Existence theorems for perturbed abstract measure differential equations

In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [B.C. Dhage, On some nonlinear alternatives of Leray–Schauder type and functional integral equations, Arch. Math. (Brno) 42 (2006) 11–23] 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 given in the above-mentioned reference, on ordered Banach spaces. Our existence results include the existence results of Sharma [R.R. Sharma, An abstract measure differential equation, Proc. Amer. Math. Soc. 32 (1972) 503–510], Joshi [S.R. Joshi, A system of abstract measure delay differential equations, J. Math. Phy. Sci. 13 (1979) 497–506] and Shendge and Joshi [G.R. Shendge, S.R. Joshi, Abstract measure differential inequalities and applications, Acta Math. Hung. 41 (1983) 53–54] as special cases under weaker continuity condition.