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Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders. [ABSTRACT FROM AUTHOR]

The concept of a β -integrated resolving function for a linear equation with a Gerasimov–Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability of the Cauchy problem to linear homogeneous and inhomogeneous equations are found in the case of the existence of a β -integrated resolving function. The necessary and sufficient conditions for the existence of such a function in terms of estimates on the resolvent of its generator are obtained. The example of a β -integrated resolving function for the Schrödinger equation is given. Thus, the paper discusses some aspects of the symmetry of the concepts of integrability and differentiability. Namely, it is shown that, in the absence of a sufficiently differentiable resolving function for a fractional differential equation, the problem of the existence of a solution can be solved by an integrated resolving function of the equation. [ABSTRACT FROM AUTHOR]

The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media. [ABSTRACT FROM AUTHOR]

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the Lipschitz continuity condition in phase variables and two types of continuity over all variables of a nonlinear operator in the equation, we obtain two versions on a theorem on the nonlocal existence of a unique solution. Two similar versions of local unique solvability of the Cauchy problem are proved under the local Lipschitz continuity condition for the nonlinear operator. The general results are used for the study of an initial boundary value problem for a generalization of the nonlinear phase field system of equations with distributed derivatives with respect to time. [ABSTRACT FROM AUTHOR]

The unique solvability for the Cauchy problem in a class of degenerate multi-term linear equations with Gerasimov–Caputo derivatives in a Banach space is investigated. To this aim, we use the condition of sectoriality for the pair of operators at the oldest derivatives from the equation and the general conditions of the other operators' coordination with invariant subspaces, which exist due to the sectoriality. An abstract result is applied to the research of unique solvability issues for the systems of the dynamics and of the thermoconvection for some viscoelastic media. [ABSTRACT FROM AUTHOR]

Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their properties are studied. In terms of the resolvent of the operator A, necessary and sufficient conditions are obtained for the existence of analytic resolving families of the equation under consideration. A perturbation theorem for such a class of operators is proved, and the Cauchy problem for the inhomogeneous equation with a distributed fractional derivative is studied. Abstract results are applied for the research of the unique solvability of initial boundary value problems for partial differential equations with a distributed derivative with respect to time. [ABSTRACT FROM AUTHOR]

Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables. [ABSTRACT FROM AUTHOR]