Boundary Value Problems with Derivatives with Respect to «Split» Measures and Monotone Nonlinearity
DOI:
https://doi.org/10.52575/2687-0959-2026-58-1-54-64Keywords:
Derivative with Respect to Measure, Boundary Value Problem, Monotone Nonlinearity, Continuous Branch, Green’s FunctionAbstract
This paper examines a continuous branch of a nonlinear spectral problem with derivatives with respect to "split" measures. Sufficient conditions for the nonemptiness of the set of nonnegative values, for each of which a nonnegative, nontrivial solution to the nonlinear spectral problem with discontinuous solutions exists, are obtained. The monotonicity of the solution with respect to the spectral parameter is demonstrated; and the convergence of the iterative sequence to the solution is proven. Difficulties arising in the analysis of a nonlinear boundary value problem with discontinuous solutions are overcome using derivatives with respect to the measure. The resulting equation is then considered as a relationship between the solution value and its derivatives up to a certain order, i.e., it becomes ordinary. This approach to treating equations with nonsmooth and discontinuous solutions was proposed by Yu.V. Pokorny. The theory of positive completely continuous operators developed by M.A. Krasnosel’skii is also used.
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