Weak Solutions for Time-Fractional Evolution Equations in Hilbert Spaces

Loreti, Paola; Sforza, Daniela

doi:10.3390/fractalfract5040138

Cited by 6 publications

(9 citation statements)

References 12 publications

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“…(3) A definition of mild solution to boundary value problems (BVPs) (2) and ( 3) is introduced and the existence and uniqueness of mild solution are investigated by using some fixedpoint theorems. (4) Theorems 2 and 3 of this article achieve some easily verifiable conditions compared with the previous works in [6,7,9,11,22,26].…”

Section: Novelties Of This Article: (1)supporting

confidence: 53%

“…where s ∈ (0, 1), Ω is an open bounded domain of R n and n > 2s. Moreover, the weak formulation of problem (6) is depicted by…”

Section: Eigenvalue Problemmentioning

confidence: 99%

“…By means of the method of eigenvalues' expansions, the differential equation is converted to the integral form. Taking Equations (6) and (7) into consideration, it demonstrates that the solution of Equation (9) is the weak solution of Equation (2).…”

Section: Green Functionmentioning

confidence: 99%

“…Byszewski [2], Byszewski and Akca [3] then went on to study the existence and uniqueness of mild solutions for different kinds of evolution equations in Banach spaces. Moreover, various existence results for evolution equations and evolution inclusions applying the fixed-point theory and semigroup theory have been achieved [4][5][6][7][8][9][10][11]. For instance, Li et al [9] learned the following fractional evolution equation in ordered Banach spaces c D q 0 u(t) + Au(t) = f (t, u(t)), t > 0, u(0) = u 0 ,…”

Section: Introductionmentioning

confidence: 99%

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Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

Zhou

Zhang

2022

Mathematics

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This paper discusses a kind of coupled nonlocal Laplacian evolution equation with Caputo time-fractional derivatives and proportional delays. Green function and mild solution are firstly established by employing the approach of eigenvalues’ expansions and Fourier analysis technique. By the properties of eigenvalues and Mittag–Leffler functions, several vital estimations of Green functions are presented. In view of these estimations and some appropriate assumptions, the existence and uniqueness of the mild solution are studied by utilizing the Leray–Schauder fixed-point theorem and the Banach fixed-point theorem. Finally, an example is provided to illustrate the effectiveness of our main results.

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Section: Novelties Of This Article: (1)supporting

confidence: 53%

“…where s ∈ (0, 1), Ω is an open bounded domain of R n and n > 2s. Moreover, the weak formulation of problem (6) is depicted by…”

Section: Eigenvalue Problemmentioning

confidence: 99%

Section: Green Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

Zhou

Zhang

2022

Mathematics

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“…For example, in paper [22], the authors study a class of a fractional differential equations in Hilbert spaces the following type…”

Section: Introductionmentioning

confidence: 99%

On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

et al. 2022

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In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation.

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