EXISTENCE OF MULTIPLE SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH NON-STANDARD BOUNDARY CONDITIONS

Abstract

This study investigates the existence and multiplicity of solutions for nonlinear boundary value problems formulated under non-standard boundary conditions, including integral, multipoint, nonlinear, and fractional structures. By transforming the differential equation into an equivalent operator equation using an appropriate Green’s function, the analysis establishes conditions ensuring compactness a

This study investigates the existence and multiplicity of solutions for nonlinear boundary value problems formulated under non-standard boundary conditions, including integral, multipoint, nonlinear, and fractional structures. By transforming the differential equation into an equivalent operator equation using an appropriate Green’s function, the analysis establishes conditions ensuring compactness and continuity of the associated operator. The results demonstrate the existence of at least one positive solution and, under strengthened assumptions, the existence of two or three distinct solutions. The study incorporates nonlinear functions such as with specific parameter choices, for example and , to validate the theoretical framework. Numerical illustrations further confirm the emergence of multiple solutions when coefficients and growth parameters vary within ranges like . The findings highlight the flexibility of the developed approach and its ability to accommodate a broad class of boundary conditions. The work also offers strong potential for extension to fractional differential equations of order between and , impulsive boundary problems, and systems of nonlinear equations.

nd continuity of the associated operator. The results demonstrate the existence of at least one positive solution and, under strengthened assumptions, the existence of two or three distinct solutions. The study incorporates nonlinear functions such as with specific parameter choices, for example and , to validate the theoretical framework. Numerical illustrations further confirm the emergence of multiple solutions when coefficients and growth parameters vary within ranges like . The findings highlight the flexibility of the developed approach and its ability to accommodate a broad class of boundary conditions. The work also offers strong potential for extension to fractional differential equations of order between and , impulsive boundary problems, and systems of nonlinear equations.