we shall prove an existence inequality for two maps on banach algebra, withan example and in sequel we have some results on r and rn spaces. this waycan be applied for generalization of some subjects of mathematics in teachingwhich how we can extend a math problem to higher level.

Different versions of Wiener’s Tauberian theorem are discussed for the generalized group algebra LI(G,A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A-valued Fourier transforms. A weak form of Wiener’s Tauberian property is introduced and it is proved that LI(G,A) is weakly Tauberian if and only if A is....

let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

in the present paper an operator-differential equation of second order in complex domain isconsidered when the coefficients have singularity of pole type at the point z=0. a theorem of existence of thesolution of the equation is proved and the spectral property of the solution is separately investigated when thecoefficients are spectral operators.