scholarly journals Constant Sign Solutions to Linear Fractional Integral Problems and Their Applications to the Monotone Method

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 156 ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equations ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Nonlinear Problems ◽  
Constant Sign ◽  
Monotone Method ◽  
Iterative Techniques ◽  
Monotone Iterative ◽  
Constant Sign Solutions ◽  
Constant Coefficients

This manuscript provides some results concerning the sign of solutions for linear fractional integral equations with constant coefficients. This information is later used to prove the existence of solutions to some nonlinear problems, together with underestimates and overestimates. These results are obtained after applying suitable modifications in the classical process of monotone iterative techniques. Finally, we provide an example where we prove the existence of solutions, and we compute some estimates.

scholarly journals From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

scholarly journals Krasnosel’skii Type Hybrid Fixed Point Theorems and Their Applications to Fractional Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
H. M. Srivastava ◽  
Sachin V. Bedre ◽  
S. M. Khairnar ◽  
B. S. Desale
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Lower Solution ◽  
Linear Spaces ◽  
Partially Ordered ◽  
Normed Linear Spaces ◽  
Monotonicity Conditions

Some hybrid fixed point theorems of Krasnosel’skii type, which involve product of two operators, are proved in partially ordered normed linear spaces. These hybrid fixed point theorems are then applied to fractional integral equations for proving the existence of solutions under certain monotonicity conditions blending with the existence of the upper or lower solution.

Constant-Sign Solutions of a System of Fredholm Integral Equations

2004 ◽  
Vol 80 (1) ◽  
pp. 57-94 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O'Regan ◽  
Patricia J. Y. Wong
Keyword(s):  
Integral Equations ◽  
Fredholm Integral Equations ◽  
Constant Sign ◽  
Constant Sign Solutions

Unilateral Global Bifurcation, Half-Linear Eigenvalues and Constant Sign Solutions for a Fractional Laplace Problem

2017 ◽  
Vol 27 (01) ◽  
pp. 1750015 ◽  
Author(s):  
Bian-Xia Yang ◽  
Hong-Rui Sun ◽  
Zhaosheng Feng
Keyword(s):  
Differential Equation ◽  
Global Bifurcation ◽  
Nonlinear Problems ◽  
Constant Sign ◽  
Bifurcation Structure ◽  
Laplace Operators ◽  
Constant Sign Solutions ◽  
Fractional Differential ◽  
Laplace Problem ◽  
The Continuum

In this paper, we are concerned with the unilateral global bifurcation structure of fractional differential equation [Formula: see text] with nondifferentiable nonlinearity [Formula: see text]. It shows that there are two distinct unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text], and two unbounded subcontinua [Formula: see text] and [Formula: see text] consisting of the continuum [Formula: see text] emanating from [Formula: see text]. As an application of this unilateral global bifurcation results, we present the existence of the principal half-eigenvalues of the half-linear fractional eigenvalue problem. Finally, we deal with the existence of constant sign solutions for a class of fractional nonlinear problems. Main results of this paper generalize the known results on classical Laplace operators to fractional Laplace operators.

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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