scholarly journals Fractional Cauchy Problems for Infinite Interval Case-II

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1165
Author(s):  
Mohammed Al Horani ◽  
Mauro Fabrizio ◽  
Angelo Favini ◽  
Hiroki Tanabe
Keyword(s):  
Existence And Uniqueness ◽  
Direct Problem ◽  
Abstract Cauchy Problem ◽  
Continuous Functions ◽  
Cauchy Problems ◽  
Fractional Powers ◽  
Uniqueness Of Solutions ◽  
Infinite Intervals ◽  
Abstract Cauchy Problems ◽  
Densely Defined

We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator B X have been investigated in the space which consists of continuous functions u on [ 0 , ∞ ) without assuming u ( 0 ) = 0 . This enables us to refine some previous results and obtain the required abstract results when the operator B X is not necessarily densely defined.

scholarly journals Complex powers of nondensely defined operators

2011 ◽  
Vol 90 (104) ◽  
pp. 47-64 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Mild Solutions ◽  
Acute Angle ◽  
Analytic Semigroups ◽  
Cauchy Problems ◽  
Growth Order ◽  
Fractional Powers ◽  
Complex Powers ◽  
Cosine Functions ◽  
Abstract Cauchy Problems ◽  
Densely Defined

The power (?A)b, b ? C is defined for a closed linear operator A whose resolvent is polynomially bounded on the region which is, in general, strictly contained in an acute angle. It is proved that all structural properties of complex powers of densely defined operators with polynomially bounded resolvent remain true in the newly arisen situation. The fractional powers are considered as generators of analytic semigroups of growth order r > 0 and applied in the study of corresponding incomplete abstract Cauchy problems. In the last section, the constructed powers are incorporated in the analysis of the existence and growth of mild solutions of operators generating fractionally integrated semigroups and cosine functions.

scholarly journals Direct and Inverse Fractional Abstract Cauchy Problems

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1016 ◽  
Author(s):  
Mohammed AL Horani ◽  
Angelo Favini ◽  
Hiroki Tanabe
Keyword(s):  
Cauchy Problem ◽  
Direct Problem ◽  
Abstract Cauchy Problem ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Cauchy Problems ◽  
Interpolation Theory ◽  
Basic Assumptions ◽  
Abstract Approach ◽  
Abstract Cauchy Problems

We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Moreover, we succeeded in handling related inverse problems, extending the treatment given by Alfredo Lorenzi. Some basic assumptions on the involved operators are also introduced allowing application of the real interpolation theory of Lions and Peetre. Our abstract approach improves previous results given by Favini–Yagi by using more general real interpolation spaces with indices θ , p, p ∈ ( 0 , ∞ ] instead of the indices θ , ∞. As a possible application of the abstract theorems, some examples of partial differential equations are given.

scholarly journals Theory of existence and uniqueness for the nonlinear Maxwell-Boltzmann equation I

1977 ◽  
Vol 16 (3) ◽  
pp. 379-414 ◽  
Author(s):  
Aleksander Glikson
Keyword(s):  
Boltzmann Equation ◽  
Existence And Uniqueness ◽  
Broad Class ◽  
Physical Space ◽  
Uniqueness Result ◽  
Cauchy Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Boltzmann Gas ◽  
Definition Of

A review of the development of the theory of existence and uniqueness of solutions to initial-value problems for mostly reduced versions of the nonlinear Maxwell-Boltzmann equation with a cut-off of intermolecular interaction, precedes the formulation and discussion of a somewhat generalized initial-value problem for the full nonlinear Maxwell-Boltzmann equation, with or without a cut-off. This is followed by a derivation of a new existence-uniqueness result for a particular Cauchy problem for the full nonlinear Maxwell-Boltzmann equation with a cut-off, under the assumption that the monatomic Boltzmann gas in the unbounded physical space X is acted upon by a member of a broad class of external conservative forces with sufficiently well-behaved potentials, defined on X and bounded from below. The result represents a significant improvement of an earlier theorem by this author which was until now the strongest obtained for Cauchy problems for the full Maxwell-Boltzmann equation. The improvement is basically due to the introduction of equivalent norms in a Banach space, the definition of which is connected with an exponential function of the total energy of a free-streaming molecule.

scholarly journals Local K-convoluted C-cosine functions and abstract cauchy problems

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2583-2598 ◽  
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Linear Operator ◽  
Abstract Cauchy Problem ◽  
Cosine Function ◽  
Equivalence Relations ◽  
Cauchy Problems ◽  
Bounded Linear ◽  
Unique Existence ◽  
Locally Integrable Function ◽  
Cosine Functions ◽  
Abstract Cauchy Problems

Let K : [0,T0)? F be a locally integrable function, and C : X ? X a bounded linear operator on a Banach space X over the field F(=R or C). In this paper, we will deduce some basic properties of a nondegenerate local K-convoluted C-cosine function on X and some generation theorems of local Kconvoluted C-cosine functions on X with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-cosine function on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem: U''(t)=Au(t)+f(t) for a.e. t ? (0, T0), u(0) = x, u'(0) = y when K is a kernel on [0, T0), C : X ? X an injection, and A : D(A) ? X ? X a closed linear operator in X such that CA ? AC. Here 0 < T0 ? ?, x,y ? X, and f ? L1,loc([0,T0),X).

scholarly journals On The Solvability of an Inverse Fractional Abstract Cauchy Problems with Conformable Derivative

2021 ◽  
Author(s):  
Lahcen Rabhi ◽  
Mohammed AL HORANI ◽  
R. Khalil
Keyword(s):  
Banach Space ◽  
Inverse Problem ◽  
Sobolev Space ◽  
Existence And Uniqueness ◽  
Parabolic Partial Differential Equations ◽  
Cauchy Problems ◽  
Conformable Derivative ◽  
Equivalent Statement ◽  
Special Cases ◽  
Abstract Cauchy Problems

In this paper, we discuss the solvability of fractional inverse problem for the conformable derivative in Banach space. We establish an equivalent statement of the existence and uniqueness of solution using fractional semigroup. Some special cases of the inverse problem are studied. An application is given to study an inverse problem in a suitable Sobolev space for fractional parabolic partial differential equations with unknown source functions.

scholarly journals Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

2021 ◽  
Vol 26 (1) ◽  
pp. 18
Author(s):  
Riccardo Fazio
Keyword(s):  
Existence And Uniqueness ◽  
Numerical Test ◽  
Transformation Method ◽  
Uniqueness Of Solutions ◽  
Boundary Problems ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Uniqueness Question ◽  
Infinite Intervals ◽  
Definition Of

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context, the numerical test is illustrated by two examples where we find meaningful numerical results.

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