scholarly journals A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Somia Khaldi ◽  
Rachid Mecheraoui ◽  
Aiman Mukheimer
Keyword(s):  
Differential Equation ◽  
Functional Derivative ◽  
Regularity Of Solutions ◽  
Initial Condition ◽  
Stability Of Solutions ◽  
Nonlocal Initial Condition ◽  
Integro Differential Equation ◽  
Fixed Point Approach ◽  
Pantograph Equation ◽  
Rassias Stability

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

scholarly journals Best approximations of the ϕ-Hadamard fractional Volterra integro-differential equation by matrix valued fuzzy control functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Safoura Rezaei Aderyani ◽  
Reza Saadati
Keyword(s):  
Differential Equation ◽  
Fuzzy Control ◽  
Normed Space ◽  
Control Function ◽  
Fuzzy Normed Space ◽  
Integro Differential Equation ◽  
Fractional Equations ◽  
Control Functions ◽  
Rassias Stability ◽  
Fuzzy Banach Space

AbstractIn this article, first, we present an example of fuzzy normed space by means of the Mittag-Leffler function. Next, we extend the concept of fuzzy normed space to matrix valued fuzzy normed space and also we introduce a class of matrix valued fuzzy control functions to stabilize a nonlinear ϕ-Hadamard fractional Volterra integro-differential equation. In this sense, we investigate the Ulam–Hyers–Rassias stability for this kind of fractional equations in matrix valued fuzzy Banach space. Finally, as an application, we investigate the Ulam–Hyers–Rassias stability using matrix valued fuzzy control function obtained through the Mittag-Leffler function.

scholarly journals On Ulam–Hyers–Rassias stability of a generalized Caputo type multi-order boundary value problem with four-point mixed integro-derivative conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Salim Ben Chikh ◽  
Abdelkader Amara ◽  
Sina Etemad ◽  
Shahram Rezapour
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Stability Of Solutions ◽  
Liouville Type ◽  
Research Article ◽  
Different Types ◽  
Computational Aspects ◽  
Fractional Differential ◽  
Rassias Stability

AbstractIn this research article, we turn to studying the existence and different types of stability such as generalized Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability of solutions for a new modeling of a boundary value problem equipped with the fractional differential equation which contains the multi-order generalized Caputo type derivatives furnished with four-point mixed generalized Riemann–Liouville type integro-derivative conditions. At the end of the current paper, we formulate two illustrative examples to confirm the correctness of theoretical findings from computational aspects.

scholarly journals Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2181
Author(s):  
Daniela Inoan ◽  
Daniela Marian
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Convolution Type ◽  
Convolution Product ◽  
Polynomial Functions ◽  
Integro Differential Equation ◽  
The Laplace Transform ◽  
The Stability ◽  
Rassias Stability

In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions. An important aspect that appears in the form of the studied equation is the symmetry of the convolution product.

scholarly journals Interior Schauder Estimates for Elliptic Equations Associated with Lévy Operators

2021 ◽  
Author(s):  
Franziska Kühn
Keyword(s):  
Differential Equation ◽  
Wide Class ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Infinitesimal Generator ◽  
Transition Density ◽  
Regularity Of Solutions ◽  
Local Regularity ◽  
Schauder Estimates ◽  
Integro Differential Equation

AbstractWe study the local regularity of solutions f to the integro-differential equation $$ Af=g \quad \text{in } U $$ A f = g in U for open sets $U \subseteq \mathbb {R}^{d}$ U ⊆ ℝ d , where A is the infinitesimal generator of a Lévy process (Xt)t≥ 0. Under the assumption that the transition density of (Xt)t≥ 0 satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions f. Our results apply for a wide class of Lévy generators, including generators of stable Lévy processes and subordinated Brownian motions.

scholarly journals Best approximation of a nonlinear fractional Volterra integro-differential equation in matrix MB-space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reza Chaharpashlou ◽  
Reza Saadati
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fractional Derivative ◽  
Stochastic Matrix ◽  
Fixed Point Method ◽  
Hilfer Fractional Derivative ◽  
Integro Differential Equation ◽  
Control Functions ◽  
Matrix Control ◽  
Rassias Stability

AbstractIn this article, we introduce a class of stochastic matrix control functions to stabilize a nonlinear fractional Volterra integro-differential equation with Ψ-Hilfer fractional derivative. Next, using the fixed-point method, we study the Ulam–Hyers and Ulam–Hyers–Rassias stability of the nonlinear fractional Volterra integro-differential equation in matrix MB-space.

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