Global existence and uniqueness theorems for nonlinear equations with Riemann-Liouville integrals of fractional order

Abstract

Using the method of monotone (in the sense of Browder-Minty) operators, global theorems on the existence, uniqueness, and estimates of the solution for various classes of inhomogeneous nonlinear integral equations are proved, in which the operators of fractional (in the sense
of Riemann-Liouville) integration enter linearly or nonlinearly, or these operators contain nonlinearity under the sign of the integral (Gummerstein-type equations). 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 play an important role, under which the fractional integration operators act continuously from the real Lebesgue space 𝐿𝑝(𝑎, 𝑏) to the
spaces conjugate to them and are strictly positive. The theorems proved 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.