Stability in Functional Integro-differential Equations of Second Order with Variable Delay

Tunç, Cemil; Gözen, Melek

doi:10.5614/j.math.fund.sci.2017.49.1.7

Cited by 8 publications

(8 citation statements)

References 18 publications

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“…The aim of this article is to generalize and improve the outcomes of Burton and Somolinos [17] for the nonlinear IDE (3) according to the range of the positive constant "a" and provide new contributions to the results mentioned above, and that can be found in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and the present literature. Hence, we have new contributions to the qualitative theory of IDEs.…”

Section: Introductionmentioning

confidence: 99%

“…We reach to the outcome of this article by defining new LKFs, which are different from those in the references of this article and that can be seen in relation to the relevant literature. Next, it can be seen that the IDE (3) is different from that in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. We should mention that the papers [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] include very interesting results on the fundamental behaviors of certain IDEs with or without delay.…”

Section: Introductionmentioning

confidence: 99%

“…Next, it can be seen that the IDE (3) is different from that in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. We should mention that the papers [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] include very interesting results on the fundamental behaviors of certain IDEs with or without delay. Our paper includes complementary results for the outcomes of [1][2][3][4][5][6] and that in the references of these sources.…”

Section: Introductionmentioning

confidence: 99%

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On the Enhanced New Qualitative Results of Nonlinear Integro-Differential Equations

Tunç

Yao

2022

Symmetry

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In this article, a class of scalar nonlinear integro-differential equations of first order with fading memory is investigated. For the considered fading memory problem, we discuss the effects of the memory over all the values of the parameter in the kernel of the equations. Using the Lyapunov–Krasovski functional method, we give various sufficient conditions of stability, asymptotic stability, uniform stability of zero solution, convergence and boundedness, and square integrability of nonzero solutions in relation to the considered scalar nonlinear integro-differential equations for various cases. As the novel contributions of this article, the new scalar nonlinear integro-differential equation with the fading memory is firstly investigated in the literature, and seven theorems, which have novel sufficient qualitative conditions, are provided on the qualitative behaviors of solutions called boundedness, convergence, stability, integrability, asymptotic stability and uniform stability of solutions. The novel outcomes and originality of this article are that the considered integro-differential equations are new mathematical models, they include former mathematical models in relation to the mathematical models of this paper as well as the given main seven qualitative results are also new. The outcomes of this paper enhance some present results and provide new contributions to the relevant literature. The results of the article have complementary properties for the symmetry of integro-differential equations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Enhanced New Qualitative Results of Nonlinear Integro-Differential Equations

Tunç

Yao

2022

Symmetry

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“…The inclusion of time delays increases the complexity of the model. Thus It is important to analyze the model's qualitative behavior, using stability or bifurcation approach [10]. In recent years, the non-linearity and sensitivity analysis of DDEs has been intensively analyzed in numerous scientific and technological fields, especially in the sense of chaotic dynamics [19,30].…”

Section: Introductionmentioning

confidence: 99%

Co-infection dynamics of two short term diseases with effect of recovery delay

Tomar¹,

Dhar²,

Kumar³

2022

J. Math. Computer Sci.

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A co-infection model with two short-term diseases with delay in recovery is proposed. Here, we consider the simultaneous transmission of infection does not happen but of simultaneous recovery from both illnesses. The system consists of four epidemiological classes populations, namely: susceptible (S), an infected class with the first disease (I 1 ), an infected class with the second disease (I 2 ), co-infected class (I 12 ). We have found all possible equilibrium states, and the basic reproduction number also examined their stability without and with delay. Analytically, we have established that the local stability of equilibrium points depends on the basic reproduction number in the absence of recovery delay. But with delay, it requires some additional conditions. We have also checked the effect of delay on stability of endemic steady state numerically and showed that beyond a critical threshold value of delay parameter, the system loses its stability, and Hopf bifurcation occurs. Finally, a numerical simulation presented supports the analytical findings.

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“…Introduction of time delays in a differential model significantly increases the complexity of the model. Therefore, studying qualitative behaviors of such models, using stability or bifurcation analysis, is necessary [7].…”

mentioning

confidence: 99%