Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations
We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type 𝑐𝒟𝛿0+𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0, 𝑡∈(0,1), 0<𝑡<1. 𝑢(1)=𝛽𝑢(𝜂)+𝜆2, 𝑢(0)=𝛼𝑢(𝜂)−𝜆1, 𝑢(0)=0, 𝑢(0)=0⋯𝑢(𝑛−1)(0)=0, where, 𝑛−1<𝛿<𝑛, 𝑛(≥3)∈ℕ,...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2010/501230 |
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| _version_ | 1832550079012012032 |
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| author | Mujeeb Ur Rehman Rahmat Ali Khan |
| author_facet | Mujeeb Ur Rehman Rahmat Ali Khan |
| author_sort | Mujeeb Ur Rehman |
| collection | DOAJ |
| description | We study existence of positive solutions to nonlinear higher-order nonlocal
boundary value problems corresponding to fractional differential equation of the type 𝑐𝒟𝛿0+𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0, 𝑡∈(0,1), 0<𝑡<1. 𝑢(1)=𝛽𝑢(𝜂)+𝜆2, 𝑢(0)=𝛼𝑢(𝜂)−𝜆1, 𝑢(0)=0, 𝑢(0)=0⋯𝑢(𝑛−1)(0)=0, where, 𝑛−1<𝛿<𝑛, 𝑛(≥3)∈ℕ, 0<𝜂,𝛼,𝛽<1, the boundary parameters 𝜆1,𝜆2∈ℝ+ and 𝑐𝐷𝛿0+ is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain
sufficient conditions for the existence and uniqueness of positive solutions to the boundary value
problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We
include examples to show the applicability of our results. |
| format | Article |
| id | doaj-art-7516229495004e6983ca73205ea7cd0f |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-7516229495004e6983ca73205ea7cd0f2025-02-03T06:07:44ZengWileyAbstract and Applied Analysis1085-33751687-04092010-01-01201010.1155/2010/501230501230Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential EquationsMujeeb Ur Rehman0Rahmat Ali Khan1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Sector H-12, Islamabad 46000, PakistanUniversity of Malakand, Chakdara Dir(L), Khyber Pakhutoonkhwa, PakistanWe study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type 𝑐𝒟𝛿0+𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡))=0, 𝑡∈(0,1), 0<𝑡<1. 𝑢(1)=𝛽𝑢(𝜂)+𝜆2, 𝑢(0)=𝛼𝑢(𝜂)−𝜆1, 𝑢(0)=0, 𝑢(0)=0⋯𝑢(𝑛−1)(0)=0, where, 𝑛−1<𝛿<𝑛, 𝑛(≥3)∈ℕ, 0<𝜂,𝛼,𝛽<1, the boundary parameters 𝜆1,𝜆2∈ℝ+ and 𝑐𝐷𝛿0+ is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.http://dx.doi.org/10.1155/2010/501230 |
| spellingShingle | Mujeeb Ur Rehman Rahmat Ali Khan Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations Abstract and Applied Analysis |
| title | Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations |
| title_full | Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations |
| title_fullStr | Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations |
| title_full_unstemmed | Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations |
| title_short | Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations |
| title_sort | positive solutions to nonlinear higher order nonlocal boundary value problems for fractional differential equations |
| url | http://dx.doi.org/10.1155/2010/501230 |
| work_keys_str_mv | AT mujeeburrehman positivesolutionstononlinearhigherordernonlocalboundaryvalueproblemsforfractionaldifferentialequations AT rahmatalikhan positivesolutionstononlinearhigherordernonlocalboundaryvalueproblemsforfractionaldifferentialequations |