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Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions

Cabada Alberto, Dimitrov Nikolay D., Jonnalagadda Jagan Mohan

Opuscula Mathematica

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2024

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Vol. 44, no. 2

167--195

EN

This article is devoted to deduce the expression of the Green’s function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green’s function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green’s function is constructed as finite sums, the Green’s function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green’s function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green’s function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.

2

Positive solutions to mixed fractional p-Laplacian boundary value problems

Guezane-Lakoud Assia, Rodríguez-López Rosana

Journal of Applied Analysis

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2023

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Vol. 29, nr 1

49--58

EN

In this paper, we discuss the existence and uniqueness of a positive solution for a p-Laplacian differential equation containing left and right Caputo derivatives. By the help of the Guo-Krasnoselskii theorem, we prove the existence of at least one positive solution. The existence of a unique positive solution is established under the assumption that the corresponding operator is α-concave and increasing. Numerical examples are given to check the obtained results.

3

On the existence of continuous positive monotonic solutions of a self-reference quadratic integral equation

El-Sayed Ahmed M.A., Ebead Hanaa R.

Journal of Mathematics and Applications

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2020

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Vol. 43

67--80

EN

In this work we study the existence of positive monotonic solutions of a self-reference quadratic integral equation in the class of continuous real valued functions. The continuous dependence of the uniquesolution will be proved. Some examples will be given.

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Fractional problems with right-handed Riemann-Lioville fractional derivatives

Jankowski T.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2016

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Vol. 20, No 1

63--83

EN

In this paper, we investigate the existence of solutions for advanced fractional differential equations containing the right-handed Riemann-Liouville fractional derivative both with nonlinear boundary conditions and also with initial conditions given at the end point T of interval [0,T]. We use both the method of successive approximations, the Banach fixed point theorem and the monotone iterative technique, as well. Linear problems are also discussed. A few examples illustrate the results