Volume 72 , issue 1 ( 2020 ) | back |

ON SOMOS'S IDENTITIES OF LEVEL TWENTY ONE AND THEIR PARTITION INTERPRETATIONS | 1$-$5 |

**Abstract**

In this paper, proofs of Somos's theta function identities of level 21 will be given. Further, we deduce certain interesting partition identities from them.

**Keywords:** Dedekind eta-function; modular equation; color partition.

**MSC:** 11F20, 11P83, 11F27

ČECH ROUGH PROXIMITY SPACES | 6$-$16 |

**Abstract**

Topology is a strong root for constructs that can be helpful to enrich the original model of approximation spaces. This paper introduces closure spaces on rough sets via a proximity relation on approximation spaces. We have used rough proximity to define the nearness between rough sets. Some results have been proved in this advanced nearness structure named Čech rough proximity. Examples are given to illustrate the proposed approach. Finally, an application of the theory is presented to demonstrate the fruitfulness of this new structure.

**Keywords:** Rough sets; proximity spaces; closure space.

**MSC:** 54A05, 54A10, 54C60, 54E05, 54E17

AN EXISTENCE THEOREM OF TRIPLED FIXED POINT FOR A CLASS OF OPERATORS ON BANACH SPACE WITH APPLICATIONS | 17$-$29 |

**Abstract**

In this paper, using the technique of measure of noncompactness, we prove some theorems on tripled fixed points for a class of operators in a Banach space. Also as an application, we discuss the existence of solution for a class of systems of nonlinear functional integral equations. Finally a concrete example illustrating the mentioned applicability is also included.

**Keywords:** Measure of noncompactness; system of integral equations; tripled fixed point.

**MSC:** 47H09, 47H10, 34A12

**Abstract**

Given any sequence $a=(a_{n})_{n\geq 1}$ of positive real numbers and any set $E$ of complex sequences, we write $E_{a}$ for the set of all sequences $x=(x_{n})_{n\geq 1}$ such that $x/a=(x_{n}/a_{n})_{n\geq 1}ın E$. We denote by $W_{a}\left( \lambda \right) $ $=\left( w_{ınfty }\left( \lambda \right) \right) _{a}$ and $W_{a}^{0}\left( \lambda \right) =\left( w_{0}\left( \lambda \right) \right) _{a}$ the sets of all sequences $x$ such that $\sup_{n}\left( \lambda _{n}^{-1}\sum_{k=1}^{n}\left\vert x_{k}\right\vert /a_{k}\right) <ınfty $ and $\lim_{n\rightarrow ınfty }\left( \lambda _{n}^{-1}\sum_{k=1}^{n}\left\vert x_{k}\right\vert /a_{k}\right) =0$, where $\lambda $ is a nondecreasing exponentially bounded sequence. In this paper we recall some properties of the Banach algebras $\left( W_{a}\left( \lambda \right) ,W_{a}\left( \lambda \right) \right) $, and $\left( W_{a}^{0}\left( \lambda \right) ,W_{a}^{0}\left( \lambda \right) \right) $, where $a$ is a positive sequence. We then consider the operator $\Delta _{\rho }$, defined by $\left[ \Delta _{\rho }x\right] _{n}=x_{n}-\rho _{n-1}x_{n-1}$ for all $n\geq 1$ with the convention $x_{0}$, $\rho _{0}=0$, and we give necessary and sufficient conditions for the operator $\Delta _{\rho }:E\rightarrow E$ to be bijective, for $E=w_{0}\left( \lambda \right) $, or $w_{ınfty }\left( \lambda \right) $. Then we consider the generalized operator of the first difference $B\left( \widetilde{r},\widetilde{s}\right) $, where $\widetilde{r% },\widetilde{s}$ are two convergent sequences, and defined by $\left[B\left( \widetilde{r},\widetilde{s}\right) x\right]_{n}=r_{n}x_{n}+s_{n-1}x_{n-1}$ for all $n\geq 1$ with the convention $x_{0},s_{0}=0$. Then we deal with the operator $B\left( \widetilde{r}, \widetilde{s}\right) $ mapping in either of the sets $W_{a}\left( \lambda \right) $, or $W_{a}^{0}\left( \lambda \right) $. We then apply the previous results to explicitly calculate the spectrum of $B\left( \widetilde{r}, \widetilde{s}\right) $ over each of the spaces $E_{a}$, where $E=w_{0}\left(\lambda \right) $, or $w_{ınfty }\left( \lambda \right) $. Finally we give a characterization of the identity $\left( W_{a}\left( \lambda \right) \right) _{B\left( r,s\right) }=W_{b}\left( \lambda \right) $.

**Keywords:** Sequence space; BK space; Banach algebra; bounded linear operator; spectrum of an operator.

**MSC:** 40C05, 46A45

LEXICOGRAPHIC PRODUCT OF VAGUE GRAPHS WITH APPLICATION | 43$-$57 |

**Abstract**

A vague graph is a generalized structure of a fuzzy graph which gives more precision, flexibility, and compatibility to a system when it is compared with the systems which are designed by using fuzzy graphs. The present study aims to introduce the notion of lexicographic min-product and max-product of two vague graphs. Then, the degree of a vertex in the lexicographic products of two vague graphs is obtained. Finally, a relationship is obtained between the lexicographic min-product and lexicographic max-product.

**Keywords:** Vague graph; lexicographic min-product; lexicographic max-product.

**MSC:** 05C72, 94C15

$I$-SECOND SUBMODULES OF A MODULE | 58$-$65 |

**Abstract**

Let $R$ be a commutative ring with identity, $I$ an ideal of $R$, and $M$ be an $R$-module. In this paper, we will introduce the concept of $I$-second submodules of $M$ as a generalization of second submodules of $M$ and obtain some related results.

**Keywords:** Second submodule; weak second submodule; $I$-prime ideal; $I$-second submodule.

**MSC:** 13C13, 13C99

NONLOCAL THREE-POINT MULTI-TERM MULTIVALUED FRACTIONAL-ORDER BOUNDARY VALUE PROBLEMS | 66$-$80 |

**Abstract**

In this paper we study a new kind of boundary value problems of multi-term fractional differential inclusions and three-point nonlocal boundary conditions. The existence of solutions is established for convex and non-convex multivalued maps by using standard theorems from the fixed point theory. We also construct some examples for demonstrating the application of the main results.

**Keywords:** Caputo fractional derivatives; existence of solutions; fixed point theorems.

**MSC:** 34A08, 34B10, 34A60

FREQUENTLY HYPERCYCLIC $\boldsymbol{C}$-DISTRIBUTION SEMIGROUPS AND THEIR GENERALIZATIONS | 81$-$94 |

**Abstract**

In this paper, we introduce the notions of $f$-frequent hypercyclicity and $F$-hypercyclicity for $C$-distribution semigroups in separable Fréchet spaces. We particularly analyze the classes of $q$-frequently hypercyclic $C$-distribution semigroups ($q\geq 1$) and frequently hypercyclic $C$-distribution semigroups, providing a great number of illustrative examples.

**Keywords:** $C$-distribution semigroups; integrated $C$-semigroups; $f$-frequent hypercyclicity; $q$-frequent hypercyclicity; Fr\' echet spaces.

**MSC:** 47A16, 47B37, 47D06