Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters
In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with <i>r</i>-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fr...
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MDPI AG
2022-04-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/4/164 |
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| author | Alexandru Tudorache Rodica Luca |
| author_facet | Alexandru Tudorache Rodica Luca |
| author_sort | Alexandru Tudorache |
| collection | DOAJ |
| description | In this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with <i>r</i>-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>, the solutions of the integral system are the fixed points of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution. |
| first_indexed | 2024-03-09T11:09:40Z |
| format | Article |
| id | doaj.art-01ba9a8f30654b06b48d66c1f5c0cfd5 |
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| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T11:09:40Z |
| publishDate | 2022-04-01 |
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| spelling | doaj.art-01ba9a8f30654b06b48d66c1f5c0cfd52023-12-01T00:48:25ZengMDPI AGAxioms2075-16802022-04-0111416410.3390/axioms11040164Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive ParametersAlexandru Tudorache0Rodica Luca1Department of Computer Science and Engineering, Gh. Asachi Technical University, 700050 Iasi, RomaniaDepartment of Mathematics, Gh. Asachi Technical University, 700506 Iasi, RomaniaIn this paper, we investigate the existence and nonexistence of positive solutions for a system of Riemann–Liouville fractional differential equations with <i>r</i>-Laplacian operators, subject to nonlocal uncoupled boundary conditions that contain Riemann–Stieltjes integrals, various fractional derivatives and positive parameters. We first change the unknown functions such that the new boundary conditions have no positive parameters, and then, by using the corresponding Green functions, we equivalently write this new problem as a system of nonlinear integral equations. By constructing an appropriate operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>, the solutions of the integral system are the fixed points of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>. Following some assumptions regarding the nonlinearities of the system, we show (by applying the Schauder fixed-point theorem) that operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> has at least one fixed point, which is a positive solution of our problem, when the positive parameters belong to some intervals. Then, we present intervals for the parameters for which our problem has no positive solution.https://www.mdpi.com/2075-1680/11/4/164Riemann–Liouville fractional differential equationsnonlocal boundary conditionspositive parameterspositive solutionsexistencenonexistence |
| spellingShingle | Alexandru Tudorache Rodica Luca Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters Axioms Riemann–Liouville fractional differential equations nonlocal boundary conditions positive parameters positive solutions existence nonexistence |
| title | Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters |
| title_full | Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters |
| title_fullStr | Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters |
| title_full_unstemmed | Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters |
| title_short | Positive Solutions for a System of Fractional Boundary Value Problems with <i>r</i>-Laplacian Operators, Uncoupled Nonlocal Conditions and Positive Parameters |
| title_sort | positive solutions for a system of fractional boundary value problems with i r i laplacian operators uncoupled nonlocal conditions and positive parameters |
| topic | Riemann–Liouville fractional differential equations nonlocal boundary conditions positive parameters positive solutions existence nonexistence |
| url | https://www.mdpi.com/2075-1680/11/4/164 |
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