spaces.

Many mathematicians defined and studied soft separation axioms and soft continuity in soft spaces by using ordinary points of a topological space X. Also, some of them studied the same concepts by using soft points. In this paper, we introduce the concepts of soft {p}_{c}-{T}_{i} and soft {p}_{c}-{T}_{i}^{\ast }, i=0,1,2 by using the concept of soft {p}_{c}-open sets in soft topological spaces. We explore several properties of such spaces. We also investigate the relationship among these spaces and provide a counter example when it is needed.

Existence of mild solution for noninstantaneous impulsive fractional order integro-differential equations with local and nonlocal conditions in Banach space is established in this paper. Existence results with local and nonlocal conditions are obtained through operator semigroup theory using generalized Banach contraction theorem and Krasnoselskii’s fixed point theorem, respectively. Finally, illustrations are added to validate derived results.

In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. Unione Mat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of Lp by constructing the suitable type of functions.