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On the asymptotic behavior of solutions of certain integro--differential equations
John R. Graef,Said R. Grace,Ercan Tunc
Keywords:Asymptotic behavior, oscillation, nonoscillation, integro--differential equations, Caputo derivative
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 \[ x(t)=O(tA(t))\textrm{ as }t\rightarrow \infty, \textrm{ \ i.e.,\ } \limsup_{t\rightarrow \infty }\frac{x(t)}{tA(t)}<\infty , \textrm{ where \ } A(t)=\int_{c}^{t}a(s)ds. \] 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.