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

Sarsenbi A. Sarsenbi A.
2023 American Institute of Mathematical Sciences

AIMS Mathematics
2023 #8 Issue 11 26275 - 26289 pp.

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.

boundary value problem , Green’s function , nonlinear equation , Schauder fixed point theorem , second-order differential equation with involution

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

Research Center of Theoretical and Applied Mathematics
Department of Mathematics and Informatics

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