In this paper we study a class of operator equations A(x, x) + B(x, x) = x in ordered Banach spaces, where A, B are two mixed monotone operators. Various theorems are established to guarantee the existence of a unique solution to the problem. In addition, associated iterative schemes have been established for finding the approximate solution converging to the fixed point of the problem. We also study the solution of the nonlinear eigenvalue equation A(x, x) + B(x, x) = λx and discuss its dependency to the parameter. Our results extend and improve many known results in this field of study. We have also successfully demonstrated the application of our results to the study of nonlinear fractional differential equations with two-point boundary conditions. c 2016 All rights reserved.Keywords: Mixed monotone operator, hypo-homogeneous mixed monotone operator, existence and uniqueness, fractional differential equation.

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In this paper, we analyze qualitative properties of the first multi-symplectic scheme for two-dimensional Maxwell's equations. We prove that the scheme is unconditionally stable and convergent, non-dissipative, and divergence-free. The numerical dispersion relation of the scheme is shown to converge to the exact dispersion relation of the Maxwell equations. We also present some numerical results to confirm our theoretical results.

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