Nonlinear partial completely continuous operators in a partially ordered Banach space and nonlinear hyperbolic partial differential equations

We prove a hybrid fixed point theorem for partial completely continuous operators in a partially ordered metric space and derive an applicable hybrid fixed point result in an ordered Banach space as a special case. As an application, we discuss a nonlinear hyperbolic partial differential equation for approximation result of local solutions by constructing the algorithms. Finally, an example is indicated to elaborate the hypotheses and abstract result of this paper.