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Since ε > 0 is arbitrary, this proves X ≤ 0. We conclude from equation (5.2) that equation (5.1) holds.
Plugging in the value of ϕ ˆ m ′ from Equation 28 in the above metric concludes to Equation 19.
If ((H_{21})) holds, we can conclude that equation (14) has at least a positive root (omega_{10}) such that equation (12) has a pair of purely imaginary root (pm iomega_{10}).
Since, we conclude that Equation 4 is valid even if the modulo 2π is removed.
Therefore, we can conclude that equation (1.5) has at least one positive T-periodic solution.
Repeating this reasoning in all arguments, we conclude that equation (2) holds.
By applying Lemma 2.1, we conclude that equation has a solution on ; that is, (1.3) has a -periodic solution with.
Therefore, by applying Lemma 2.1, we can conclude that equation (1.3) has at least one positive T-periodic solution.
Therefore, it can be concluded that Equation 19 gives, for the present case, a valid estimation of nanofluid viscosity.
(We therefore conclude that equation is not equivalent to MMVI (1.3), as claimed by Noor [27].) Now take the initial guess for.
According to the construction of (psi_{2k}(xi)) and Lemma 3.5, we conclude that equation (1.1) admits infinitely many discontinuous traveling wave entropy solutions.
