Let \(X\) be a Banach space with a partial ordering introduced by a cone \(K\) and let \(X_1, X_2\) be subspaces of \(X\) such that \(X_1\cup X_2= X\), \(X_1\cap X_2= \{0\}\). If \(F:X\to X\) is an operator in \(X\) then denote by \(F_i\) the term \(\pi_i\circ F\) (where \(\pi_i\) are the projections on \(X_i\)) for \(i\in \{1,2\}\). \(F\) is called a skew-increasing operator of first (respectively second) kind if \(F_1, F_2\) are both quasidecreasing (respectively \(F_1\) is quasi-increasing and \(F_2\) is quasi-decreasing or conversely). The main results of the paper are fixed point and coupled fixed point theorems for skew-increasing operators of first and second kind. These results generalize some fixed point theorems for increasing operators given by K. Deimling and D. Guo. An application to a Cauchy problem is also given.