On Fixed Points of Multivalued Mappings in Spaces with a Vector-Valued Metric

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

“…Lemmas 3 and 4 allow for applying Theorem 1 to Equations (11) and (13). Thus, we obtain the following statement.…”

“…Due to the regularity of the cone L + , in the set of solutions to Equation ( 13) belonging to the interval [0, ς] L , there exists a minimal element; denote this element by ς * and prove that it is the smallest element in the set of all solutions to Equation (13). Suppose the assertion is not true and there exists a solution ς ∈ L + such that ς * ≰ ς.…”

“…Then, the map P M Y is called a vector metric or, for shortness, a v-metric, and a pair (Y, P M Y ) is a vector metric (v-metric) space. For similar definitions of vector metric spaces, as well as for theorems on fixed points of maps acting in such spaces, see, e.g., [12][13][14].…”

“…It is easy to see that if the norm in E is monotone, then a v-metric P E X defines a "classical" metric via ∥P E X ∥ E . However, as it is noted in [13] (Remark 1), using a v-metric instead of the corresponding "classical" one allows for, for example, obtaining less burdensome conditions of the existence of solutions and more accurate estimates for them. Moreover, in the case of C > 1, the map defined as ∥P E X ∥ E cannot be considered a metric (it does not satisfy the triangle inequality).…”

“…For t ∈ T, from the fact that the function φ is nonincreasing with respect to the second argument, we obtain Kη)(t), (S h η)(t), η(t) = φ t, ( Kη)(t), (S h η)(t), ς * (t) ≥ φ t, ( Kς * )(t), (S h ς * )(t), ς * (t) = 0.So, for the function η, the inequality φ t, ( Kη)(t), (S h η)(t), η(t) ≥ 0 holds on the entire interval [0, T]. But according to the proof above, there should exist a solution ν ∈ [0, η] L to Equation(13) for which the inequalities ν ≤ η < ς * should hold. And this contradicts the fact that ς * is a minimal element in the solution set.…”