BOUNDARY VALUE PROBLEMS OF SINGULARLY PERTURBED INTEGRO－DIFFERENTIAL EQUATIONS

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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′ _i s are integral operators defined onC[0,1]: $$T_i :g(t) to T_i g{text{ = }}varphi _{text{i}} (t,varepsilon ) + int_0^t {K_i } (t,xi ,varepsilon )g(xi )dxi .$$ 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: $$left{ {begin{array}{*{20}c} {varepsilon ^2 y^{(n)} = f(t,y,y', cdot cdot cdot ,y^{(n - 2)} ,varepsilon ),} {y^{(j)} (0,varepsilon ) = alpha _j (varepsilon ),j = 0,1, cdot cdot cdot ,n - 3,} {alpha _i y^{^{(n - 2)} } (i,varepsilon ) - ( - 1)^i beta _i y^{(n - 1)} (i,varepsilon ) = A_i (varepsilon ),i = 0,1.} end{array} } right.$$