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New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. [ABSTRACT FROM AUTHOR]

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]

We investigate the unique solvability of a class of linear inverse problems with a time‐independent unknown coefficient in an evolution equation in Banach space, which is resolved with respect to the fractional Riemann‐Liouville derivative. We assume that the operator in the right‐hand side of the equation generates a family of resolving operators for the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. It is shown that the inverse problem is well‐posed with respect to the graph norm of the generating operator only. A well‐posedness criterion is found. The obtained abstract results are applied to the unique solvability study of an inverse problem for a class of time‐fractional partial differential equations. An example, in particular, shows that in the case of an unbounded generating operator, the inverse problem can be ill‐posed with respect to the norm of the whole space. [ABSTRACT FROM AUTHOR]

The local unique solvability of the Cauchy-type problem to a semilinear equation in a Banach space, which is solved with respect to the highest order Riemann–Liouville derivative, is proved. A linear unbounded operator at the unknown function in the equation generates an analytic in a sector resolving the family of operators of the linear homogeneous fractional-order equation. This result is applied to the study of initial-boundary value problems for a class of nonlinear partial differential equations, in particular, containing a nonlinear superdiffusion equation. Besides, it is used for the investigation of the local unique solvability of the Showalter–Sidorov type problem to a semilinear Sobolev-type equation in a Banach space with a sectorial pair of operators and with Riemann–Liouville derivatives. For this aim, we essentially use two types of condition on the nonlinear operator: the condition on the image of this operator or the condition of its independence of the elements of the degeneration subspace. These abstract results are demonstrated on examples of initial-boundary value problems for Sobolev-type nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The criteria of the well-posedness is obtained for an inverse problem to a class of fractional order in the sense of Caputo degenerate evolution equations with a relatively bounded pair of operators and with the generalized Showalter–Sidorov initial conditions. It is formulated in terms of the relative spectrum of the pair and of the characteristic function of the problem. Sufficient conditions of the unique solvability are obtained for a similar problem with the Cauchy initial condition. For these purposes the unique solvability of the same inverse problem was studied for the equation with a bounded operator near an unknown function, which is solved with respect to the fractional derivative. General results are applied to the inverse problem research for the time fractional system of equations describing the dynamics of a viscoelastic fluid in the weakly degenerate and the strongly degenerate cases. [ABSTRACT FROM AUTHOR]

Unique solvability and well-posedness issues are studied for linear inverse problems with a constant unknown parameter to fractional order differential equations with Riemann – Liouvlle derivative in Banach spaces. Firstly, well-posedness criteria for the inverse problem with the Cauchy type initial conditions to the differential equation in a Banach space that solved with respect to the fractional derivative is obtained. This result is applied to search of sufficient conditions for the unique solution existence of the inverse problem for equation with linear degenerate operator at the Riemann – Liouville fractional derivative. It is shown that the presence of the matching conditions for the data of the problem excludes the possibility of the well-posedness consideration for the degenerate inverse problem with the Cauchy type condition. But for the inverse problem with the Showalter – Sidorov type conditions it is found the criteria of the well-posedness. Abstract results are used to the search of conditions of the unique solvability for an inverse problem to a class of partial differential equations of time-fractional order with polynomials of elliptic differential operators with respect to the spatial variables. [ABSTRACT FROM AUTHOR]

The existence of a unique solution for the identification problem to linear fractional differential equations in Banach spaces were proved in the cases of the nondegenerate equation and of the equation with degenerate operator at the Caputo derivative. The degenerate case was studied for the overdetermination on the phase space and on the degeneracy subspace of the corresponding homogeneous equation. Abstract results are used to the investigation of the identification problem for time-fractional order Sobolev's system of equations in the cases of the overdetermination on the fluid velocity and on the fluid pressure gradient. [ABSTRACT FROM AUTHOR]

We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter–Sidorov conditions which are natural for degenerate type equations. We prove the results of local unique solvability by using, mainly, the method of contraction mappings. The obtained theory via its abstract results is applied to the research of initial-boundary value problems for both Scott–Blair and modified Sobolev systems of equations with delays. [ABSTRACT FROM AUTHOR]