Equations in abstract spaces see also 34gxx, 35r09, 35r10, 47jxx secondary. Boundary value problems for linear higherorder equations 35k65. We translate the results obtained previously for a single equation to the systems of various types and illustrate them with. Second order linear differential equations in banach. Favini, on the behaviour of singular semigroups in intermediate and interpolation spaces and its applications to maximal regularity for degenerate integro differential evolution equations, submitted. Wellposedness of degenerate integro differential equations in function spaces rafael aparicio, valentin keyantuo communicated by jerome a. We study an inhomogeneous linear fractional differential equation in a banach space with a degenerate operator under the derivative. Alhuthali faculty of science king abdulaziz university jeddah saudi arabia rajab. The main instrument of research is the technique of distributions in banach spaces.
Kjzdk201900504, the university young core teacher foundation of chongqing 020603011714 and the program of chongqing. The boundary value problems for degenerate anisotropic differential operator equations with variable coefficients are studied. Pdf periodic solutions of degenerate differential equations in. The theory of resolving systems has been applied in section 2 for proving the grobmanhartman theorem for differential equations in banach spaces with a degenerate operator at the derivative see. This work was supported by the nsf of china grant nos. Second order linear differential equations in banach spaces. Apr 25, 2012 under consideration is the first order linear inhomogeneous differential equation in an abstract banach space with a degenerate operator at the derivative, a relatively pradial operator at the unknown function, and a continuous delay operator. Oct 02, 2010 regular degenerate separable differential operators and applications shakhmurov, veli 20101002 00. S u p e r s m o o t h m a p p i n g s we have given p. Linear differential equations in banach space, american mathematical society, providence, ri.
Several applications to partial differential equations and systems will be given in a subsequent paper to. Nonlinear degenerate evolution equations and partial. Under consideration is the first order linear inhomogeneous differential equation in an abstract banach space with a degenerate operator at the derivative, a relatively pradial operator at the unknown function, and a continuous delay operator. Differential and integral equations project euclid. We obtain conditions of unique solvability of the cauchy problem and the showalter problem by means of degenerate semigroup theory methods. Calculus in this section, xand ywill be banach space and uwill be an open subset of x. Systems of degenerate differential equations in banach. Quasilinear degenerate evolution equations in banach. Treves, basic linear partial differential equations, acad. Systems of degenerate differential equations in banach spaces.
Equations in banach spaces mohammed al horani university of jordan angelo favini university of bologna abstract in this talk we will extend our previous results and solve the problem not only for. Favinian identification problem for firstorder degenerate differential equations. On firstorder ordinary differential equations in banach spaces. Degenerate differential equations in banach spaces. On the ulam stability of a class of banach space valued. Recently, many researchers paid attention to existence results of solution of fractional differential equations see15. Several conditions for the separability, fredholmness and resolvent estimates in l p spaces are given.
On firstorder ordinary differential equations in banach. Ordinary di erential equations in banach spaces rob kipka western michigan university robert. Secondorder degenerate differential equations in banach spaces. Periodic solutions of second order degenerate differential.
Favini, perturbation methods for inverse problems on degenerate differential equations, submitted. Donato, 540126 bologna, italy bdepartment of mechanics and mathematics, kharkov national university, kharkov, ukraine. Solutions of third order degenerate equations with. Miller, solution in banach algebras of differential equations with irregular singular point acta math. Differential calculus on banach spaces pdf free download.
We obtain conditions of unique solvability of the cauchy problem and the showalter problem by means. Inhomogeneous degenerate sobolev type equations with delay. This form of degeneration may be strong and some convenient. Osipov, an equation in a banach space with degenerate operator for derivative, in. Degenerate differential equations in banach spaces crc press book this work presents a detailed study of linear abstract degenerate differential equations, using both the semigroups generated by multivalued linear operators and extensions of the operational method from da prato and grisvard. Finally, we provide applications to degenerate differential equations that. Degenerate differential equations in banach spaces crc. We use operatorvalued fourier multipliers to obtain characterizations for wellposedness of a large class of degenerate integrodi erential equations of second order in time in banach. On banach fixed point theorem solving a special kind of. Hilbert space methods for partial differential equations.
In addition to the classical heat and wave equations with standard boundary conditions, the applications in these chapters include a multitude of nonstandard problems such as equations of pseudoparabolic, sobolev, viscoelasticity, degenerate or mixed type. The principal parts of the appropriate generated differential operators are nonselfadjoint. Second order linear differential equations in banach spaces can be used for modelling such second order equations of mathematical physics as the wave equation, the kleingordon equation, et al. Volume of papers of graduate students in mathematics and mechanics in russian, voronezh univ.
We prove the ulam stability of a class of banach space valued second order linear differential equations p x y. Read quasilinear degenerate evolution equations in banach spaces, journal of evolution equations on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Some applications to ordinary and partial differential equations with delay are described. It is not our goal to study differential equations or partial differential equations. Degenerate nonstationary differential equations with. We are concerned with both regular and degenerate firstorder identification problems related to systems of differential equations of weakly parabolic type in banach spaces.
Quasilinear degenerate evolution equations in banach spaces. In this way, a unified treatment can be given to subjects such as growth of solutions, singular perturbation of parabolic, hyperbolic and schrodinger. Coercive boundary value problems for regular degenerate. Wellposedness of fractional degenerate differential equations in. The existence of a unique strong solution for the cauchy problem to semilinear nondegenerate fractional differential equation and for the generalized showaltersidorov problem to semilinear fractional differential equation with degenerate operator at the caputo derivative in banach spaces is proved. Abstract integral equations, integral equations in abstract spaces secondary. M are closed linear operators in a complex banach space x satisfying da. Besov maximal regularity for a class of degenerate integro.
Degenerate differential equations in banach spaces 1st. Sep 11, 2008 we consider systems of degenerate differential equations in banach spaces of a special form. Ir on a banach space e, we characterize the regular admissibility. Tzanko donchev, shamas bilal, ovidiu carja, nasir javaid, alina i.
Kuznetsov, differentiable solutions to degenerate systems of ordinary equations funct. Using operatorvalued fourier multipliers theorems, we obtain. We observe that in the besov space context, the results are applicable to the more familiar scale of periodic vectorvalued h older spaces. At a for all iel, the theory of linear autonomous differential equations is based on the investigation of spectral properties of the operator a see e.
Evolution inclusions in banach spaces under dissipative. Degenerate anisotropic differential operators and applications. Clearly, being hilbert spaces uniformly convex, all the results involving uniformly convex banach spaces can be read in terms of hilbert spaces. Chernyshovas, a linear differential equation with a fredholm operator acting on the derivative, differential equations and their applications, 14, vilnius 1976 pp. Let x be a banach space endowed with the norm \vert \cdot \vert. A generalization of the operator method by grisvard is used to ensure weak and strict solutions to some degenerate differential equations with delay in banach spaces, whose operator coefficients. Request pdf periodic solutions of second order degenerate differential equations with delay in banach spaces we give necessary and sufficient. Differential equations of fractional orderoccur more frequently in different research such as physics, control of dynamical systems, engineering, chemistry etc. Wellposedness of degenerate integrodifferential equations in function spaces rafael aparicio, valentin keyantuo communicated by jerome a. Perturbative methods for inverse problems on degenerate. Linear differential equation in a banach space encyclopedia.
Semigroups of nonlinear contractions in banach spaces 98 1. We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in banach spaces. Some tools existence theorem references and resources statement of the main theorem 4 let x be a banach space and x 0 2x be given. Solutions of third order degenerate equations with infinite.
Contributions using interpolation theory and the semigroup method will be particularly appreciated. Source degenerate identification problems with smoothing. Pdf let a and m be closed linear operators defined on a complex banach space x. Volterra integral equations see also 34a12 citation. The nonlocal boundary value problems for regular degenerate differential operator equations with the parameter are studied. Nonlinear semigroups and differential equations in banach. Sharp estimates for resolvent, discreetness of spectrum, and completeness of root elements of the corresponding differential operators are obtained. Degenerate nonstationary differential equations with delay. It is not our goal to study differential equations or partial differential equations in this module 321. Mmodified resolvent set ofa, as is the fourier transform of aat s.
Abstracta generalization of the operator method by grisvard is used to ensure weak and strict solutions to some degenerate differential equations with delay in banach spaces, whose operator coefficients are time depending. Differential equations192 200393110 degenerate nonstationary differential equations with delay in banach spaces angelo favinia,1 and larisa vlasenkob adipartimento di matematica, universita. Japan journal of industrial and applied mathematics 9. We consider systems of degenerate differential equations in banach spaces of a special form. Ordinary differential equations in a banach space let xbe a banach space, u. Let b, m, l be three closed linear operators on the banach space e satisfying h1h3. Several applications to partial differential equations and systems will be given in a subsequent paper to show the fullness of our abstract results. Regular degenerate separable differential operators and applications shakhmurov, veli 20101002 00. Nonlinear operators and nonlinear equations of evolu tion in banach spaces.
Equations in banach spaces with a degenerate operator under a. Several conditions for the separability and fredholmness in banach valued spaces are given. Awawdeh, differential equations in banach spaces with applications, ph. Request pdf degenerate nonstationary differential equations with delay in banach spaces a generalization of the operator method by grisvard is used to ensure weak and strict solutions to some. We treat periodic vectorvalued lebesgue, besov and trieblellizorkin spaces. U,x z is to be interpreted as a time dependent vector. Favini, on the behaviour of singular semigroups in intermediate and interpolation spaces and its applications to maximal regularity for degenerate integrodifferential evolution equations, submitted. Differential equations in a banach space springerlink. The nonlocal boundary value problems for regular degenerate differentialoperator equations with the parameter are studied. The obtained results are then applied to provide maximal time regularity for the solutions to a wide class of degenerate integro and nonintegro differential evolution equations in banach spaces. Fixed point theorems and applications univerzita karlova.
Wellposedness of fractional degenerate differential equations in banach. Banach space, a is an operator valued function taking t into a bounded linear operator at acting on x. Buy second order linear differential equations in banach spaces on free shipping on qualified orders. A generalization of the operator method by grisvard is used to ensure weak and strict solutions to some degenerate differential equations with delay in banach spaces, whose operator coefficients are time depending. Regular degenerate separable differential operators and. Our result may be regarded as a generalization of the above mentioned results by in ponce 16 and bu 8 when a 0. Sep 01, 2004 read quasilinear degenerate evolution equations in banach spaces, journal of evolution equations on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Kjzdk201900504, the university young core teacher foundation of chongqing 020603011714 and the program of chongqing innovation research group project in. Initial value problems for linear higherorder equations 35g15. Degenerate differential equations in banach spaces presents a study of linear abstract degenerate differential equations, using the semigroups generated by multivalued linear operators and extensions of the operational method from da prato and grisvard. Evolution inclusions in banach spaces under dissipative conditions author. Degenerate firstorder inverse problems in banach spaces. Ir on a banach space e, we characterize the regular admissibility of a subspace of bucir,e in terms of solvability of. On firstorder ordinary differential equations in banach spaces by madeaha mabrouk alghanmi a this thesis submitted for the requirements of the degree of master of science mathematics differential equations supervised by dr.
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