Boundary value problems of singularly perturbed integro-differential equations

In this paper, the boundary value problem for the integro-differential equation with a small parameter ε>0: $$\left\{ {\begin{array}{*{20}c} {\varepsilon ^2 x'' = f(t,T_1 x, \cdot \cdot \cdot ,T_m x,x,\varepsilon ),} \\ {\alpha _i x(i,\varepsilon ) - ( - 1)^i \beta _i x'(i,\varepsilon ) = A_i (\varepsilon ),i = 0,1} \\ \end{array} } \right.$$ is discussed, whereT′_is are integral operators defined onC[0,1]: $$T_i :g(t) \to T_i g{\text{ = }}\varphi _{\text{i}} (t,\varepsilon ) + \int_0^t {K_i } (t,\xi ,\varepsilon )g(\xi )d\xi .$$ Using the differential inequality technique, the existence of solutions is proved and the estimate of solutions is obtained as well. In particular, this result applied to the high-order (n≥3) boundary value problem for ordinary differential equations with a small parameter ε>0: $$\left\{ {\begin{array}{*{20}c} {\varepsilon ^2 y^{(n)} = f(t,y,y', \cdot \cdot \cdot ,y^{(n - 2)} ,\varepsilon ),} \\ {y^{(j)} (0,\varepsilon ) = \alpha _j (\varepsilon ),j = 0,1, \cdot \cdot \cdot ,n - 3,} \\ {\alpha _i y^{^{(n - 2)} } (i,\varepsilon ) - ( - 1)^i \beta _i y^{(n - 1)} (i,\varepsilon ) = A_i (\varepsilon ),i = 0,1.} \\ \end{array} } \right.$$