Generalized Almost Periodicity in Lebesgue Spaces with Variable Exponents

Abstract

In this paper, we introduce and analyze Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We investigate the invariance of these types of generalized almost-periodicity in Lebesgue spaces with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract semilinear integro-differential inclusions in Banach spaces.

1. Introduction and Preliminaries

The class of almost-periodic functions was introduced by a Danish mathematician Harald Bohr [1,2,3], (1924–1926), the younger brother of the Nobel Prize-winning physicist Niels Bohr, and later generalized by many others (cf. the research monographs [4,5,6], for the basic theory of almost-periodic functions; for some applications given, see the research monographs [7,8,9,10]). The theory of almost-periodic functions is still a very active field of investigations of numerous authors, full of open problems, conjectures, hypotheses and possibilities for further expansions.

The notion of (asymptotical) Stepanov almost-periodicity and the notion of (asymptotical) Stepanov almost automorphicity in Lebesgue spaces with variable exponents have been introduced in the papers [11,12] by Diagana and Zitane. Recently, we have reconsidered this notion in our joint papers with Diagana [13,14]. Furthermore, in [15,16,17], we have recently analyzed the classes of (Stepanov) uniformly recurrent functions and several various classes of (asymptotically) Weyl almost-periodic functions in Lebesgue spaces with variable exponents. The main aim of this paper is to continue the analysis raised in the above-mentioned papers by introducing and investigating the classes of Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We also consider the corresponding classes of functions depending on two parameters and clarify a great number of composition principles in this context.

The organization of paper can be briefly described as follows. In Section 1, we recall the basic definitions and results from the theory of Lebesgue spaces with variable exponents. Section 2 recollects the basic definitions and results from the theory of vector-valued almost-periodic functions which we need later. Section 3 investigates the Stepanov uniformly recurrent functions in Lebesgue spaces with variable exponents. The proofs of structural results in this section can be given by employing the slight modifications of the corresponding results from [13] (see also [15]) and therefore omitted. Our main contributions are given in Section 4 and Section 5, where we introduce and analyze several various classes of Doss almost-periodic (uniformly recurrent) functions in Lebesgue spaces with variable exponents and the invariance of generalized Doss almost-periodicity under the actions of convolution products. The final section of paper is reserved for applications of our abstract theoretical results to the abstract semilinear integro-differential inclusions in Banach spaces. In addition to the above, we provide several illustrative examples, remarks and comments about the material presented. We feel it is our duty to say that the papers of Kostić [15,16,17] are still not published, unfortunately. We will use only a few definitions from these papers and prove here any result which has any connection with some of the results established in these papers, for which we cannot yet tell that are fully true.

We use the standard notation throughout the paper. Unless specified otherwise, we assume that $(X,\parallel ·\parallel )$ is a complex Banach space. If $(Y,\parallel ·{\parallel }_Y)$ is another Banach space over the field of complex numbers, then the shorthand $L(X,Y)$ stands for the Banach algebra of all bounded linear operators from X into Y with $L(X,X)$ being denoted $L\left(X\right)$. By $L^p\_loc(I:X)$, $C(I:X),$ $C\_b$(and $C\_0(I:X)$ we denote the vector spaces consisting of all p-locally integrable functions $f:I\to X$, all continuous functions $f:I\to X,$ all bounded continuous functions $f:I\to X$ and all continuous functions $f:I\to X$ satisfying that ${lim}_{\left|t\right|\to +\infty }\parallel f\left(t\right)\parallel =0,$ respectively ($1\le p<\infty$). If $f:\mathbb{R}\to X$, then we define $\check{f}:\mathbb{R}\to X$ by $\check{f}\left(x\right):=f(-x),$ $x\in \mathbb{R}.$ Suppose, finally that $f:[a,b]\to \mathbb{R}$ is a non-negative Lebesgue-integrable function, where $a,b\in \mathbb{R},$ $a<b,$ and $\varphi :[0,\infty )\to \mathbb{R}$ is a convex function. Let us recall that the Jensen integral inequality states that

$${\displaystyle \varphi \left(\frac{1}{b-a}{\int }_a^{b}f\left(x\right)dx\right)\le \frac{1}{b-a}{\int }_a^b\varphi \left(f\left(x\right)\right)dx.}$$

Lebesgue Spaces with Variable Exponents $L^{p\left(x\right)}$

Let $\varnothing \ne \mathsf{\Omega }\subseteq \mathbb{R}$ be a nonempty subset and let $M(\mathsf{\Omega }:X)$ stand for the collection of all measurable functions $f:\mathsf{\Omega }\to X;$ $M\left(\mathsf{\Omega }\right):=M(\mathsf{\Omega }:\mathbb{R}).$ By $\mathcal{P}\left(\mathsf{\Omega }\right)$ we denote the vector space of all Lebesgue measurable functions $p:\mathsf{\Omega }\to [1,\infty ].$ For any $p\in \mathcal{P}\left(\mathsf{\Omega }\right)$ and $f\in M(\mathsf{\Omega }:X),$ set

$${\phi }_{p\left(x\right)}\left(t\right):=\left\{\begin{array}{c}{t}^{p\left(x\right)},t\ge 0,1\le p\left(x\right)<\infty ,\hfill \\ 0,0\le t\le 1,p\left(x\right)=\infty ,\hfill \\ \infty ,t>1,p\left(x\right)=\infty \hfill \end{array}\right.$$

and

$$\rho \left(f\right):={\int }_{\mathsf{\Omega }}{\phi }_{p\left(x\right)}(\parallel f\left(x\right)\parallel )dx.$$

We define the Lebesgue space $L^{p\left(x\right)}(\mathsf{\Omega }:X)$ with variable exponent as follows,

$$L^{p\left(x\right)}(\mathsf{\Omega }:X):=\left\{f\in M(\mathsf{\Omega }:X):\underset{\lambda \to 0+}{lim}\rho \left(\lambda f\right)=0\right\}$$

equivalently

$$\begin{array}{c}\hfill L^{p\left(x\right)}(\mathsf{\Omega }:X)=\left\{f\in M(\mathsf{\Omega }:X):\mathrm{there} \mathrm{exists} \lambda >0 \mathrm{such} \mathrm{that} \rho \left(\lambda f\right)<\infty \right\};\end{array}$$

see e.g., ([18], p. 73). For every $u\in L^{p\left(x\right)}(\mathsf{\Omega }:X),$ we introduce the Luxemburg norm of $u(·)$ by

$${\parallel u\parallel }_{p\left(x\right)}:={\parallel u\parallel }_{L^{p\left(x\right)}(\mathsf{\Omega }:X)}:=inf\left\{\lambda >0:\rho (f/\lambda )\le 1\right\}.$$

Equipped with the above norm, the space $L^{p\left(x\right)}(\mathsf{\Omega }:X)$ becomes a Banach space (see e.g., ([18], Theorem 3.2.7) for the scalar-valued case), coinciding with the usual Lebesgue space $L^p(\mathsf{\Omega }:X)$ in the case that $p\left(x\right)=p\ge 1$ is a constant function. For any $p\in M\left(\mathsf{\Omega }\right),$ we set

$$p^{-}:={\mathrm{essinf}}_{x\in \mathsf{\Omega }}p\left(x\right)\mathrm{and}{p}^{+}:={\mathrm{esssup}}_{x\in \mathsf{\Omega }}p\left(x\right).$$

Define

$$C_{+}\left(\mathsf{\Omega }\right):=\left\{p\in M\left(\mathsf{\Omega }\right):1<p^{-}\le p\left(x\right)\le p^{+}<\infty \mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega }\right\}$$

and

$$D_{+}\left(\mathsf{\Omega }\right):=\left\{p\in M\left(\mathsf{\Omega }\right):1\le p^{-}\le p\left(x\right)\le p^{+}<\infty \mathrm{for}\mathrm{a}.\mathrm{e}.x\in \mathsf{\Omega }\right\}.$$

For $p\in D_{+}\left([0,1]\right),$ the space $L^{p\left(x\right)}(\mathsf{\Omega }:X)$ behaves nicely, with almost all fundamental properties of the Lebesgue space with constant exponent $L^p(\mathsf{\Omega }:X)$ being retained; in this case, we know that

$$L^{p\left(x\right)}(\mathsf{\Omega }:X)=\left\{f\in M(\mathsf{\Omega }:X):\mathrm{for} \mathrm{all} \lambda >0\mathrm{we} \mathrm{have} \rho \left(\lambda f\right)<\infty \right\}.$$

We will use the following lemma (see e.g., ([18], Lemma 3.2.20, (3.2.22); Corollary 3.3.4; p. 77) for the scalar-valued case):

Lemma 1.

(i) 

(The Hölder inequality) Let $p,q,r\in \mathcal{P}\left(\mathsf{\Omega }\right)$ such that

$$\frac{1}{q\left(x\right)}=\frac{1}{p\left(x\right)}+\frac{1}{r\left(x\right)},x\in \mathsf{\Omega }.$$

Then, for every $u\in L^{p\left(x\right)}(\mathsf{\Omega }:X)$ and $v\in L^{r\left(x\right)}\left(\mathsf{\Omega }\right),$ we have $uv\in L^{q\left(x\right)}(\mathsf{\Omega }:X)$ and

$$\begin{array}{c}\hfill {\parallel uv\parallel }_{q\left(x\right)}\le {2\parallel u\parallel }_{p\left(x\right)}{\parallel v\parallel }_{r\left(x\right)}.\end{array}$$

(ii) 

Let Ω be of a finite Lebesgue’s measure $m\left(\mathsf{\Omega }\right)<\infty$ and let $p,q\in \mathcal{P}\left(\mathsf{\Omega }\right)$ such $q\le p$ a.e. on $\mathsf{\Omega }.$ Then $L^{p\left(x\right)}(\mathsf{\Omega }:X)$ is continuously embedded in $L^{q\left(x\right)}(\mathsf{\Omega }:X),$ with the constant of embedding less than or equal to $2(1+m(\mathsf{\Omega }\left)\right).$

(iii) 

Let $f\in L^{p\left(x\right)}(\mathsf{\Omega }:X),$ $g\in M(\mathsf{\Omega }:X)$ and $0\le \parallel g\parallel \le \parallel f\parallel$ a.e. on $\mathsf{\Omega }.$ Then $g\in L^{p\left(x\right)}(\mathsf{\Omega }:X)$ and ${\parallel g\parallel }_{p\left(x\right)}\le {\parallel f\parallel }_{p\left(x\right)}.$

For additional details upon Lebesgue spaces with variable exponents $L^{p\left(x\right)},$ we refer the reader to [11,12,18].

2. Generalized Almost-Periodic Type Functions in Banach Spaces

Let I be either $\mathbb{R}$ or $[0,\infty ),$ and let $f:I\to X$ be a given continuous function. Given $\epsilon >0,$ we call $\tau >0$ an $\epsilon$-period for $f(·)$ if and only if $\parallel f(t+\tau )-f\left(t\right)\parallel \le \epsilon ,$ $t\in I.$ The set consisting of all $\epsilon$-periods for $f(·)$ is denoted by $\vartheta (f,\epsilon ).$ The function f is said to be almost periodic if and only if for each $\epsilon >0$ the set $\vartheta (f,\epsilon )$ is relatively dense in $[0,\infty ),$ which means that there exists $\ell >0$ such that any subinterval of the interval $[0,\infty )$ of length ℓ intersects $\vartheta (f,\epsilon )$. The almost-periodic functions form a vector space which will be denoted by $AP(I:X).$ The function $f:I\to X$ is said to be asymptotically almost periodic if and only if there exists an almost-periodic function $h:I\to X$ and a function $\varphi \in C_0(I:X)$ such that $f\left(t\right)=h\left(t\right)+\varphi \left(t\right)$ for all $t\in I.$ Following Haraux and Souplet [19], we say that the function $f(·)$ is uniformly recurrent if and only if there exists a strictly increasing sequence $\left({\alpha }_n\right)$ of positive real numbers such that ${lim}_{n\to +\infty }{\alpha }_n=+\infty$ and

$$\underset{n\to \infty }{lim}\underset{t\in \mathbb{R}}{sup}∥f(t+{\alpha }_n)-f\left(t\right)∥=0.$$

(1)

It is well known that any almost-periodic function is uniformly recurrent, while the converse statement is not true in general. The collection of uniformly recurrent functions will be denoted by $UR(I:X).$ A function $f\in C(I:X)$ is said to be asymptotically uniformly recurrent if and only if there exists a uniformly recurrent function $h:I\to E$ and a function $\varphi \in C_0(I:E)$ such that $f\left(t\right)=h\left(t\right)+\varphi \left(t\right)$ for all $t\in I.$ The collection of asymptotically uniformly recurrent functions will be denoted by $AUR(I:X).$

Further on, suppose that $1\le p<\infty .$ A function $f\in L_{loc}^p(I:X)$ is said to be Stepanov p-bounded if and only if

$${\parallel f\parallel }_{S^p}:=\underset{t\in I}{sup}{\left({\int }_t^{t+1}{\parallel f\left(s\right)\parallel }^{p}ds\right)}^{1/p}=\underset{t\in I}{sup}{\left({\int }_0^1{\parallel f(s+t)\parallel }^{p}ds\right)}^{1/p}<\infty .$$

Endowed with the above norm, the space $L_S^p(I:X)$ consisting of all $S^p$-bounded functions, becomes a Banach space. A function $f\in L_S^p(I:X)$ is said to be Stepanov p-almost periodic if and only if the function $\widehat{f}:I\to L^p([0,1]:X),$ defined by

$$\widehat{f}\left(t\right)\left(s\right):=f(t+s),t\in I,s\in [0,1],$$

is almost periodic; a function $f\in L_{loc}^p(I:X)$ is said to be Stepanov p-uniformly recurrent if and only if the function $\widehat{f}:I\to L^p([0,1]:X)$ is uniformly recurrent. We say that a function $f\in L_{loc}^p(I:X)$ is asymptotically Stepanov p-uniformly recurrent if and only if there exist a Stepanov p-uniformly recurrent function $h(·)$ and a function $q\in L_S^p(I:X)$ such that $f\left(t\right)=h\left(t\right)+q\left(t\right),$ $t\in I$ and $\widehat{q}\in C_0(I:L^p([0,1]:X)).$

The following notion of Stepanov $p\left(x\right)$-boundedness differs from the one introduced by Diagana and Zitane in ([11], Definition 3.10) and ([12], Definition 4.5), where the authors have used the condition $p\in C_{+}\left(\mathbb{R}\right):$

Definition 1

(see [13]). Let $p\in \mathcal{P}\left(\right[0,1\left]\right).$ A function $f\in M(I:X)$ is said to be Stepanov $p\left(x\right)$-bounded (or $S^{p\left(x\right)}$-bounded), if and only if $f(·+t)\in L^{p\left(x\right)}([0,1]:X)$ for all $t\in I,$ and ${sup}_{t\in I}{\parallel f(·+t)\parallel }_{p\left(x\right)}<\infty$, that is,

$${\parallel f\parallel }_{S^{p\left(x\right)}}:=\underset{t\in I}{sup}inf\left\{\lambda >0:{\int }_0^1{\phi }_{p\left(x\right)}\left(\frac{\parallel f(x+t)\parallel }{\lambda }\right)dx\le 1\right\}<\infty .$$

The collection of such functions will be denoted by $L_S^{p\left(x\right)}(I:X)$.

Equipped with the norm ${\parallel ·\parallel }_{S^{p\left(x\right)}}$, the space $L_S^{p\left(x\right)}(I:X)$ consisting of all $S^{p\left(x\right)}$-bounded functions is a Banach space, which is continuously embedded in $L_S^1(I:X),$ for any $p\in \mathcal{P}\left(\right[0,1\left]\right).$

In [13], we have recently introduced the concept of (asymptotical) $S^{p\left(x\right)}$-almost-periodicity as follows (see also ([13], Proposition 4.12) for case $I=[0,\infty )$):

Definition 2.

(i) 

Let $p\in \mathcal{P}\left(\right[0,1\left]\right).$ A function $f\in L_S^{p\left(x\right)}(I:X)$ is said to be Stepanov $p\left(x\right)$-almost periodic if and only if the function $\widehat{f}:I\to L^{p\left(x\right)}([0,1]:X)$ is almost periodic. The collection of such functions will be denoted by $AP{S}^{p\left(x\right)}(I:X)$.

(ii) 

Let $p\in \mathcal{P}\left(\right[0,1\left]\right).$ A function $f\in L_S^{p\left(x\right)}(I:X)$ is said to be asymptotically Stepanov $p\left(x\right)$-almost periodic if and only if there are two Stepanov $p\left(x\right)$-bounded functions $g:I\to X$ and $q:I\to X$ satisfying the following conditions:

(a) 

$g(·)$ is Stepanov $p\left(x\right)$-almost periodic,

(b) 

$\widehat{q}(·)$ belongs to the class $C_0(I:L^{p\left(x\right)}([0,1]:X)),$

(c) 

$f\left(t\right)=g\left(t\right)+q\left(t\right)$ for a.e. $t\in I.$

The space of Stepanov $p\left(x\right)$-bounded functions is translation invariant, the space of (asymptotically) Stepanov $p\left(x\right)$-almost-periodic functions is translation invariant and contained in the space of (asymptotically) Stepanov almost-periodic functions. Let us note that in [15], we have constructed several concrete examples of bounded, uniformly recurrent, uniformly continuous functions which are not almost periodic. Taking into account the Bochner theorem (see e.g., [9]), it follows that these functions cannot be Stepanov $p\left(x\right)$-almost periodic. However, after introducing the corresponding notion, it will become clear that these functions are Stepanov p-uniformly recurrent for any finite exponent $p\ge 1.$

Further on, by $C_0(I\times Y:X)$ we denote the space of all continuous functions $H:I\times Y\to X$ such that ${lim}_{\left|t\right|\to +\infty }H(t,y)=0$ uniformly for y in any compact subset of $Y.$ Let us recall that a continuous function $F:I\times Y\to X$ is said to be uniformly continuous on bounded sets, uniformly for $t\in I$ if and only if for every $ϵ>0$ and every bounded subset B of Y there exists a number ${\delta }_{ϵ,B}>0$ such that $\parallel F(t,x)-F(t,y)\parallel \le ϵ$ for all $t\in I$ and all $x,y\in B$ satisfying that $\parallel x-y\parallel \le {\delta }_{ϵ,B}.$ If $F:I\times Y\to X,$ then we define the function $\widehat{F}:I\times Y\to L^p([0,1]:X)$ by $\widehat{F}(t,y):=F(t+·,y),$ $t\ge 0,$ $y\in Y.$

The following definitions have recently been introduced in [16]:

Definition 3.

(i) 

A continuous function $F:I\times Y\to X$ is called uniformly recurrent if and only if for every $ϵ>0$ and every compact $K\subseteq Y$ there exists a strictly increasing sequence $\left({\alpha }_n\right)$ of positive reals tending to plus infinity such that

$$\underset{n\to +\infty }{lim}\underset{t\in I}{sup}∥F(t+{\alpha }_n,y)-F(t,y)∥=0,y\in K.$$

(2)

The collection of all two-parameter uniformly recurrent functions will be denoted by $UR(I\times Y:X).$

(ii) 

A continuous function $F:I\times Y\to X$ is said to be asymptotically uniformly recurrent if and only if $f(·)$ admits a decomposition $F=G+Q,$ where $G\in UR(I\times Y:X)$ and $Q\in C_0(I\times Y:X).$

Definition 4.

Let $1\le p<\infty .$

(i) 

A function $F:I\times Y\to X$ is called Stepanov p-uniformly recurrent if and only if the function $\widehat{F}:I\times Y\to L^p([0,1]:X)$ is uniformly recurrent.

(ii) 

We say that $F:I\times Y\to X$ is asymptotically Stepanov p-uniformly recurrent if and only if there exist two functions $G:I\times Y\to X$ and $Q:I\times Y\to X$ satisfying that for each $y\in Y$ the functions $G(·,y)$ and $Q(·,y)$ are locally p-integrable, as well as that the following holds:

(a) 

$\widehat{G}:I\times Y\to L^p([0,1]:X)$ is uniformly recurrent,

(b) 

$\widehat{Q}\in C_0(I\times Y:L^p([0,1]:X)),$

(c) 

$F(t,y)=G(t,y)+Q(t,y)$ for all $t\in I$ and $y\in Y.$

3. Stepanov Uniform Recurrence in Lebesgue Spaces with Variable Exponents

First of all, we will introduce the concept of (asymptotical) $S^{p\left(x\right)}$-uniform recurrence (case $p\left(x\right)\equiv p\in [1,\infty )$ has been considered in [15]):

Definition 5.

Let $p\in \mathcal{P}\left(\right[0,1\left]\right),$ and let $f:I\to X$ be such that $f\in L^{p\left(x\right)}(K:X)$ for any compact set $K\subseteq I.$

(i) 

We say that $f(·)$ is Stepanov $p\left(x\right)$-uniformly recurrent if and only if the function $\widehat{f}:I\to L^{p\left(x\right)}([0,1]:X)$ is uniformly recurrent. The collection of such functions will be denoted by $UR{S}^{p\left(x\right)}(I:X)$ ($UR{S}^p(I:X),$ if $p\left(x\right)\equiv p\in [1,\infty )$).

(ii) 

We say that $f(·)$ is asymptotically Stepanov $p\left(x\right)$-uniformly recurrent if and only if there exist a Stepanov $p\left(x\right)$-uniformly recurrent function $h(·)$ and a function $q\in L_S^{p\left(x\right)}(I:X)$ such that $f\left(t\right)=h\left(t\right)+q\left(t\right),$ $t\in I$ and $\widehat{q}\in C_0(I:L^{p\left(x\right)}([0,1]:X)).$ The collection of such functions will be denoted by $AURP{S}^{p\left(x\right)}(I:X)$ ($AUR{S}^p(I:X),$ if $p\left(x\right)\equiv p\in [1,\infty )$).

The spaces $UR{S}^{p\left(x\right)}(I:X)$ and $AUR{S}^{p\left(x\right)}(I:X)$ are translation invariant, as it can be easily approved. Furthermore, we have the following proposition which can be deduced by using the same argumentation as in the proofs of corresponding structural results concerning Stepanov almost-periodicity with variable exponent (see [13]):

Proposition 1.

(i) 

Suppose $p\in \mathcal{P}\left(\right[0,1\left]\right)$. Then $UR{S}^{p\left(x\right)}(I:X)\subseteq UR{S}^1(I:X),$ $AUR{S}^{p\left(x\right)}(I:X)\subseteq AUR{S}^1(I:X),$ $UR(I:X)\subseteq UR{S}^{p\left(x\right)}(I:X)\subseteq UR{S}^1(I:X)$ and $AUR(I:X)\subseteq AUR{S}^{p\left(x\right)}(I:X)\subseteq AUR{S}^1(I:X).$

(ii) 

Suppose $p\in D_{+}\left([0,1]\right)$ and $1\le p^{-}\le p\left(x\right)\le p^{+}<\infty$ for a.e. $x\in [0,1]$. Then we have $UR{S}^{p^{+}}(I:X)\subseteq UR{S}^{p\left(x\right)}(I:X)\subseteq UR{S}^{p^{-}}(I:X)$ and $AUR{S}^{p^{+}}(I:X)\subseteq AUR{S}^{p\left(x\right)}(I:X)\subseteq AUR{S}^{p^{-}}(I:X).$

(iii) 

Assume that $p,q\in \mathcal{P}\left(\right[0,1\left]\right)$ and $p\le q$ a.e. on $[0,1].$ Then we have: $UR{S}^{q\left(x\right)}(I:X)\subseteq UR{S}^{p\left(x\right)}(I:X)$ and $AUR{S}^{q\left(x\right)}(I:X)\subseteq AUR{S}^{p\left(x\right)}(I:X).$

(iv) 

If $p\in D_{+}\left([0,1]\right),$ then

$$\begin{array}{c}\hfill L^{\infty }(I:X)\cap UR{S}^{p\left(x\right)}(I:X)=L^{\infty }(I:X)\cap UR{S}^1(I:X)\end{array}$$

and

$$\begin{array}{c}\hfill L^{\infty }(I:X)\cap AUR{S}^{p\left(x\right)}(I:X)=L^{\infty }(I:X)\cap AUR{S}^1(I:X).\end{array}$$

We continue by providing two illustrative examples.

Example 1.

Let us recall that H. Bohr and E. Folner have constructed, for any given number $p>1$, a Stepanov almost-periodic function defined on the whole real axis that is Stepanov p-bounded and not Stepanov p-almost periodic (see ([20], Example, pp. 70–73)). In [9], we have shown that the function $f(·)$ cannot be Stepanov p-almost automorphic and we want to observe here that the same argumentation yields that the function $f(·)$ cannot be Stepanov p-uniformly recurrent. Strictly speaking, let us consider case $h_1=2$ in the afore-mentioned example. If we suppose the contrary, then the mapping $\widehat{f}:\mathbb{R}\to L^p([0,1]:X)$ is uniformly recurrent, which in particular implies that for each number $ϵ>0$ there exists an arbitrarily large positive real number $\tau >0$ such that

$${\int }_{-3/2}^{3/2}{\left|f(s+\tau )-f\left(s\right)\right|}^{p}ds<2{ϵ}^p,$$

which is in contradiction with the estimate ${\int }_{-3/2}^{3/2}{\left|f(s+\tau )-f\left(s\right)\right|}^{p}ds\ge 2^{-p}$ (see ([20], p. 73, l.-9–l.-4)).

Example 2.

Define sign$\left(0\right):=0,$ $f\left(x\right):=sinx+sin\sqrt{2}x,$ $x\in \mathbb{R}$ and $p\left(x\right):=1-lnx,$ $x\in [0,1].$ We know that the function sign$f(·)$ is neither Stepanov $p\left(x\right)$-almost periodic nor Stepanov $p\left(x\right)$-almost automorphic [13,14]. Moreover, we have proved that for every real numbers $\lambda \in (0,2/e),$ $l>0$, every interval $I\subseteq \mathbb{R}\setminus \left\{0\right\}$ of length l and every number $\tau \in I,$ there exists a number $t\in \mathbb{R}$ such that

$$\begin{array}{cc}\hfill {\int }_0^1{\left(\frac{1}{\lambda }\right)}^{1-lnx}& |sign\left[sin(x+t+\tau )+sin\sqrt{2}(x+t+\tau )\right]\hfill \\ & -sign\left[sin(x+t)+sin\sqrt{2}(x+t)\right]{|}^{1-lnx}dx=\infty .\hfill \end{array}$$

This implies that the function sign$f(·)$ cannot be Stepanov $p\left(x\right)$-uniformly recurrent, as well.

Now we will state two results about the invariance of uniform recurrence under the actions of infinite convolution products. The first result slightly extends ([9], Proposition 2.6.11); the proof can be given by using the same arguments as in the proof of above-mentioned proposition, with the appealing to the Hölder inequality in Lemma 1(i):

Proposition 2.

Suppose that $q\in \mathcal{P}\left(\right[0,1\left]\right),$ $1/p\left(x\right)+1/q\left(x\right)=1$ and ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(X,Y)$ is a strongly continuous operator family satisfying that $M:={\sum }_{k=0}^{\infty }{\parallel R(·+k)\parallel }_{L^{q\left(x\right)}[0,1]}<\infty .$ If $\check{f}:\mathbb{R}\to X$ is $S^{p\left(x\right)}$-bounded, $S^{p\left(x\right)}$-uniformly recurrent and the function $\widehat{\check{f}}:\mathbb{R}\to L^{p\left(x\right)}([0,1]:X)$ is uniformly continuous, then the function $F:\mathbb{R}\to Y,$ given by

$$F\left(t\right):={\int }_{-\infty }^{t}R(t-s)f\left(s\right)ds,t\in \mathbb{R},$$

(3)

is well defined and uniformly recurrent.

Using a similar argumentation and the proof of ([13], Proposition 3.2), we can clarify the following result in which we do not require that the function $\check{f}:\mathbb{R}\to X$ is $S^{p\left(x\right)}$-bounded:

Proposition 3.

Suppose that $q\in \mathcal{P}\left(\right[0,1\left]\right),$ $1/p\left(x\right)+1/q\left(x\right)=1$ and ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(X,Y)$ is a strongly continuous operator family satisfying that $M:={\sum }_{k=0}^{\infty }{\parallel R(·+k)\parallel }_{L^{q\left(x\right)}[0,1]}<\infty .$ If $\check{f}:\mathbb{R}\to X$ is $S^{p\left(x\right)}$-uniformly recurrent, the function $\widehat{\check{f}}:\mathbb{R}\to L^{p\left(x\right)}([0,1]:X)$ is uniformly continuous,

$${∥\check{f}(·-t)∥}_{L^{p\left(x\right)}[0,1]}\le P\left(t\right),t\in \mathbb{R}$$

and ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(X,Y)$ is a strongly continuous operator family satisfying that for each $t\in \mathbb{R}$ we have

$$\sum _{k=0}^{\infty }{\parallel R(·+k)\parallel }_{L^{q\left(x\right)}[0,1]}P(t-k)<\infty ,$$

then the function $F:\mathbb{R}\to Y,$ given by (3), is well defined and uniformly recurrent.

Now we will introduce the notion of (asymptotical) Stepanov $p\left(x\right)$-uniform recurrence for the functions depending on two parameters; this notion extends the notion introduced in Definition 3 and Definition 4, where we have considered the constant coefficient $p\left(x\right)\equiv p\in [p,\infty )$:

Definition 6.

Let $p\in \mathcal{P}\left(\right[0,1\left]\right).$

(i) 

A function $f:I\times Y\to X$ is called Stepanov $p\left(x\right)$-uniformly recurrent if and only if $\widehat{f}:I\times Y\to L^{p\left(x\right)}([0,1]:X)$ is uniformly recurrent.

(ii) 

A function $f:[0,\infty )\times Y\to X$ is said to be asymptotically $S^{p\left(x\right)}$-uniformly recurrent if and only if $\widehat{f}:[0,\infty )\times Y\to L^{p\left(x\right)}([0,1]:X)$ is asymptotically uniformly recurrent.

A great number of composition principles established for Stepanov $p\left(x\right)$-almost-periodic functions can be extended for Stepanov $p\left(x\right)$-uniformly recurrent functions. For example, in ([13], Theorem 5.4), we have slightly improved the important composition principle attributed to Long and Ding ([21], Theorem 2.2). Using Lemma 1(i)–(iii) and the arguments employed in the proof of the last-mentioned theorem, we may deduce the following statement concerning $S^{p\left(x\right)}$-uniform recurrence:

Theorem 1.

Let $p\in \mathcal{P}\left(\right[0,1\left]\right).$ Suppose that the following conditions hold:

(i) 

The function $F:I\times Y\to X$ is Stepanov $p\left(x\right)$-uniformly recurrent, and there exist a function $r\in \mathcal{P}\left(\right[0,1\left]\right)$ and a function $L_f\in L_S^{r\left(x\right)}\left(I\right)$ such that $r(·)\ge max\left(p\right(·),p(·)/(p(·)-1\left)\right)$ and

$$\parallel F(t,x)-F(t,y)\parallel \le L_F\left(t\right){\parallel x-y\parallel }_Y,t\in I,x,y\in Y.$$

(4)

(ii) 

The function $f:I\to Y$ is Stepanov $p\left(x\right)$-uniformly recurrent and there exists a set $\mathrm{E}\subseteq I$ with $m\left(\mathrm{E}\right)=0$ such that $K:=\left\{f\right(t):t\in I\setminus \mathrm{E}\}$ is relatively compact in $Y.$

(iii) 

For every compact set $K\subseteq Y,$ there exists a strictly increasing sequence $\left({\alpha }_n\right)$ of positive real numbers tending to plus infinity such that

$$\underset{n\to +\infty }{lim}\underset{t\in I}{sup}\underset{u\in K}{sup}{\parallel F(t+s+{\alpha }_n,u)-F(t+s,u)\parallel }_{L^{p\left(s\right)}([0,1]:X)}=0$$

(5)

and (1) holds with the function $f(·)$ and the norm $\parallel ·\parallel$ replaced respectively by the function $\widehat{f}(·)$ and the norm ${\parallel ·\parallel }_{L^{p\left(x\right)}([0,1]:X)}$ therein.

Then $q\left(x\right):=p\left(x\right)r\left(x\right)/\left(p\right(x)+r(x\left)\right)\in [1,p(x\left)\right)$ and $F(·,f(·\left)\right)$ is Stepanov $q\left(x\right)$-uniformly recurrent. Furthermore, the assumption that $F(·,0)$ is Stepanov $q\left(x\right)$-bounded implies that the function $F(·,f(·\left)\right)$ is Stepanov $q\left(x\right)$-bounded, as well.

It is not so difficult to reformulate the statements of ([9], Proposition 2.7.3–Proposition 2.7.4) and ([13], Proposition 5.5) for the asymptotical Stepanov $p\left(x\right)$-uniform recurrence. Details can be left to the interested readers.

4. Doss Almost-Periodicity and Doss Uniform Recurrence in Lebesgue Spaces with Variable Exponents

Throughout this section, we assume the following general condition:

$\left(A\right)$ $I=\mathbb{R}$ or $I=[0,\infty ),$ $\varphi :[0,\infty )\to [0,\infty )$ is measurable, $p\in \mathcal{P}\left(I\right)$ and $F:(0,\infty )\to (0,\infty ).$

The notion of Doss-$p\left(x\right)$-almost-periodicity has not been introduced so far. Following the approach from our recent research paper [17], where we have analyzed the classes of (equi-)Weyl-$(p,\varphi ,F)$-almost-periodic functions and (equi-)Weyl-${(p,\varphi ,F)}_i$-almost-periodic functions ($i=1,2$), we introduce the following notion for Doss classes:

Definition 7.

Suppose that condition (A) holds, $f:I\to X$ and $\varphi (\parallel f(·+\tau )-f(·)\parallel )\in L^{p\left(x\right)}\left(K\right)$ for any $\tau \in I$ and any compact subset K of $I.$

(i) 

A function $f(·)$ is said to be Doss-$(p,\varphi ,F)$-almost periodic if and only if for every $ϵ>0$, the set of numbers $\tau \in I$ for which

$$\underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\left[\varphi {\left(∥f(·+\tau )-f(·)∥\right)}_{L^{p\left(x\right)}[-t,t]}\right]\right]<ϵ,$$

(6)

in the case that $I=\mathbb{R},$ resp.,

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\left[\varphi {\left(∥f(·+\tau )-f(·)∥\right)}_{L^{p\left(x\right)}[0,t]}\right]\right]<ϵ,\end{array}$$

in the case that $I=[0,\infty ),$ is relatively dense in $I.$

(ii) 

A function $f(·)$ is said to be Doss-$(p,\varphi ,F)$-uniformly recurrent if and only if there exists a strictly increasing sequence $\left({\alpha }_n\right)$ of positive real numbers such that ${lim}_{n\to +\infty }{\alpha }_n=+\infty$ and

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\left[\varphi {\left(∥f(·+{\alpha }_n)-f(·)∥\right)}_{L^{p\left(x\right)}[-t,t]}\right]\right]=0,\end{array}$$

in the case that $I=\mathbb{R},$ resp.,

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\left[\varphi {\left(∥f(·+{\alpha }_n)-f(·)∥\right)}_{L^{p\left(x\right)}[0,t]}\right]\right]=0,\end{array}$$

in the case that $I=[0,\infty ),$ is relatively dense in $I.$

Definition 8.

Suppose that condition (A) holds, $f:I\to X$ and $\parallel f(·+\tau )-f(·)\parallel \in L^{p\left(x\right)}\left(K\right)$ for any $\tau \in I$ and any compact subset K of $I.$

(i) 

A function $f(·)$ is said to be Doss-${(p,\varphi ,F)}_1$-almost periodic if and only if for every $ϵ>0$, the set of numbers $\tau \in I$ for which

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\varphi \left[{\left(∥f(·+\tau )-f(·)∥\right)}_{L^{p\left(x\right)}[-t,t]}\right]\right]<ϵ,\end{array}$$

in the case that $I=\mathbb{R},$ resp.,

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\varphi \left[{\left(∥f(·+\tau )-f(·)∥\right)}_{L^{p\left(x\right)}[0,t]}\right]\right]<ϵ,\end{array}$$

in the case that $I=[0,\infty ),$ is relatively dense in $I.$

(ii) 

A function $f(·)$ is said to be Doss-${(p,\varphi ,F)}_1$-uniformly recurrent if and only if there exists a strictly increasing sequence $\left({\alpha }_n\right)$ of positive real numbers such that ${lim}_{n\to +\infty }{\alpha }_n=+\infty$ and

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\varphi \left[{\left(∥f(·+{\alpha }_n)-f(·)∥\right)}_{L^{p\left(x\right)}[-t,t]}\right]\right]=0,\end{array}$$

in the case that $I=\mathbb{R},$ resp.,

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)\left[\varphi {\left(∥f(·+{\alpha }_n)-f(·)∥\right)}_{L^{p\left(x\right)}[0,t]}\right]\right]=0,\end{array}$$

in the case that $I=[0,\infty ),$ is relatively dense in $I.$

Definition 9.

Suppose that condition (A) holds, $f:I\to X$ and $\parallel f(·+\tau )-f(·)\parallel \in L^{p\left(x\right)}\left(K\right)$ for any $\tau \in I$ and any compact subset K of $I.$

(i) 

A function $f(·)$ is said to be Doss-${(p,\varphi ,F)}_2$-almost periodic if and only if for every $ϵ>0$, the set of numbers $\tau \in I$ for which

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim\; sup}\left[\varphi \left[F\left(t\right){\left(∥f(·+\tau )-f(·)∥\right)}_{L^{p\left(x\right)}[-t,t]}\right]\right]<ϵ,\end{array}$$

in the case that $I=\mathbb{R},$ resp.,

$$\begin{array}{c}\hfill \underset{t\to +\infty }{lim\; sup}\left[\varphi \left[F\left(t\right){\left(∥f(·+\tau )-f(·)∥\right)}_{L^{p\left(x\right)}[0,t]}\right]\right]<ϵ,\end{array}$$

in the case that $I=[0,\infty ),$ is relatively dense in $I.$

(ii) 

A function $f(·)$ is said to be Doss-${(p,\varphi ,F)}_2$-uniformly recurrent if and only if there exists a strictly increasing sequence $\left({\alpha }_n\right)$ of positive real numbers such that ${lim}_{n\to +\infty }{\alpha }_n=+\infty$ and

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\underset{t\to +\infty }{lim\; sup}\left[\varphi \left[F\left(t\right){\left(∥f(·+{\alpha }_n)-f(·)∥\right)}_{L^{p\left(x\right)}[-t,t]}\right]\right]=0,\end{array}$$

in the case that $I=\mathbb{R},$ resp.,

$$\begin{array}{c}\hfill \underset{n\to +\infty }{lim}\underset{t\to +\infty }{lim\; sup}\left[\varphi \left[F\left(t\right){\left(∥f(·+{\alpha }_n)-f(·)∥\right)}_{L^{p\left(x\right)}[0,t]}\right]\right]=0,\end{array}$$

in the case that $I=[0,\infty ),$ is relatively dense in $I.$

Case in which $\varphi \left(x\right)\equiv x$ and $\psi \left(t\right)\equiv {\left(2t\right)}^{(-1)/p},$ $t>0$ if $I=\mathbb{R},$ resp. $\psi \left(t\right)\equiv t^{(-1)/p},$ $t>0$ if $I=[0,\infty ),$ leads to the usual class of Doss p-almost-periodic functions [9,22]. The notion introduced in the above three definitions is rather general; for example, in the case that $p\left(x\right)\equiv p\in [1,\infty )$ and $\sigma >0,$ then any essentially bounded function $f(·)$ is Doss-$(p,x,t^{-(1+\sigma )/p})$-almost periodic.

Example 3.

(i) 

Suppose that $\varphi \left(0\right)=0.$ Then any continuous periodic function $f:I\to X$ is Doss-${(p,\varphi ,F)}_i$-almost periodic for $i=1,2;$ furthermore, if $\varphi (·)$ is locally bounded, then the function $f(·)$ is Doss-$(p,\varphi ,F)$-almost periodic.

(ii) 

Suppose that $f:I\to E$ is almost periodic. Then $f(·)$ is Doss-$(p,\varphi ,F)$-almost periodic [Doss-${(p,\varphi ,F)}_1$-almost periodic/Doss-${(p,\varphi ,F)}_2$-almost periodic] if $\varphi (·)$ is continuous, monotonically increasing and ${F(·)\parallel 1\parallel }_{L^{p\left(x\right)}[-·,·]}\in L^{\infty }\left((0,\infty )\right)$ [$\varphi (·)$ is monotonically increasing, there exists a continuous function $\phi :[0,\infty )\to [0,\infty )$ such that $\varphi \left(xy\right)\le \phi \left(x\right)\varphi \left(y\right),$ $x,y\ge 0$ and ${F(·)\parallel 1\parallel }_{L^{p\left(x\right)}[-·,·]}\in L^{\infty }\left((0,\infty )\right)$/$\varphi (·)$ is monotonically increasing, there exists a continuous function $\phi :[0,\infty )\to [0,\infty )$ such that $\varphi \left(xy\right)\le \phi \left(x\right)\varphi \left(y\right),$ $x,y\ge 0$ and ${\varphi (F(·)\parallel 1\parallel }_{L^{p\left(x\right)}[-·,·]})\in L^{\infty }\left((0,\infty )\right)$].

Example 4.

(see ([19], Theorem 1.1) and ([15], Theorem 1.2)) The function $f:\mathbb{R}\to \mathbb{R},$ given by

$$\begin{array}{c}\hfill f\left(t\right):=\sum _{n=1}^{\infty }\frac{1}{n}{sin}^2\left(\frac{t}{2^n}\right)dt,t\in \mathbb{R},\end{array}$$

is uniformly continuous, uniformly recurrent and Besicovitch unbounded (see [9] for the notion). It is known that for each number $\tau \in \mathbb{R}$ we have

$$\underset{t\to +\infty }{lim}\frac{1}{t}{\int }_0^t{|f(s+\tau )-f\left(s\right)|}^{p}ds=0,p\ge 1,$$

so that the function $f(·)$ is Doss $p\left(x\right)$-almost periodic for any function $p\in D_{+}\left(\mathbb{R}\right).$

To ensure the translation invariance of generalized Weyl spaces of almost-periodic functions in [17], we have analyzed the classes of (equi-)Weyl-$[p,\varphi ,F]$-almost-periodic functions and (equi-)Weyl-${[p,\varphi ,F]}_i$-almost-periodic functions ($i=1,2$). In this paper, we will follow a slightly different approach. First, for any ${\tau }_0\in I$ we set $p_{{\tau }_0}(·):=p(·+{\tau }_0).$ Then we have the following:

Theorem 2.

Suppose that $F_1(·)$ is monotonically decreasing, there exists a function $F_0:(0,\infty )\to (0,\infty )$ such that $F\left(xy\right)\le F_0\left(x\right)·F\left(y\right),$ $x,y>0,$ ${\tau }_0\in I$ and

$$\underset{t\to +\infty }{lim\; sup}{F}_0\left(\frac{t}{t+{\tau }_0}\right)<\infty .$$

(7)

Define $f_{{\tau }_0}(·):=f(·+{\tau }_0).$ Then the following hold:

(i) 

Suppose that $f(·)$ is Doss-$(p,\varphi ,F)$-almost periodic, resp. Doss-$(p,\varphi ,F)$-uniformly recurrent. Then $f_{{\tau }_0}(·)$ is Doss-$(p_{{\tau }_0},\varphi ,F_1)$-almost periodic, resp. Doss-$(p_{{\tau }_0},\varphi ,F_1)$-uniformly recurrent.

(ii) 

Suppose that $f(·)$ is Doss-${(p,\varphi ,F)}_1$-almost periodic, resp. Doss-${(p,\varphi ,F)}_1$-uniformly recurrent, and $\varphi (·)$ is monotonically increasing. Then $f_{{\tau }_0}(·)$ is Doss-${(p_{{\tau }_0},\varphi ,F_1)}_1$-almost periodic, resp. Doss-${(p_{{\tau }_0},\varphi ,F_1)}_1$-uniformly recurrent.

(iii) 

Suppose that $f(·)$ is Doss-${(p,\varphi ,F)}_2$-almost periodic, resp. Doss-${(p,\varphi ,F)}_2$-uniformly recurrent, $\varphi (·)$ is monotonically increasing, there exists a function ${\varphi }_0:[0,\infty )\to [0,\infty )$ such that $\varphi \left(xy\right)\le {\varphi }_0\left(x\right)·\varphi \left(y\right),$ $x,y\ge 0$ and, in place of condition (7),

$$\underset{t\to +\infty }{lim\; sup}\left({\varphi }_0\circ F_0\right)\left(\frac{t}{t+{\tau }_0}\right)<\infty .$$

(8)

Then $f_{{\tau }_0}(·)$ is Doss-${(p_{{\tau }_0},{\varphi }_1,F_1)}_2$-almost periodic, resp. Doss-${(p_{{\tau }_0},{\varphi }_1,F_1)}_2$-uniformly recurrent.

Proof. 

We will consider only Doss almost-periodic functions with variable exponent. Suppose that $\tau \in I$ and (6) holds. We need to prove first that $\varphi (\parallel f(·+\tau +{\tau }_0)-f(·+{\tau }_0)\parallel )\in L^{p_{{\tau }_0}\left(x\right)}\left(K\right)$ for any $\tau \in I$ and any compact subset K of $I.$ However, this directly follows from the corresponding definitions of the space $L^{p_{{\tau }_0}\left(x\right)}\left(K\right),$ the function $p_{{\tau }_0}(·)$ and an elementary substitution $·\mapsto ·+{\tau }_0.$ The statement (i) then follows from the next computation:

$$\begin{array}{c}\underset{t\to +\infty }{lim\; sup}\left[F_1\left(t\right)inf\left\{\lambda >0:{\int }_0^t{\phi }_{p_{{\tau }_0}\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau +{\tau }_0)-f(x+{\tau }_0)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ =\underset{t\to +\infty }{lim\; sup}\left[F_1\left(t\right)inf\left\{\lambda >0:{\int }_{{\tau }_0}^{t+{\tau }_0}{\phi }_{p_{{\tau }_0}(x-{\tau }_0)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ =\underset{t\to +\infty }{lim}\underset{y\ge t}{sup}\left[F_1\left(y\right)inf\left\{\lambda >0:{\int }_{{\tau }_0}^{y+{\tau }_0}{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ \le \underset{t\to +\infty }{lim\; sup}\underset{y\ge t}{sup}[F_1\left(\frac{t}{t+{\tau }_0}(y+{\tau }_0)\right)\hfill \\ \times inf\left\{\lambda >0:{\int }_{{\tau }_0}^{y+{\tau }_0}{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}]\hfill \\ \le \underset{t\to +\infty }{lim\; sup}\underset{y\ge t}{sup}[F_0\left(\frac{t}{t+{\tau }_0}\right)\hfill \\ \times F(y+{\tau }_0)inf\left\{\lambda >0:{\int }_{{\tau }_0}^{y+{\tau }_0}{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}]\hfill \\ \le \underset{t\to +\infty }{lim\; sup}{F}_0\left(\frac{t}{t+{\tau }_0}\right)\times \hfill \\ \underset{t\to +\infty }{lim\; sup}\underset{y\ge t}{sup}\left[F(y+{\tau }_0)inf\left\{\lambda >0:{\int }_{{\tau }_0}^{y+{\tau }_0}{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ \le \underset{t\to +\infty }{lim\; sup}{F}_0\left(\frac{t}{t+{\tau }_0}\right)\times \hfill \\ \underset{t\to +\infty }{lim\; sup}\underset{y\ge t}{sup}\left[F(y+{\tau }_0)inf\left\{\lambda >0:{\int }_0^{y+{\tau }_0}{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ \le \underset{t\to +\infty }{lim\; sup}{F}_0\left(\frac{t}{t+{\tau }_0}\right)\times \hfill \\ \underset{t\to +\infty }{lim\; sup}\underset{y\ge t+{\tau }_0}{sup}\left[F\left(y\right)inf\left\{\lambda >0:{\int }_0^y{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ =\underset{t\to +\infty }{lim\; sup}{F}_0\left(\frac{t}{t+{\tau }_0}\right)\times \hfill \\ \underset{t\to +\infty }{lim\; sup}\underset{y\ge t}{sup}\left[F\left(y\right)inf\left\{\lambda >0:{\int }_0^y{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ =\underset{t\to +\infty }{lim\; sup}{F}_0\left(\frac{t}{t+{\tau }_0}\right)\times \hfill \\ \underset{t\to +\infty }{lim\; sup}\left[F\left(t\right)inf\left\{\lambda >0:{\int }_0^t{\phi }_{p\left(x\right)}\left(\frac{\varphi \left(\parallel f(x+\tau )-f\left(x\right)\parallel \right)}{\lambda }\right)dx\le 1\right\}\right]\hfill \\ \le \underset{t\to +\infty }{lim\; sup}{F}_0\left(\frac{t}{t+{\tau }_0}\right)·ϵ.\hfill \end{array}$$

(9)

The proof of (ii) is similar because then we can start from the term

$$\underset{t\to +\infty }{lim\; sup}\left[F_1\left(t\right)\varphi \left(inf\left\{\lambda >0:{\int }_0^t{\phi }_{p_{{\tau }_0}\left(x\right)}\left(\frac{\parallel f(x+\tau +{\tau }_0)-f(x+{\tau }_0)\parallel }{\lambda }\right)dx\le 1\right\}\right)\right],$$

use the same computation and the assumption that $\varphi (·)$ is monotonically increasing. The proof of (iii) is also similar because, with the obvious change of computation caused using different notion, we can use the same computation and the inequality (see also (8) and (9))

$$\varphi \left(F_0\left(\frac{t}{t+{\tau }_0}\right)·F(y+{\tau }_0)\right)\le {\varphi }_0\left(F_0\left(\frac{t}{t+{\tau }_0}\right)\right)·{\varphi }_1\left(F(y+{\tau }_0)\right).$$

 □

The following result is very similar to ([17], Proposition 2.5); we will include the proof for the sake of completeness.

Proposition 4.

Suppose that $p\left(x\right)\equiv 1,$ $f:I\to X,$ $\parallel f(·+\tau )-f(·)\parallel \in L^1\left(K\right)$ for any $\tau \in I$ and any compact subset K of $I,$ as well as condition

(B): 

$\varphi (·)$ is convex and there exists a function $\phi :[0,\infty )\to (0,\infty )$ such that $\varphi \left(tx\right)\le \phi \left(t\right)\varphi \left(x\right)$ for all $t\ge 0$ and $x\ge 0.$

Set $F_1\left(t\right):=F\left(t\right)t{\left[\phi \left(t\right)\right]}^{-1},$ $t>0,$ $F_2\left(t\right):={\left(2t\right)}^{-1}\phi \left(2F\left(t\right)t\right),$ $t>0$ provided that $I=\mathbb{R},$ and $F_2\left(t\right):=t^{-1}\phi \left(F\left(t\right)t\right),$ $t>0$ provided that $I=[0,\infty ).$ Then we have:

(i) 

If $f(·)$ is Doss-$(1,\varphi ,F)$-almost periodic, resp. Doss-$(1,\varphi ,F)$-uniformly recurrent, then $f(·)$ is Doss-${(1,\varphi ,F_1)}_1$-almost periodic, resp. Doss-${(1,\varphi ,F_1)}_1$-uniformly recurrent.

(ii) 

If $f(·)$ is Doss-$(1,\varphi ,F_2)$-almost periodic, resp. Doss-$(1,\varphi ,F_2)$-uniformly recurrent, then $f(·)$ is Doss-${(1,\varphi ,F)}_2$-almost periodic, resp. Doss-${(1,\varphi ,F)}_2$-uniformly recurrent.

Proof. 

We will consider only Doss almost-periodic functions with variable exponent and case $I=[0,\infty ).$ To prove (i), we can use the assumption (B) and the Jensen integral inequality ($\tau >0$):

$$\begin{array}{cc}\hfill \varphi (& {\parallel f(·+\tau )-f(·)\parallel }_{L^1[0,t]})=\varphi \left(t·t^{-1}{\parallel f(·+\tau )-f(·)\parallel }_{L^1[0,t]}\right)\hfill \\ \hfill \le & \phi \left(t\right)\varphi \left(t^{-1}{\parallel f(·+\tau )-f(·)\parallel }_{L^1[0,t]}\right)\le \phi \left(t\right)t^{-1}{\left[\varphi \left(\parallel f(·+\tau )-f(·)\parallel \right)\right]}_{L^1[0,t]}.\hfill \end{array}$$

This simply yields that $f(·)$ is Doss-${(1,\varphi ,F_1)}_1$-almost periodic. To prove (ii), suppose that $f(·)$ is Doss-$(1,\varphi ,F_2)$-almost periodic. Then the assumption (B) and the Jensen integral inequality together imply ($\tau >0$):

$$\begin{array}{cc}\hfill \varphi (& {F\left(t\right)\parallel f(·+\tau )-f(·)\parallel }_{L^1[0,t]})=\varphi \left(F\left(t\right)t·t^{-1}{\parallel f(·+\tau )-f(·)\parallel }_{L^1[0,t]}\right)\hfill \\ \hfill \le & \phi \left(F\left(t\right)\right)t^{-1}{\left[\varphi \left(\parallel f(·+\tau )-f(·)\parallel \right)\right]}_{L^1[0,t]}.\hfill \end{array}$$

This simply yields that $f(·)$ is Doss-${(1,\varphi ,F)}_2$-almost periodic. □

Remark 1.

(i) 

It is clear that if $f(·)$ is Doss-$(p,\varphi ,F)$-almost periodic [Doss-$(p,\varphi ,F)$-uniformly recurrent], resp. Doss-${(p,\varphi ,F)}_1$-almost periodic [Doss-${(p,\varphi ,F)}_1$-uniformly recurrent], and $F\left(t\right)\ge F_1\left(t\right)$ for every $t\in I,$ then $f(·)$ is Doss-$(p,\varphi ,F_1)$-almost periodic [Doss-$(p,\varphi ,F_1)$–uniformly recurrent], resp. Doss-${(p,\varphi ,F_1)}_1$-almost periodic [Doss-${(p,\varphi ,F_1)}_1$-uniformly recurrent]. Furthermore, if $f(·)$ is Doss-${(p,\varphi ,F)}_2$-almost periodic [Doss-${(p,\varphi ,F)}_2$-uniformly recurrent], then $f(·)$ is Doss-${(p,\varphi ,F_1)}_2$-almost periodic [Doss-${(p,\varphi ,F_1)}_2$-uniformly recurrent] provided that $F\left(t\right)\ge F_1\left(t\right)$ for every $t\in I$ and $\varphi (·)$ is monotonically increasing, or $F\left(t\right)\le F_1\left(t\right)$ for every $t\in I$ and $\varphi (·)$ is monotonically decreasing.

(ii) 

If $f(·)$ is Doss-$(p,\varphi ,F)$-almost periodic [Doss-$(p,\varphi ,F)$-uniformly recurrent], resp. Doss-${(p,\varphi ,F)}_i$-almost periodic [Doss-${(p,\varphi ,F)}_i$-uniformly recurrent], ${\varphi }_1(·)$ is measurable and $0\le {\varphi }_1\le \varphi ,$ then Lemma 1(iii) yields that $f(·)$ is Doss-$(p,{\varphi }_1,F)$-almost periodic [Doss-$(p,{\varphi }_1,F)$-uniformly recurrent], resp. Doss-${(p,{\varphi }_1,F)}_i$-almost periodic [Doss-${(p,{\varphi }_1,F)}_i$-uniformly recurrent], where $i=1,2.$

Example 5.

(i) 

Let $p\left(x\right)\equiv p\in [1,\infty )$ and $f\left(x\right):={\chi }_{[0,1/2]}\left(x\right),$ $x\in \mathbb{R}.$ Then it can be simply shown that for each real number τ such that $\left|\tau \right|>1$ we have

$${\int }_{-t}^t{|f(x+\tau )-f\left(x\right)|}^{p}dx\le \frac{1}{2}+2{\int }_0^{1/2}{\left|f\left(x\right)\right|}^{p}dx,t\in \mathbb{R}.$$

This implies that $f(·)$ is Doss-$(p,x,t^{-\sigma })$-almost periodic for each real number $\sigma >0.$

(ii) 

Let $p\left(x\right)\equiv p\in [1,\infty )$ and $f\left(x\right):={\chi }_{[0,\infty )}\left(x\right),$ $x\in \mathbb{R}.$ Then it can be simply shown that for each real number τ we have

$${\int }_{-t}^t{|f(x+\tau )-f\left(x\right)|}^{p}dx={\int }_{-t+\tau }^{\tau }{\left|f\left(x\right)\right|}^{p}dx+{\int }_{\tau }^{t+\tau }{\left|f\left(x\right)\right|}^{p}dx,t\in \mathbb{R}.$$

Hence,

$${\int }_{-t}^t{|f(x+\tau )-f\left(x\right)|}^{p}dx\le 2\left|\tau \right|,provided \tau \in \mathbb{R},t\ge \left|\tau \right|,$$

and $f(·)$ is Doss-$(p,x,t^{-\sigma })$-almost periodic for each real number $\sigma >0.$

Concerning embeddings between different Doss almost-periodic type spaces with variable exponent, we would like to state the following result:

Proposition 5.

Let $p,q\in \mathcal{P}\left(I\right)$ and let $1\le q\left(x\right)\le p\left(x\right)$ for a.e. $x\in I.$

(i) 

Suppose that a function $f(·)$ is Doss-$(p,\varphi ,F)$-almost periodic, resp. Doss-$(p,\varphi ,F)$-uniformly recurrent, and $F_1\left(t\right):=F\left(t\right)/t,$ $t>0.$ Then $f(·)$ is Doss-$(q,\varphi ,F_1)$-almost periodic, resp. Doss-$(q,\varphi ,F_1)$-uniformly recurrent.

(ii) 

Suppose that a function $f(·)$ is Doss-${(p,\varphi ,F)}_1$-almost-periodic, resp. Doss-${(p,\varphi ,F)}_1$-uniformly recurrent, $\varphi (·)$ is monotonically increasing, there exists a function $\phi :[0,\infty )\to (0,\infty )$ such that $\varphi \left(xy\right)\le \phi \left(x\right)\varphi \left(y\right),$ $x,y\ge 0$ and $F_1\left(t\right):=F\left(t\right)/\phi \left(2(1+2t)\right),$ $t>0$ provided $I=\mathbb{R},$ resp. $F_1\left(t\right):=F\left(t\right)/\phi \left(2(1+t)\right),$ $t>0$ provided $I=[0,\infty ).$ Then $f(·)$ is Doss-${(q,\varphi ,F_1)}_1$-almost periodic, resp. Doss-${(q,\varphi ,F_1)}_1$-uniformly recurrent.

(iii) 

Suppose that a function $f(·)$ is Doss-${(p,\varphi ,F)}_2$-almost periodic, resp. Doss-${(p,\varphi ,F)}_2$-uniformly recurrent, there exists a function $\phi :[0,\infty )\to [0,\infty )$ such that $\varphi \left(xy\right)\le \phi \left(x\right)\varphi \left(y\right),$ $x,y\ge 0$ and

$$\begin{array}{cc}\hfill \phi (& \frac{2F_1(·)(1+2·)}{F(·)})\in L^{\infty }\left((0,\infty )\right),ifI=\mathbb{R},\hfill \\ & resp.\phi \left(\frac{2F_1(·)(1+·)}{F(·)}\right)\in L^{\infty }\left((0,\infty )\right),ifI=[0,\infty ).\hfill \end{array}$$

Then $f(·)$ is Doss-${(q,\varphi ,F_1)}_2$-almost periodic, resp. Doss-${(q,\varphi ,F_1)}_2$-uniformly recurrent.

Proof. 

We will prove only (iii), for the class of Doss-${(p,\varphi ,F)}_2$-almost-periodic functions defined on the interval $I=[0,\infty ).$ Let the numbers $t,\tau >0$ be given. Then the conclusion simply follows from the calculation

$$\begin{array}{cc}\hfill \varphi (& F_1\left(t\right){\parallel f(·+\tau )-f(·)\parallel }_{L^{q\left(x\right)}[0,t]})\hfill \\ & \le \varphi \left(2F_1\left(t\right)(1+t){\parallel f(·+\tau )-f(·)\parallel }_{L^{p\left(x\right)}[0,t]}\right)\hfill \\ & =\varphi \left(\frac{2F_1\left(t\right)(1+t)}{F\left(t\right)}F\left(t\right){\parallel f(·+\tau )-f(·)\parallel }_{L^{p\left(x\right)}[0,t]}\right)\hfill \\ & \le \phi \left(\frac{2F_1\left(t\right)(1+t)}{F\left(t\right)}\right)\varphi \left({F\left(t\right)\parallel f(·+\tau )-f(·)\parallel }_{L^{p\left(x\right)}[0,t]}\right),\hfill \end{array}$$

where we have used Lemma 1(ii), and the corresponding definition of Doss-${(q,\varphi ,F_1)}_2$- almost periodicity. □

5. Invariance of Generalized Doss Almost-Periodicity with Variable Exponent under the Actions of Convolution Products

In this section, we will investigate the invariance of three types of generalized Doss almost-periodicity introduced in the previous section under the actions of infinite convolution products (for the sake of simplicity, we will not consider here the finite convolution products).

In ([22], Theorem 2.1), we have analyzed the invariance of Doss p-almost-periodicity under the actions of infinite convolution products, provided that the function $f(·)$ in (3) is Stepanov p-bounded ($1\le p<\infty$). In the formulation of the subsequent result, which is not satisfactory to a certain extent (let us only note that the above-mentioned theorem, which is a unique result in the existing literature concerning this problematic, cannot be deduced from Theorem 3), we will not use this condition:

Theorem 3.

Suppose that $\phi :[0,\infty )\to [0,\infty )$ is a function and $\varphi :[0,\infty )\to [0,\infty )$ is a convex monotonically increasing function satisfying $\varphi \left(xy\right)\le \phi \left(x\right)\varphi \left(y\right)$ for all $x,y\ge 0$ and $p\in \mathcal{P}\left(\mathbb{R}\right).$ Suppose, further, $\check{f}:\mathbb{R}\to X$ is Doss-$(p,\varphi ,F)$-almost periodic, resp. Doss-$(p,\varphi ,F)$-uniformly recurrent, and measurable, $F_1:(0,\infty )\to (0,\infty ),$ $q\in \mathcal{P}\left(\mathbb{R}\right),$ $1/p\left(x\right)+1/q\left(x\right)=1,$ ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(X,Y)$ is a strongly continuous operator family and for every real number $x\in \mathbb{R}$ we have

$${\int }_{-x}^{\infty }\parallel R(v+x)\parallel \parallel \check{f}\left(v\right)\parallel dv<\infty .$$

(10)

Suppose that for each $ϵ>0$ there exist an increasing sequence $\left(a_m\right)$ of positive real numbers tending to plus infinity and a number $t_0\left(ϵ\right)>0$ satisfying that for every $t\ge t_0\left(ϵ\right),$ we have

$${\int }_{-t}^t{\phi }_{p\left(x\right)}\left(2\phi \left(a_m\right)a_m^{-1}{F}_1\left(t\right)\underset{m\to +\infty }{lim\; sup}\left[{\left[\phi (\parallel R(·+x)\parallel )\right]}_{L^{q(·)}[-x,-x+a_m]}F{\left(t+a_m\right)}^{-1}\right]\right)dx\le 1.$$

(11)

Then the function $F:\mathbb{R}\to Y,$ given by (3), is well defined and Doss-$(p,\varphi ,F_1)$-almost periodic, resp. Doss-$(p,\varphi ,F_1)$-uniformly recurrent.

Proof. 

We will consider only the class of Doss-$(p,\varphi ,F)$-almost-periodic functions because the proof for the class of Doss-$(p,\varphi ,F)$-uniformly recurrent functions can be deduced quite analogously. Since $F\left(x\right)={\int }_{-x}^{\infty }R(v+x)\check{f}\left(v\right)dv,$ $x\in \mathbb{R},$ the validity of condition (10) yields that the function $F(·)$ is well defined as well as that the integrals in definitions of $F\left(x\right)$ and $F(x+\tau )-F\left(x\right)$ converge absolutely ($x\in \mathbb{R}$). Let $ϵ>0$ be fixed, and let the sequences $\left(t_n\right),$ $\left(t_n^{\prime }\right)$ and $\left(a_m\right)$ satisfy the prescribed requirements. Using the fact that the function $\varphi (·)$ is continuous and the function ${\phi }_{p\left(x\right)}(·)$ is monotonically increasing, we have ($x\in \mathbb{R},$ $\lambda ,\tau >0$):

$$\begin{array}{cc}\hfill {\phi }_{p\left(x\right)}& \left(\frac{\varphi (\parallel F(x+\tau )-F(x)\parallel )}{\lambda }\right)\hfill \\ & \le {\phi }_{p\left(x\right)}\left(\frac{\varphi \left({\int }_{-x}^{\infty }\parallel R(v+x)\parallel \parallel \check{f}(v+\tau )-\check{f}\left(v\right)\parallel dv\right)}{\lambda }\right)\hfill \\ & \le {\phi }_{p\left(x\right)}\left(\underset{m\to +\infty }{lim}\frac{\varphi \left({\int }_{-x}^{-x+a_m}\parallel R(v+x)\parallel \parallel \check{f}(v+\tau )-\check{f}\left(v\right)\parallel dv\right)}{\lambda }\right)\hfill \\ & ={\phi }_{p\left(x\right)}\left(\underset{m\to +\infty }{lim}\frac{\varphi \left({\int }_{-x}^{-x+a_m}{a}_m{a}_m^{-1}\parallel R(v+x)\parallel \parallel \check{f}(v+\tau )-\check{f}\left(v\right)\parallel dv\right)}{\lambda }\right)\hfill \\ & \le {\phi }_{p\left(x\right)}\left(\underset{m\to +\infty }{lim\; sup}\frac{\phi \left(a_m\right)a_m^{-1}{\int }_{-x}^{-x+a_m}\varphi (\parallel R(v+x)\parallel \parallel \check{f}(v+\tau )-\check{f}\left(v\right)\parallel )dv}{\lambda }\right)\hfill \\ & \le {\phi }_{p\left(x\right)}(\underset{m\to +\infty }{lim\; sup}\frac{2\phi \left(a_m\right)a_m^{-1}{\left[\phi (\parallel R(v+x)\parallel )\right]}_{L^{q\left(v\right)}[-x,-x+a_m]}}{\lambda }\hfill \\ & \times \frac{{\left[\varphi (\parallel \check{f}(v+\tau )-\check{f}\left(v\right)\parallel )\right]}_{L^{p\left(v\right)}[-x,-x+a_m]}}{\lambda }),\hfill \end{array}$$

where we have also used the Jensen integral inequality and the Hölder inequality. Let $ϵ>0$ be fixed and let $\tau >0$ be such that (6) holds, i.e., there exists $t_1(ϵ,\tau )\ge 0$ such that

$$\left[F\left(t\right)\left[\varphi {\left(∥\check{f}(·+\tau )-\check{f}(·)∥\right)}_{L^{p\left(x\right)}[-t,t]}\right]\right]<ϵ,t\ge t_1(ϵ,\tau ).$$

(12)

Suppose that $t\ge max(t_0\left(ϵ\right),t_1(ϵ,\tau )).$ Then for each $x\in [-t,t]$ and $m\in \mathbb{N}$ we have $[-x,-x+a_m]\subseteq [-(t+a_m),t+a_m]$ so that the above calculation and (12) give

$${\phi }_{p\left(x\right)}\left(\frac{\varphi (\parallel F(x+\tau )-F(x)\parallel )}{\lambda }\right)\le {\phi }_{p\left(x\right)}\left(\underset{m\to +\infty }{lim\; sup}\frac{2\phi \left(a_m\right)a_m^{-1}{\left[\phi (\parallel R(v+x)\parallel )\right]}_{L^{q\left(v\right)}[-x,-x+a_m]}ϵ/F(t+a_m)}{\lambda }\right).$$

(13)

Integrating this estimate over the interval $[-t,t]$ and using (11) we get that the inequality

$${\int }_{-t}^t{\phi }_{p\left(x\right)}\left(\frac{\varphi (\parallel F(x+\tau )-F(x)\parallel )}{\lambda }\right)dx\le 1$$

holds with $\lambda =ϵ/F_1\left(t\right),$ which completes the proof in a routine manner. □

We can similarly prove the following results for Doss-${(p,\varphi ,F)}_1$-almost-periodic functions, resp. Doss-${(p,\varphi ,F)}_1$-uniformly recurrent functions, and Doss-${(p,\varphi ,F)}_2$-almost-periodic functions, resp. Doss-${(p,\varphi ,F)}_2$-uniformly recurrent functions; for the sake of brevity, we will only provide descriptions of the proofs since they are very similar to the proof of Theorem 3 above (see also ([17], Theorem 4.6, Theorem 4.9)):

Theorem 4.

Suppose that $\varphi :[0,\infty )\to [0,\infty )$ is a continuous monotonically increasing bijection and $p\in \mathcal{P}\left(\mathbb{R}\right).$ Suppose, further, $\check{f}:\mathbb{R}\to X$ is Doss-${(p,\varphi ,F)}_1$-almost periodic, resp. Doss-${(p,\varphi ,F)}_1$-uniformly recurrent, and measurable, $F_1:(0,\infty )\to (0,\infty ),$ $q\in \mathcal{P}\left(\mathbb{R}\right),$ $1/p\left(x\right)+1/q\left(x\right)=1,$ ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(X,Y)$ is a strongly continuous operator family and, for every real number $x\in \mathbb{R},$ we have (10). Suppose that for each $ϵ>0$ there exist an increasing sequence $\left(a_m\right)$ of positive real numbers tending to plus infinity and a number $t_0\left(ϵ\right)>0$ satisfying that for every $t\ge t_0\left(ϵ\right),$ we have

$${\int }_{-t}^t{\phi }_{p\left(x\right)}\left(\frac{{lim\; sup}_{m\to +\infty }\left[2{\left[\phi (\parallel R(·+x)\parallel )\right]}_{L^{q(·)}[-x,-x+a_m]}{\varphi }^{-1}(ϵ/F(t+a_m))\right]}{{\varphi }^{-1}(ϵ/F_1\left(t\right))}\right)dx\le 1.$$

(14)

Then the function $F:\mathbb{R}\to Y,$ given by (3), is well defined and Doss-${(p,\varphi ,F_1)}_1$-almost periodic, resp. Doss-${(p,\varphi ,F_1)}_1$-uniformly recurrent.

Proof. 

As in the proof of Theorem 3 above, we have that the function $F(·)$ is well defined as well as that the integrals in definitions of $F\left(x\right)$ and $F(x+\tau )-F\left(x\right)$ converge absolutely ($x\in \mathbb{R}$). Let $ϵ>0$ be fixed. Then it suffices to show that for every $t\ge t_0\left(ϵ\right),$ we have ($x\in \mathbb{R},$ $\lambda ,\tau >0$)

$${∥R\left(s\right)\left[F\right(x+t+\tau -s)-F(x+t-s\left)\right]ds∥}_{L^{p\left(x\right)}[-t,t]}\le {\varphi }^{-1}\left(ϵ/F_1\left(t\right)\right).$$

However, we can repeat the arguments used in the proof of the above-mentioned theorem, with $\varphi \left(x\right)\equiv x,$ to see that:

$$\begin{array}{cc}\hfill {\phi }_{p\left(x\right)}& \left(\frac{\parallel F(x+\tau )-F\left(x\right)\parallel }{\lambda }\right)\hfill \\ & \le \frac{2{\left[\phi (\parallel R(v+x)\parallel )\right]}_{L^{q\left(v\right)}[-x,-x+a_m]}{\left[\parallel \check{f}(v+\tau )-\check{f}\left(v\right)\parallel \right]}_{L^{p\left(v\right)}[-x,-x+a_m]}}{\lambda })\hfill \\ & \le \frac{2{\left[\phi (\parallel R(v+x)\parallel )\right]}_{L^{q\left(v\right)}[-x,-x+a_m]}{\varphi }^{-1}(ϵ/F(t+a_m))}{\lambda }).\hfill \end{array}$$

The rest of proof is clear because we can take $\lambda ={\varphi }^{-1}(ϵ/F_1\left(t\right))$ and use condition (14). □

Theorem 5.

Suppose that $\varphi :[0,\infty )\to [0,\infty )$ is a continuous monotonically increasing bijection and $p\in \mathcal{P}\left(\mathbb{R}\right).$ Suppose, further, $\check{f}:\mathbb{R}\to X$ is Doss-${(p,\varphi ,F)}_2$-almost periodic, resp. Doss-${(p,\varphi ,F)}_2$-uniformly recurrent, and measurable, $F_1:(0,\infty )\to (0,\infty ),$ $q\in \mathcal{P}\left(\mathbb{R}\right),$ $1/p\left(x\right)+1/q\left(x\right)=1,$ ${\left(R\left(t\right)\right)}_{t>0}\subseteq L(X,Y)$ is a strongly continuous operator family and, for every real number $x\in \mathbb{R},$ we have (10). Suppose that for each $ϵ>0$ there exist an increasing sequence $\left(a_m\right)$ of positive real numbers tending to plus infinity and a number $t_0\left(ϵ\right)>0$ satisfying that for every $t\ge t_0\left(ϵ\right),$ we have

$${\int }_{-t}^t{\phi }_{p\left(x\right)}\left(2F_1\left(t\right)\underset{m\to +\infty }{lim\; sup}\frac{{\left[\phi (\parallel R(·+x)\parallel )\right]}_{L^{q(·)}[-x,-x+a_m]}}{F\left(t+a_m\right)}\right)dx\le 1.$$

(15)

Then the function $F:\mathbb{R}\to Y,$ given by (3), is well defined and Doss-${(p,\varphi ,F_1)}_2$-almost periodic, resp. Doss-${(p,\varphi ,F_1)}_2$-uniformly recurrent.

Proof. 

We can use the same trick as above, with $\lambda ={\varphi }^{-1}\left(ϵ\right)/F_1\left(t\right)$ and use condition (15). □

Remark 2.

(i) 

Suppose that $p\left(x\right)\equiv p\in [1,\infty ).$ Then we can use the usual Hölder inequality to see that the estimates (11)–(15) can be modified by removing the multiplication with the number 2 therein.

(ii) 

Although we will not define the notion of Besicovitch-Doss almost-periodicity with variable exponent here, we would like to note that the statement of ([9], Theorem 2.13.7) and the corresponding part of this result which considers the Doss almost-periodicity cannot be so easily reexamined in our framework.

Concerning the convolution invariance of generalized almost-periodicity introduced in this paper, we will only state and prove the following result (see also ([9], Theorem 3.11.26)):

Proposition 6.

Suppose that $\psi \in L^1\left(\mathbb{R}\right),$ $-\infty <a<b<+\infty ,$ $supp\left(\psi \right)\subseteq [a,b],$ $\phi :[0,\infty )\to [0,\infty ),$ $\varphi :[0,\infty )\to [0,\infty )$ is a convex monotonically increasing function satisfying $\varphi \left(xy\right)\le \phi \left(x\right)\varphi \left(y\right)$ for all $x,y\ge 0,$ $p,q\in \mathcal{P}\left(\mathbb{R}\right)$ and $1/p\left(x\right)+1/q\left(x\right)=1.$ Suppose, further, that the function $f:\mathbb{R}\to X$ is Doss-$(p,\varphi ,F)$-almost periodic, resp. Doss-$(p,\varphi ,F)$-uniformly recurrent, and essentially bounded. Then the function

$$x\mapsto (\psi \ast f)\left(x\right):={\int }_{-\infty }^{+\infty }\psi (x-y)f\left(y\right)dy,x\in \mathbb{R}$$

(16)

is well defined and essentially bounded. Furthermore, if $p_1\in \mathcal{P}\left(\mathbb{R}\right),$ $F_1:(0,\infty )\to (0,\infty )$ and if, for every $ϵ>0$ there exists a positive real number $t_1\left(ϵ\right)>0$ such that

$${\int }_{-t}^t{\phi }_{p_1\left(x\right)}\left(2F_1\left(t\right)\phi (b-a)\frac{{∥\phi \left(\left|\psi \right(x-z\left)\right|\right)∥}_{L^{q\left(z\right)}[x-b,x-a]}}{(b-a)F(t+c)}\right)dx\le 1,$$

(17)

where $c=max\left(\right|a|,|b\left|\right),$ then the function $\psi \ast f(·)$ is Doss-$(p_1,\varphi ,F_1)$-almost periodic, resp. Doss-$(p_1,\varphi ,F_1)$-uniformly recurrent.

Proof. 

We will give the main detail of proof for the class of Doss-$(p,\varphi ,F)$-almost-periodic functions, only. For every $x\in \mathbb{R}$ and $\tau \in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill \varphi & \left(\parallel (\psi \ast f)(x+\tau )-(\psi \ast f)\left(x\right)\parallel \right)\hfill \\ & \le \varphi \left((b-a){(b-a)}^{-1}{\int }_a^b\left|\psi \left(y\right)\right|·\parallel f(x+\tau -y)-f(x-y)\parallel dy\right)\hfill \\ & \le \frac{\phi (b-a)}{b-a}{\int }_a^b\varphi \left(\left|\psi \right(y\left)\right|·\parallel f(x+\tau -y)-f(x-y)\parallel \right)dy\hfill \\ & =\frac{\phi (b-a)}{b-a}{\int }_{x-b}^{x-a}\varphi \left(\left|\psi \right(x-z\left)\right|·\parallel f(z+\tau )-f\left(z\right)\parallel \right)dz\hfill \\ & \le 2\frac{\phi (b-a)}{b-a}{∥\phi \left(\left|\psi \right(x-z\left)\right|\right)∥}_{L^{q\left(z\right)}[x-b,x-a]}{\parallel f(z+\tau )-f\left(z\right)\parallel }_{L^{p\left(z\right)}[x-b,x-a]},\hfill \end{array}$$

where we have used the Jensen integral inequality and the Hölder inequality. The proof can be completed as it has been done in the final part of the proof of Theorem 3. □

Composition principles for Besicovitch almost-periodic functions have been investigated by Ayachi and Blot in [23]. We will consider composition principles for Doss almost-periodic functions with variable exponents somewhere else.

6. An Application to the Abstract Semilinear Integro-Differential Inclusions

It is clear that our results on the invariance of generalized almost-periodicity under the actions of infinite convolution products, clarified in Proposition 2–Proposition 3 and Theorem 3–Theorem 5, can be used in the qualitative analysis of solutions to the following abstract integral inclusion

$$\begin{array}{c}\hfill u\left(t\right)\in \mathcal{A}{\int }_{-\infty }^{t}a(t-s)u\left(s\right)ds+f\left(t\right),t\in \mathbb{R},\end{array}$$

where $a\in L_{loc}^1\left([0,\infty )\right),$ $a\ne 0,$ $\check{f}:\mathbb{R}\to X$ is Stepanov $p\left(x\right)$-uniformly recurrent (Doss-$(p,\varphi ,F)$-almost-periodic/Doss-${(p,\varphi ,F)}_i$-almost-periodic, $i=1,2$) and $\mathcal{A}$ is a closed multivalued linear operator on $X;$ furthermore, the same results can be applied in the qualitative analysis of solutions for numerous various classes of abstract Volterra integro-differential equations and inclusions, like

$$D^{\alpha }u\left(t\right)=Au\left(t\right)+{\int }_{-\infty }^{t}a(t-s)Au\left(s\right)ds+f(t,u\left(t\right)),t\in \mathbb{R},$$

where $D^{\alpha }u\left(t\right)$ denotes the Weyl-Liouville fractional derivative of order $\alpha >0,$ $a\in L_{loc}^1\left([0,\infty )\right)$ is a scalar-valued kernel, the function $f(·,·)$ enjoys some properties and A generates a non-degenerate $\alpha$-resolvent operator family on E (equations of this type arise in the study of heat flow in materials with memory and certain models in population dynamics and viscoelasticity; see [9] for more details).

In the following application to the semilinear fractional Cauchy inclusions in finite-dimensional spaces, we will continue our recent analysis from ([16], Section 5) (concerning fractional calculus and fractional differential equations, we can recommend for the reader [24,25,26,27] and references cited therein). Consider the space $X:={\mathbb{C}}^n,$ where $n\ge 2.$ Suppose that $c>0$, $A,B\in {\mathbb{C}}^{n,n}$ (the space of all complex matrices of format $n\times n$), the matrix B is not invertible, as well as that the degree of complex polynomial $P\left(\lambda \right):=\det (\lambda B-A),$ $\lambda \in \mathbb{C}$ is equal to n and its roots belong to the region $\{\lambda \in \mathbb{C}:\Re \lambda <-c(\left|\Im \lambda \right|+1\left)\right\}.$ We know that the multivalued linear operator $\mathcal{A}=A{B}^{-1}$ generates an exponentially decaying strongly continuous degenerate semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ which can be analytically extended to a sector around positive real axis (cf. [27] for the notion and more details). Let $0<\gamma <1$ and $\nu >-1.$ Define

$$T_{\gamma ,\nu }\left(t\right)x:=t^{\gamma \nu }{\int }_0^{\infty }{s}^{\nu }{\mathsf{\Phi }}_{\gamma }\left(s\right)T\left(s{t}^{\gamma }\right)xds,t>0,x\in X,$$

$$S_{\gamma }\left(t\right):=T_{\gamma ,0}\left(t\right),P_{\gamma }\left(t\right):=\gamma T_{\gamma ,1}\left(t\right)/t^{\gamma },t>0,$$

where ${\mathsf{\Phi }}_{\gamma }(·)$ denotes the Wright function, and $R_{\gamma }\left(t\right):=t^{\gamma -1}{P}_{\gamma }\left(t\right),$ $t>0.$ Then we know that

$$\parallel R_{\gamma }\left(t\right)\parallel =O\left(t^{\gamma -1}+t^{-\gamma -1}\right),t>0.$$

(18)

Of concern is the following abstract fractional inclusion

$$D_{+}^{\gamma }\overrightarrow{u}\left(t\right)\in -\mathcal{A}\overrightarrow{u}\left(t\right)+F(t,\overrightarrow{u}\left(t\right)),t\in \mathbb{R},$$

(19)

where $D_{+}^{\gamma }u\left(t\right)$ denotes the Weyl-Liouville fractional derivative of order $\gamma$ and $F:\mathbb{R}\times X\to X;$ after the usual substitution $\overrightarrow{v}\left(t\right)\in B^{-1}\overrightarrow{u}\left(t\right),$ $t\in \mathbb{R},$ this inclusion becomes

$$D_{+}^{\gamma }\left[B\overrightarrow{v}\left(t\right)\right]=-A\overrightarrow{v}\left(t\right)+F\left(t,B\overrightarrow{v}\left(t\right)\right),t\in \mathbb{R}.$$

We say that a continuous function $u:\mathbb{R}\to X$ is a mild solution of (19) if and only if

$$\overrightarrow{u}\left(t\right)={\int }_{-\infty }^t{R}_{\gamma }(t-s)F\left(s,\overrightarrow{u}\left(s\right)\right)ds,t\in \mathbb{R}.$$

Fix now a strictly increasing sequence $\left({\alpha }_n\right)$ of positive reals tending to plus infinity, and set

$$BU{R}_{\left({\alpha }_n\right)}(\mathbb{R}:X):=\left\{\overrightarrow{u}\in UR(\mathbb{R}:X);\overrightarrow{u}(·)\mathrm{is} \mathrm{bounded} \mathrm{and} \left(1\right) \mathrm{holds} \mathrm{with} f=\overrightarrow{u}\right\}.$$

Equipped with the metric $d(·,·):={\parallel ·-·\parallel }_{\infty },$ $BU{R}_{\left({\alpha }_n\right)}(\mathbb{R}:X)$ is a complete metric space.

Now we can state the following result, which is very similar to ([16], Theorem 5.1):

Theorem 6.

Suppose that the function $F:\mathbb{R}\times X\to X$ satisfies that for each bounded subset B of X there exists a finite real constant $M_B>0$ such that ${sup}_{t\in \mathbb{R}}{sup}_{y\in B}\parallel F(t,y)\parallel \le M_B.$ Suppose, further, that $p,r\in \mathcal{P}\left(\right[0,1\left]\right),$ the function $F:\mathbb{R}\times X\to X$ is Stepanov $p\left(x\right)$-uniformly recurrent, $r(·)\ge max\left(p\right(·),p(·)/(p(·)-1\left)\right)$ and there exists a function $L_F\in L_S^{r\left(x\right)}\left(I\right)$ is such that $q\left(x\right):=p\left(x\right)r\left(x\right)/\left(p\right(x)+r(x\left)\right)>1$ for a.e. $x\in \mathbb{R}$ and (4) holds with $I=\mathbb{R}.$ If there exist a positive real number $q^{\prime }>0$ and an integer $n\in \mathbb{N}$ such that $(\gamma -1)q^{\prime }>-1$ and $q\left(x\right)/(q\left(x\right)-1)\le q^{\prime }$ for a.e. $x\in \mathbb{R},$ and $M_n<1,$ where

$$\begin{array}{cc}\hfill M_n:=& \underset{t\ge 0}{sup}{\int }_{-\infty }^t{\int }_{-\infty }^{x_n}···{\int }_{-\infty }^{x_2}∥R_{\gamma }(t-x_n)∥\hfill \\ & \times \prod _{i=2}^n∥R_{\gamma }(x_i-x_{i-1})∥\prod _{i=1}^n{L}_F\left(x_i\right)d{x}_{1}d{x}_2···d{x}_n,\hfill \end{array}$$

and for every compact set $K\subseteq Y,$ (5) holds, then the abstract fractional Cauchy inclusion (19) has a unique bounded uniformly recurrent solution.

Proof. 

We will only outline the main details of proof. Define $\mathsf{{\rm Y}}:BU{R}_{\left({\alpha }_n\right)}(\mathbb{R}:X)\to BU{R}_{\left({\alpha }_n\right)}(\mathbb{R}:X)$ by

$$\left(\mathsf{{\rm Y}}\overrightarrow{u}\right)\left(t\right):={\int }_{-\infty }^t{R}_{\gamma }(t-s)F(s,\overrightarrow{u}\left(s\right))ds,t\in \mathbb{R}.$$

Suppose that $\overrightarrow{u}\in BU{R}_{\left({\alpha }_n\right)}(\mathbb{R}:X).$ Then $R\left(\overrightarrow{u}\right)=B$ is a bounded set, so that the mapping $t\mapsto F(t,\overrightarrow{u}\left(t\right)),$ $t\in \mathbb{R}$ is bounded. Applying Theorem 1, we have that the function $F(·,\overrightarrow{u}(·))$ is Stepanov $q\left(x\right)$-uniformly recurrent. Define $q^{\prime }\left(x\right):=q\left(x\right)/(q\left(x\right)-1)$ for a.e. $x\in \mathbb{R}$. Then (18) and the prescribed assumptions imply that $\parallel R_{\gamma }(·)\parallel \in L^{q^{\prime }\left(x\right)}[0,1]$ and ${\sum }_{k=0}^{\infty }{\parallel R_{\gamma }(·)\parallel }_{L^{q^{\prime }\left(x\right)}[k,k+1]}<\infty .$ Applying Proposition 2, we get that the function $t\mapsto {\int }_{-\infty }^t{R}_{\gamma }(t-s)F(s,\overrightarrow{u}\left(s\right))ds,$ $t\in \mathbb{R}$ is uniformly recurrent. It can be simply verified that this function is also bounded continuous so that $\mathsf{{\rm Y}}\overrightarrow{u}\in BU{R}_{\left({\alpha }_n\right)}(\mathbb{R}:X)$ and the mapping $\mathsf{{\rm Y}}(·)$ is well defined. A simple calculation shows that

$${∥\left({\mathsf{{\rm Y}}}^n{\overrightarrow{u}}_1\right)-\left({\mathsf{{\rm Y}}}^n{\overrightarrow{u}}_2\right)∥}_{\infty }\le M_n{∥\overrightarrow{u_1}-\overrightarrow{u_2}∥}_{\infty },\overrightarrow{u_1},\overrightarrow{u_2}\in BU{R}_{\left({\alpha }_n\right)}(\mathbb{R}:X),n\in \mathbb{N}.$$

Since we have assumed that $M_n<1,$ the well-known extension of the Banach contraction principle shows that the mapping $\mathsf{{\rm Y}}(·)$ has a unique fixed point. This completes the proof of theorem. □

7. Conclusions

In this paper, we have analyzed generalized almost-periodicity in Lebesgue spaces with variable exponents. The classes of Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents are introduced and thoroughly investigated. Basically, our theoretical results regarding the invariance of generalized almost-periodicity with variable exponents can be applied at any place where the generalized variation of parameters formula can be employed. An interesting application to the abstract fractional semilinear Cauchy inclusions is provided.