fWeakly Nonlinear Boundary-Value Problems for Systems of Impulsive Integrodifferential Equations. Critical Case of the S
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WEAKLY NONLINEAR BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF IMPULSIVE INTEGRODIFFERENTIAL EQUATIONS. CRITICAL CASE OF THE SECOND ORDER I. A. Bondar
UDC 517.929
We establish conditions for the existence of solutions of an impulsive weakly nonlinear boundary-value problem in the critical case of the second order and determine the structure of these solutions. By using the theory of orthoprojectors and pseudoinverse Moore–Penrose matrices, we study sufficient conditions for the existence of solutions of problems of this kind and propose an iterative algorithm for their construction. It is shown that the existence of solution of the original boundary-value problem depends on the conditions obtained with regard for nonlinearity and the second approximation to the required solution. It is proposed to consider the impulsive problem as an inner boundary-value problem.
Introduction Consider a weakly nonlinear system of integrodifferential equations with impulsive actions at fixed times
x.t P / � ˆ.t /
Zb a
ŒA.s/x.s/ C B.s/x.s/çds P D f .t / C "
Zb
K.t; s/Z.x.s; "/; s; "/ds;
a
ÅEi xj t D⌧i WD Si x.⌧i � 0/ C �i C "J1 .x.�; "/; "/; t ¤ ⌧i ;
t 2 Œa; bç;
⌧i 2 .a; b/;
`x.�/ D ˛ C "J2 .x.�; "/; "/;
(1)
(2)
i D 1; 2; : : : ; p;
˛ 2 Rq :
(3)
In what follows, we use the following assumptions and notation taken from the works [1–5]: A.t /; B.t /; ˆ.t /; and K.t; s/ are .m ⇥ n/-; .m ⇥ n/-; .n ⇥ m/-; and .n ⇥ n/-dimensional matrices, respectively, whose components belong to the space L2 Œa; bçI the vector columns of the matrix ˆ.t / are linearly independent on Œa; bç; f .t / is an n-dimensional vector function from L2 Œa; bçI Ei and Si are .ki ⇥ n/-dimensional matrices, �i is a ki -dimensional vector column of constants, and rank .Ei C Si / D ki ; i D 1; 2; : : : ; p: Here, the pulses are specified not for all components of the unknown n-dimensional vector function x.t / D col .x1 .t /; x2 .t /; : : : ; xki .t /; : : : ; xn .t //: In fact, they are specified only for some ki components. Moreover, ` is a bounded linear vector functional defined in D2 ŒaI bç; i.e., ` D col .`1 ; `2 ; `3 ; : : : ; `p /W D2 ŒaI bç ! Rp I Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivs’ka Str., 3, Kyiv, 01024, Ukraine; e-mail: [email protected]. Translated from Neliniini Kolyvannya, Vol. 22, No. 2, pp. 147–164, April–June, 2019. Original article submitted February 20, 2019; revision submitted May 1, 2019. 1072-3374/20/2494–0553
c 2020 �
Springer Science+Business Media, LLC
553
I. A. B ONDAR
554
˛ D col .˛1 ; ˛2 ; ˛3 ; : : : ; ˛p / 2 Rp I Z.x.t; "/; t; "/ is an n-dimensional vector function nonlinear with respect to the first component, continuously differentiable with respect to x in the vicinity of the generating solution, square-integrable with respect to t; and continuous in ": ⇤ ⇥ Z.�; t; "/ 2 C 1 kx � x0 k � ;
Z.x.�; "/; �; "/ 2 L2 Œa; bç;
Z.x.t; �/; t; �/ 2 C Œ0; "0 çI
and J1 .x.�; "/; "/ and J2 .x.�; "/; "/ are nonlinear and bounded p- and q-dimensional vector functionals, respectively,
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