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On a quadratic integral equation with supremum involving Erdélyi‐Kober fractional order
Author(s) -
Darwish Mohamed Abdalla,
Sadarangani Kishin
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400063
Subject(s) - mathematics , infimum and supremum , monotonic function , integral equation , quadratic equation , measure (data warehouse) , order (exchange) , fixed point theorem , mathematical analysis , computer science , geometry , finance , database , economics
We introduce Erdélyi‐Kober fractional quadratic integral equation with supremum, namely x ( t ) = f ( t ) + α ( T x ) ( t ) Γ ( β )∫ 0 ts α − 1u ( t , s , x ( s ) , max [ 0 , σ ( s ) ]| x ( τ ) | )t α − s α1 − βd s , 0 ≤ t ≤ 1 ,α > 0 , β ∈ ( 0 , 1 ) . This equation contains as special cases numerous integral equations studied by other authors. We show that there exists at least one monotonic solution belonging to C [0, 1] of our equation. The main tools in our analysis are Darbo fixed point theorem and the measure of noncompactness related to monotonicity which was introduced by Banaś and Olszowy. Finally, we present an example for illustrating the natural realizations of our abstract results.