Existence and uniqueness of solutions of differential equations with
respect to non-additive measures
Abstract
By taking Sugeno-derivative into account, firstly, we investigate the
existence of solutions to the initial value problems (IVP) of
first-order differential equations with respect to non-additive measure
(more precisely, distorted Lebesgue measure). It particularly occurs in
the mathematical modeling of biology. We begin by expressing the
differential equation in terms of ordinary derivative and the derivative
with respect to the distorted Lebesgue measure. Then, by using the fixed
point theorem on cones, we construct an operator and prove the existence
of positive increasing solutions on cones in semi-order Banach spaces.
In addition, we also use Picard’s-Lindel\”of theorem to
prove the existence and uniqueness of the solution of the equation.
Second, we investigate the existence of a solution to the boundary value
problem (BVP) on cones with integral boundary conditions of a mix-order
differential equation with respect to non-additive measures. Moreover,
the Krasnoselskii fixed point theorem is also applied to both BVP and
IVP and obtains at least one positive increasing solution. Examples with
graphs are provided to validate the results.