Investigation of nonlinear fractional delay differential equation via singular fractional operator

The present research work is basically devoted to construction of a fractional order differential equation with time delay. Initially, integral representation is given to solution of the underline problem. Afterwards, operator form of solution is studied under some auxiliary hypothesis. Since uniqueness of solution is required, therefore we also provide results for exploring the uniqueness of solution for the underlying model. Using Lebesgue dominated convergence theorem and some other results from analysis, this work provides results devoted to existence of at least one solution. Also, for investigating the nature of solution for the proposed model, we study different kind of stability analysis. These stability related results show, how the solution behave with time. At the end of the article, we illustrate the obtained results via some examples.

On Fuzzy Henstock-Kurzweil-Stieltjes-♦-Double Integral on Time scales

We introduce and study some properties of fuzzy Henstock-Kurzweil-Stietljes-$ \Diamond $-double integral on time scales. Also, we state and prove the uniform convergence theorem, monotone convergence theorem and dominated convergence theorem for the fuzzy Henstock-Kurzweil Stieltjes-$\Diamond$-double integrable functions on time scales.

Bicomplex Version of Lebesgue's Dominated Convergence Theorem and Hyperbolic Invariant Measure

In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue's dominated convergence theorem and some other results related to this theorem. Also we have proved the bicomplex version of Lebesgue-Radon-Nikodym theorem. Finally we have introduced the idea of hyperbolic version of invariant measure.AMS Subject Classification (2010) : 28E99, 30G35.

A dominated convergence theorem for Eisenstein series

Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

Necessary and Sufficient Conditions for Oscillation of Solutions to Second-Order Neutral Differential Equations with Impulses

In this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system\left\{ {\matrix{{{{\left( {r\left( t \right){{\left( {z'\left( t \right)} \right)}^\gamma }} \right)}^\prime } + q\left( t \right){x^\alpha }\left( {\sigma \left( t \right)} \right) = 0,} \hfill & {t \ge {t_0},\,\,\,t \ne {\lambda _k},} \hfill \cr {\Delta \left( {r\left( {{\lambda _k}} \right){{\left( {z'\left( {{\lambda _k}} \right)} \right)}^\gamma }} \right) + h\left( {{\lambda _k}} \right){x^\alpha }\left( {\sigma \left( {{\lambda _k}} \right)} \right) = 0,} \hfill & {k \in \mathbb{N}} \hfill \cr } } \right. are established, where z\left( t \right) = x\left( t \right) + p\left( t \right)x\left( {\tau \left( t \right)} \right)Under the assumption \int {^\infty {{\left( {r\left( \eta \right)} \right)}^{ - 1/\alpha }}d\eta = \infty } two cases when γ>α and γ<α are considered. The main tool is Lebesgue’s Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem.

Employing Kuratowski measure of non-compactness for positive solutions of system of singular fractional q-differential equations with numerical effects

In this work, we investigate the existence of solutions for the system of two singular fractional q-differential equations under integral boundary conditions via the concept of Caputo fractional q-derivative and fractional Riemann-Liouville type q-integral. Some new existence results are obtained by applying Krattowski measure of non-compactness. Also, the Darbo?s fixed point theorem and the Lebesgue dominated convergence theorem are the main tools in deriving our proofs. Lastly, we present an example illustrating the primary effects.

EXISTENCE OF NONOSCILLATORY RELATIVELY BOUNDED SOLUTIONS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS

In this paper some sufficient conditions are obtained to insure the existence of positive solutions which is relatively bounded from one side for nonlinear neutral differential equations of second order.Weused the Krasnoselskii’s fixed point theorem and Lebesgue’s dominated convergence theorem to obtain new sufficient conditions for the existence of a Nonoscillatoryone side relatively boundedsolutions.These conditions are more applicable than some known results in the references. Three examples included to illustrate the results obtained.

A new fuzzy-valued integral and its convergence theorems

In this paper, we generalize Kluv?anek-Lewis type set-valued integral to fuzzy number measures in Banach spaces. Firstly, we introduce the new fuzzy-valued integral with some natural properties. And then, We establish Vitali type convergence theorem and dominated convergence theorem for this kind of integral.