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Existence of solution to fractional differential equation with fractional integral type boundary conditions
Author(s) -
Ali Anwar,
Sarwar Muhammad,
Zada Mian Bahadur,
Shah Kamal
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6864
Subject(s) - mathematics , fractional calculus , boundary value problem , type (biology) , mathematical analysis , nonlinear system , order (exchange) , biology , physics , finance , quantum mechanics , economics , ecology
This paper is devoted by developing sufficient condition required for the existence of solution to a nonlinear fractional order boundary value problemD γ u ( ℓ ) = ψ ( ℓ , u ( λ ℓ ) ) , ℓ ∈ Z = [ 0 , 1 ] ,with fractional integral boundary conditionsp 1 u ( 0 ) + q 1 u ( 1 ) = 1 Γ ( γ )∫ 0 1( 1 − ρ ) γ − 1g 1 ( ρ , u ( ρ ) ) d ρ , andp 2u ′ ( 0 ) + q 2u ′ ( 1 ) = 1 Γ ( γ )∫ 0 1( 1 − ρ ) γ − 1g 2 ( ρ , u ( ρ ) ) d ρ , where γ ∈ (1, 2], 0 < λ < 1, D denotes the Caputo fractional derivative (in short CFD), ψ , g 1 , g 2 : Z × R → R are continuous functions andp i , q i ( i = 1 , 2 ) are positive real numbers. Using topological degree theory sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, a concrete example is presented in the end.