Jñānābha, Vol. 50 (2) (2020), (167-178)
Jñānābha, Vol. 50 (2) (2020), (167-178)
NONLINEAR ABSTRACT MEASURE HYBRID DIFFERENTIAL EQUATIONS WITH A LINEAR PERTURBATION OF SECOND TYPE
By
Bapurao C. Dhage
‘Kasubai’, Gurukul Colony, Thodga Road,
Ahmedpur - 413515, Distr. Latur
Maharashtra, India,
bcdhage@gmail.com
(Received : July 22, 2020 ; Revised: September 17, 2020)
DOI: https://doi.org/10.58250/jnanabha.2020.50221
In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [4] under the mixed generalized Lipschitz and Carathéodory conditions. The existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage [4] on ordered Banach spaces. Our existence results include the existence results of Sharma [23], Joshi [19] and Shendge and Joshi [25] as special cases under weaker continuity condition.
2010 Mathematics Subject Classifications: 34K10, 47H10
Keywords and phrases: Abstract measure differential equation; Dhage fixed point theorem; Existence theorem; Extremal solutions.