Fixed Points of Non-Smooth Functions on Finite Dimensional Ordered Banach Spaces via Clarke Generalized Jacobian

Abstract

Considering Lipschitz functions which are not necessarily Fréchet differentiable, we obtain a non-smooth version of Lakshmikantham's theorem in finite dimensional ordered Banach spaces . We also present an application of the obtained result in dynamical Coulomb friction problem. Recently, Lakshmikantham et al. [14] have proved some fixed point theorems in ordered Banach space X for a Fréchet differentiable mapping T : X → X. They showed the applications of their results in ODE initial value problems and semilinear parabolic initial boundary value problems. Vijesh and Kumar [19] and Mouhadjer and Benahmed [16] obtained some generalizations of Lakshmikantham's fixed point theorems.