Integral equations of fractional order with a variable external coefficient and monotonic nonlinearity

Abstract

Under reasonably easy-to-observe restrictions on the nonlinearities, without assuming that they satisfy the Lipschitz condition, global theorems on the existence, uniqueness, and estimates of the solution for three different classes of inhomogeneous nonlinear integral equations are proved by the method of monotone (in the sense of Browder – Minty) operators. In these equations, the operators of fractional (in the sense of Riemann – Liouville) integration with
a variable external coefficient enter linearly or nonlinearly, or these operators contain a nonlinearity under the sign of the integral (Hammerstein-type equation). In the latter case, the existence and uniqueness of the solution are established without the coercivity condition on the nonlinearity. In all cases, the conditions found in the work under which the fractional integration operators with a variable external coefficient act continuously from the real Lebesgue
space 𝐿𝑝(𝑎, 𝑏) to the spaces conjugate to them and are strictly positive play an important role. The proved theorems within the framework of the space 𝐿2(𝑎, 𝑏) cover the corresponding linear equations with integrals of fractional order. From the obtained estimates, in particular,
it directly follows that under the conditions of the proved theorems, the corresponding homogeneous linear and nonlinear integral equations have only a trivial (zero) solution.