In this paper, we establish some fixed point theorems with T-contractions and w-distances in partially ordered metric spaces. The main tool used in our proof is a generalized altering distance function. Our results can be applied directly to study multidimensional fixed point which covers the concepts of coupled, tripled, quadruple fixed point etc. Moreover, a Fredholm integral equation and an initial-value problem for partial differential equation of parabolic type are also discussed.

In this paper, by using the topological degree and fixed point index theory, the existence of three kinds of solutions (i.e., sign-changing solutions, positive solutions, and negative solutions) for asymptotically linear operator equations is discussed. The abstract results obtained here are applied to nonlinear integral and differential equations.