ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF CERTAIN INTEGRO-DIFFERENTIAL EQUATIONS

Said R. Grace; John R. Graef; Ercan Tunç

doi:10.11948/2156-907X.20180226

2019 Volume 9 Issue 4

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Said R. Grace, John R. Graef, Ercan Tunç. ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF CERTAIN INTEGRO-DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1305-1318. doi: 10.11948/2156-907X.20180226

Citation:

Said R. Grace, John R. Graef, Ercan Tunç. ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF CERTAIN INTEGRO-DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1305-1318. doi: 10.11948/2156-907X.20180226

ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF CERTAIN INTEGRO-DIFFERENTIAL EQUATIONS

1.
Cairo University, Department of Engineering Mathematics, Faculty of Engineering, Orman, Giza 12221, Egypt
2.
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
3.
Gaziosmanpasa University, Department of Mathematics, Faculty of Arts and Sciences, 60240, Tokat, Turkey

Corresponding author: Email address:John-Graef@utc.edu (J. R. Graef)
Fund Project: The research of J. R. Graef was supported in part by a University of Tennessee at Chattanooga SimCenter - Center of Excellence in Applied Computational Science and Engineering (CEACSE) grant

Abstract

Abstract

The authors present conditions under which every positive solution $x(t)$ of the integro-differential equation $x^{\prime \prime }(t)=a(t)+\int_{c}^{t}(t-s)^{\alpha-1}[e(s)+k(t, s)f(s, x(s))]ds, \quad c>1, \ \alpha >0, $ satisfies $ x(t)=O(tA(t))\textrm{ as }t\rightarrow \infty, $ i.e, $\limsup_{t\rightarrow \infty }\frac{x(t)}{tA(t)} < \infty, \textrm{where} \ A(t)=\int_{c}^{t}a(s)ds.$ From the results obtained, they derive a technique that can be applied to some related integro-differential equations that are equivalent to certain fractional differential equations of Caputo type of any order.
- Asymptotic behavior /
- oscillation /
- nonoscillation /
- integro-differential equations /
- Caputo derivative /
- fractional differential equations
MSC: 34E10, 34A34

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References

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