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Now by applying for in inequality (1.5), we have (2.4).

Now, by applying Lemma 2.1, we show that (i) holds.

Now, by applying Eq. (22), one obtains Eq. (13).

Now by applying Theorem 2.1 to the strictly increasing function, we have (2.23).

Now, by applying (5.1) and our Theorem 3, condition (5.2) holds.

Now by applying inequality (2.26), and (3.6) and (3.7) thus we establish Theorem 3.2.

The proof is now completed by applying Lemma 2.5.

The rest of assertions from the statement now follow by applying Theorem 3.1, the metric space ( K ( X ), h ) being complete.

The assertion of the lemma now follows by applying the homotopy invariance of the fixed point index to P on V. □.

The statement of the lemma now follows by applying the W-invariant differential operator (D_ell ) to the formula for (mathrm {ord}(W_lambda ) J varphi ) above.

Let be a solution of the system (5.2). then, from which we get that, and the proof now follows by applying Lemma 2.3.