Volterra integral equation in Banach space

Abstract

Despite the significant results achieved in the study of operator equations (including Volterra equations) in normalized Banach spaces, fundamental research in this field of mathematics attracts the attention of a huge number of mathematicians around the world. The solutions of the Volterra equation describe many important processes in various fields of science and technology. Studies of various inverse problems, experimental or experimental data processing problems related to the study of spherical or axisymmetric plasma formations, numerous mathematical models of the existence of various biological systems lead to the consideration and solution of this type of integral equations. A great contribution to the development of this theory was made by N.A. Magnitsky, L.I. Panov, A.N. Tikhonov, M.M. Lavrentiev and others. Fundamental results were obtained in the study of multiple operator equations with singularities in various functional spaces. Solutions depending on many parameters were constructed for the above equations. Currently, such problems are considered in spaces of arbitrary dimension and with coefficients having a derivative of finite order. In this paper, a finite set of solutions in a certain functional space is constructed for an integral equation of the first kind. The kernel of the integral operator has a finite order and is sufficiently differentiable near zero. The integral equation under consideration is reduced to an integro-differential equation representing two terms. For the first term, it is possible to solve the corresponding inhomogeneous equation and obtain a set of solutions in some functional normalized space. For the second term, we obtain an equation with an operator whose norm in some operator space is arbitrarily small near zero. Such splitting of the integral operator makes it possible to construct a partial and general solution of the corresponding integro-differential equation in the form of convergent equations. Applying modern methods of functional analysis, it is possible, by studying two separate equations, to construct a multiparametric family of solutions with values in some Banach space with weight for the original equation under consideration.