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For a Tychonoff space $X$, we denote by $(C(X), \tau_k, \tau_p)$ the bitopological space of all real-valued continuous functions on $X$ where $\tau_k$ is the compact-open topology and $\tau_p$ is the topology of pointwise convergence. In papers [5, 6, 13] variations of selective separability and tightness in $(C(X), \tau_k, \tau_p)$ were investigated. In this paper we continued to study the selective properties and the corresponding topological games in the space $(C(X), \tau_k, \tau_p)$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1805.04363

The paper studies the open-hereditary property of semi-separation axioms and applies it to the study of digital topological spaces such as an n-dimensional Khalimsky topological space, a Marcus-Wyse topological space and so on. More precisely, we study various properties of digital topological spaces related to low-level and semi-separation axioms such as T1/2 , semi-T1/2 , semi-T1, semi-T2, etc. Besides, using the finite or the infinite product property of the semi-Ti-separation axiom, i ? {1,2}, we prove that the n-dimensional Khalimsky topological space is a semi-T2-space. After showing that not every subspace of the digital topological spaces satisfies the semi-Ti-separation axiom, i ?{1,2}, we prove that the semi-Tiseparation property is open-hereditary, i ? {1,2}. All spaces in the paper are assumed to be nonempty and connected.

‎In this ‎paper,‎ we study two-stage stochastic linear ‎programming‎ ‎(‎SLP) ‎problems with fixed ‎recourse‎. The problem is often large scale as the objective function involves an expectation over a discrete set of scenarios. This paper presents a ‎parallel ‎implementation ‎of ‎the ‎augmented Lagrangian ‎method ‎for ‎solving ‎SLPs.‎ ‎‎‎‎‎‎‎‎‎‎Our parallel method is based on a modified version of the L-shaped method and reducing linear master and recourse programs to unconstrained‎ maximization of concave differentiable piecewise quadratic functions‎. ‎The maximization problem is solved using the generalized Newton method‎. ‎The parallel method is implemented in ‎Matl‎ab‎. ‎Large scale SLP with several millions of variables and several hundreds of‎ ‎thousands of constraints ‎a‎re solved‎. ‎The ‎‎results of uniprocessor and multiprocessor computations are‎ ‎presented ‎which‎‎‎ show that the parallel algorithm is ‎effective.‎

Up to now there is no homotopy for Marcus-Wyse (for short $M$-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short $M$-) topological spaces on ${\bf Z}^2$, we need to generalize the $M$-topology on ${\bf Z}^2$ to higher dimensional spaces $X \subset {\bf Z}^n$, $n \geq 3$ \cite{HL1}. Hence the present paper establishes a new topology on ${\bf Z}^n, n \in {\bf N}$, where ${\bf N}$ is the set of natural numbers. It is called the {\it generalized Marcus-Wyse} (for short $H$-) topology and is denoted by $({\bf Z}^n, \gamma^n)$. Besides, we prove that $({\bf Z}^3, \gamma^3)$ induces only $6$- or $18$-adjacency relations. Namely, $({\bf Z}^3, \gamma^3)$ does not support a $26$-adjacency, which is quite different from the Khalimsky topology for $3$D digital spaces. After developing an $H$-adjacency induced by the connectedness of $({\bf Z}^n, \gamma^n)$, the present paper establishes topological graphs based on the $H$-topology, which is called an $HA$-space in the paper, so that we can establish a category of $HA$-spaces. By using the $H$-adjacency, we propose an $H$-topological graph homomorphism (for short $HA$-map) and an $HA$-isomorphism. Besides, we prove that an $HA$-map ({\it resp.} an $HA$-isomorphism) is broader than an $H$-continuous map ({\it resp.} an $H$-homeomorphism) and is an $H$-connectedness preserving map. Finally, after investigating some properties of an $HA$-isomorphism, we propose both an $HA$-retract and an extension problem of an $HA$-map for studying $HA$-spaces.

The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology.

Recently Srivastava et al. (26) introduced the incomplete Pochhammer symbols by means of the incomplete gamma functions (s; x) and (s; x), and defined incomplete hypergeometric functions whose a number of interesting and fundamental properties and characteristics have been investigated. Further, Cetinkaya (6) introduced the incomplete second Appell hypergeometric functions and studied many interesting and fundamental properties and characteristics. In this paper, motivated by the above- mentioned works, we introduce two incomplete Srivastava's triple hypergeometric functions H and H by using the incomplete Pochhammer symbols and investigate certain properties, for example, their various integral representations, derivative formula, reduction formula and recurrence relation. Various (known or new) special cases and consequences of the results presented here are also considered. 1. Introduction, Definitions and Preliminaries Throughout this paper, N, Z , and C denote the sets of positive integers, negative integers, complex numbers, respectively, and N0 := N(f0g and Z 0 := Z (f0g. The familiar incomplete Gamma functions (s; x) and (s; x) defined by

Since a locally finite topological structure plays an important role in the fields of pure and applied topology, the paper studies a special kind of locally finite spaces, so called a space set topology (for brevity, SST) and further, proves that an SST is an Alexandroff space satisfying the separation axiom T0. Unlike a point set topology, since each element of an SST is a space, the present paper names the topology by the space set topology. Besides, for a connected topological space (X,T) with |X| = 2 the axioms T0, semi-T1/2 and T1/2 are proved to be equivalent to each other. Furthermore, the paper shows that an SST can be used for studying both continuous and digital spaces so that it plays a crucial role in both classical and digital topology, combinatorial, discrete and computational geometry. In addition, a connected SST can be a good example showing that the separation axiom semi-T1/2 does not imply T1/2.

The aim of this paper is to introduce a topological model of fractals. Self similar fractals will be approached as inverse limit of finite one dimensional topological spaces with alpha continuous bonding functions. The second approach is to investigate topological graphs in terms nano topological spaces for Lellis Thivagar. From these approximations, the dynamics of Julia sets as a special type of self similar fractals will be studied and some physical properties of fractals through their nano topological graphs will be applied.

The aim of the article is to establish the structure of partial cone b-metric spaces over Banach algebras. Topological and structural properties are investigated of the new spaces. We also define generalized Lipschitz mappings and give their application in fixed point theory. The results presented in this paper substantially extend and strengthen the results of the literature. Few examples are provided to support our conclusions and as an application we establish the existence and uniqueness of a solution to a class of system of nonlinear integral equations.

In this paper, a new pseudo-random number generator (PRNG) based on improved onedimensional discrete-space chaotic map is proposed. Like the original, the improved map relies on bijective mapping of permutations and natural numbers. Instead of using standard Lehmer code, we use a mapping computable in linear time, which significantly speeds up the PRNG. Results of NIST 800-22 test suite and TestU01 test suite confirm that the proposed approach can be used for generation of pseudo-random numbers. Due to discrete nature of used chaotic map, the proposed PRNG is not influenced by dynamical degradation and has virtually unlimited key space. Proposed approach has much better ratio between required memory and security level than previous secure one-dimensional discrete-space chaotic PRNGs. Also, proposed PRNG is much faster than other secure PRNGs of the same type. Satisfactory speed and small memory requirements indicate that proposed PRNG has properties desirable for use in devices with limited memory space, such as wireless sensor networks.

In this paper, we introduce a new family of distributions extending the odd family of distributions. A new tuning parameter is introduced, with some connections to the well-known transmuted transformation. Some mathematical results are obtained, including moments, generating function and order statistics. Then, we study a special case dealing with the standard loglogistic distribution and the modifiedWeibull distribution. Its main features are to have densities with flexible shapes where skewness, kurtosis, heavy tails and modality can be observed, and increasing-decreasing-increasing, unimodal and bathtub shaped hazard rate functions. Estimation of the related parameters is investigated by the maximum likelihood method. We illustrate the usefulness of our extended odd family of distributions with applications to two practical data sets.

© 2019, University of Nis. All rights reserved. In this paper we introduced the concept of strong probabilistic metric spaces (sPM spaces) and we show some of its basic properties. In this frame we present several fixed point results for mappings of contractive type. Our results generalize and unify several fixed point theorems in literature. Finally, we give some possible applications of our results.

In this paper, we define an involute and an evolving involute of order k of a null Cartan curve in Minkowski space En1 for n ? 3 and 1 ? k ? n-1. In relation to that, we prove that if a null Cartan helix has a null Cartan involute of order 1 or 2, then it is Bertrand null Cartan curve and its involute is its Bertrand mate curve. In particular, we show that Bertrand mate curve of Bertrand null Cartan curve can also be a non-null curve and find the relationship between the Cartan frame of a null Cartan curve and the Frenet or the Cartan frame of its non-null or null Cartan involute of order 1 ? k ? 2. We show that among all null Cartan curves in E31 , only the null Cartan cubic has two families of involutes of order 1, one of which lies on B-scroll. We also give some relations between involutes of orders 1 and 2 of a null Cartan curve in Minkowski 3-space. As an application, we show that involutes of order 1 of a null Cartan curve in E31 , evolving according to null Betchov-Da Rios vortex filament equation, generate timelike Hasimoto surfaces.

In this paper, we present a unique polar-like decomposition theorem for rectangular complex matrices. Applying this decomposition, we define on the set of rectangular matrices a new partial ordering called WL(weak L?wner) partial order ? an extension of the GL(generalized L?wner) partial order, and derive some basic properties of the new partial ordering.

In this paper, we present a new generalization to compute determinants and inverses of r-circulant matrices Qn = circr ((b/a)?(2)/2 q1,(b/a)?(3)/2 q2,..., (b/a)?(n+1)/2 qn) and Ln = circr ((b/a)?(1)/2 l1,(b/a)?(2)/2 l2,..., (b/a)?(n)/2 ln) whose entries are the biperiodic Fibonacci and the biperiodic Lucas numbers, respectively. Also, we express determinants of the matrices Qn and Ln by using only the biperiodic Fibonacci and the biperiodic Lucas numbers.

In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of appropriate inequalities, as well as allows direct establishment of the dependence between (the exponent of) some functions that occur as bounds of the approximation and the interval in which the corresponding inequality holds true.

Very recently, Shahzad et al. [RACSAM 111 (2017) 307-324] introduced the notion of (A,S)- contractions which unifies several well known nonlinear type contractions (e.g. R-contractions, Z-contractions, L-contractions etc.) in one go. In this paper, we introduce the notion of generalized (A,S)f-contractions and utilize the same to present some coincidence and common fixed point results for a pair of self-mappings (g,f)defined on a metric space endowed with a binary relation S. In this course, we ought to introduce some new notions namely: (I,S)-continuity, (I,S)-compatibility and local (g,f)-transitivity. Consequently, several results involving R-contractions and Z-contractions are deduced. Finally, we furnish illustrative examples to demonstrate the utility of our results.

Classes of circulant graphs play an important role in modeling interconnection networks in parallel and distributed computing. They also find applications in modeling quantum spin networks supporting the perfect state transfer. It has been noticed that unitary Cayley graphs as a class of circulant graphs possess many good properties such as small diameter, mirror symmetry, recursive structure, regularity, etc. and therefore can serve as a model for efficient interconnection networks. In this paper we go a step further and analyze some other characteristics of unitary Cayley graphs important for the modeling of a good interconnection network. We show that all unitary Cayley graphs are hamiltonian. More precisely, every unitary Cayley graph is hamiltonian-laceable (up to one exception for X6) if it is bipartite, and hamiltonianconnected if it is not. We prove this by presenting an explicit construction of hamiltonian paths on Xnm using the hamiltonian paths on Xn and Xm for gcd(n,m) = 1. Moreover, we also prove that every unitary Cayley graph is bipancyclic and every nonbipartite unitary Cayley graph is pancyclic.

In the present paper, firstly, we review the notion of the SO-complete metric spaces. This notion let us to consider some fixed point theorems for single-valued mappings in incomplete metric spaces. Secondly, as motivated by the recent work of H. Baghani et al.(A fixed point theorem for a new class of set-valued mappings in R-complete (not necessarily complete) metric spaces, Filomat, 31 (2017), 3875-3884), we obtain the results of Ansari et al. [J. Fixed Point Theory Appl. (2017), 1145-1163] with very much weaker conditions. Also, we provide some examples show that our main theorem is a generalization of previous results. Finally, we give an application to the boundary value system for our results.

In this paper, categorical properties of L-fuzzifying convergence spaces are investigated. It is shown that (1) the category L-FYC of L-fuzzifying convergence spaces is a strong topological universe; (2) the category L-FYKC of L-fuzzifying Kent convergence spaces, as a bireflective and bicoreflective subcategory of L-FYC, is also a strong topological universe; (3) the category L-FYLC of L-fuzzifying limit spaces, as a bireflective subcategory of L-FYKC, is a topological universe.

In this paper, weobtained best proximity coincidence point theorems for ?-Geraghty contractions in the setting of complete metric spaces by using weak P-property. Also we presented some examples to prove the validity of our results. Our results extended and unify many existing results in the literature. Moreover, in the last section as applications of our main results, we can apply some coincidence best proximity point and coupled coincidence proximity point on metric spaces endowed with an arbitrary binary relation.

In this paper we study the uniform approximation of functions by Favard-Sz?sz-Mirakyan operators of max-product type in some exponential weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity.

The aim of a decision support system (DSS) is to enhance the course of decision-making by furnishing a decision episode and support during the decision-making and enforcing procedure. Inspired by management information systems, the DSS assists policy makers to reach a decision by man-machine interaction through data, models and knowledge. Knowledge fusion is an effective means of enhancing the efficiency of a DSS. Thus, this paper presents a three-layer, six-step knowledge fusion pattern from the perspective of interaction and the three procedures used to arrive at a decision. For better understanding, an evaluation of the knowledge state with knowledge objects in the fusion pattern is also presented. Furthermore, we briefly examine a case study of a Fire Rescue DSS(FRDSS).

This paper introduces the definition of a Lie rough group as a natural development of the concepts of a smooth manifold and a rough group on an approximation space. Furthermore, the properties of Lie rough groups are discussed. It is shown that every Lie rough group is a topological rough group, and that the product of two Lie rough groups is again a Lie rough group. We define the concepts of Lie rough subgroups and Lie rough normal subgroups. Finally, our aim is to give an example by using definition of Lie rough homomorphism sets G and H.

In this paper, we propose a new lifetime model as a discrete version of the continuous weighted exponential distribution which is called discrete weighted exponential distribution (DWED). This model is a generalization of the discrete exponential distribution which is originally introduced by Chakraborty (2015). We present various statistical indices/properties of this distribution including reliability indices, moment generating function, probability generating function, survival and hazard rate functions, index of dispersion, and stress-strength parameter. We first present a numerical method to compute the maximum likelihood estimations (MLEs) of the models parameters, and then conduct a simulation study to further analyze these estimations. The advantages of the DWED are shown in practice by applying it on two real world applications and compare it with some other well-known lifetime distributions.

Tauberian theorem provides a criterion for the convergence of non convergent (summable) sequences. In this paper, we established a Tauberian theorem for nth real sequences via Ces?ro summability by using de la Vall?e Poussin mean and slow oscillation. The discussion and findings are capable to unify several useful concepts in the literature, and should also provide nontrivial extension of several results. Some examples are incorporated in support of our definitions and results. The findings are further expected to be helpful in designing and study several other interesting problems in summability theory and applications.

In this paper, a new type of graph contractive multi-valued mappings in a metric space with a directed graph is introduced and studied. A common fixed point theorem of those two multi-valued mappings is established under some appropriate conditions. Moreover, some examples illustrating our main result are also given. The obtained result extends and generalizes several fixed point results of multivalued mappings in the literature. We apply our main result to obtain common fixed point results for two multi-valued mappings in ?-chainable complete metric spaces and two cyclic contraction multi-valued mappings.

The main aim of the present paper is to define new soft separation axioms which lead us, first, to generalize existing comparable properties via general topology, second, to eliminate restrictions on the shape of soft open sets on soft regular spaces which given in [22], and third, to obtain a relationship between soft Hausdorff and new soft regular spaces similar to those exists via general topology. To this end, we define partial belong and total non belong relations, and investigate many properties related to these two relations. We then introduce new soft separation axioms, namely p-soft Ti-spaces (i = 0,1,2,3,4), depending on a total non belong relation, and study their features in detail. With the help of examples, we illustrate the relationships among these soft separation axioms and point out that p-soft Ti-spaces are stronger than soft Ti-spaces, for i = 0,1,4. Also, we define a p-soft regular space, which is weaker than a soft regular space and verify that a p-soft regular condition is sufficient for the equivalent among p-soft Ti-spaces, for i = 0,1,2. Furthermore, we prove the equivalent among finite p-soft Ti-spaces, for i = 1,2,3 and derive that a finite product of p-soft Ti-spaces is p-soft Ti, for i = 0,1,2,3,4. In the last section, we show the relationships which associate some p-soft Ti-spaces with soft compactness, and in particular, we conclude under what conditions a soft subset of a p-soft T2-space is soft compact and prove that every soft compact p-soft T2-space is soft T3-space. Finally, we illuminate that some findings obtained in general topology are not true concerning soft topological spaces which among of them a finite soft topological space need not be soft compact.

The Randic index R(G) of a graph G is the sum of the weights (dudv)-1/2 of all edges uv in G, where du denotes the degree of vertex u. Du and Zhou [On Randic indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760-2770] determined the n-vertex trees with the third for n ? 7, the fourth for n ? 10, the fifth and the sixth for n ? 11 maximum Randic indices. Recently, Li et al. [The Randic indices of trees, unicyclic graphs and bicyclic graphs, Ars Comb. 127 (2016), 409-419] obtained the n-vertex trees with the seventh, the eighth, the ninth and the tenth for n ? 11 maximum Randic indices. In this paper, we correct the ordering for the Randic indices of trees obtained by Li et al., and characterize the trees with from the seventh to the sixteenth maximum Randic indices. The obtained extremal trees are molecular and thereby the obtained ordering also holds for molecular trees.

In this paper, we study the existence and uniqueness of best proximity points for cyclic Meir-Keeler contraction mappings in metric spaces with the property W-WUC. Also, the existence of best proximity points for set-valued cyclic Meir-Keeler contraction mappings in metric spaces with the property WUC are obtained

In this paper, we introduce the concept of fuzzy soft filters and study some of their properties. Also, we study the notion of convergence of fuzzy soft filters in fuzzy soft topological spaces. We prove the existence of product fuzzy soft filters.

In this paper, we introduce the concept of a modified F-contraction via ?-admissible mappings and propose some theorems that guarantee the existence and uniqueness of fixed point for such mappings in the frame of complete metric spaces. We also provide some illustrative examples. Moreover, we consider an application solving an integral equation.

In this work, our intention is to introduce the notion of rational (?-?-FG)-contraction mapping in b-metric-like spaces, and produce relevant fixed point and periodic point results for weakly ?-admissible mappings. Ulam-Hyers stability of this problem is also investigated. To illustrate our results, we give throughout the paper some examples, in particular in order to justify the use of rational terms. As an application, we obtain sufficient conditions for the existence of solutions for Cantilever Beam Problem.

In view of the relationship with the Kr\"{a}tzel function, we derive a new series representation for the $\lambda$-generalized Hurwitz-Lerch Zeta function introduced by H.M. Srivastava [Appl. Math. Inf. Sci. 8 (2014) 1485--1500] and determine the monotonicity of its coefficients. An integral representation of the Mathieu $\left(\textbf{a},\bm{\lambda}\right)$-series is rederived by applying the Abel's summation formula (which provides a slight modification of the result given by Pog\'{a}ny [Integral Transforms Spec. Funct. 16 (8) (2005) 685--689]) and this modified form of the result is then used to obtain a new integral representation for the $\lambda$-generalized Hurwitz-Lerch Zeta function. Finally, by making use of the various results presented in this paper, we establish two sets of two-sided inequalities for the $\lambda$-generalized Hurwitz-Lerch Zeta function.

This paper extends and generalizes results of Mukheimer [(?,?,?)-contractive mappings in ordered partial b-metric spaces, J. Nonlinear Sci. Appl. 7(2014), 168-179]. A new concept of (?-?1-?2)-contractive mapping using two altering distance functions in ordered b-metric-like space is introduced and basic fixed point results have been studied. Useful examples are illustrated to justify the applicability and effectiveness of the results presented herein. As an application, the existence of solution of fourth-order two-point boundary value problems is discussed and rationalized by a numerical example.

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

In this paper, we interest with deriving the sufficient and necessary conditions for optimal solution of special classes of Programming. These classes involve generalized E-[0,1] convex functions. Characterization of efficient solutions for E-[0,1] convex multi-objective Programming are obtained. Finally, sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution are derived.

[EN] In this paper we introduce and study the concept of strong convergence in fuzzy metric spaces (X, M,*) in the sense of George and Veeramani. This concept is related with the condition Lambda M-t > 0(x, y, t) > 0, which frequently is required or missing in this context. Among other results we characterize the class of s-fuzzy metrics by the strong convergence defined here and we solve partially the question of finding explicitly a compatible metric with a given fuzzy metric., Valentn Gregori acknowledges the support of the Spanish Ministry of Economy and Competitiveness under Grant MTM2015-64373-P (MINECO/FEDER, UE).

The purpose of this paper is to present some fixed point results in b-metric spaces using a contractive condition of Hardy-Rogers type with respect to the functional H. The data dependence of the fixed point set, the well-posedness of the fixed point problem, as well as, the Ulam-Hyres stability are also studied.

In this paper, the authors investigate a majorization problem for certain subclasses of multivalent meromorphic functions defined in the punctured unit disk U* having a pole of order p at origin. The subclasses under investigation are defined through iterations and combinations of the Liu-Srivastava operator and a meromorphic analogue of the Cho-Kwon-Srivastava operator for normalized analytic function. Several consequences of the main results in form of corollaries are also pointed out.

In this paper, we consider an initial - boundary value problem for a nonlinear wave equation containing a nonlocal term. Using a high order iterative scheme, the existence of a unique weak solution is proved. Furthermore, the sequence established here converges to a unique weak solution at a rate of order N (N≥2).

In this paper, we propose a method based on generating functions for constructing q-analogues of some discrete type positive linear operators (e.g., q-Lupas, q-Bernstein, q-Meyer-Konig and Zeller and ¨ q-Bleimann-Butzer-Hahn operators etc.). In other words, general operators of the discrete type are constructed, and their respective formulae for central moments are thereby obtained. Finally, through the use of specific generating functions, we are able to provide some relevant exemplary applications of general operators.

In this paper, inspired the very interesting results of Ciric [20], we investigate the existing non-unique fixed points of certain operators in the context of b-metric spaces. Our main results unify and cover several existing results on the topic in the literature.

The classical power sum and alternating power sum identities can be stated as ?m,i=0 sn(i)= 1/n+1 (Bn+1(m+1)- Bn+1), ?m,i=0(-1)i sn (i) = 1/2 ((-1)m En(m+1) + En), where sn(x)=xn is the simplest possible Appell polynomial for the Sheffer pair (1,t). The impetus for this research starts from the question that what if we replace sn(x)=xn by any Appell polynomial. In this paper, we give a generalization of power and alternating power sums to any Appell polynomials.

In this paper existence and uniqueness of fixed points for a general class of contractive and nonexpansive mappings on modular metric spaces is discussed. As an application of the theoretical results, the existence of a solution of anti-periodic boundary value problems for nonlinear first order differential equations of Carath?odory?s type is considered in the framework of modular metric spaces.

In this paper, we associate a partial g-hypergroupoid with a given g-hypergraph and analyze the properties of this hyperstructure. We prove that a g-hypergroupoid may be a commutative hypergroup without being a join space. Next, we define diagonal direct product of g-hypergroupoids. Further, we construct a sequence of g-hypergroupoids and investigate some relationships between it?s terms. Also, we study the quotient of a g-hypergroupoid by defining a regular relation. Finally, we describe fundamental relation of an Hv-semigroup as a g-hypergroupoid.

In this paper, the concept of (?-?)-generalized rational contraction multivalued operator is introduced and then the existence of common fixed points of such mapping in complete dislocated quasi bmetric spaces is obtained. Some examples are presented to show that the results proved herein are potential generalization and extension of comparable existing results in the literature. We also study Ulam-Hyers stability of fixed point problems of (?-?)-generalized rational contraction multivalued operator. We also obtain some common fixed point results for single and multivalued mappings in a complete dq b-metric space endowed with a partial order. As an application, the existence of a continuous solution of an integral equation under appropriate assumptions is obtained.

In this paper, we study the existence and uniqueness theorem for Lp (1 < p < 2) solutions to a class of infinite time interval backward doubly stochastic differential equations (BDSDEs). Furthermore, we obtain the comparison theorem for 1-dimensional infinite time interval BDSDEs in Lp.

In the paper, we present new semi-analytical approach for FDE?s consisting in combination of the method of steps and a technique called differential transformation method (DTM). This approach reduces the original Cauchy problem for delayed or neutral differential equation to Cauchy problem for ordinary differential equation for which DTM is convenient and efficient method. Moreover, there is no need of any symbolic calculations or initial approximation guesstimates in contrast to methods like the homotopy analysis method, the homotopy perturbation method, the variational iteration method or the Adomian decomposition method. The efficiency of the proposed method is shown on certain classes of FDE?s with multiple constant delays including FDE of neutral type. We also compare it to the current approach of using DTM and the Adomian decomposition method where Cauchy problem is not well posed.

In this paper we provide procedures for testing the existence of various types of generalized inverses in involutive residuated semigroups and involutive quantales defined by the Moore-Penrose equations, and for computing the extreme inverses whenever they exist. We also determine certain instances when a? = a*, whenever a? exists. The obtained results can be applied to a wide class of quantales of fuzzy relations and fuzzy matrices, as well as to Gelfand quantales.