### A computational investigation of an integro-differential inequality with periodic potential.

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In this paper, we develop a generalized quasilinearization technique for a nonlinear second order periodic boundary value problem and obtain a sequence of approximate solutions converging uniformly and quadratically to a solution of the problem. Then we improve the convergence of the sequence of approximate solutions by establishing the convergence of order $k$$(...$

The method of quasilinearization is a well–known technique for obtaining approximate solutions of nonlinear differential equations. This method has recently been generalized and extended using less restrictive assumptions so as to apply to a larger class of differential equations. In this paper, we use this technique to nonlinear differential problems.

Recently, we have developed the necessary and sufficient conditions under which a rational function $F\left(hA\right)$ approximates the semigroup of operators $exp\left(tA\right)$ generated by an infinitesimal operator $A$. The present paper extends these results to an inhomogeneous equation ${u}^{\text{'}}\left(t\right)=Au\left(t\right)+f\left(t\right)$.