Fractional Retarded Dynamic Equations on Time Scales with Δ-HKP Integral

1. Introduction

Differential equations constitute one of the principal mathematical tools for describing dynamic phenomena arising in natural sciences, engineering, economics, and many interdisciplinary areas. In numerous realistic models, however, the evolution of a system depends not only on its current state but also on its past history. Such memory effects naturally occur in population dynamics, where the growth rate depends on maturation time, in epidemiology through incubation periods, in economics where investment decisions and business cycles are influenced by previous values of macroeconomic indicators, and in control theory where delayed feedback plays a fundamental role in the system dynamics. These observations have led to the extensive development of the theory of retarded differential equations, which provides an appropriate mathematical framework for modeling hereditary systems and processes with aftereffects. During the last decades, fractional calculus has become an increasingly important tool for modeling phenomena involving memory and hereditary properties. In contrast to classical integer-order derivatives, fractional derivatives incorporate information from the entire history of a process and therefore provide a more realistic description of many physical and engineering systems. Applications of fractional differential equations include anomalous diffusion, viscoelastic materials, thermal processes, electrical circuits, biological systems, and financial mathematics [1,7,8,12,38]. Owing to their nonlocal character, fractional models have proved to be more accurate than classical models in describing complex dynamical behavior. The study of fractional differential equations in Banach spaces has significantly broadened the applicability of fractional calculus to infinite-dimensional systems. In particular, Salem and co-authors initiated a systematic investigation of fractional differential equations under weak topological assumptions, employing methods of functional analysis and measures of weak noncompactness to establish existence results [32]. Their paper stimulated extensive research on weak solutions and pseudosolutions for various classes of fractional differential and integrodifferential equations. Another important direction of research is provided by the theory of time scales, introduced by Hilger to unify differential and difference equations within a common mathematical framework. The theory of time scales enables the simultaneous treatment of continuous, discrete, and hybrid dynamical systems and has found numerous applications in engineering, biology, economics, and control theory [3,4,5,10,11,13,14,23,24,26,31,34,35]. As a consequence, fractional calculus on time scales has emerged as a natural extension of classical fractional analysis, allowing the description of memory effects in systems evolving on hybrid temporal structures. In many practical applications, however, memory effects arise from two independent mechanisms. The first one is represented by the fractional derivative, reflecting the hereditary nature of the process, whereas the second originates from explicit delays describing the dependence of the present state on past values of the unknown function. Such phenomena occur in neural networks with transmission delays, physiological control systems, epidemiological models involving incubation periods, ecological systems with maturation delays, and engineering systems with delayed feedback. Consequently, fractional retarded dynamic equations on time scales provide a natural mathematical model for systems possessing hereditary properties, explicit delays, and hybrid continuous-discrete temporal evolution simultaneously. From the analytical point of view, the investigation of such equations presents substantial difficulties. Nonlinear mappings arising in applications frequently satisfy only weak sequential continuity assumptions and fail to possess compactness properties required by classical fixed point methods. Furthermore, many vector-valued functions occurring in practical models are not absolutely integrable, which limits the applicability of Lebesgue or Bochner integration theories. In this context, the Henstock–Kurzweil integral and its Banach-space extension, namely the Henstock–Kurzweil–Pettis integral, provide an effective analytical framework for studying vector-valued equations under considerably weaker regularity assumptions [14,15]. In recent years, fractional delay differential equations have attracted considerable attention. Numerous existence, uniqueness, and stability results have been established for various classes of equations involving Caputo and Riemann–Liouville derivatives, including both linear and nonlinear models. Particular emphasis has been placed on asymptotic stability, Hyers–Ulam stability, and qualitative properties of solutions for systems involving multiple delays. These equations have also found important applications in biological, epidemiological, and physical models where memory and delayed interactions coexist [6,9,18,19,22,36,37]. Recent studies have demonstrated a growing interest in fractional dynamic equations on time scales, including existence, stability, and qualitative properties of solutions under various assumptions [29,33]. Furthermore, abstract dynamic equations in Banach spaces and generalized integro-dynamic models on time scales have recently been investigated using fixed point techniques and measures of noncompactness [2,30]. Nevertheless, the simultaneous incorporation of explicit delays, Caputo fractional dynamics, the $\Delta$-Henstock–Kurzweil–Pettis integral, and weak topological methods has not yet been addressed. In particular, the theory of fractional retarded dynamic equations on time scales involving the $\Delta$-HKP integral and formulated in Banach spaces endowed with the weak topology remains essentially undeveloped. Despite considerable progress in both theories, their simultaneous combination remains largely unexplored. Motivated by this gap in the existing literature, we investigate the fractional retarded dynamic equation on a time scale $\mathbb{T}$. In this paper, we study a class of fractional retarded dynamic equations on time scales in a Banach space E of the form

$${}_T{}^C\Delta ^{\alpha }x\left(t\right)=f(t,x_t),$$

(1.1)

with the initial condition

$$x\left(\theta \right)=\phi \left(\theta \right),\theta \in I_r=[-r,0]\cap \mathbb{T}.$$

where $x_t\left(\theta \right)=x(t+\theta )$, $r>0$ is a fixed real number, $\phi$ is a given function, and $\mathbb{T}$ denotes a time scale. Moreover, we assume that $f:I_a\times C(I_r,E)\to E$ is $\Delta$-HKP integrable. These equations incorporate memory effects resulting from both the delay and the fractional order of the derivative, as well as a hybrid time structure. The analysis is carried out in the weak topology using the De Blasi measure of weak noncompactness and fixed point methods for weakly sequentially continuous mappings. The main objective of this paper is to establish sufficient conditions for the existence of pseudosolutions to the considered problem. The obtained results extend existing contributions and contribute to the development of the modern theory of dynamic equations, providing tools for the analysis of complex systems with memory, delay, and continuous-discrete structure. The main contributions of this paper can be summarized as follows. We investigate a class of fractional retarded dynamic equations on time scales involving the $\Delta$-Henstock–Kurzweil–Pettis integral in Banach spaces endowed with the weak topology. The considered model combines three important features of modern mathematical models, namely fractional dynamics, explicit delay effects, and hybrid continuous–discrete temporal structures, within a unified analytical framework. Our existence result is established under weak sequential continuity assumptions by employing the De Blasi measure of weak noncompactness together with Kubiaczyk’s fixed point theorem. This approach avoids the compactness assumptions commonly required in classical existence theories and allows the treatment of a broader class of nonlinear operators. Consequently, the obtained results substantially extend several previously independent research directions and provide a unified framework for the analysis of hybrid dynamical systems with hereditary effects and explicit delays.

2. Preliminaries

Let $(E,\parallel ·\parallel )$ be a Banach space and let $E^{*}$ be its dual space. Denote by $(C(I_a,E),\omega )$ the space of all continuous functions from $I_a$ to E endowed with the topology

$$\sigma \left(C(I_a,E),C{(I_a,E)}^{*}\right),$$

and by $C_{rd}(I_a,E)$ denote the space of all rd-continuous functions from the time scale interval $I_a$ to E. By ${\mu }_{\Delta }$ we denote the Lebesgue measure on the time scale $\mathbb{T}$. For a precise definition and basic properties of this measure, we refer the reader to [13].

We now gather some well-known definitions and results from the literature, which we will use throughout this paper.

I

For the covenience of the reader and to facilitate the exposition, we introduce some preliminary definitions and notations related to time scales, commonly found in the literature (see references [10,11,16,20,21] and the cited papers therein).

A time scale $\mathbb{T}$ is a nonempty closed subset of real numbers $\mathbb{R}$, with the subspace topology inherited from the standard topology of $\mathbb{R}$.

By an interval $I_a$ we mean the time scale interval

$$I_a=[0,a]\cap \mathbb{T}=\{t\in \mathbb{T}:0\le t\le a\}={[0,a]}_{\mathbb{T}}.$$

Throughout this paper, all intervals are understood as intervals on a time scale.

Definition 1.

The forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ and the backward jump operator $\rho :\mathbb{T}\to \mathbb{T}$ as $\sigma \left(t\right)=inf\{s\in \mathbb{T}:s>t\}$ and $\rho \left(t\right)=sup\{s\in \mathbb{T}:s<t\}$, respectively. We put $inf\varnothing =inf\mathbb{T}$ (i.e. $\rho \left(m\right)=m$ if $\mathbb{T}$ has a minimum m).

The jump operators $\sigma$ and $\rho$ allow the classification of points in time scale in the following way: t is called right dense, right scattered, left dense, left scattered, dense and isolated if $\sigma \left(t\right)=t,\sigma \left(t\right)>t,\rho \left(t\right)=t,\rho \left(t\right)<t,\rho \left(t\right)=t=\sigma \left(t\right),\rho \left(t\right)<t<\sigma \left(t\right),$ respectively.

Definition 2.

We say that a function k is right-dense continuous (rd-continuous) if k is continuous at every right-dense point $t\in \mathbb{T}$ and

$$\underset{s\to t^{-}}{lim}k\left(s\right)$$

exists and is finite at every left-dense point $t\in \mathbb{T}$.

Next, we define the so-called $\Delta$-derivative.

Definition 3.

Fix $t\in \mathbb{T}$. Let $f:I_a\to E$. Then we define Δ-derivative of f by

$$f^{\Delta }\left(t\right)=\underset{s\to t}{lim}{\displaystyle \frac{f\left(\sigma \right(t\left)\right)-f\left(s\right)}{\sigma \left(t\right)-s}}.$$

The function f is called Δ-differentiable on $\mathbb{T}$, if for each $t\in \mathbb{T}$ there exists $f^{\Delta }\left(t\right)$.

Note that

(1)

$f^{\Delta }=f^{\prime }$ is the usual derivative if $\mathbb{T}=\mathbb{R}$,

(2)

$f^{\Delta }=\Delta f$ is the usual forward difference operator if $\mathbb{T}=\mathbb{Z}$,

(3)

$f^{\Delta }=D_{q}f={\displaystyle \frac{f\left(qt\right)-f\left(t\right)}{(q-1)t}}$ is the q-derivative or the delta derivative on the q-time scale if $\mathbb{T}=q^{{\mathbb{N}}_0}=\{q^t:t\in {\mathbb{N}}_0,q>1\}$.

Hence, the time scale allows us to consider the unification of differential, difference and q-difference equations as particular cases. However, our results also hold for more exotic time scales, which appear in fields such as mathematical biology or economics (see [10,11], for instance).

II

As in classical case, we need to introduce vector-valued Henstock–Kurzweil $\Delta$-integrals. Definitions and basic properties of non absolute integrals were presented in [14].

We will use the notation

$$\eta \left(t\right):=\sigma \left(t\right)-t$$

where $\eta$ is called the graininess function and

$$\nu \left(t\right):=t-\rho \left(t\right),$$

where $\nu$ is called the left-graininess function.

We say that $\delta =({\delta }_L,{\delta }_R)$ is a $\Delta$-gauge for time scale interval $[a,b]$ provided

$${\delta }_L\left(t\right)>0\mathrm{on}(a,b],$$

$${\delta }_R\left(t\right)>0\mathrm{on}[a,b),$$

$${\delta }_L\left(t\right)\ge 0,{\delta }_R\left(t\right)\ge 0$$

and

$${\delta }_R\left(t\right)\ge \eta \left(t\right)$$

for all $t\in [a,b)$.

We say that a partition D for a time scale interval $[a,b]$ given by

$$D=\{a=t_0\le {\xi }_1\le t_1\le \dots \le t_{n-1}\le {\xi }_n\le t_n=b\}$$

with $t_i>t_{i-1}$, for $1\le i\le n$ and $t_i,{\xi }_i\in \mathbb{T}$ is $\delta$-fine if

$${\xi }_i-{\delta }_L\left({\xi }_i\right)\le t_{i-1}<t_i\le {\xi }_i+{\delta }_R\left({\xi }_i\right),$$

for $1\le i\le n$.

Definition 4.

A function

$$f:{[a,b]}_{\mathbb{T}}\to E$$

is the Henstock–Kurzweil Δ-integrable on ${[a,b]}_{\mathbb{T}}$ (Δ-HK integrable in short) if there exists a function

$$F:{[a,b]}_{\mathbb{T}}\to E,$$

defined on the subintervals of ${[a,b]}_{\mathbb{T}}$, satisfying the following property: given $\epsilon >0$ there exists a positive function δ on ${[a,b]}_{\mathbb{T}}$ such that $D=\{{[u,v]}_{\mathbb{T}},\xi \}$ is δ-fine division of a ${[a,b]}_{\mathbb{T}}$, we have

$$∥\sum _D\left(f\left(\xi \right)(v-u)-\left(F\right(v)-F(u\left)\right)\right)∥<\epsilon .$$

Definition 5.

The function $f:I_a\to E$ is Δ–Henstock–Kurzweil–Pettis integrable (Δ–HKP integrable for short) if

(1.)

$\forall x^{*}\in E^{*},x^{*}f$ is Henstock–Kurzweil Δ–integrable on $I_a$,

(2.)

$\forall t\in I_a\forall x^{*}\in E^{*},x^{*}g\left(t\right)=(\Delta -\mathrm{HK}){\int }_0^t{x}^{*}f\left(s\right)\Delta s.$

The function g will be called a primitive of f and by $g\left(t\right)=(\Delta -\mathrm{HKP}){\int }_0^{t}f\left(s\right)\Delta s$ we will denote the Δ-Henstock–Kurzweil–Pettis integral of f on the interval $I_a$.

In [14] the author gives examples of $\Delta$–Henstock–Kurzweil–Pettis integrable functions which are not integrable in the sense of Pettis and Henstock–Kurzweil on time scales.

Theorem 1.

Suppose that $f,f_n:[a,b]\to E,n=1,2,\dots$, are Δ-HKP integrable functions. Let $F_n$ be a primitive of $f_n$. If one assumes that:

(1.)

$\forall x^{*}\in E^{*},x^{*}{f}_n\left(x\right)\to x^{*}f\left(x\right){\mu }_{\Delta }-\mathrm{almost}\mathrm{everywhere}\mathrm{on}{I}_a$,

(2.)

$\forall x^{*}\in E^{*}$ the family $G=\{x^{*}{F}_n:n=1,2,\dots \}$ is uniformly $AC{G}^{*}$ on $I_a$ (i.e., weakly uniformly $AC{G}^{*}$ on $I_a$),

(3.)

$\forall x^{*}\in E^{*}$ the set G is equicontinuous on $I_a$,

then f is Δ-HKP integrable on $I_a$ and ${\int }_0^t{f}_n\left(s\right)\Delta s$ tends weakly in E to ${\int }_0^{t}f\left(s\right)\Delta s$ for each $t\in I_a$.

Theorem 2.

[Mean Value Theorem] For each Δ-subinterval $[c,d]\subset [a,b]$, if the integral

$$(\Delta -\mathrm{HKP}){\int }_c^{d}y\left(s\right)\Delta s$$

exists then we have

$$(\Delta -\mathrm{HKP}){\int }_c^{d}y\left(s\right)\Delta s\in {\mu }_{\Delta }\left([c,d]\right)·\overline{conv}y\left([c,d]\right),$$

where

$$\overline{conv}y\left([c,d]\right)$$

denotes the close convex hull of the set $y\left(\right[c,d\left]\right)$.

For completeness we introduce the definitions of the Caputo derivative of fractional order.

Definition 6.

Suppose that $\mathbb{T}$ is a time scale. The Caputo fractional derivative of g is defined by

$${}_{\mathbb{T}}{}^C\Delta ^{\alpha }g\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_0^t{(t-s)}^{n-\alpha -1}{g}^{{\Delta }^n}\left(s\right)\Delta s,t\in I_a,$$

where $n=\left[\alpha \right]+1$ and $\left[\alpha \right]$ denote the integer part of α and integral is taken in the sense of Δ-HKP, Γ is the Gamma function.

Definition 7.

Suppose that $\mathbb{T}$ is a time scale, $g:I\to E$ is Δ-HKP integrable function. The fractional Δ-HKP integral of the order $\alpha \in {\mathbb{R}}^{+}$ of g is defined by

$$I^{\alpha }g\left(t\right)={\int }_a^t{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}g\left(s\right)\Delta s,$$

where integral is taken in the sense of Δ-HKP and Γ is the Gamma function.

III

Our fundamental tool is the deBlasi measure of weak noncompactness $\beta \left(A\right)$.

The deBlasi measure of weak noncompactness $\beta \left(A\right)$ is defined by

$$\beta \left(A\right)=inf\{t>0:\mathrm{there}\mathrm{exists}C\in K^{\omega }\mathrm{such}\mathrm{that}A\subset C+t{B}_0\},$$

where $K^{\omega }$ is the set of weakly compact subsets of E and $B_0$ is the norm unit ball in E.

The properties of the measure of noncompactness $\beta \left(A\right)$ are as follows:

(i)

if $A\subset B$ then $\beta \left(A\right)\le \beta \left(B\right)$;

(ii)

$\beta \left(A\right)=0$ if and only if A is relatively weakly compact;

(iii)

$\beta (A\cup B)=max\left\{\beta \right(A),\beta (B\left)\right\}$;

(iv)

$\beta \left({\overline{A}}^{\omega }\right)=\beta \left(A\right)$, where ${\overline{A}}^{\omega }$ denotes the weak closure of A;

(v)

$\beta \left(\lambda A\right)=\left|\lambda \right|\beta \left(A\right)$, $(\lambda \in \mathbb{R})$;

(vi)

$\beta (A+B)\le \beta \left(A\right)+\beta \left(B\right)$;

(vii)

$\beta (conv(A\left)\right)=\beta \left(A\right)$, where $conv\left(A\right)$ denotes the convex extension of A.

Theorem 3.

[26] Let $H\subset C(I_a,E)$ be a family of strongly equicontinuous functions. Let $H\left(t\right)=\left\{h\right(t)\in E:h\in H\},$ for $t\in I_a$ and $H\left(I_a\right)={\bigcup }_{t\in I_a}H\left(t\right)$. Then

$${\beta }_C\left(H\right)=\underset{t\in I_a}{sup}\beta \left(H\left(t\right)\right)=\beta \left(H\left(I_a\right)\right),$$

where ${\beta }_C\left(H\right)$ denotes the measure of noncompactness in $C(I_a,E)$, and the function $t\mapsto \beta \left(H\right(t\left)\right)$ is continuous.

Definition 8.

A function $f:I_a\to E$ is said to be weakly continuous if it is continuous from $I_a$ to E endowed with its weak topology. A function $g:E\to E_1$ where E and $E_1$ are Banach spaces, is said to be weakly sequentially continuous if for each weakly convergent sequence $\left(x_n\right)$ in E, the sequence $\left(g\left(x_n\right)\right)$ is weakly convergent in $E_1$.

When the sequence $x_n$ tends weakly to $x_0$ in E, we will write $x_n\stackrel{\omega }{\to }{x}_0$.

Definition 9.

[21] A family $\mathcal{F}$ of functions F is said to be uniformly absolutely continuous in the restricted sense on A or in short uniformly $A{C}_{*}\left(A\right)$, if for every $\epsilon >0$ there is $\eta >0$, such that for every F in $\mathcal{F}$ and for every finite or infinite sequence of non-overlapping intervals $\left\{[a_i,b_i]\right\}$ with $a_i,b_i\in A$, and satisfying ${\sum }_i|b_i-a_i|<\eta ,$ we have ${\sum }_i\omega (F,[a_i,b_i])<\epsilon ,$ where ω denotes the oscillation of F over $[a_i,b_i]$.

A family $\mathcal{F}$ of functions F is said to be uniformly generalized absolutely continuous in the restricted sense on $[a,b]$ or uniformly $AC{G}_{*}\left([a,b]\right)$ if $[a,b]$ is the union of a sequence of closed sets $A_i$ such that on each $A_i$ the function F is uniformly $A{C}_{*}\left(A_i\right)$.

In the proof of the main theorem we will apply the following fixed point theorem.

Theorem 4.

[27] Let X be a metrizable locally convex topological vector space. Let D be a closed convex subset of X, and let F be a weakly-weakly sequentially continuous map from D into itself. If for some $x\in D$ the implication that

$$\overline{V}=\overline{conv}(\left\{x\right\}\cup F\left(V\right))⟹V\mathrm{is}\mathrm{relatively}\mathrm{weakly}\mathrm{compact}$$

(2.1)

holds for every subset V of D, then F has a fixed point.

3. Main Problem

Let $r,a$ be nonnegative real numbers, $I_a={[0,a]}_{\mathbb{T}}$, $a>0$. Let x be some function defined on ${[-r,a]}_{\mathbb{T}}$. For any $t\in I_a$, the function $x_t$ is defined as $x_t\left(\theta \right)=x(\theta +t),$ where $\theta \in I_r={[-r,0]}_{\mathbb{T}}.$ Here $\theta$ may be a function involving t.

Let $f:I_a\times C(I_r,E)\to E$ and

$${}_{\mathbb{T}}{}^C\Delta ^{\alpha }x\left(t\right)=f(t,x_t),$$

$$x\left(\theta \right)=\phi \left(\theta \right),$$

where $\alpha \in (0,1]$ and $\phi$ is some specified function.

We will consider the problem

$$x\left(t\right)=\phi \left(\theta \right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f(s,x_s)\Delta s,t\in I_a,$$

(3.1)

where the integral is taken in the sense of fractional $\Delta$-HKP integrals.

Fix $x^{*}\in E^{*}$ and consider the problem

$${}_{\mathbb{T}}{}^C\Delta ^{\alpha }\left(x^{*}x\right)\left(t\right)=x^{*}\left(f(t,x_t)\right),x\left(\theta \right)=\phi \left(\theta \right),t\in I_a.$$

Let us introduce a definition.

Definition 10.

Let $F:I\to E$ and let $A\subset I$. The function $f:A\to E$ is a fractional pseudo Δ-derivative of F on A if for each $x^{*}\in E^{*}$ the real-valued function $x^{*}F$ is ${}_{\mathbb{T}}{}^C\Delta ^{\alpha }$-differentiable ${\mu }_{\Delta }$ almost everywhere on A and ${}_{\mathbb{T}}{}^C\Delta ^{\alpha }\left(x^{*}F\right)=x^{*}f{\mu }_{\Delta }-\mathrm{almost}\mathrm{everywhere}\mathrm{on}A.$

Regarding the above definition it is clear that the left-hand side can be rewritten to the form $x^{*}\left({}_{\mathbb{T}}{}^C\Delta ^{\alpha }x\left(t\right)\right),$ where ${}_{\mathbb{T}}{}^C\Delta ^{\alpha }$ denotes the fractional pseudo $\Delta$-derivative.

To obtain the existence result for our problem it is necessary to define a notion of a solution.

Definition 11.

A function $x:I_a\to E$ is said to be a pseudosolution of the problem (1) if it satisfies the following conditions:

1.

$x(·)$ is an $AC{G}^{*}$ function,

2.

$x\left(\theta \right)=\phi \left(\theta \right)$,

3.

for each $x^{*}\in E^{*}$ there exists a set $A\left(x^{*}\right)$ with ${\mu }_{\Delta }$ measure zero, such that for each $t\notin A\left(x^{*}\right)$,

$${}_{\mathbb{T}}{}^C\Delta ^{\alpha }\left(x^{*}x\right)\left(t\right)=x^{*}\left(f(t,x_t)\right).$$

Definition 12.

A continuous function $x:I_a\to E$ is said to be a solution to problem (3) if it satisfies (3) for every $t\in I_a$.

Now we prove an existence theorem for the problem (3).

Two functions ${\phi }_1,{\phi }_2$ which are $\Delta$-HKP integrable on some interval ${[u,v]}_{\mathbb{T}}$ are said to belong to the same equivalence class if

$${\phi }_1\left(t\right)={\phi }_2\left(t\right)$$

almost everywhere in ${[u,v]}_{\mathbb{T}}$.

Let $H[u,v]$ denote the space of equivalence classes of functions which are $\Delta$-HKP integrable on ${[u,v]}_{\mathbb{T}}$. The norm ${\parallel ·\parallel }_H$ on $H[u,v]$ is defined as follows: for $P\in H[u,v]$,

$${\parallel P\parallel }_H=\underset{t\in [u,v]\cap \mathbb{T}}{sup}\parallel \Phi \left(t\right)\parallel ,$$

where

$$\Phi \left(t\right)={\int }_u^t\varphi \left(s\right)\Delta s$$

for any $\varphi \in P$.

Let $\phi$ be some function fixed in $H[-r,0]$, where $r>0$. The sets ${\Omega }_b$ and $R_{a,b}$ are defined as

$${\Omega }_b={\{x\in H[-r,0]:\parallel x-\phi \parallel }_H\le b\},$$

$$R_{a,b}=I_a\times {\Omega }_b,$$

where $a,b$ are positive numbers.

Continuity here is understood in the sense that if $\left\{x_n\right\}$, $n=1,2,\dots$, is a sequence in ${\Omega }_b$ and $x_n\left(s\right)$ converges uniformly on ${[-r,0]}_{\mathbb{T}}$ to some $x_0\in {\Omega }_b$ as $n\to \infty$, then for almost all $t\in I_a$, $f(t,x_n)$ converges to $f(t,x_0)$ as $n\to \infty$.

It is convenient here to introduce an auxiliary function $\widehat{x}$: if x is defined on $I_c=[0,c]\cap \mathbb{T}$ with $x\left(0\right)=\phi \left(0\right)$, the function $\widehat{x}$ is defined as:

$${\widehat{x}}_t=\left\{\begin{array}{cc}x\left(t\right),\hfill & t\in I_c,\hfill \\ \phi \left(t\right),\hfill & t\in I_r.\hfill \end{array}\right.$$

The set

$$A(\phi ,a)=\left\{x\in (C(I_a,E),\omega ):x\left(0\right)=\phi \left(0\right),\parallel x\parallel \le b+\parallel \phi \left(0\right)\parallel ,{\widehat{x}}_t\in {\Omega }_b\right\}.$$

It is easy to see that the set $A(\phi ,a)$ is bounded, closed and convex.

Let

$$F:(C(I_a,E),\omega )\to (C(I_a,E),\omega )$$

be defined by

$$F\left(x\right)\left(t\right)=\phi \left(\theta \right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f(s,{\widehat{x}}_s)\Delta s,$$

for $t\in I_a$ and $x\in A(\phi ,a)$, where the integral is taken in the sense of $\Delta$-HKP.

Moreover, let

$$K=\{F\left(x\right)\in (C(I_a,E),\omega ):x\in A(\phi ,a)\}.$$

Theorem 5.

Let $\phi \in H[-r,0]$ be a fixed function. Let E be a Banach space with norm $\parallel ·\parallel$. Assume that for every $AC{G}^{*}$ function $x:I_a\to E$:

1.

the function $f(t,x_t)$ is Δ-HKP integrable;

2.

for each $t\in I_a$, the mapping $f(t,·)$ is weakly–weakly sequentially continuous on $R_{a,b}$;

3.

for every bounded set $X\subset E$ and every time scale interval $I\subset I_a$,

$$\beta \left(f\right(I,X\left)\right)\le L\beta \left(X\right),$$

(3.2)

where β denotes the De Blasi measure of weak noncompactness and the constant $L>0$ satisfies

$${\displaystyle \frac{L}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\Delta s<1.$$

(3.3)

Suppose additionally that the set K is equicontinuous and uniformly $AC{G}^{*}$ on $I_a$. Then there exists a pseudo-solution of the problem (3) on the interval $I_c={[0,d]}_{\mathbb{T}}$, for some $0<d\le a$.

Proof. 

We will prove, in fact, the existence of a solution for the problem (3) because each solution of the problem (3) is the solution of the problem (1).

Fix an arbitrary $b\ge 0$. Recall, that the set K of continuous function $F\left(x\right)\in K$ defined on a time scale interval $I_a$ is equicontinuous on $I_a$ if for each $\epsilon >0$ there exists $\delta >0$ such that

$$\parallel F\left(x\right)\left(t\right)-F\left(x\right)\left(\tau \right)\parallel <\epsilon$$

for all $x\in A(\phi ,a)$ whenever $|t-\tau |<\delta$, $t,\tau \in I_a$, for each $F\left(x\right)\in K$.

Thus, for each $\epsilon >0$ there exists $\delta >0$ such that

$$∥{\int }_{\tau }^t{(t-s)}^{\alpha -1}f(s,x\left(s\right))\Delta s∥<\epsilon ,$$

for all $x\in A(\phi ,a)$, whenever $|t-\tau |<\delta$ and $t,\tau \in I_a$.

As a result, there exists a number d, $0<d\le a$, such that

$$∥{\int }_0^t{(t-s)}^{\alpha -1}f(s,{\widehat{x}}_s)\Delta s∥\le b,$$

$${\int }_{-\tau }^{\tau }{(t-s)}^{\alpha -1}[\phi \left(0\right)-\phi \left(s\right)]\Delta s<k,$$

$${\int }_{-\tau }^{\tau }{\int }_0^{t+s}{(t-p)}^{\alpha -1}f(p,{\widehat{x}}_p)\Delta p\Delta s<l,$$

$$k+l=b,t\in I_d,x\in A(\phi ,a).$$

We will show that the operator F is well defined and maps $A(\phi ,a)$ into $A(\phi ,a)$.

To see this note for any $x^{*}\in E^{*}$, such that $\parallel x^{*}\parallel \le 1$, for any $x\in A(\phi ,a)$ and $t\in I_d$ we have:

$$\begin{array}{cc}\hfill |x^{*}F\left(x\right)\left(t\right)|& =|x^{*}\phi \left(0\right)|+\left|x^{*}\left({\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f(s,{\widehat{x}}_s)\Delta s\right)\right|\hfill \\ & \le \parallel x^{*}\parallel \parallel \phi \left(0\right)\parallel +\parallel x^{*}\parallel ∥{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{x}^{*}\left(f(s,{\widehat{x}}_s)\right)\Delta s∥\hfill \\ & \le \parallel x_0\parallel +\left|{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\right|b\le \parallel x_0\parallel +b.\hfill \end{array}$$

So

$$sup\left\{\right|x^{*}F\left(x\right)\left(t\right)|:x^{*}\in E^{*},\parallel x^{*}\parallel \le 1\}\le \parallel x_0\parallel +b$$

and as a result

$$\parallel F\left(x\right)\left(t\right)\parallel \le \parallel x_0\parallel +b.$$

Moreover,

$$\begin{array}{cc}\hfill \parallel \widehat{F}\left(x_t\right){-\phi \parallel }_H& =\underset{\tau \in I_r}{sup}∥{\int }_{-r}^{\tau }{(t-s)}^{\alpha -1}[\widehat{F}\left(x_t\right)\left(s\right)-\phi \left(s\right)]\Delta s∥\hfill \\ & =\underset{\tau \in I_r}{sup}∥{\int }_{-r}^{\tau }{(t-s)}^{\alpha -1}[\widehat{F}\left(x\right)(t+s)-\phi \left(s\right)]\Delta s∥\hfill \\ & =\underset{\tau \in I_r}{sup}∥{\int }_{-r}^{\tau }{(t-s)}^{\alpha -1}\left[\phi \left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{t+s}{(t-p)}^{\alpha -1}f(p,{\widehat{x}}_p)\Delta p-\phi \left(s\right)\right]\Delta s∥\hfill \\ & \le \underset{\tau \in I_r}{sup}∥{\int }_{-r}^{\tau }{(t-s)}^{\alpha -1}[\phi \left(0\right)-\phi \left(s\right)]\Delta s∥\hfill \\ & +\underset{\tau \in I_r}{sup}\left|{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\right|∥{\int }_{-r}^{\tau }{\int }_0^{t+s}{(t-p)}^{\alpha -1}f(p,{\widehat{x}}_p)\Delta p\Delta s∥\hfill \\ & \le k+l=b.\hfill \end{array}$$

We will show, that the operator F is weakly–weakly sequentially continuous. Assume that

$$x_n\to x\mathrm{in}C(I_d,E).$$

By Lemma 9 of [17], this means

$$x_n\left(t\right)\to x\left(t\right),\mathrm{for}\mathrm{every}t\in I_d.$$

Since $f(t,·)$ is weakly–weakly sequentially continuous,

$$f(s,{\widehat{x}}_{n,s})\to f(s,{\widehat{x}}_s),\mathrm{for}\mathrm{every}s\in I_d.$$

Hence, for every $x^{*}\in E^{*}$,

$$x^{*}\left(f(s,{\widehat{x}}_{n,s})\right)\to x^{*}\left(f(s,{\widehat{x}}_s)\right)$$

for ${\mu }_{\Delta }$ a.e. $s\in I_d$.

Moreover, by the assumptions of Theorem 5, the family of primitives $F_n\left(x\right)\left(t\right)$ is uniformly $AC{G}^{*}$ and equicontinuous on $I_d$.

Therefore all assumptions of Theorem 1 are fulfilled. Consequently,

$${\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f(s,{\widehat{x}}_{n,s})\Delta s\to {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f(s,{\widehat{x}}_s)\Delta s$$

weakly in E for every $t\in I_d$.

Thus

$$F\left(x_n\right)\left(t\right)\to F\left(x\right)\left(t\right),t\in I_d,$$

and hence

$$F\left(x_n\right)\to F\left(x\right)\mathrm{in}C(I_d,E).$$

Therefore F is weakly–weakly sequentially continuous.

Suppose that $V\subset A(\phi ,a)$ satisfies the condition

$$\overline{V}=\overline{conv}(\left\{x\right\}\cup F\left(V\right)).$$

We will prove that V is relatively weakly compact in $A(\phi ,a)$ and so (2) is satisfied.

Let, for $t\in I_a$,

$$V\left(t\right)=\left\{v\right(t)\in E:v\in V\}.$$

Put

$$\left\{{\int }_0^t{(t-s)}^{\alpha -1}f(s,{\widehat{x}}_s)\Delta s:x\in V\right\}={\int }_0^t{(t-s)}^{\alpha -1}f(s,V_s)\Delta s,$$

where

$$V_s=\{{\widehat{x}}_s:x\in V\},$$

and

$$F\left(V\left(t\right)\right)=\phi \left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}f(s,V_s)\Delta s.$$

Since $V\subset A(\phi ,a)$, $F\left(V\right)\subset K$. Then

$$V\subset \overline{V}=\overline{conv}(\left\{x\right\}\cup F\left(V\right))$$

is equicontinuous. By Theorem 3,

$$t\mapsto v\left(t\right)=\beta \left(V\right(t\left)\right)$$

is continuous on $I_d$.

For fixed $t\in I_d$ we divide the interval $[0,t]$ into m parts in the following way:

$$t_0=0,$$

$$t_1=\underset{s\in I_a}{sup}\{s:s\ge t_0,s-t_0<\delta \},$$

$$t_2=\underset{s\in I_a}{sup}\{s:s\ge t_1,s-t_1<\delta \},$$

$$\dots ,$$

$$t_n=\underset{s\in I_a}{sup}\{s:s\ge t_{n-1},s-t_{n-1}<\delta \}.$$

Since $\mathbb{T}$ is closed we have $t_i\in I_a$. If some $t_{i+1}=t_i$, then

$$t_{i+2}=inf\{t\in \mathbb{T}:t\ge t_{i+1}\}.$$

By the Theorem 2,

$$\begin{array}{cc}\hfill F\left(x\right)\left(t\right)& =\phi \left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left[{\displaystyle \sum _{i=0}^{m-1}}{\int }_{J_i}{(t_i-s)}^{\alpha -1}f(s,{\widehat{x}}_s)\Delta s\right]\hfill \\ & \in \phi \left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=0}^{m-1}}{\mu }_{\Delta }\left(J_i\right){(t_i-s)}^{\alpha -1}\overline{conv}f(J_i,V\left(J_i\right)),\hfill \end{array}$$

where

$$J_i=[t_i,t_{i+1}],i=0,1,\dots ,m-1.$$

Using (4) and properties of the measure of weak noncompactness we obtain

$$\begin{array}{cc}\hfill \beta \left(F\right(V\left(t\right)\left)\right)& \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=0}^{m-1}}{\mu }_{\Delta }\left(J_i\right){(t_i-s)}^{\alpha -1}\beta \left(f(J_i,V_s\left(J_i\right))\right)\hfill \\ & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=0}^{m-1}}{\mu }_{\Delta }\left(J_i\right){(t_i-s)}^{\alpha -1}L\beta \left(V_s\left(J_i\right)\right)\hfill \\ & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle \sum _{i=0}^{m-1}}{\mu }_{\Delta }\left(J_i\right){(t_i-s)}^{\alpha -1}L\beta \left(V_s\left(I_d\right)\right)\hfill \\ & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}d{(d-s)}^{\alpha -1}L\beta \left(V_s\left(I_d\right)\right).\hfill \end{array}$$

Since $V\subset \overline{V}=\overline{conv}(\left\{x\right\}\cup F\left(V\right))$, we have $\beta \left(V\left(t\right)\right)\le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}d{(d-s)}^{\alpha -1}L\beta \left(V_s\left(I_d\right)\right),t\in I_d$.

Taking the supremum over $t\in I_d$ and applying Theorem 3 yields

$$\beta \left(V\left(I_d\right)\right)=\underset{s\in I_d}{sup}\beta \left(V\left(t\right)\right)\le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}d{(d-s)}^{\alpha -1}L\beta \left(V_s\left(I_d\right)\right).$$

Since $0<{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}d{(d-s)}^{\alpha -1}L<1$ we have $v\left(t\right)=\beta \left(V\right(t\left)\right)=0$, $t\in I_d$.

Using Ascoli’s theorem we have that V is relatively weakly compact. Therefore, condition (2) of Kubiaczyk’s fixed point theorem is satisfied. Consequently, the operator F possesses a fixed point in $A(\phi ,a)$, which is a pseudosolution of the equivalent integral equation (3). This means that there exists a pseudosolution of the problem (1).    □

Remark 1.

The main existence result can be interpreted as a fixed point principle applied to a nonlocal dynamic system combining delay and fractional effects on time scales.

The considered equation incorporates two distinct sources of memory: the delay term, which depends on the past trajectory $\left(x_t\right)$, and the fractional derivative, which accounts for hereditary effects distributed over the entire history of the system. As a consequence, the problem is intrinsically nonlocal both in time and in state.

To overcome these difficulties, the original problem is reformulated as an equivalent integral equation involving a Volterra-type operator. This transformation allows one to interpret the solution as a fixed point of an operator acting on a suitable function space. The operator aggregates past states weighted by a singular kernel of fractional type, reflecting the long-memory behavior of the system.

The assumptions imposed on the nonlinear term ensure that this operator is well-defined and weakly sequentially continuous. Moreover, the use of the De Blasi measure of weak noncompactness provides a mechanism to control the possible dispersion of trajectories in infinite-dimensional spaces. In particular, the condition involving the constant L guarantees that the accumulation of memory effects does not dominate the system, preventing loss of compactness.

From an analytical perspective, the result shows that even in the absence of strong compactness or classical integrability assumptions, e.g., Bochner or Pettis integrability, the problem still admits a solution in a generalized sense. This is achieved by working within the framework of the Henstock–Kurzweil–Pettis integral, which significantly enlarges the admissible class of functions.

Consequently, the theorem establishes that the interplay between delay, fractional dynamics, and generalized integration can be handled within a unified functional-analytic framework, ensuring the existence of pseudo-solutions under relatively weak assumptions.

4. An Illustrative Example

In this section, we present a concrete example illustrating the applicability of Theorem 5.

Let

$$\mathbb{T}=\mathbb{Z},I_a={[0,3]}_{\mathbb{T}}=\{0,1,2,3\},$$

and consider the Banach space

$$E=l^2,$$

consisting of all square-summable real sequences equipped with the standard norm

$${\parallel x\parallel }_{l^2}={\left({\displaystyle \sum _{n=1}^{\infty }}{\left|x_n\right|}^2\right)}^{1/2}.$$

Assume that

$$0<\alpha <1,r=1,$$

and define

$$x_t\left(\theta \right)=x(t+\theta ),\theta \in I_r={[-1,0]}_{\mathbb{T}}.$$

Thus, for every instant t, the state of the system depends not only on its current value but also on its value one time unit earlier.

Consider the fractional retarded dynamic equation

$${}_{\mathbb{T}}{}^C\Delta ^{\alpha }x\left(t\right)=f(t,x_t),t\in I_a,$$

with the initial condition

$$x\left(\theta \right)=\phi \left(\theta \right),\theta \in I_r.$$

Define the linear operator

$$A:l^2\to l^2$$

by

$$A(y_1,y_2,\dots )=\left({\displaystyle \frac{y_1}{2}},{\displaystyle \frac{y_2}{2^2}},{\displaystyle \frac{y_3}{2^3}},\dots \right).$$

The operator A is compact since its diagonal coefficients converge to zero. Consequently, it maps bounded sets into relatively compact subsets of $l^2$, which plays an essential role in verifying the assumptions involving the De Blasi measure of weak noncompactness.

Next, define the forcing term

$$g\left(t\right)=\left({\displaystyle \frac{sint}{2}},{\displaystyle \frac{sint}{2^2}},{\displaystyle \frac{sint}{2^3}},\dots \right),$$

which is a bounded vector-valued function on the finite time scale interval $I_a$.

Finally, let

$$f(t,\psi )=\lambda sin\left(\langle \psi (-1),e_1\rangle \right)A\psi (-1)+g\left(t\right),0<\lambda <1,$$

where

$$e_1=(1,0,0,\dots )$$

denotes the first element of the canonical basis of $l^2$.

Notice that

$$\langle \psi (-1),e_1\rangle ={\psi }_1(-1),$$

that is, the scalar product simply extracts the first coordinate of the delayed state. Hence, the nonlinear coefficient

$$sin\left(\langle \psi (-1),e_1\rangle \right)$$

depends only on the first component of the previous state and remains uniformly bounded by one. Therefore, the nonlinearity introduces a bounded state-dependent feedback without affecting the growth properties of the operator.

The corresponding fractional retarded dynamic equation takes the form

$${}_{\mathbb{T}}{}^C\Delta ^{\alpha }x\left(t\right)=\lambda sin\left(\langle x(t-1),e_1\rangle \right)Ax(t-1)+g\left(t\right).$$

The presence of the term $x(t-1)$ models an explicit delay, meaning that the evolution of the system at time t depends on its previous state, while the fractional Caputo derivative describes hereditary effects distributed over the whole history of the process. Consequently, the equation combines two independent mechanisms of memory within a single hybrid continuous-discrete framework.

Since the interval $I_a$ is finite, every E-valued mapping defined on $I_a$ is $\Delta$-HKP integrable. Therefore, for every fixed history function $\psi$, the mapping

$$t⟼f(t,\psi )$$

is $\Delta$-HKP integrable.

Assume now that

$${\psi }_n\to \psi \mathrm{weakly}\mathrm{in}C(I_r,E).$$

Then

$${\psi }_n(-1)\to \psi (-1)\mathrm{weakly}\mathrm{in}E.$$

Since the operator A is compact, it transforms weakly convergent sequences into strongly convergent ones. Moreover, the scalar function

$$y⟼sin\left(\langle y,e_1\rangle \right)$$

is continuous, because the mapping

$$y⟼\langle y,e_1\rangle$$

is a continuous linear functional on $l^2$.

Hence,

$$f(t,{\psi }_n)\to f(t,\psi )$$

for every $t\in I_a$, proving that $f(t,·)$ is weakly-weakly sequentially continuous.

Furthermore, for every bounded set $X\subset E$, the image $A\left(X\right)$ is relatively weakly compact, implying

$$\beta \left(A\right(X\left)\right)=0.$$

Since the set $g\left(I_a\right)$ is finite, it is relatively compact as well, and therefore

$$\beta \left(g\left(I_a\right)\right)=0.$$

Consequently,

$$\beta \left(f(I_a,X)\right)=0\le L\beta \left(X\right),$$

for every bounded subset $X\subset E$, where $L>0$ is arbitrary.

Finally, if

$${\displaystyle \frac{L}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\Delta s<1,$$

then all assumptions of Theorem 5 are fulfilled.

Therefore, the fractional retarded dynamic equation

$${}_{\mathbb{T}}{}^C\Delta ^{\alpha }x\left(t\right)=\lambda sin\left(\langle x(t-1),e_1\rangle \right)Ax(t-1)+g\left(t\right)$$

admits at least one pseudosolution on some interval

$$I_c={[0,c]}_{\mathbb{T}},0<c\le 3.$$

This example represents an infinite-dimensional nonlinear delayed dynamical system in which the future evolution depends simultaneously on explicit delays and hereditary effects described by the fractional derivative. The compact operator A models dissipative interactions between infinitely many components of the state vector, while the bounded nonlinear factor provides a state-dependent feedback mechanism. Consequently, the example demonstrates that Theorem 5 applies to genuinely nonlinear fractional retarded dynamic equations on time scales formulated in Banach spaces.