Dynamics of approximate solutions to a class of evolution equations in Banach spaces

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

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Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

Advances in Difference Equations ◽

10.1186/s13662-021-03430-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Denghao Pang ◽

Wei Jiang ◽

Azmat Ullah Khan Niazi ◽

Jiale Sheng

Keyword(s):

Banach Spaces ◽

Mild Solution ◽

Measure Of Noncompactness ◽

Evolution Equations ◽

Well Posedness ◽

Optimal Controls ◽

Lagrange Problem ◽

Fractional Differential ◽

Fractional Evolution Systems ◽

Definition Of

AbstractIn this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.

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Fast Reaction Limits via $$\Gamma $$-Convergence of the Flux Rate Functional

Journal of Dynamics and Differential Equations ◽

10.1007/s10884-021-10024-2 ◽

2021 ◽

Author(s):

Mark A. Peletier ◽

D. R. Michiel Renger

Keyword(s):

Evolution Equations ◽

Detailed Balance ◽

Approximate Solutions ◽

Reaction Rates ◽

Convergence Result ◽

Rate Function ◽

Jump Processes ◽

Kolmogorov Equations ◽

Markov Jump ◽

Fast Reaction

AbstractWe study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as $$1/\epsilon $$ 1 / ϵ , and we prove the convergence in the fast-reaction limit $$\epsilon \rightarrow 0$$ ϵ → 0 . We establish a $$\Gamma $$ Γ -convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the $$\Gamma $$ Γ -convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

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Passivity and admissibility for evolution equations in Banach spaces

Nonlinear Analysis ◽

10.1016/0362-546x(82)90091-8 ◽

1982 ◽

Vol 6 (3) ◽

pp. 225-236 ◽

Cited By ~ 3

Author(s):

A.G. Kartsatos ◽

J. Toro

Keyword(s):

Banach Spaces ◽

Evolution Equations

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Iterative methods for generalized split feasibility problems in Banach spaces

Carpathian Journal of Mathematics ◽

10.37193/cjm.2017.01.02 ◽

2017 ◽

Vol 33 (1) ◽

pp. 09-26

Author(s):

QAMRUL HASAN ANSARI ◽

◽

AISHA REHAN ◽

◽

Keyword(s):

Weak Convergence ◽

Banach Spaces ◽

Iterative Methods ◽

Hilbert Spaces ◽

Equilibrium Problems ◽

Iterative Algorithms ◽

Approximate Solutions ◽

Split Feasibility Problems ◽

Convergence Results ◽

Split Feasibility

Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205–221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, SetValued Var. Anal., 23 (2015), 205–221].

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