We establish a criterion for the existence of solutions of linear inhomogeneous boundary-value problems in a Banach space. We obtain conditions for the normal solvability of such problems and consider their special cases, namely, countable-dimensional boundary-value problems. Statement of the ProblemIn a Banach space B 1 , we consider the differential equationwhere1 ) is the Banach space of vector functions continuous on [a; b], and A(t) is an operator function that acts from the Banach space B 1 into itself, is strongly continuous [1, p. 141], and has the norm |||A||| = sup t∈[a;b]Then a solution x(t) of the equationds is continuously differentiable at every point t ∈ [a; b] and satisfies Eq. (1) everywhere on [a; b]. Thus, we seek a solution x(t) of Eq. (1) in the space C 1 ([a; b], B 1 ) of functions continuously differentiable on [a; b] and taking values in the Banach space B 1 . Together with the operator equation (1), we consider the boundary condition

UDC 517.9We obtain necessary and sufficient conditions for the existence of solutions of weakly nonlinear boundaryvalue problems for differential equations in a Banach space. A convergent iterative procedure is proposed for the determination of solutions. We also establish a relationship between necessary and sufficient conditions. Statement of the Problem and Preliminary ResultsIn a Banach space B 1 ; we consider a boundary-value problem for a nonlinear differential equation with small nonnegative parameter " of the form(1)x. / D˛C "J.x. ; "/; "/;where the vector function f .t / acts from a segment OEaI b into the Banach space B 1 ; i.e.,t2OEaIb kf .t/k ½ ; C.OEaI b; B 1 / is the Banach space of vector functions continuous on OEaI b; the operator function A.t/ acts from the Banach space B 1 into itself for every t 2 OEaI b; is strongly continuous [1, p. 141], and has the norm jjjAjjj D sup t 2OEaIbkA.t/k < 1;Z.x; t; "/ is a nonlinear vector function continuously differentiable with respect to x in the neighborhood of a generating solution and continuous in t and "; i.e., Z. ; t; "/ 2 C 1 OEkx x 0 k Ä q; Z.x; ; "/ 2 C .OEaI b; B 1 / ; Z.x; t; / 2 C OE0I " 0 ;q and " 0 are sufficiently small constants,˛is an element of the space B 2 ; i.e.,˛2 B 2 ; and J.x. ; "/; "/ is a nonlinear bounded vector functional continuously differentiable in the sense of Fréchet with respect to x and

We consider boundary-value problems for a system of ordinary differential equations with a small parameter " in the equations and boundary conditions. We establish conditions for the bifurcation of solutions of a weakly perturbed linear boundary-value problem in a Banach space. Statement of the Problem and Preliminary ResultsIn a Banach space B 1 ; we consider the differential equationwhere the vector function f .t / acts from a segment OEaI b into the Banach spaceis the Banach space of vector functions continuous on OEaI b; the operator functions A.t / and A 1 .t/ acting from the Banach space B 1 into itself are strongly continuous [1, p. 141] and have the norms jjjAjjj D sup t 2OEaIbkA.t /k < 1 and jjjA 1 jjj D sup t 2OEaIb kA 1 .t/k < 1;and " 1 is a small parameter. Together with the operator equation (1), we consider the boundary-value problemx. / D˛C "`1x. /;(2)where`and`1 are linear operators continuous on OEaI b and acting from the space C.OEaI b; B 1 / into a Banach space B 2 :`W C.OEaI b; B 1 / ! B 2 ;`1W C.OEaI b; B 1 / ! B 2 ; and˛is an element of the space B 2 W˛2 B 2 : A solution of Eq.(1) is understood as a solution x.t/ D x.t; "/ of the integral equation

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