The Parabolic Transform and Some Singular Integral Evolution Equations

In the present paper, we consider solvability conditions for the Cauchy problem for ordinary differential equations of fractional order α> 1 in Banach spaces with so-called “worsening” righthand sides. For integer-order (first- and higher-order) equations with “worsening” operators, there are classical results due to Nagumo and Ovsyannikov (e.g., see [1–6]), based on the analysis of the convergence of the successive approximation method in scales of continuously embedded Banach spaces. A number of further results in this direction was obtained in [7–9]. Similar results for fractionalorder equations with “worsening” operators were considered in [10]. Nazarov [11, 12] developed a new method for first-order differential equations in scales of compactly embedded Banach spaces on the basis of Tonelli approximations. In the present paper, we suggest a modification of his method, which results in a number of assertions on the (nonlocal!) solvability of the Cauchy problem for differential equations of integer or fractional order strictly greater than unity.

V.A. Trenogin’s results on the stability of solutions to the Cauchy problem for differential equations in Banach spaces with respect to linear approximation in the case of degenerate linearization are presented from the viewpoint of branching equation in the root subspace.

In this short note, we want to give the main parts of the verification of the ordered Prabhakar fractional order differential operator and the Cauchy problem for an equation with singular coefficients. That is, we present the exact solution of the Cauchy problem for the Prabhakar fractional derivative partial differential equation. Using the Fourier method, we find the general solution of the given equation. The unknown coefficient is found using the Hankel transformation. We have also presented important statements the bivariate MittagLeffler function E2 (x; y) and the Bessel function. The main objective is the Cauchy problem for a singular coefficient equation involving ordered Prabhakar derivatives. We present here the proof algorithm and a brief description of the main steps.

The initial value problems of first order impulsive differential equations in Banach spaces

This paper concerns the dissipativity of nonlinear Volterra functional differential equations (VFDEs) in Banach space and their numerical discretization. We derive sufficient conditions for the dissipativity of nonlinear VFDEs. The general results provide a unified theoretical treatment for dissipativity analysis to ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type appearing in practice. Then the dissipativity property of the backward Euler method for VFDEs is investigated. It is shown that the method can inherit the dissipativity of the underlying system. The close relationship between the absorbing set of the numerically discrete system generated by the backward Euler method and that of the underlying system is revealed.

Stochastic differential equations in a Banach space driven by the cylindrical Wiener process

In this paper, we consider the problem of the exact and approximate solutions of certain differential equations with variational derivatives of the first and second orders. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations with the first variational derivatives are given. An interpolation method for solving ordinary differential equations with variational derivatives is demonstrated. The general scheme of an approximate solution of the Cauchy problem for nonlinear differential equations with variational derivatives of the first order, based on the use of the operator interpolation apparatus, is presented. The exact solution of the differential equation of the hyperbolic type with variational derivatives, similar to the classical Dalamber solution, is obtained. The Hermite interpolation problem with the conditions of coincidence in the nodes of the interpolated and interpolation functionals, as well as their variational derivatives of the first and second orders, is considered for functionals defined on the sets of differentiable functions. The found explicit representation of the solution of the given interpolation problem is based on an arbitrary Chebyshev system of functions. This solution is generalized for the case of interpolation of functionals on one out of two variables and applied to construct an approximate solution of the Cauchy problem for the differential equation of the hyperbolic type with variational derivatives. The description of the material is illustrated by numerous examples.

Existence and uniqueness of solutions for initial value problems of second order ordinary differential equations in Banach spaces

Existence theorems of solutions for two-point boundary value problem of second order ordinary differential equations in Banach spaces

In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm-Liouville boundary value problems for nonlinear second order impulsive differential equations in Banach spaces.

The analytic and numerical stability of stiff impulsive differential equations in Banach space

SECOND ORDER DIFFERENTIAL EQUATIONS IN BANACH SPACE

The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.

In this article, we are concerned with the existence of extremal solutions to the initial value problem of impulsive differential equations in ordered Banach spaces. The existence and uniqueness theorem for the solution of the associated linear impulsive differential equation is established. With the aid of this theorem, the existence of minimal and maximal solutions for the initial value problem of nonlinear impulsive differential equations is obtained under the situation that the nonlinear term and impulsive functions are not monotone increasing by using perturbation methods and monotone iterative technique. The results obtained in this paper improve and extend some related results in abstract differential equations. An example is also given to illustrate the feasibility of our abstract results.

In this paper, we deal with the nonlocal Cauchy problem for a class of two-term time fractional differential equations in Banach spaces. By constructing a suitable measure of noncompactness on the space of solutions, we prove the existence of a compact set containing decay mild solutions to the mentioned problem.