Malkhaz Ashordia ON THE FREDHOLM PROPERTY FOR GENERAL LINEAR BOUNDARY VALUE PROBLEMS FOR IMPULSIVE SYSTEMS WITH SINGULARITIES Dedicated to the blessed memory of Professor T. Chanturia

A general linear singular boundary value problem dxi dt = Pi(t) · x3−i + qi(t) (i = 1, 2), xi(τk+)− xi(τk−) = Gi(k) · x3−i(τk) + hi(k) (i = 1, 2; k = 1, 2, . . .); li(x1, x2) = ci (i = 1, 2) is considered, where Pi ∈ Lloc(]a, b[ ,Rni×n3−i), qi ∈ Lloc(]a, b[ ,Ri), Gi : {1, 2, . . .} → Rni×n3−i , hi : {1, 2, . . .} → Ri , ci ∈ Ri , and li is a linear bounded operator (i = 1, 2). The singularity is understood in the sense that Pi 6∈ L([a, b],Rni×n3−i), qj 6∈ L([a, b],Rj ) or ∞ ∑ k=1 ‖Gi(k)‖+ ‖hj(k)‖ ) = +∞ for some i, j ∈ {1, 2}. The conditions are established under which this problem is uniquely solvable if and only if the corresponding homogeneous boundary value problem has only the trivial solution. Analogous problems for similar impulsive systems with small parameters are also considered. îâäæñéâ. àŽêýæèñèæŽ äëàŽáæ ïŽýæï ûîòæãæ æéìñèïñîæ ïŽïŽäôãîë ŽéëùŽêŽ dxi dt = Pi(t) · x3−i + qi(t) (i = 1, 2), xi(τk+)− xi(τk−) = Gi(k) · x3−i(τk) + hi(k) (i=1, 2; k = 1, 2, . . .); li(x1, x2) = ci (i = 1, 2), ïŽáŽù Pi ∈ Lloc(]a, b[ ,Rni×n3−i), qi ∈ Lloc(]a, b[ ,Ri), Gi : {1, 2, . . .} → Rni×n3−i , hi : {1, 2, . . .} → Ri , ci ∈ Ri , ýëèë li (i = 1, 2) ûîòæãæ öâéëïŽäôãîñèæ ëìâîŽðëîæŽ. Reported on the Tbilisi Seminar on Qualitative Theory of Differential Equations on December 13, 2010.