In this article, we obtain Lyapunov-type inequalities for certain odd order linear boundary-value problems. Our inequalities involve integrals of both q+(t) and q−(t) in addition to that of |q(t)|. The Green’s function for even order boundary-value problems plays a key role in our proofs. Also, using the Fredholm alternative theorem, we obtain a criterion for the existence and uniqueness of sol...

1. Complex differentiation 2. Exponentials, trigonometric functions 3. Differentiating power series: Abel’s theorem 4. Path integrals 5. Cauchy’s theorem 6. Cauchy’s formula 7. Power series expansions, Morera’s theorem 8. Identity principle 9. Liouville’s theorem: bounded entire functions are constant 10. Laurent expansions around singularities 11. Residues and evaluation of integrals 12. Logar...

*Correspondence: [email protected] 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK Full list of author information is available at the end of the article Abstract We establish analogues of the mean value theorem and Taylor’s theorem for fractional differential operators defined using a Mittag–Leffler kernel. We formulate a new model for the fract...

in this paper, a vector version of the intermediate value theorem is established. the main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{on fixed point theorems for monotone increasing vector valued mappings via scalarizing}, positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

In this paper, a new version of mean value theorem for interval-valued functions on time scales is established. Meantime, some basic concepts and results associated with semigroups operators are presented. As an application operators, under certain conditions, we consider the initial problem differential equations scales. Finally, two issues worthy further discussion

We show that the nth derivative of Riemann zeta function, when summed over non-trivial zeros zeta, is real and positive/negative in mean for n odd/even, respectively. this by giving a full asymptotic expansion these sums.