In this paper we study the existence of integrable solutions for initial value problem for implicit fractional order functional differential equations with infinite delay. Our results are based on Schauder type fixed point theorem and the Banach contraction principle fixed point theorem.

In the setting of spaces of homogeneous-type, we define the Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi$, where $\phi$ is a function of positive lower type and upper type less than $1$, and show that $I_{\phi }$ and $D_{\phi }$ are bounded from Lipschitz spaces ${\Lambda }^{\xi }$ to ${\Lambda }^{\xi \phi }$ and ${\Lambda }^{\xi /\phi }$ respectively, with suitable restrictions on the quasi-increasing function $\xi$ in each case. We also prove that $I_{\phi }$ and $D_{\phi }$ are bounded from the generalized Besov ${\dot{B}}_p^{\psi ,q}$, with $1\le p,q<\infty$, and Triebel-Lizorkin spaces ${\dot{F}}_p^{\psi ,q}$, with $1<p,q<\infty$, of order $\psi$ to those of order $\phi \psi$ and $\psi /\phi$ respectively,...

Using the generalized Erdélyi-Kober fractional integrals, an attempt is made to establish certain new fractional integral inequalities, related to the weighted version of the Chebyshev functional. The results given earlier by Purohit and Raina (2013) and Dahmani et al. (2011) are special cases of results obtained in present paper.

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valued measurable functions is considered.MML identifier: MESFUN6C, version: 7.9.01 4.101.1015

Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.The method of integral transforms based on using a fractional generalization of the Fourier transform and the classical Laplace transform is applied for solving Cauchy-type problem for the time-space fractional diffusion equation expressed in terms of the Caputo time-fractional derivative and a generalized Riemann-Liouville space-fractional derivative.

Let ${ℬ}_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty$. Let ${ℬ}_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty$. Define ${𝒜}_c^n$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in ${ℬ}_c$. Similarly with ${𝒜}_r^n$ from ${ℬ}_r$. A type of integral is defined on distributions in ${𝒜}_c^n$ and ${𝒜}_r^n$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb{N}$, the spaces...

Conditions guaranteeing Pettis integrability of a Gelfand integrable multifunction and a decomposition theorem for the Henstock-Kurzweil-Gelfand integrable multifunctions are presented.

Let $m$ be a natural number. Let $f,g$ and $Q$ be real polynomials such that $\{degf,degg\}\subset \{1,2\},degQ<mdegf,g$ is not a square and $f$ has imaginary roots, if it is not linear. Effective methods for the integration of $Q/(f^m\sqrt{g}$ are exhibited.