Functional Equation fx pfx − 1 − qfx − 2 and Its Hyers-Ulam Stability
2.Moreover,asubstantialpartofproofofTheorem 2.1canbederivedfromtheoremspresentedinthebooks 19,20 .However,the theorems in 19, 20 deal with solutions of the linear equation under some regularityconditions, for example, the continuity, convexity, diﬀerentiability, analyticity and so on,while Theorem 2.1 deals with the general solution of 1.7 without regularity conditions.
A Note on Stability of an Operator Linear Equation of the Second Order
We prove some Hyers-Ulam stability results for an operator linear equation of the second order that is patterned on the difference equation, which defines the Lucas sequences (and in particular the Fibonacci numbers). In this way, we obtain several results on stability of some linear functional and differential and integral equations of the second order and some fixed point results for a particular (not necessarily linear) operator.
Effect of Aeration and Agitation Regimes on Lipase Production by Newly Isolated Rhodotorula mucilaginosa–MTCC 8737 in Stirred Tank Reactor Using Molasses as Sole Production Medium
The influence of media and process parameters (aeration and agitation) on fermentation broth rheology and biomass formation has been studied in 1.5-l stirred tank reactor for lipase production using Rhodotorula mucilaginosa MTCC 8737. Molasses, as sole production medium, is used for lipase production by varying aeration (1, 2, and 3 vvm) and agitation speeds (100, 200, and 300 rpm). Maximum lipase activity of 72 U/ml was obtained during 96 h of fermentation at 2 vvm, 200 rpm, pH 7, and 25 ± 2°C temperature. Lipase production kinetics with respect to dry cell weight of biomass showed YP/S of 25.71 U/mg, specific product formation of 10.9 U/mg DC, and YX/S 2.35 mg/mg. Maximum lipase activity (MC 2) of 56 U/ml was observed at 1% molasses, and a further increase in the molasses concentration of (%) 1.5 and 2 inhibited the product formation of lipase with 15 and 8.5 U/ml, respectively. The production kinetics of molasses media showed YP/X was 14 U/mg DC, YP/S 16 U/mg, and YX/S 1.14 mg/mg during 96 h of bioreactor operation. The kLa values for all batches (MC 1–MC 4) at 96 h of fermentation were 32, 28, 21, and 19/h, and the |oxygen transfer rate were 54.4, 56, 35.7, and 17.29 mg/l h, respectively. Increase in molasses concentration resulted in decreased lipase activity by increase in viscosity of the fermentation broth.