Weakly nonlinear boundary-value problems for differential equations in a Banach space in the critical case

Abstract

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

Weakly Nonlinear Boundary-Value Problems for Operator Equations in Banach Spaces

“…Theorem 2. Suppose that the boundary-value problem ( 2), ( 3) satisfies conditions (6) and has a solution…”

“…a nonlinear operator satisfying the conditions (a 2 ) and (a 3 ) in (6); `1W l 1 .I; B 1 / ⇥ I ⇥ I " 0 ! B ⇥ I " 0 is a nonlinear vector functional satisfying the conditions (a 6 ) and (a 7 ) in (6).…”

“…For systems of ordinary differential equations and functional-differential equations with impulsive action in the general Noetherian case, these problems were studied in [3][4][5].These problems were generalized to the case of Banach spaces in which a finite-dimensional Euclidean space of values of a function is replaced by a Banach space. In the case where L is an ordinary differential operator acting in a Banach space, weakly nonlinear boundary-value problems (1) were investigated in [6]. As an important specific feature of these problems, we can mention the fact that the equation Lz D f of the linear generating boundary-value problem is solvable for any right-hand side, i.e., the operator L is everywhere solvable [7].In the case where L is an integral Fredholm operator with degenerate kernel, which is not everywhere solvable, the boundary-value problem (1) was considered in [8].At present, there is no general approach to the investigation of weakly nonlinear boundary-value problems (1) whose linear part contains an operator, which is not everywhere solvable.…”

“…These problems were generalized to the case of Banach spaces in which a finite-dimensional Euclidean space of values of a function is replaced by a Banach space. In the case where L is an ordinary differential operator acting in a Banach space, weakly nonlinear boundary-value problems (1) were investigated in [6]. As an important specific feature of these problems, we can mention the fact that the equation Lz D f of the linear generating boundary-value problem is solvable for any right-hand side, i.e., the operator L is everywhere solvable [7].…”

“…`z0 .�/ D ˛ (5) obtained from the boundary-value problem (2), (3) for " D 0: By analogy with similar problems for systems of ordinary differential equations [2][3][4]6], the boundary-value problem (4), (5) is called a generating boundary-value problem for the boundary-value problem (2), (3) and the solutions of the boundary-value problem (4), (5) are called generating.…”