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Volume 9, Number 4, 2019, Pages 1305-1318                                                                DOI:10.11948/2156-907X.20180226
On the asymptotic behavior of solutions of certain integro-differential equations
Said R. Grace,John R. Graef,Ercan Tunc
Keywords:Asymptotic behavior, oscillation, nonoscillation, integro--differen\-tial equations, Caputo derivative, fractional differential equations.
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.
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