This paper is concerned with oscillation of the second-order half-linear dynamic equation
(
r
(
t
)
(
x
Δ
)
γ
)
Δ
+
p
(
t
)
x
γ
(
t
)
=
0
,
on a time scale
T
where
γ is the quotient of odd positive ...integers,
r
(
t
)
and
p
(
t
)
are positive rd-continuous functions on
T
. Our results solve a problem posed by R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085–1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375–387 and our results in the special cases when
T
=
R
and
T
=
Z
involve and improve some oscillation results for second-order differential and difference equations; and when
T
=
h
Z
,
T
=
q
N
0
and
T
=
N
0
2
, etc., our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.
It is the purpose of this paper to give oscillation criteria for the third order nonlinear delay dynamic equation
(
a
(
t
)
{
r
(
t
)
x
Δ
(
t
)
Δ
}
γ
)
Δ
+
f
(
t
,
x
(
τ
(
t
)
)
)
=
0
,
on a time ...scale
T
, where
γ
≥
1
is the quotient of odd positive integers,
a
and
r
are positive
r
d
-continuous functions on
T
, and the so-called delay function
τ
:
T
→
T
satisfies
τ
(
t
)
≤
t
for
t
∈
T
and
lim
t
→
∞
τ
(
t
)
=
∞
and
f
∈
C
(
T
×
R
,
R
)
. Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when
T
=
R
and
T
=
N
involve and improve some oscillation results for third order delay differential and difference equations; when
T
=
h
N
,
T
=
q
N
0
and
T
=
N
2
our oscillation results are essentially new. Some examples are given to illustrate the main results.
Abstract
The conventional Metal cutting fluids (MCFs) used are mineral-based petroleum oils that perform well but are toxic and difficult to dispose of; therefore, these are hazardous to human health ...as well as the environment. This issue can be solved by using natural vegetable oil-based MCF, which are readily available, environment and human-friendly, and renewable. Therefore, we synthesized various types of emulsifiers (anionic, and nonionic with different ethylene oxide units as well as mono and gemini cationic surfactants as corrosion inhibitors and biocides) based on recycled vegetable oil (RO) from spent bleaching earth (SBE), and elucidated their chemical structures by different spectroscopic techniques. The individually synthesized emulsifiers (anionic, and nonionic with different ethylene oxide units) at different ratios (8–15 by wt.%) and mixed emulsifiers (anionic/nonionic, nonionic/nonionic with different degrees of ethylene oxide) at different ratios (8–12 by wt.%) were utilized as additives in the preparation of different vegetable residual oil-based MCF formulations. The mixed emulsifiers at different ratios of nonionic/nonionic with hydrophilic-lipophilic balance (HLB) value 10 (Formulas I, II, III, and IV), and anionic/nonionic (Formula V, and VI) exhibited stable emulsions compared to individual emulsifiers. Formulas (I and VI) displayed good protection effectiveness in corrosion tests. Formula VI had better wettability (25.22 on CS, 23.68 on Al, and 22.28 on WC) and a smaller particle size (63.97 nm). Tribological properties of Formula VI were also performed. The results exhibit that Formula VI is consistent with the commercial sample. As a result, this study contributed to the resolution of one of the industry's problems
The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation
(r(t)(xΔ(t))γ)Δ+f(t,x(g(t)))=0,
on an arbitrary time scale T, where γ is a quotient of odd ...positive integers and r is a positive rd-continuous function on T. The function g:T→T satisfies g(t)⩾t and limt→∞g(t)=∞ and f∈C(T×R,R). We establish some new sufficient conditions such that the above equation is oscillatory by using generalized Riccati transformation. Our results in the special cases when T=R and T=N involve and improve some oscillation results for second-order differential and difference equations; and when T=hN,T=qN0 and T=N2 our oscillation results are essentially new. Some examples illustrating the importance of our results are included.
In this work, we concentrate on the existence of the solutions set of the following problem cDqασ(t)∈F(t,σ(t),cDqασ(t)),t∈I=0,Tσ0=σ0∈E, as well as its topological structure in Banach space E. By ...transforming the problem posed into a fixed point problem, we provide the necessary conditions for the existence and compactness of solutions set. Finally, we present an example as an illustration of main results.
The objective of this article is to examine the oscillatory behavior of a class of quasilinear second-order dynamic equations on time scales. Our focus will be on the noncanonical case, which has ...received relatively less attention compared to the more commonly studied canonical dynamic equations. Our approach involves transforming the noncanonical equation into a corresponding canonical equation. By utilizing this transformation and a range of techniques, we develop new, more efficient, and precise oscillation criteria. Finally, we demonstrate the significance and usefulness of our results by applying them to specific cases within the equation being studied.
In this paper, we investigate the asymptotic properties of third-order nonlinear neutral differential equations with mixed nonlinearities using the comparison principle. Our results not only vastly ...improve upon but also broadly generalize many previously known ones. Examples demonstrating the applicability and efficacy of our results are provided.
This paper employed the well-known Riccati transformation method to deduce a Kneser-type oscillation criterion for second-order dynamic equations. These results are considered an extension and ...improvement of the known Kneser results for second-order differential equations and are new for other time scales. We have included examples to highlight the significance of the results we achieved.
In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficients an of bi-univalent functions in the family of analytic functions in the open unit disk. Estimation ...of the bound value for the initial coefficients of the functions in these classes is also established.
In this study, novel criteria are derived to ensure the oscillation of solutions in nonlinear advanced noncanonical dynamic equations. The obtained results are reminiscent of the criteria proposed by ...Hille and Ohriska for canonical dynamic equations. Additionally, this paper addresses a previously unresolved issue found in numerous existing works in the literature on advanced dynamic equations. This study provides a range of illustrative examples to showcase the precision, practicality, and adaptability of the obtained findings.
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