On the existence of the solution of Hammerstein integral equations and fractional differential equations

2021 ◽  
Author(s):  
Mudasir Younis ◽  
Deepak Singh
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Hammerstein Integral Equations ◽  
Fractional Differential

scholarly journals Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1093
Author(s):  
Daniel Cao Labora
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Initial Value ◽  
Major Question ◽  
Initial Values ◽  
Fractional Differential

One major question in Fractional Calculus is to better understand the role of the initial values in fractional differential equations. In this sense, there is no consensus about what is the reasonable fractional abstraction of the idea of “initial value problem”. This work provides an answer to this question. The techniques that are used involve known results concerning Volterra integral equations, and the spaces of summable fractional differentiability introduced by Samko et al. In a few words, we study the natural consequences in fractional differential equations of the already existing results involving existence and uniqueness for their integral analogues, in terms of the Riemann–Liouville fractional integral. In particular, we show that a fractional differential equation of a certain order with Riemann–Liouville derivatives demands, in principle, less initial values than the ceiling of the order to have a uniquely determined solution, in contrast to a widely extended opinion. We compute explicitly the amount of necessary initial values and the orders of differentiability where these conditions need to be imposed.

Positive solutions for a system of nonlocal fractional boundary value problems

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Johnny Henderson ◽  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Existence Results ◽  
Hammerstein Integral Equations ◽  
Existence Of Positive Solutions ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

AbstractWe investigate the existence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations, subject to multipoint boundary conditions. Existence results for systems of nonlinear Hammerstein integral equations are also presented. Some nontrivial examples are included.

scholarly journals Properties for ψ-Fractional Integrals Involving a General Function ψ and Applications

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 517
Author(s):  
Jin Liang ◽  
Yunyi Mu
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Integrodifferential Equations ◽  
Fractional Integrals ◽  
General Function ◽  
Fractional Differential ◽  
Hadamard Fractional Integrals

In this paper, we are concerned with the ψ-fractional integrals, which is a generalization of the well-known Riemann–Liouville fractional integrals and the Hadamard fractional integrals, and are useful in the study of various fractional integral equations, fractional differential equations, and fractional integrodifferential equations. Our main goal is to present some new properties for ψ-fractional integrals involving a general function ψ by establishing several new equalities for the ψ-fractional integrals. We also give two applications of our new equalities.

Improvements in a method for solving fractional integral equations with some links with fractional differential equations

2018 ◽  
Vol 21 (1) ◽  
pp. 174-189 ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Source Term ◽  
Initial Conditions ◽  
Standard Procedure ◽  
Relaxation Equation ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders. So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders. Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.

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