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)

 

 

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.