THE ASYMPTOTIC SOLUTION OF A SINGULARLY PERTURBED CAUCHY PROBLEM FOR THE FIRST-ORDER EQUATION IN A BANACH SPACE
The article is devoted singularly perturbed Cauchy problem for differential equations in Banach space with a small parameter at the derivative. The coefficient under the unknown function is a Fredholm operator with zero index and the kernel of any dimension. Chains associated elements, corresponding to the zero eigenvalues have different lengths. We construct an asymptotic expansion of the solution of this problem by Vasilyeva-Vishik-Lyusternik method. When calculating the components of the regular part of the expansion of the solution cascade decomposition method of equations is used. Asymptoticity of this expansion is proved. The conditions regular degeneracy are found. Conditions solution behavior are formulated as the parameter to zero.