JOURNAL Series A - Pure and Applied Mathematics

44 At. Georgieva, St. Kostadinov2 Statement of the problemLet X be an arbitrary real Banach space with norm |.| and identity operator I. Weconsider the impulse differential equationdxdt = f(t, x), (t ≠ t n) (1)x(t + n ) = Q n (x(t n )), (2)where f : [a, b] × X → X is a continuous function, the operatorsQ n : X → X (n = 1, 2, ...p) are continuous and a < t 1 < t 2 < ... < t p ≤ b. Weassume that all functions are left continuity at t n .Let X ∗ be the dual space of X andF x = {x ∗ ∈ X ∗ : x ∗ (x) = |x| 2 = |x ∗ | 2 }(x ∈ X).Using F we introduce the semi-scalar product [6], [8](x, y) − = inf{y ∗ (x) : y ∗ ∈ F y}. (3)In [6] many well-known properties of the semi-scalar product are shown. Here,we remark only that if X is a Hilbert space, then the semi-scalar product in (3) isequal to the ordinary scalar product.Remark 1. If X is a complex Banach space, we can introduce a semi-scalar productby the formula(x, y) − = Re(x, y) − + iIm(x, y) − ,whereRe(x, y) − = inf{Re y ∗ (x) : y ∗ (y) ∈ R and y ∗ (y) = |y| 2 },Im(x, y) − = inf{Im y ∗ (x) : y ∗ (y) ∈ R and y ∗ (y) = |y| 2 }.It is easy to show that Lemma 3.2 [6] is valid in the complex case too.We setc = sup |f(t, 0)|.a≤t≤bDefinition 1. [6] We say that X is a strictly convex space if x ≠ y and|x| = |y| = 1 imply |λx + (1 − λ)y| < 1 for λ ∈ (0, 1).Let ω : (a, b] × R + → R be an arbitrary function.We introduce the conditions.H1. The inequality ω(t, 0) ≥ 0 (a < t ≤ b) holds.