Bloch-Type Periodic Functions

The following sections are included:

• Preface
• About the Authors
• Acknowledgments
• Contents

Let ℝ (ℂ) be the set of all real (complex) numbers and ℝ₊ be the collection of all nonnegative real numbers. Let ℕ denote the set of all positive integers and ℂ₊ = {λ ∈ ℂ : Reλ > 0}. Let (X, ‖ · ‖) be a Banach space and C(ℝ,X) (C(ℝ₊,X)) denotes the space of all continuous functions f : ℝ → X (f : ℝ₊ → X). The notation C₀(ℝ₊,X) is the space of all continuous functions g : ℝ₊ → X such that lim_t→∞ ‖g(t)‖ = 0. Let BC(ℝ,X) (BC(ℝ₊,X)) stand for the Banach space formed by all bounded continuous functions f : ℝ → X (f : ℝ₊ → X) with sup-norm $\parallel f{\parallel }_{\infty }={\sup }_{t\in ℝ}\parallel f\left(t\right)\parallel \left(\parallel f{\parallel }_{\infty }={\sup }_{t\in {ℝ}_{+}}\parallel f\left(t\right)\parallel \right)$. The space BC(ℝ × X, X) (BC(ℝ₊ × X,X)) is the set consisting of all functions f : ℝ × X → X (f : ℝ₊ × X → X) such that f(·, x) ∈ BC(ℝ,X) (f(·, x) ∈ BC(ℝ₊,X)) uniformly for each x in any compact subset of X. We denote by $\mathcal{B}$(X, Y ) the space of all bounded linear operators from X to Y and will be abbreviated by $\mathcal{B}$(X) if X = Y…

It is well known that solutions to some equations from propagation of heat or waves in solid state physics, such as wave functions satisfying the Schödinger equation with the potential field having crystal lattice periodicity, often behave Bloch type periodicity (or (ω, k)-Bloch periodicity), see for instance (Bloch, 1929; Kittel, 2005; Ren, 2006). It can be found that Bloch type periodicity has classical ω-periodicity and ω-antiperiodicity as special cases. In Hasler and N’Guérékata (2014), Hasler and N’Guérékata established theories on the space of Bloch type periodic functions such as completeness, convolution and composite properties from the viewpoint of mathematics, and studied the existence and uniqueness of Bloch type periodic solutions to a semilinear integrodifferential equation in Banach spaces. The notion of asymptotically Bloch type periodic function was also presented in Hasler and N’Guérékata (2014), which can be seen as the Bloch type periodic function with a perturbation vanishing at infinity and a natural generalization of asymptotically ω-periodic function (see Gao et al., 2006) and asymptotically ω-antiperiodic function (see Chen and Wang, 2015). Kostić and Velinov (2017) analyzed the existence and uniqueness of asymptotically Bloch type periodic solutions to an abstract fractional nonlinear differential inclusions with piecewise constant argument in Banach spaces. As is known, the ω-periodicity or ω-antiperiodicity have some other significant generalizations such as (weighted) pseudo ω-periodicity and (weighted) pseudo ω-antiperiodicity, which has been considerably applied to investigate the existence (and uniqueness) of (weighted) pseudo ω-periodic/ω-antiperiodic solutions to various evolution equations in Banach spaces, see for instance Al-Islam et al. (2012), Alvarez et al. (2015), Cao et al. (2012), Liu (2010) and Xia (2014) and references therein. On the other side, Henríquez, Pierri and Táboas studied the S-asymptotically ω-periodic function (see Henríquez et al. 2008a, 2008b; Pierri and Nicola, 2009), which is different in ergodicity from the asymptotically ω-periodic function. For more results on S-asymptotic ω-periodicity and its applications to some different differential equations, please refer to Andrade and Cuevas (2010), Andrade et al. (2021), Bedi et al. (2021), Brindle and N’Guérékata (2019a), Brindle and N’Guérékata (2019b), Brindle and N’Guérékata (2019c), Cuevas and Souza (2010), Cuevas and Lizama (2010), Henríquez et al. (2013), Hernández and Pierri (2018), Oueama-Guengai and N’Guérékata (2018), Oueama-Guengai (2018), Pierri (2012), and Shu et al. (2015) and references cited therein. The concept of pseudo S-asymptotic ω-periodicity was presented in Pierri and Rolnik (2013), which can be seen as an extension of the aforementioned S-asymptotic ω-periodicity. Moreover, Stepanov-like S-asymptotically ω-periodic function was presented by Henríquez (2013), which was further developed as Stepanov-like weighted pseudo S-asymptotically periodic function by Xia (2015b). The more development and applications of (pseudo) S-asymptotically ω-periodic function can be referred to Cuevas et al. (2014), Dimbour (2020), He et al. (2020), Henríquez et al. (2016), Xia (2015a) and Yang and Wang (2019) and references therein. We can also refer to Alvarez et al. (2018), Chaouchi et al. (2020), Mophou and N’Guérékata (2020), and Khalladi et al. (2021) for a more generalized notion of an ω-periodic function. However, there are few results on pseudo S-asymptotic ω-antiperiodicity corresponding to the ω-antiperiodicity yet. It is also seen from Hasler and N’Guérékata (2014) that a Bloch-type periodic function may not be Bloch-type periodic with a perturbation vanishing at infinity, but it can be asymptotically Bloch-type periodic. Thus, it is meaningful to extend the Bloch-type periodic function such that it can remain certain quasi Bloch-type periodic under differently small perturbations. In this chapter, we introduce some generalized Bloch-type periodic functions and present some fundamental properties on spaces of such functions, most of which can be found in details in Chang and Wei (2021a, 2021b, 2022), and Wei and Chang (2022).

In this chapter, we mainly consider the following semilinear integrodifferential equation

This chapter is mainly concerned with the existence of generalized Blochtype periodic mild solutions to the following semilinear multi-term fractional evolution equation

In this chapter, we consider the existence and uniqueness of generalized Bloch-type periodic mild solutions to the following semilinear fractional differential equation of Sobolev type

In this chapter, we mainly consider the existence of generalized Bloch-type periodic solutions to the following semilinear fractional integrodifferential equation:

In this chapter, we mainly consider the following damped evolution equation:

This chapter is mainly concerned with the existence of pseudo S-asymptotically Bloch-type periodic mild solutions to the following partial integrodifferential equation:

This chapter is mainly concerned with the existence of generalized Bloch-type periodic mild solutions to the following semilinear integral equation:

In this chapter, we list basic results on norm continuity (continuity in B(X)) and characterizations of compactness for some fractional resolvent operator families which appear in Chapters 4 and 5.

The following sections are included:

• Bibliography
• Index