For higher-order nonlinear differential equations with deviating arguments and with nonintegrable singularities with respect to the time variable, we establish sharp sufficient conditions for the Cauchy problem to be solvable and well-posed.

Classical theorems on differential inequalities [Coddington and Levinson, Theory of ordinary differential equations, Mc Graw-Hill Book Company, Inc., 1995, Hartman, Ordinary differential equations, John Wiley & Sons, 1964, Walter, Differential and integral inequalities, Springer-Verlag, 1970] are generalized for initial value problems of the kind and where ƒ : C([a, b]; Rn ) → Lloc (]a, b]; Rn ) is a singular Volterra operator, c 0 ∈ Rn , h : [a, b] → [0, +∞[ is continuous and positive on ]a, b], ‖ · ‖ is a norm in Rn , and [u]+ and [u]– are respectively the positive and the negative part of the vector u ∈ Rn .

Sufficient conditions are found for the global solvability of the weighted Cauchy problem where 𝑓 : 𝐶([𝑎, 𝑏]; 𝑅𝑛) → 𝐿𝑙𝑜𝑐(]𝑎, 𝑏]; 𝑅𝑛) is a singular Volterra operator, 𝑐0 ∈ 𝑅𝑛, 𝑕 : [𝑎, 𝑏] → [0, +∞[ is a function continuous and positive on ]𝑎, 𝑏], and ∥ · ∥ is the norm in 𝑅𝑛.

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Impact Factor:ISRA (India) = 1.344 ISI (Dubai, UAE) = 0.829 GIF (Australia) = 0.564 JIF = 1.500 SIS (USA) = 0.912 РИНЦ (Russia) = 0.207 ESJI (KZ) = 4.102 SJIF (Morocco) = 2.031 ICV (Poland) = 6.630 PIF (India) = 1.940 IBI (India) = 4.260 Philadelphia, USA 122Abstract: The paper describes the construction of a shape of the orthopedic boot-tree print by means of the solution to differential equation with deviating argument. The obtained solutions to the second-order differential equation with deviating argument allow for describing the shapes of the orthopedic boot-tree print with high degree of accuracy. It also allows for varying the shapes of the orthopedic boot-tree print when moving from the one size to the second one in an unlimited number that is of particular relevance in the production of orthopedic shoes.