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Existence and uniqueness for ψ ‐Hilfer fractional differential equation with nonlocal multi‐point condition
Author(s) -
Borisut Piyachat,
Kumam Poom,
Ahmed Idris,
Jirakitpuwapat Wachirapong
Publication year - 2021
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.6092
Subject(s) - mathematics , uniqueness , fixed point theorem , fractional calculus , mathematical analysis , order (exchange) , stability (learning theory) , banach fixed point theorem , differential equation , finance , economics , machine learning , computer science
In this paper, we study and investigate the ψ −Hilfer fractional differential equation with nonlocal multi‐point condition of the form:Da +q , p ; ψ u ( t ) = f ( t , u ( t ) , Da +q , p ; ψ u ( t ) ) , t ∈ [ a , b ] ,Ia +1 − r ; ψ u ( a ) =∑ i = 1 mβ i u ( η i ) , q ≤ r = q + p − q p < 1 ,η i ∈ [ a , b ] ,where 0 < q < 1 , 0 ≤ p ≤ 1 , m ∈ N ,β i ∈ R , i =1,2,..., m , − ∞ < a < b < ∞ ,Da +q , p ; ψis the ψ − Hilfer fractional derivative, f : [ a , b ] × R × R → R is a continuous function, andIa +1 − r ; ψis the ψ ‐Riemann‐Liouville fractional integral of order 1− r . By using Schaefer's and Banach fixed point theorems, we prove the existence, uniqueness, and stability analysis of this problem. An example is given to illustrate the applicability of our results.