Boundary value problems for a second-order differential equation with involution in the second derivative and their solvability

Abdissalam Sarsenbi; Abdizhahan Sarsenbi; Abdissalam Sarsenbi; Abdizhahan Sarsenbi

doi:10.3934/math.20231340

AIMS Mathematics

2023, Volume 8, Issue 11: 26275-26289. doi: 10.3934/math.20231340

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Boundary value problems for a second-order differential equation with involution in the second derivative and their solvability

Abdissalam Sarsenbi ^1,2,
Abdizhahan Sarsenbi ^{1
,
,}

1.
Research Center of Theoretical and Applied Mathematics, Department of Mathematics, M. Auezov South Kazakhstan University, Shymkent, Kazakhstan
2.
Department of Mathematics and Informatics, Tashenev University, Shymkent, Kazakhstan

Received: 05 July 2023 Revised: 28 August 2023 Accepted: 06 September 2023 Published: 13 September 2023
MSC : 34A34, 34B27, 34K10

We consider the two-point boundary value problems for a nonlinear one-dimensional second-order differential equation with involution in the second derivative and in lower terms. The questions of existence and uniqueness of the classical solution of two-point boundary value problems are studied. The definition of the Green's function is generalized for the case of boundary value problems for the second-order linear differential equation with involution, indicating the points of discontinuities and the magnitude of discontinuities of the first derivative. Uniform estimates for the Green's function of the linear part of boundary value problems are established. Using the contraction mapping principle and the Schauder fixed point theorem, theorems on the existence and uniqueness of solutions to the boundary value problems are proved. The results obtained in this paper cover the boundary value problems for one-dimensional differential equations with and without involution in the lower terms.
- second-order differential equation with involution,
- Green's function,
- nonlinear equation,
- boundary value problem,
- Schauder fixed point theorem
Citation: Abdissalam Sarsenbi, Abdizhahan Sarsenbi. Boundary value problems for a second-order differential equation with involution in the second derivative and their solvability[J]. AIMS Mathematics, 2023, 8(11): 26275-26289. doi: 10.3934/math.20231340

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Abstract

We consider the two-point boundary value problems for a nonlinear one-dimensional second-order differential equation with involution in the second derivative and in lower terms. The questions of existence and uniqueness of the classical solution of two-point boundary value problems are studied. The definition of the Green's function is generalized for the case of boundary value problems for the second-order linear differential equation with involution, indicating the points of discontinuities and the magnitude of discontinuities of the first derivative. Uniform estimates for the Green's function of the linear part of boundary value problems are established. Using the contraction mapping principle and the Schauder fixed point theorem, theorems on the existence and uniqueness of solutions to the boundary value problems are proved. The results obtained in this paper cover the boundary value problems for one-dimensional differential equations with and without involution in the lower terms.

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