Then by Theorem 4.2 and the identities ( A q ′, p ′ + ) k = ( B p, q + ) k and ( A q ′, p ′ - ) k = ( B p, q - ) k, the boundedness of the operator defined by (2) from l q ′, u - 1 into l p ′, v - 1 is equivalent to the conditions of the statement (i) of Theorem 3.1.
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The notion of coloring here comes from Frege who distinguished in "The thought" between the truth condition of a statement and those aspects of the statement which might provide information about the state of mind of the speaker or hearer but do not contribute to its truth conditions.
In this paper, we notice that the techniques used in [37, 38] are valid if the contraction condition in the statement of the theorem can be expressed in two variables.
However, Karapinar and Agarwal [55] noticed that the techniques used in [54, 63] are valid if the contraction condition in the statement of the theorem can be expressed in two variables and they proved some theorems on the existence and uniqueness of a common fixed point for which the techniques of the papers [54, 63] are not applicable.
In this paper, we note that the approach of Samet et al. (Int. J. Anal. 2013:917158, 2013) and Jleli-Samet (Fixed Point Theory Appl. 2012 210, 2012) is inapplicable unless the contraction condition in the statement of the theorem can be reduced into two variables.
As an application of Corollary 3.2 and Theorem 3.3, the result of Redjel's theorem (adding condition (U1) to the statement of Theorem 1.5, the uniqueness of the fixed point can be obtained.
Validity of Statement 1.3 follows from condition (i), the Statement 1.4 from condition (vii).
It does not say anything about whether, for a particular theory T satisfying the conditions of the theorem, the statement "T is consistent" can be proved in the sense of being shown to be true by a conclusive argument, or by a proof generally acceptable for mathematicians.
Obviously, the condition given in the statement of this theorem is sufficient.
