Singular Initial-Value and Boundary-Value Problems for Integrodifferential Equations in Dynamical Insurance Models with Investments

Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.

doi:10.1007/s10958-016-3037-1

Cited by 3 publications

(5 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Cauchy problem for functional differential equations in the non-Volterra case [1] ( § 2.2.3, p. 50) has been studied quite intensively in recent years [1][2][3][4][5][6][7][8][9][10]. We consider the Cauchy problem for the linear second order functional differential equation ẍ(t) = (T + x)(t) − (T − x)(t) + f (t), t ∈ [0, 1],…”

Section: Preliminariesmentioning

confidence: 99%

“…First of all, Cauchy problem (1) and ( 2) has the Fredholm property and the Fredholm alternative is valid (Lemma 1). Consideration of the homogeneous problem (7) can be reduced to the study of the Cauchy problem for simpler equations using the following lemma.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Suppose problem ( 1) and ( 2) does not possess the property A 1 . Then some homogeneous problem (7) has a non-trivial solution y. Let s 1 be a point of the minimum of the solution y on the interval [0, 1], s 2 a point of the maximum of y.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Conversely, let problem (11) under conditions ( 12) and ( 13) have a non-trivial solution. Then there exists a measurable function ξ (7) has the same solution for the linear positive operators…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Moreover, problem ( 25) is a problem (7) for some linear positive operators T + , T − : C[0, 1] → L satisfying conditions (3). Thus, from the assumption that in the family of problems (7) under conditions (3) there is a problem with a non-trivial solution we conclude that in this family there is a problem with a non-trivial solution that takes a maximum value at the point t = 1. Therefore, in both Lemmas 5 and 6 we can put t 2 = 1.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 4 more Smart Citations

On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

Bravyi

2023

Mathematics

View full text Add to dashboard Cite

Cauchy problems are considered for families of, generally speaking, non-Volterra functional differential equations of the second order. For each family considered, in terms of the parameters of this family, necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family are obtained. Such necessary and sufficient conditions are obtained for the following four kinds of families: integral restrictions are imposed on positive and negative functional operators, namely, operator norms are specified; pointwise restrictions are imposed on positive and negative functional operators in the form of values of operators’ actions on the unit function; an integral constraint is imposed on a positive functional operator, a pointwise constraint is imposed on a negative functional operator; a pointwise constraint is imposed on a positive functional operator, an integral constraint is imposed on a negative functional operator. In all cases, effective conditions for the solvability of the Cauchy problem for all equations of the family are obtained, expressed through some inequalities regarding the parameters of the families. The set of parameters of families of equations for which Cauchy problems are uniquely solvable can be easily calculated approximately with any accuracy. The resulting solvability conditions improve the solvability conditions following from the Banach contraction principle. An example of the Cauchy problem for an equation with a coefficient changing sign is given. Taking into account various restrictions for the positive and negative parts of functional operators allows us to significantly improve the known solvability conditions.

show abstract