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end{aligned} As F is continuous from the right, there exists a real number (h>1) such that Fbigl(hH(Tx_{0},Tx_{1} bigr)leq Fbigl(H(Tx_{0},Tx_{1}) bigr)+tau.

(2.25) Since F is continuous from the right, there exists a real number (h>1) such that Fbigl(hH(Tx_{n},Tx bigr)< Fbigl(H(Tx_{n},Tx bigr)+tau.

Since (F inmathcal{F}_{s}) is continuous from the right, there exists a real number (h>1) such that Fbigl(hsH(Tx_{0},Tx_{1} bigr) < Fbigl(sH(Tx_{0},Tx_{1}) bigr) + tau.

From the assumption, we have 2tau+Fbigl(H(Tx_{0},Tx_{1} bigr)leq F bigl(d(x_{0},x_{1} bigr). Since F is continuous from the right, there exists a real number (h>1) such that Fbigl(hH(Tx_{0},Tx_{1} bigr)leq Fbigl(H(Tx_{0},Tx_{1}) bigr)+tau.

end{aligned} Since F is continuous from the right, so there exists a real number (h>1) such that Fbigl(hH(Tx_{n},Tx bigr)< Fbigl(H(Tx_{n},Tx bigr)+tau.

In this case, (t_{1}) must be right-scattered, for otherwise if (t_{1}) is right-dense, there exists (epsilon_{1}) sufficiently small so that (x t)

If (q_{alpha}) is right-dense, then there exists a hollow right neighborhood (mathring{U}_ ( q_{alpha} ) ) of (q_{alpha}) such that (u_{2} ( t ) >0), for (tinmathring{U}_ ( q_{alpha} ) ).

If (t_{alpha}) is right-dense, then there exists a hollow right neighborhood (mathring{U}_ ( t_{alpha} ) ) of (t_{alpha}) such that (u_{1} ( t ) <0), for (tinmathring{U}_ ( t_{alpha} ) ).

end{aligned} (20) In view of the fact that θ is right continuous, then there exists (delta^{prime}>0) such that (theta(2varepsilon+delta^{prime})

Indeed, suppose (overline{lambda}right, that is, there exists (varepsilon>0) such that, for any (lambdain overline{lambda},overline{lambda }+varepsilon)), wle{{w}_{lambda}}.

Hence it is α-right-regular, then there exists a subsequence ({x_{n k)}} ) of ({x_{n}}) such that (alpha(x_{n k)},u geq1) for all k.