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In Theorem 1, setting ϕ = I, the identity mapping, we deduce the following corollary.

Since (T Cto C) is a generalized Lipschitz continuous, hemicontinuous, and pseudocontractive mapping, we deduce that F is a generalized Lipschitz continuous, hemecontinuous, and monotone operator.

In Theorem 1, if S : X → X is a continuous map, we deduce the following corollary.

Due to the classical contractive mapping theorem, we deduce that the map (mathcal{H}_{epsilon}^{k_{1}}) has a unique fixed point denoted by (omega_{k_{1}}^{d} tau,m,epsilon)) (i.e. (mathcal{H}_{epsilon}^{k_{1}} omega_{k_{1}}^{d} tau,m,epsilon))= omega _{k_{1}}^{d} tau,m,epsilon))) in (bar{B}(0,varpi)), for all (epsilonin D 0,epsilon_{0})).

Due to the classical contractive mapping theorem, we deduce that the map (mathcal{H}_{epsilon}^{k_{2}}) has a unique fixed point denoted by (omega_{k_{2}}^{d} tau,m,epsilon)) (i.e. (mathcal{H}_{epsilon}^{k_{2}}(omega_{k_{2}}^{d} tau,m,epsilon))= omega _{k_{2}}^{d} tau,m,epsilon))) in (bar{B}(0,upsilon)), for all (epsilonin D 0,epsilon_{0})).

By the open mapping theorem we deduce that,, is continuous too.

Hence, by the Banach contraction mapping principle, we deduce that A has a fixed point which is the unique solution of problem (1.1).

Therefore, by the Banach contraction mapping principle, we deduce that (mathcal{A}) has a fixed point which is the unique solution of the problem (3.6).

Therefore, by the Banach contraction mapping principle, we deduce that (mathcal{Q}) has a fixed point which is the unique solution of problem (1.1).

Thus, by Banach's contraction mapping principle, we deduce that the operator (mathcal{H}) has a fixed point, which equivalently implies that problem (1.1) has a unique solution on ([0,T]).

If R : X → X is the identity mapping, we can deduce easily the following common fixed point results.