New oscillation criteria of special type second-order non-linear dynamic equations on time scales

Negi, Shekhar Singh; Abbas, Syed; Malik, Muslim; Xia, Yonghui

doi:10.1007/s40096-018-0241-9

Cited by 17 publications

(12 citation statements)

References 33 publications

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“…From the year 2000 till now, a rapidly growing attention has been paid in the subject of various studies on time-scales; for instance, the reader may consult the papers [13][14][15][16][17][18][19][20][21][22][23] and references cited therein. On the other hand, solutions of the various types of the first-order dynamic equation on time scale have been discussed by several authors; for instance, the literature.…”

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confidence: 99%

A generalized delta derivative on time scale with applications

Negi

Abbas

Malik

2020

Math Methods in App Sciences

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Inspired by a well‐known “Hilger‐derivative” on time scale introduced by Stephan Hilger, we introduce a new derivative on the time scale, called “black delta (or in notation ▴)‐derivative.” It is an adaptive unified approach of derivative of continuous and discrete calculus. Several important nontrivial results concerning this derivative are obtained. Also, a necessary and sufficient condition for the existence of this new derivative and its connection with Hilger‐derivative are discussed. In addition, a new general a‐exponential function (for fixed a ≥ 1) is introduced to obtain the unique solution of the initial value problem on time scales with this derivative. Furthermore, a relation between the a‐exponential function with the existing exponential function on the time scale is obtained. We elucidate the results by considering some interesting examples. We expect that this derivative will be applicable for future investigations on various studies in the field of mathematical modeling, engineering applications.

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Section: Figure 1 Life Span Of Pharaoh Cicadamentioning

confidence: 99%

A generalized delta derivative on time scale with applications

Negi

Abbas

Malik

2020

Math Methods in App Sciences

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“…Several excellent monographs() summarized some important work in the field. Some recent results on oscillation and nonoscillation for second‐order linear and nonlinear dynamic equations can be found in the previous studies() and references therein.…”

Section: Introductionmentioning

confidence: 99%

Necessary and sufficient conditions for oscillation of second‐order dynamic equations on time scales

Zhou

Ahmad

Alsaedi

2019

Math Methods in App Sciences

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“…Several excellent monographs [1][2][3][4][5] on the topic indeed reflect its popularity. Some recent results on oscillation and existence of nonoscillatory solutions for dynamic equations can be found in the articles [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Oscillation and nonoscillation theorems of neutral dynamic equations on time scales

2019

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We present the oscillation criteria for the following neutral dynamic equation on time scales: $$ \bigl(y(t)-C(t)y(t-\zeta )\bigr)^{\Delta }+P(t)y(t-\eta )-Q(t)y(t-\delta )=0, \quad t\in {\mathbb{T}}, $$ ( y ( t ) − C ( t ) y ( t − ζ ) ) Δ + P ( t ) y ( t − η ) − Q ( t ) y ( t − δ ) = 0 , t ∈ T , where $C, P, Q\in C_{\mathit{rd}}([t_{0},\infty ),{\mathbb{R}}^{+})$ C , P , Q ∈ C rd ( [ t 0 , ∞ ) , R + ) , ${\mathbb{R}} ^{+}=[0,\infty )$ R + = [ 0 , ∞ ) , $\gamma , \eta , \delta \in {\mathbb{T}}$ γ , η , δ ∈ T and $\gamma >0$ γ > 0 , $\eta >\delta \geq 0$ η > δ ≥ 0 . New conditions for the existence of nonoscillatory solutions of the given equation are also obtained.

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New oscillation criteria of special type second-order non-linear dynamic equations on time scales

Abstract: By the Riccati transformation technique, we study some new oscillatory properties for the second-order dynamic equation on an arbitrary time scale T: We also establish the Kamenev-type and Philos-type oscillation criteria. At the end, we give examples which illustrate our main results.

Cited by 17 publications

References 33 publications

A generalized delta derivative on time scale with applications

A generalized delta derivative on time scale with applications

Necessary and sufficient conditions for oscillation of second‐order dynamic equations on time scales

Oscillation and nonoscillation theorems of neutral dynamic equations on time scales

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