A Priori Uniform Bounds as Measure-Theoretic Tools: Long-Term Analysis via Classical-Enhanced Synthesis

Abstract

This work presents a systematic study of nonlinear differential equations within Sobolev spaces, focusing on mild solutions and their qualitative properties. An iterative reconstruction method is developed to obtain uniform a priori bounds, which ensure both the existence and tightness of invariant measures. Furthermore, uniqueness of these measures is established under appropriate structural conditions. The results provide a rigorous foundation for analyzing the asymptotic behavior of nonlinear dynamical systems.

1. Introduction

The primary focus of this study lies in investigating nonlinear systems of differential equations:

$$\begin{array}{c}du=[\Psi u+f\left(u_x\right)]dx+\tau \left(u_x\right)d\Phi \left(x\right)\mathrm{in}\Omega ,x>0,\\ u(x,y)=\gamma (x,y),x\in [-h,0),u(0,y)={\lambda }_0\left(y\right)\mathrm{in}\Omega ,\\ u(x,y)=0,y\in \partial \Omega ,x\ge 0,\end{array}$$

(1)

where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$. This class of models captures the behavior of dynamical systems influenced by stochastic perturbations together with time-delay effects, operating within a confined spatial domain. Such formulations arise naturally in describing physical or biological processes, for instance, thermal conduction with hereditary memory, constrained population growth, or the transmission of signals in bounded media subject to random disturbances such as noise or defects. The boundary condition $u(x,y)=0$ on $\partial \Omega$ corresponds to physically relevant situations: insulated boundaries in thermal physics, prescribed displacements in elasticity, or absorbing interfaces in ecological systems.

The associated problem in the complete space can be expressed as

$$\begin{array}{c}du=[\Psi u+f\left(u_x\right)]dx+\tau \left(u_x\right)d\Phi \left(x\right)\mathrm{in}{\mathbb{R}}^n,x>0,\\ u(x,y)=\gamma (x,y),x\in [-h,0),u(0,y)={\lambda }_0\left(y\right)\mathrm{in}{\mathbb{R}}^n,\end{array}$$

(2)

where $u_x=u(x+\zeta )$ and $\zeta \in [-h,0]$. Here, $[-h,0]$ represents the delay interval. We define the elliptic operator $\Psi$ by

$$\Psi =\Psi \left(y\right)=\sum _{i=1}^n{a}_i\left(y\right){\displaystyle \frac{\partial }{\partial y_i}}+\sum _{i,j=1}^n{b}_{ij}\left(y\right){\displaystyle \frac{{\partial }^2}{\partial y_i\partial y_j}}+c\left(y\right).$$

(3)

A generalized setting extends the framework from finite regions to the entire space, thereby addressing phenomena such as diffusion, wave propagation, and transport without spatial restrictions. In this broader context, one encounters models for fluid motion under stochastic influences, electromagnetic waves traversing random media, or financial processes evolving in unbounded domains. Stochastic forcing in these equations accounts for uncertainties including turbulent fluctuations, market variability, or randomly varying external excitations.

Nonlinear systems of differential equations, specifically of the forms (1) and (2), were originally introduced in prior studies [1,2,3] as effective mathematical representations for processes that exhibit a strong reliance on their historical states. Memory effects of this type manifest across numerous scientific and engineering disciplines. In population dynamics and epidemiology, delays represent gestation or incubation intervals that modify reproductive or infection rates [4]. Within viscoelastic materials, current stress–strain relations depend explicitly on prior deformation history [5]. In control theory, feedback mechanisms are typically subject to signal delays that critically impact stability [6]. In thermal science, generalized conduction models with memory kernels describe non-Fourier heat transfer [7]. Likewise, in neuroscience, synaptic delays are decisive in shaping the dynamical patterns of neuronal assemblies [8]. These diverse scenarios underscore how delay-driven nonlinear systems provide realistic representations of complex temporal behavior.

The study of nonlinear differential systems with memory and delay has a substantial history, both in bounded and unbounded domains. Foundational contributions include the delay-dependent stability criteria in [9], the rigorous treatment of hereditary effects in nonlinear evolution equations developed in [10,11,12], and the deployment of functional-analytic methods in [13,14]. Collectively, these works established core methodologies for analyzing systems in which the present state depends fundamentally on the past. Distinct from earlier studies, the present work broadens the scope to encompass a wider variety of nonlinearities within second-order systems, formulated both on bounded domains and on ${\mathbb{R}}^n$. The approach integrates operator-theoretic techniques with iterative constructions to establish existence, uniqueness, and qualitative dynamics. This dual formulation and methodological synthesis provide a unified perspective on problems that were traditionally treated in narrower contexts.

The primary objective of this current study is to demonstrate the existence and uniqueness of invariant measures for systems (1) and (2) based on the Phragmén–Lindelöf method [15]. Specifically, we will employ the Phragmén–Lindelöf theorem proposed by Li and Chen [15], which comprises the following key steps: Our first focus is on establishing the existence of the solution for systems (1) or (2). To achieve this, we will explore a specific functional space where the corresponding transition semigroup exhibits Feller properties. This analysis will provide valuable insights into the long-term behavior and stability of the system. In addition to investigating the existence of the solution, we will also delve into the compactness of the semigroup $\mathcal{D}\left(x\right)$ generated by $\Psi$. Through this investigation, we will gain a deeper understanding of the stability and convergence properties of the semigroup, which are crucial for comprehending the overall behavior of the system. Furthermore, we will explore the boundedness in probability of the relevant equation under appropriate initial conditions. By examining the behavior of the solution within a certain probability range, we can ascertain its reliability and predictability under various circumstances.

Lately, this method has been utilized to ascertain the presence of an unchanging measure across various categories of partial differential equations (PDEs). Notable examples include fluid equations, nonlinear Schrödinger systems, quasilinear elliptic systems, and nonlocal partial differential equations. For instance, Alghanmi et al. [16] explored the solvability of coupled systems of nonlinear implicit differential equations incorporating $\varrho$-fractional derivatives subject to anti-periodic boundary conditions, thereby contributing to the refinement of fractional-order modeling. Hao et al. [17] introduced a companion-based multi-level finite element framework for identifying multiple solutions of nonlinear differential equations, thus advancing computational methodologies within nonlinear analysis. Jiao et al. [18] carried out a detailed investigation of solution structures for complex nonlinear partial differential difference equations in ${\mathbb{C}}^2$, which deepened the understanding of discrete–continuous hybrid systems. Lan [19] derived both existence and uniqueness theorems for nonlinear Cauchy-type problems governed by first-order fractional differential equations, offering a rigorous mathematical foundation for fractional dynamical systems. Moreover, Xu et al. [20] established novel solution results for various classes of product-type nonlinear PDEs in ${\mathbb{C}}^3$, thereby expanding analytic techniques applicable to higher-dimensional complex domains.

This study examines the existence of an invariant measure in the phase space ${\mathbb{R}}^n\times L^2(-h,0;{\mathbb{R}}^n)$ and focuses on the mild solutions of system (2) in the phase space $L_{\varpi }^2\left({\mathbb{R}}^n\right)\times L^2(-h,0;L_{\varpi }^2\left({\mathbb{R}}^n\right))$, where $L_{\varpi }^2\left({\mathbb{R}}^n\right)$ is a weighted space. Previous research has mainly investigated the systems (1) and (2) in $C([-h,0];L_{\varpi }^2\left({\mathbb{R}}^n\right))$, which offers a simpler problem. However, in order to apply the Phragmén–Lindelöf method, it is necessary to work in $L_{\varpi }^2\left({\mathbb{R}}^n\right)\times L^2(-h,0;L_{\varpi }^2\left({\mathbb{R}}^n\right))$, which is the primary focus of our research. Additionally, we establish the uniqueness and existence of the solution. The existing literature has explored the conditions for the existence and uniqueness of the solution, as well as the Feller and Markov properties in the aforementioned spaces. However, our study expands upon this work by investigating the more complex problem in the phase space $L_{\varpi }^2\left({\mathbb{R}}^n\right)\times L^2(-h,0;L_{\varpi }^2\left({\mathbb{R}}^n\right))$ and demonstrating the existence and uniqueness of the solution.

To enhance the clarity and coherence of this article, the content is structured as follows: In Section 2 we introduce the notations. In Section 3, we formulate the main results. Section 4, Section 5, Section 6, Section 7 and Section 8 are dedicated to completing the proofs of them, namely Theorems 1, 2, and 4–6. Notably, in Theorem 5, we not only prove the existence of an invariant measure but also present an insightful example illustrating the application of Theorem 5 to nonlinear systems of integral differential equations. Section 9 provides a conclusive summary.

2. Preliminaries

In what follows, we consider the domain $\Omega$, which can either be a bounded domain with a boundary $\partial \Omega$ that satisfies the Lyapunov condition or $\Omega$ is equal to ${\mathbb{R}}^n$. The parameters $\zeta$ and $\overline{\zeta }$ are introduced through the weight functions

$$\varpi \left(y\right):={\displaystyle \frac{1}{1+{\left|y\right|}^{\zeta }}},\overline{\varpi }\left(y\right):={\displaystyle \frac{1}{1+{\left|y\right|}^{\overline{\zeta }}}}.$$

(4)

These weight functions are introduced to regulate the asymptotic behavior of solutions and measures in the unbounded setting $\Omega ={\mathbb{R}}^n$, ensuring proper decay or integrability within the associated functional framework. More specifically:

• The parameter $\zeta$ determines the primary decay imposed on functions in weighted Sobolev or Banach spaces. Larger values of $\zeta$ correspond to a steeper decay in the weight $\varpi \left(y\right)$, leading to stricter integrability conditions for solutions and invariant measures.
• The auxiliary exponent $\overline{\zeta }$ arises in defining dual or comparison weights, such as $\overline{\varpi }\left(y\right)$, and naturally appears in duality arguments or estimates involving test functions. Its role is to guarantee the finiteness of integrals where both $\varpi$ and $\overline{\varpi }$ are present.
• The inequality

  $$\zeta >n+\overline{\zeta }$$

  highlighted in Theorem 4 plays a decisive role. Since n represents the spatial dimension, this condition ensures that the imposed decay is sufficiently strong to balance the volume growth of ${\mathbb{R}}^n$ as $\left|y\right|\to \infty$ together with the weaker contribution from $\overline{\varpi }\left(y\right)$. Consequently, it guarantees the tightness of measure families and compactness of embeddings in weighted spaces, both of which are indispensable for proving the existence of invariant measures.

Next, we present the following spaces such as

$$\mathcal{H}:=L^2(\Omega ),{\mathcal{A}}_0^{\varpi }:=L_{\varpi }^2(\Omega ),{\mathcal{A}}_1^{\varpi }:=L^2(-h,0;L_{\varpi }^2(\Omega )),{\mathcal{A}}^{\varpi }:={\mathcal{A}}_0^{\varpi }\times {\mathcal{A}}_1^{\varpi }$$

with

$$\begin{array}{c}{\parallel u\parallel }_{{\mathcal{A}}_{\varpi }^0}^2:={\parallel u(·)\parallel }_{\varpi }^2:={\int }_{\Omega }{u}^2\left(y\right)\varpi \left(y\right)dy,\\ {\parallel u(\zeta ,·)\parallel }_{{\mathcal{A}}_{\varpi }^1}^2:={\int }_{-h}^0{\int }_{\Omega }{u}^2(\zeta ,y)\varpi \left(y\right)dyd\zeta ,\\ \parallel (u(·),u_1(\zeta ,·)){\parallel }_{{\mathcal{A}}^{\varpi }}^2={\parallel u\left(y\right)\parallel }_{\varpi }^2+\parallel u_1{(\zeta ,y)\parallel }_{{\mathcal{A}}_{\varpi }^1}^2,{\parallel u(·)\parallel }_{\mathcal{H}}^2={\int }_{\Omega }{u}^2\left(y\right)dy.\end{array}$$

The coefficients $b_{ij}$ of $\Psi$, as defined in (3), exhibit Hölder continuity with $\alpha \in (0,1)$. These coefficients are also symmetric, bounded, and satisfy the ellipticity condition

$$\sum _{i,j=1}^n{b}_{i,j}{\varrho }_i{\varrho }_j\ge d_0\left|\varrho \right|,\mathrm{for}\mathrm{all}\varrho \in {\mathbb{R}}^n,d_0>0.$$

Here, $d_0$ is explicitly introduced as a fixed positive constant ensuring uniform ellipticity. It is part of the underlying hypothesis rather than a free parameter.

We enforce homogeneous Dirichlet boundary conditions on the boundary of $\Omega$ in the case that $\Omega$ is bounded. In this scenario, $\Omega (\Psi )={\mathcal{H}}^2(\Omega )\cap {\mathcal{H}}_0^1(\Omega ).$ If $\Omega ={\mathbb{R}}^n$, then one has that $\Omega (\Psi )={\mathcal{H}}^2\left({\mathbb{R}}^n\right).$ Let $\mathcal{G}(x,y,z)$ represent the fundamental solution for ${\displaystyle \frac{\partial }{\partial x}}-\Psi$. It can be deduced from references such as [21], p. 311, that there are two positive constants $d_1\left(\eta \right)$ and $d_2\left(\eta \right)$ such that

$$0\le \mathcal{G}(x,y,z)\le d_1\left(\eta \right)x^{-n/2}{e}^{-d_2\left(\eta \right){\displaystyle \frac{{|y-z|}^2}{x}}}$$

(5)

holds for $x\in [0,\eta ]$ and $y,z\in \Omega$.

Notably, in inequality (5), the two positive constants $d_1$ and $d_2$ rely not only $\eta$, but on the constants $d_0$, n, $\eta$, the Hölder constants, and maximum values of the coefficients of $\Psi$. The estimates display various properties when the operator is represented in the divergence form $\Psi u=\mathrm{div}(\mathrm{a}\nabla \mathrm{u})$, as demonstrated in [22], p. 202, denoted as

$${\theta }_1(x,y-z)\le \mathcal{G}(x,y,z)\le {\theta }_2(x,y-z),$$

(6)

where

$${\theta }_i(x,y)=\mathcal{E}(d_0,n)x^{-n/2}{e}^{-\mathcal{E}(d_0,n){\displaystyle \frac{{\left|y\right|}^2}{x}}},x\ge 0,i=1,2,x,y\in {\mathbb{R}}^n.$$

Here, the quantity $\mathcal{E}(d_0,\eta )>0$ depends solely on the ellipticity constant $d_0$, the time horizon $\eta$, and the Hölder/supremum norms of the coefficients of $\Psi$, but remains independent of the spatial variable x. Since the same $\mathcal{E}(d_0,\eta )$ arises for $i=1$ and $i=2$, it follows that ${\theta }_1(x,y)={\theta }_2(x,y)$ holds.

Lemma 1.

Given any positive value of $\eta >0$, there exists a positive constant $d(\zeta ,\eta )>0$ such that

$${\int }_{\Omega }\mathcal{G}(x,y,z)\varpi \left(z\right)dz\le d(\zeta ,\eta )\varpi \left(y\right),x\in [0,\eta ]$$

holds true.

Proof. 

Notice that (4) yields that

$${\displaystyle \frac{\varpi \left(y\right)}{\varpi \left(z\right)}}\le {d\left(\zeta \right)(1+|y-z|}^{\zeta })$$

for a given positive $d\left(\zeta \right)$. So

$$\begin{array}{cc}\hfill {\int }_{\Omega }\varpi \left(z\right)\mathcal{G}(x,y,z)dz& \le d\left(\zeta \right){\int }_{\Omega }{\varpi }^{-1}(y-z)\mathcal{G}(x,y,z)\varpi \left(y\right)dz\hfill \\ \hfill & \le d\left(\zeta \right)d_1\left(\eta \right){\int }_{{\mathbb{R}}^n}{x}^{-n/2}{e}^{-d_2\left(\eta \right){\displaystyle \frac{{\left|y\right|}^2}{x}}}{(1+|y|}^{\zeta })dz\varpi \left(y\right)\hfill \\ \hfill & \le d(\zeta ,\eta )\varpi \left(y\right).\hfill \end{array}$$

□

Put

$$\left(\mathcal{D}\left(x\right)\lambda \right)\left(y\right):={\int }_{\Omega }\lambda \left(z\right)\mathcal{G}(x,y,z)dz,x>0,y\in \Omega ,\lambda \in L^2(\Omega ),$$

(7)

and $\mathcal{D}\left(0\right)=\mathcal{N}$, where $\mathcal{N}$ denotes the identity mapping.

It is a semigroup defined on the function space $L^2(\Omega )$, and its generator is denoted as $\Psi$. By applying Lemma 1, it follows for all $\lambda \in L^2(\Omega )$ and for any $x\in [0,\eta ]$,

$$\begin{array}{cc}\hfill {\parallel \mathcal{D}\left(x\right)\lambda \parallel }_{{\mathcal{A}}_{\varpi }^0}^2& ={\int }_{\Omega }{\left({\int }_{\Omega }\mathcal{G}(x,y,z)\lambda \left(z\right)dz\right)}^2\varpi \left(y\right)dy\hfill \\ \hfill & \le {\int }_{\Omega }\varpi \left(y\right)\left({\int }_{\Omega }\mathcal{G}(x,y,z)dz\right)\left({\int }_{\Omega }{\lambda }^2\left(z\right)\mathcal{G}(x,y,z)dz\right)dy\hfill \\ \hfill & \le C{\int }_{\Omega }\left({\int }_{\Omega }\mathcal{G}(x,y,z){\displaystyle \frac{\varpi \left(y\right)}{\varpi \left(z\right)}}dy\right)\varpi \left(z\right){\lambda }^2\left(z\right)dz\hfill \\ \hfill & \le d_{\varpi }\left(\eta \right){\parallel \lambda \parallel }_{{\mathcal{A}}_{\varpi }^0}^2\hfill \end{array}$$

(8)

holds. The aforementioned estimation facilitates the extension of the semigroup $\mathcal{D}\left(x\right)$ into a linear mapping that operates from ${\mathcal{A}}_0^{\varpi }$ to itself. Since $L^2(\Omega )$ is densely packed within ${\mathcal{A}}_0^{\varpi }$, it can be inferred that $\mathcal{D}\left(x\right)$ demonstrates strong continuity within ${\mathcal{A}}_0^{\varpi }$.

Let $b_i\ge 0$, ${\sum }_{i=1}^{\infty }{b}_i<\infty$, and $\mathcal{H}$ be equipped with an orthonormal basis $e_n$. Here, it is assumed that $e_n\in L^{\infty }(\Omega )$ and ${sup}_n{\left|e_n\right|}_{L^{\infty }(\Omega )}<\infty$. We introduce the operator $\mathcal{F}$ as an element of $\mathcal{L}\left(\mathcal{H}\right)$. The operator $\mathcal{F}$ satisfies the conditions of being non-negative, $\eta \zeta \left(\mathcal{F}\right)<\infty$, and $\mathcal{F}{e}_n=b_n{e}_n$. Let $(\Gamma ,\mathcal{E},P)$ be a complete probability space. We define

$$\Phi \left(x\right):=\sum _{i=1}^{\infty }\sqrt{b_i}{e}_i\left(y\right){\alpha }_i\left(x\right),x\ge 0,$$

which denotes a $\mathcal{F}$-Wiener process with values in $L^2\left(\mathcal{F}\right)$ on the interval $x\ge 0$. Here, ${\alpha }_i\left(x\right)$ represents standard, one-dimensional, mutually independent Wiener processes. Additionally, we consider a normal filtration denoted by $F_x,x\ge 0$ that satisfies the following conditions:

• $\Phi (x+h)-\Phi \left(x\right)$ is independent of ${\mathcal{E}}_x$ for all $h\ge 0,x\ge 0$.
• $\Phi \left(x\right)$ is ${\mathcal{E}}_x$-measurable.

Let $\psi ={\mathcal{F}}^{{\displaystyle \frac{1}{2}}}\left(\mathcal{H}\right)$ be denoted. According to [23], [Lemma 6.2.2], it is concluded that $\psi \in L^{\infty }(\Omega )$. In line with [24], we introduce the multiplication operator denoted as $\wedge :\psi \to {\mathcal{A}}_0^{\varpi }$, which can be defined as follows: for a fixed $\lambda \in {\mathcal{A}}_0^{\varpi }$, $\wedge \left(\rho \right)=\lambda \rho$ for any $\rho \in \psi$. As both $\lambda \in {\mathcal{A}}_0^{\varpi }$ and $\lambda \in L^{\infty }(\Omega )$ hold, this operator is well defined, resulting in $\wedge \circ {\mathcal{F}}^{1/2}:L^2(\Omega )\to {\mathcal{A}}_0^{\varpi }$ being a Volterra–Fredholm operator. Moreover, the operator ∧ also satisfies the properties of being a Volterra–Fredholm operator, as stated in the following condition:

$$\begin{array}{cc}\hfill \parallel \wedge \circ {\mathcal{F}}^{1/2}{\parallel }_{{\mathcal{L}}_2}^2& :=\sum _{n=1}^{\infty }{\parallel \wedge \circ {\mathcal{F}}^{1/2}{e}_n\parallel }_{{\mathcal{A}}_0^{\varpi }}^2\hfill \\ \hfill & =\sum _{n=1}^{\infty }{b}_n{\int }_{\Omega }{\lambda }^2\left(y\right)e_n^2\left(y\right)\varpi \left(y\right)dy\hfill \\ \hfill & \le \eta \zeta \left(\mathcal{F}\right)\underset{n}{sup}\parallel e_n{\parallel }_{\infty }^2{\parallel \lambda \parallel }_{\varpi }^2,\hfill \end{array}$$

(9)

where $\eta \zeta \left(\mathcal{F}\right)={\sum }_{n=1}^{\infty }{b}_n=a$. So if $\wedge :\Gamma \times [0,\eta ]\to \mathcal{L}(\psi ,{\mathcal{A}}_0^{\varpi })$ is a predictable process such that

$$\Lambda {\int }_0^{\eta }{\parallel \wedge \circ {\mathcal{F}}^{1/2}\parallel }_{{\mathcal{L}}_2}^{2}dt<\infty ,$$

then one has that

$${\int }_0^x\vee \left(t\right)d\Phi \left(t\right)\in {\mathcal{A}}_0^{\varpi }$$

with

$${\int }_0^x\vee \left(t\right)d\Phi \left(t\right)=\sum _{i=1}^{\infty }\sqrt{b_i}{\int }_0^x\wedge (t,·)e_i(·)d{\alpha }_i\left(t\right).$$

Moreover,

$$\Lambda \parallel {\int }_0^x\vee \left(t\right)d\Phi \left(t\right){\parallel }_{{\mathcal{A}}_{\varpi }^0}^2\le a\underset{n}{sup}\parallel e_n{\parallel }_{\infty }^2{\int }_0^x\Lambda {\parallel \vee (t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}dt.$$

(10)

Let us make the assumptions that f and $\tau$ satisfy the following conditions:

(i)

There exists a positive constant $\delta$ such that

$$\parallel f\left({\lambda }_1\right)-f\left({\lambda }_2\right){\parallel }_{{\mathcal{A}}_0^{\varpi }}+\parallel \tau \left({\lambda }_1\right)-\tau \left({\lambda }_2\right){\parallel }_{{\mathcal{A}}_0^{\varpi }}\le \delta {\parallel {\lambda }_1-{\lambda }_2\parallel }_{{\mathcal{A}}_1^{\varpi }}$$

holds for any ${\lambda }_1,{\lambda }_2\in {\mathcal{A}}_1^{\varpi }$.

(ii)

Both functionals f and $\tau$ map elements from ${\mathcal{A}}_1^{\varpi }$ to ${\mathcal{A}}_0^{\varpi }$.

We define a stochastic process $u(x,·)\in {\mathcal{A}}_0^{\varpi }$ to be a mild solution of (1) or (2) if it satisfies the following equation:

$$u(x,·)=\mathcal{D}\left(x\right)\lambda (0,·)+{\int }_0^{x}f\left(u_t\right)\mathcal{D}(x-t)dt+{\int }_0^x\tau \left(u_t\right)\mathcal{D}(x-t)d\Phi \left(t\right)$$

(11)

subject to the given conditions:

$$\begin{array}{c}\hfill u(x,·)=\lambda (x,·)\in {\mathcal{A}}_1^{\varpi },u(0,·)=\lambda (0,·)\in {\mathcal{A}}_0^{\varpi },x\in [-h,0].\end{array}$$

Hence, the phase space of the problem can be represented as the Sobolev space ${\mathcal{A}}^{\varpi }$. In this scenario, the variable $z\left(x\right)$ belongs to ${\mathcal{A}}^{\varpi }$ if it can be expressed as $z\left(x\right)=(u(x,·),u_x)$, where $u(x,·)$ is an element of ${\mathcal{A}}_0^{\varpi }$ and $u_x$ is an element of ${\mathcal{A}}_1^{\varpi }$. Here, $u_x$ is defined as $u(x+\zeta ,·)$, with $\zeta$ taking values in the interval $[-h,0]$.

3. Main Results

Theorem 1.

Assume that f and τ fulfill conditions (i) and (ii), while $\lambda (x,·)$ is a ${\mathcal{E}}_0$ measurable stochastic process, where $x\in [-h,0]$. This process is independent of Φ and satisfies the conditions

$${\Lambda \parallel \lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p<\infty ,\Lambda {\parallel \lambda (·,·)\parallel }_{{\mathcal{A}}_1^{\varpi }}^p<\infty ,p\ge 2.$$

Under these conditions, a unique mild solution of the system (1) (or (2)) exists on the interval $[0,\eta ]$. Furthermore, the following inequality holds for any $x\in [0,\eta ]$:

$${\Lambda \parallel z\left(x\right)\parallel }_{{\mathcal{A}}^{\varpi }}^p\le {\mathcal{E}\left(\eta \right)(1+\Lambda \parallel z\left(0\right)\parallel }_{{\mathcal{A}}^{\varpi }}^p).$$

(12)

Theorem 2.

For any $\gamma \in {\mathcal{A}}_1^{\varpi }$ such that $\gamma (0,·)\in {\mathcal{A}}_0^{\varpi }$, and any ${\gamma }_1\in {\mathcal{A}}_1^{\varpi }$ such that ${\gamma }_1(0,·)\in {\mathcal{A}}_0^{\varpi }$, we define the functions $z\left(x\right)$ and $z_1\left(x\right)$ as follows:

$$z\left(x\right)=z(x,\gamma )=\left(\begin{array}{c}u(x,\gamma )\\ u_x\left(\gamma \right)\end{array}\right),z_1\left(x\right)=z(x,{\gamma }_1)=\left(\begin{array}{c}u(x,{\gamma }_1)\\ u_x\left({\gamma }_1\right)\end{array}\right).$$

Given the conditions specified in Theorem 1, there exists a constant $d\left(\eta \right)$ such that the following inequality holds:

$$\underset{x\in [0,\eta ]}{sup}\Lambda \parallel z\left(x\right)-z_1{\left(x\right)\parallel }_{{\mathcal{A}}^{\varpi }}^2\le d\left(\eta \right)\Lambda {\parallel \gamma \left(x\right)-{\gamma }_1\left(x\right)\parallel }_{{\mathcal{A}}^{\varpi }}^2.$$

The subsequent proposition demonstrates the continuity of the trajectories of the solution $u(x,·)$.

Proposition 1.

Given that $u(x,·)$ is a mild solution of either (1) or (2), and taking into account the conditions stipulated in Theorem 1, we can deduce that $u_x$ exhibits probability continuity at $x=0$ associated with the norm ${|·|}_{{\mathcal{A}}_1^{\varpi }}$. This can be expressed as follows:

$$\parallel u_x-u_0{\parallel }_{{\mathcal{A}}_1^{\varpi }}^2={\int }_{-h}^0\Lambda {\parallel u(x+\zeta )-\lambda \left(\zeta \right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \to 0,t\to 0.$$

Proof. 

Notice that

$$\Lambda \parallel u_x-u_0{\parallel }_{{\mathcal{A}}_1^{\varpi }}^2\le {\int }_{-h}^{-x}{\Lambda \parallel \lambda (x+\zeta )-\lambda \left(\zeta \right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta +{\int }_{-x}^0\Lambda {\parallel u(x+\zeta )-\lambda \left(\zeta \right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta .$$

In [25], [Theorem 1.4.2], Pazy finds the density of $C([-h,0],{\mathcal{A}}_0^{\varpi }\times L^2(\Gamma ))$ in $L^2([-h,0],{\mathcal{A}}_0^{\varpi }\times {\mathcal{A}}_0^{\varpi }\times L^2(\Gamma ))$, and the first term can be shown to converge to zero. Likewise, the second term converges to zero as x tends to zero, owing to the boundedness of the integrand.  □

The Banach space ${\mathcal{A}}_b\left({\mathcal{A}}^{\varpi }\right)$ is defined to include bounded real Borel functions from ${\mathcal{A}}^{\varpi }$ to $\mathbb{R}$. The existence of the solution for all $x\ge 0$ is ensured through the arbitrary selection of $\eta >0$ in Theorem 1, which implies the corresponding existence of $z\left(x\right)$ for $x\ge 0$. By substituting $[-h,0]$ with $[-h+t,t]$ for $t\ge 0$ as the initial interval, we can establish the uniqueness and existence of solutions for $x\ge t\ge 0$, which is denoted as $u(x,t,\lambda )$. Likewise,

$$u_x(t,\lambda )=u(x+\zeta ,t,\lambda ),\zeta \in [-h,0]$$

represents a shift of the solution $u(x,\lambda )$, with the property that $u_t(t,\lambda )=u(t+\zeta ,t,\lambda )=\lambda \left(\zeta \right)$, and for $\zeta =0$, $\lambda (0,·)\in {\mathcal{A}}_0^{\varpi }$.

Following the work of [26], we introduce the family of shift operators as

$${\psi }_t^x\lambda :=u(x+\zeta ,t,\lambda )=u_x(t,\lambda ).$$

Let ${\mathcal{E}}_t^x(d\Phi )$ denote the minimal $\tau$-algebra that includes $\Phi \left(\kappa \right)-\Phi \left(t\right),\kappa \in [t,x]$. It should be emphasized that the independence of $u_x(t,\lambda )$ from the $\tau$-algebra ${\mathcal{G}}^x$, defined as the minimal sigma-algebra including $\Phi \left(\kappa \right)-\Phi \left(x\right)$ for $\kappa \ge x$, is worth noting.

For any nonstochastic $\lambda \in {\mathcal{A}}^{\varpi }$ with $t\ge 0$ and $x\ge s$, ${\psi }_t^x\lambda :=u_x(t,\lambda )$ is an ${\mathcal{E}}_t^x(d\Phi )$-measurable stochastic function that takes values in ${\mathcal{A}}_1^{\varpi }$, where $u(x,t,\lambda )\in {\mathcal{A}}_0^{\varpi }$ for $\zeta =0$. By defining $z(x,t,\lambda )=(u(t,x,\lambda ),u_x(t,\lambda ))$, we establish that y maps ${\mathcal{A}}^{\varpi }$ into itself. Theorem 1 yields the following result.

Proposition 2.

Scientifically speaking, the family of the operators (3) satisfies

$${\psi }_{\kappa }^x{\psi }_t^{\kappa }\lambda ={\psi }_t^x\lambda ,$$

where $x\ge \kappa \ge t\ge 0$ and $\lambda \in {\mathcal{A}}^{\varpi }$.

Let $\Omega$ represent a $\tau$-algebra consisting of Borel subsets of ${\mathcal{A}}^{\varpi }$. Therefore, $z(x,t,\lambda )$ naturally denotes the probability measure ${\mu }_x$ defined on $\Omega$ as follows:

$${\mu }_x(\Psi )=\mathcal{I}\{z(x,t,\lambda )\in \Psi \}=\mathcal{I}\{{\psi }_t^x\lambda \in \Psi \}=\mathcal{I}(t,\lambda ,x,\Psi ).$$

(13)

This measure $\mu$ can be regarded as the transition function associated with the stochastic process $z(x,t,\lambda )$. Similar to the finite dimensional case discussed in [24], p. 47, we can substantiate that it fulfills the criteria of a transition probability. In this manner, we obtain the following result.

Theorem 3.

Considering the assumptions of Theorem 1, we can conclude that the process $z(x,t,\lambda )\in {\mathcal{A}}^{\varpi }$ functions as a Markov process on ${\mathcal{A}}^{\varpi }$. The transition function, denoted by $\mathcal{I}(t,\lambda ,x,\Psi )$, is given by (13).

Proposition 3.

For any $x\ge t\ge 0$, one has that

$$\mathcal{I}(t,\lambda ,x,\Psi )=\mathcal{I}(0,\lambda ,x-t,\Psi ).$$

Proof. 

If we denote $\tilde{u}\left(x\right)=u(t+x,t,\lambda )$, then one has that $\tilde{u}\left(0\right)=\lambda (0,·)$ and ${\tilde{u}}_0=u(t+\zeta ,t,\lambda )=\lambda (\zeta ,·)$. On the other hand,

$$\begin{array}{cc}\hfill \tilde{u}\left(x\right)& =u(t+x,t,\lambda )=\lambda (0,·)\mathcal{D}\left(x\right)+{\int }_t^{s+x}f\left(u_{\kappa }\right)\mathcal{D}(t+x-\kappa )d\kappa \hfill \\ \hfill & +{\int }_t^{s+x}\tau \left(u_{\kappa }\right)\mathcal{D}(t+x-\kappa )d\Phi \left(\kappa \right)=\lambda (0,·)\mathcal{D}\left(x\right)\hfill \\ \hfill & +{\int }_0^{x}f\left(u_{\kappa +t}\right)\mathcal{D}(x-\kappa )d\kappa +{\int }_0^x\tau \left(u_{\kappa +t}\right)\mathcal{D}(x-\kappa )d\tilde{\Phi }\left(\kappa \right),\hfill \end{array}$$

where $\tilde{\Phi }\left(\kappa \right):=\Phi (t+\kappa )-\Phi \left(t\right)$ represents a $\mathcal{F}$-Wiener process once more. In this way, the function $\tilde{u}$ solves

$$\tilde{u}\left(x\right)=\mathcal{D}\left(x\right)\lambda (0,·)+{\int }_0^x\mathcal{D}(x-\kappa )f\left(\tilde{u}\left(\kappa \right)\right)d\kappa +{\int }_0^x\mathcal{D}(x-\kappa )\tau \left(\tilde{u}\left(\kappa \right)\right)d\tilde{\Phi }\left(\kappa \right).$$

(14)

The equation $u(x,0,\lambda )$ satisfies the same conditions, where $u(0,0,\lambda )=\lambda (0,·)$ and $u_0=\lambda (\zeta ,·)$. The only distinction is that $u(x,0,\lambda )$ is a solution to (14). But, due to the identical distribution of $\Phi$ and $\tilde{\Phi }$, the distribution of $u(t+x,t,\lambda )$ is equal to that of $u(x,0,\lambda )$ and is therefore independent of t. Consequently, the distribution of $u(x+\zeta ,t,\lambda )=u_x(t,\lambda )=u(x-t+\zeta ,t\lambda )$ fits into the distribution of $u_{x-t}(0,\lambda )=u(x-t+\zeta ,0,\lambda )$. This yields the desired result:

$$\begin{array}{cc}\hfill \mathcal{I}(t,\lambda ,x,\Psi )& =\mathcal{I}\{u_x(t,\lambda )\in \Psi \}=\mathcal{I}\{u(x+\zeta ,t,\lambda )\in \Psi \}\hfill \\ \hfill & =\mathcal{I}\{u(x-t+\zeta ,0,\lambda )\in \Psi \}=\mathcal{I}\{u_{x-t}(0,\lambda )\in \Psi \}.\hfill \end{array}$$

□

For $\theta \in {\mathcal{A}}_b\left({\mathcal{A}}^{\varpi }\right)$, $\lambda \in {\mathcal{A}}^{\varpi }$, and $x\ge t\ge 0$, we define

$${\mathcal{I}}_{t,x}\left(\lambda \right):=\Lambda \theta \left(z(x,t,\lambda )\right).$$

Based on Proposition 3, we obtain the expression ${\mathcal{I}}_{0,x-t}\left(\lambda \right)$, which can be denoted as ${\mathcal{I}}_x\lambda ={\mathcal{I}}_{0,x}\left(\lambda \right)$. Taking into account Proposition 1 and Theorem 2, the subsequent result is derived.

Proposition 4.

Assuming conditions of Theorem 1 hold, we can conclude that ${\mathcal{I}}_x,x\ge 0$ possesses the Feller property and is stochastically continuous. Specifically, it satisfies the following property:

$${\mathcal{I}}_x:d_b\left({\mathcal{A}}^{\varpi }\right)\to d_b\left({\mathcal{A}}^{\varpi }\right),\underset{t\to 0}{lim}{\mathcal{I}}_x\lambda \left(\zeta \right)=\lambda \left(\zeta \right).$$

If we introduce the function $\overline{\varpi }\left(y\right)={(1+|y|}^{\overline{\zeta }}{)}^{-1}$, then the central result of our study is presented in the subsequent theorem.

Theorem 4.

If we suppose that the conditions of Theorem 1 are satisfied and (11) possesses a solution in ${\mathcal{A}}^{\overline{\varpi }}$ that exhibits boundedness in probability for $x\ge 0$ satisfying

$$\zeta >n+\overline{\zeta },$$

(15)

we can conclude the existence of an invariant measure μ on ${\mathcal{A}}^{\varpi }$, denoted as $\mu \in {\mathcal{A}}^{\varpi }$. This measure satisfies the following equality:

$${\int }_{{\mathcal{A}}^{\varpi }}{\mathcal{I}}_x\lambda \left(y\right)d\mu \left(y\right)={\int }_{{\mathcal{A}}^{\varpi }}\lambda \left(y\right)d\mu ,\mathit{for}\mathit{all}x\ge 0\mathit{and}\lambda \in d_b\left({\mathcal{A}}^{\varpi }\right).$$

Remark 1.

It is clear that condition (15) is equivalent to the following inequality:

$${\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\varpi \left(y\right)}{\overline{\varpi }\left(y\right)}}dy<\infty .$$

Theorem 5.

Given the following assumptions:

• $\Omega ={\mathbb{R}}^n$ and $n\ge 3$;
• The conditions stated in Theorem 1 are satisfied;
• For a certain ${\tau }_0>0$, $\tau \left(u\right)$ is bounded above by ${\tau }_0$, where $u\in {\mathcal{A}}_1^{\varpi }$;
• There exists a function $\vee \in L^1\left({\mathbb{R}}^n\right)\cap L^{\infty }\left({\mathbb{R}}^n\right)$ such that $\left|f\right(u(·)\left)\right|\le \vee (·)$, where $u\in {\mathcal{A}}_1^{\varpi }$;
• The functions $u(x,·)=\lambda (x,·)$ satisfy $u(x,·)=\lambda (x,·),x\in [-h,0]$, $u(0,·)=\lambda (0,y)$, and meet the conditions:

  $$\Lambda {\int }_{{\mathbb{R}}^n}{\left|\lambda (0,y)\right|}^{2}dy<\infty \mathit{and}\Lambda {\int }_{{\mathbb{R}}^n}{\int }_{-h}^0{\left|\lambda (\zeta ,y)\right|}^{2}dyd\zeta <\infty .$$

Consequently, we have the following result:

$$\underset{x\ge 0}{sup}\Lambda {\parallel z\left(x\right)\parallel }_{{\mathcal{A}}^{\varpi }}^2<\infty ,$$

which serves as a sufficient condition for probability boundedness.

In this section, we finally consider the weight $\varpi \equiv 1$. Therefore, we define the following spaces:

$${\mathcal{A}}_0:=L^2(\Omega ),{\mathcal{A}}_1:=L^2(-h,0;{\mathcal{A}}_0),\mathcal{A}:={\mathcal{A}}_0\times {\mathcal{A}}_1.$$

The semigroup (7) now exhibits an exponential estimate given by:

$$\parallel \mathcal{D}\left(x\right)u_0{\parallel }_{{\mathcal{A}}_0}^2\le e^{-2{\iota }_{1}t}{\parallel u_0\parallel }_{{\mathcal{A}}_0}^2,$$

where ${\iota }_1>0$ represents the principal eigenvalue of $-\Psi$. We may conventionally extend the $\mathcal{F}$-Wiener process $Phi\left(x\right)$ to $x\in \mathbb{R}$ as

$$\Phi \left(x\right)=\left\{\begin{array}{c}\Phi \left(x\right),x\ge 0;\hfill \\ \mathcal{M}(-x),x\le 0.\hfill \end{array}\right.$$

Here, $\mathcal{M}$ refers to a separate $\mathcal{F}$-Wiener process that is unrelated to $\Phi$.

Definition 1.

A process $u\left(x\right)$, with values in ${\mathcal{A}}_0$, is considered to be a mild solution of the system (1) for $x\in \mathbb{R}$ if the following conditions hold:

• $u\left(x\right)$ is ${\mathcal{E}}_x$ measurable for $x\in \mathbb{R}$;
• $${\Lambda \parallel u\left(x\right)\parallel }_{{\mathcal{A}}_0}^2<\infty ,$$

  where $x\in \mathbb{R}$.
• One has that

  $$u\left(x\right)=\mathcal{D}(x-x_0)u\left(x_0\right)+{\int }_{x_0}^x\mathcal{D}(x-t)f\left(u_t\right)dt+{\int }_{x_0}^x\mathcal{D}(x-t)\tau \left(u_t\right)d\Phi \left(t\right)$$

  for all $-\infty <x_0<t<\infty$ with probability 1.

Theorem 6.

Given a sufficiently small Lipschitz constant δ (see (31) for the precise requirement), the system (1) has a unique solution designated as $u^{*}(x,y)$, which is specified for $x\in \mathbb{R}$. Furthermore, it satisfies the inequality stated as

$$\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel u^{*}\left(x\right)\parallel }_{\mathcal{A}}^2<\infty .$$

In addition to this, the solution possesses an exponential attraction property. This implies the existence of positive quantities $\mathcal{E}$ and ϑ for which the condition

$${\Lambda \parallel u(·,x)-\varrho (·,x)\parallel }_{\mathcal{A}}^2\le \mathcal{E}{e}^{-\vartheta (x-x_0)}\Lambda {\parallel u(·,x_0)-\varrho (·,x_0)\parallel }_{\mathcal{A}}^2$$

holds true, for any chosen initial conditions $x_0\in \mathbb{R}$ and $x>x_0+h$, and for any alternative solution $\varrho \left(x\right)$ with the initial conditions $\varrho \left(x_0\right)\in {\mathcal{A}}_0$ and ${\varrho }_{x_0}\in {\mathcal{A}}_1$.

4. Proofs of Theorem 1

We will prove Theorem 1 in this section.

Let ${\mathcal{A}}_{p,\eta }$, where $p\ge 2$, be the space of ${\mathcal{E}}_x$-measurable processes for all $x\in [0,\eta ]$. It is equipped with the norm ${\parallel \wedge \parallel }_{{\mathcal{A}}_{p,\eta }}^p$, defined as $\Lambda$ multiplied by the integral of ${\parallel \wedge (x,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p$ over the interval $[-h,\eta ]$. Now, we define the expression $\vee \wedge (x,·)$ for $x\in [0,\eta ]$ as follows:

$$\begin{array}{cc}\hfill \vee \wedge (x,·)& :=\mathcal{D}\left(x\right)\wedge (0,·)+{\int }_0^x\mathcal{D}(x-t)f(\wedge (t+\zeta ,·))dt\hfill \\ \hfill & +{\int }_0^x\mathcal{D}(x-t)\tau (\wedge (t+\zeta ,·))d\Phi \left(t\right).\hfill \end{array}$$

while setting $\vee \wedge (x,·)=\lambda (x,·)$ for $x\in [-h,0]$, with $\vee \wedge (0,·)=\lambda (0,·)$. Using this definition, we can derive the following inequality:

$$\begin{array}{cc}\hfill {\vee \wedge (x,·)\parallel }_{{\mathcal{A}}_{p,\eta }}^p& \le \Lambda {\int }_{-h}^0{\parallel \lambda (x,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dx+3^{p-1}\Lambda {\int }_0^{\eta }{\parallel \mathcal{D}\left(x\right)\lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dx\hfill \\ \hfill & +3^{p-1}\Lambda {\int }_0^{\eta }\parallel {\int }_0^x\mathcal{D}(x-t)f(\wedge (t+\zeta ,·))dt{\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dx\hfill \\ \hfill & +3^{p-1}\Lambda {\int }_0^{\eta }\parallel {\int }_0^x\mathcal{D}(x-t)\tau (\wedge (t+\zeta ,·))d\Phi \left(t\right){\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dx\hfill \\ \hfill & \le d_1\left(\eta \right)+3^{p-1}({\phi }_1+{\phi }_2+{\phi }_3).\hfill \end{array}$$

By (8), one has that

$${\phi }_1\le d_{\varpi }^p\left(\eta \right){\int }_0^{\eta }\Lambda {\parallel \lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dx<\infty .$$

It follows from conditions (i) and (ii) for f that

$$\begin{array}{cc}\hfill {\phi }_2& \le d_{\varpi }^p\left(\eta \right){\int }_0^{\eta }{\eta }^{p-1}\left(\Lambda {\int }_0^x{\parallel f({\wedge }_t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dt\right)dx\hfill \\ \hfill & \le d_2{\int }_0^{\eta }dx{\int }_0^x\left(1+\Lambda \parallel {\wedge }_t{\parallel }_{{\mathcal{A}}_1^{\varpi }}^p\right)dt\hfill \\ \hfill & \le d_3+d_2{\int }_0^{\eta }{\int }_0^x\Lambda {\left({\int }_{-h}^0{\parallel \wedge (t+\zeta ,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \right)}^{p/2}dtdx\hfill \\ \hfill & \le d_3+d_4\Lambda {\int }_{-h}^{\eta }{\parallel \wedge (x,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dx<\infty .\hfill \end{array}$$

(16)

In order to estimate ${\phi }_3$, we make use of (10) and [27], [Lemma 4.1]. By applying the definition of the Sobolev–Schmidt norm as given in (9), we obtain the following expression:

$$\begin{array}{cc}\hfill {\phi }_3& \le d\left(p\right){\int }_0^{\eta }\Lambda {\left({\int }_0^x{\parallel \tau \left({\wedge }_t(·)\right)\mathcal{D}(x-t)\parallel }_{{\mathcal{L}}_2}^{2}dt\right)}^{p/2}dx\hfill \\ \hfill & \le d\left(p\right)a^p\underset{n}{sup}{\parallel e_n\parallel }_{\infty }^p{\int }_0^{\eta }\Lambda {\left({\int }_0^x{\parallel \tau \left({\wedge }_t(·)\right)\mathcal{D}(x-t)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}dt\right)}^{p/2}dx\hfill \\ \hfill & \le d_4+d_5{\int }_0^{\eta }{\int }_0^x\Lambda {\parallel {\wedge }_t(·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dtdx<\infty ,\hfill \end{array}$$

(17)

which is similar to the estimation in (16). By combining these estimates, we obtain the following expression: $\vee :{\mathcal{A}}_{p,\eta }\to {\mathcal{A}}_{p,\eta }$.

To demonstrate the contractive property of ∨, we consider arbitrary $\wedge ,\tilde{\wedge }\in {\mathcal{A}}_{p,x}$. Let us proceed with the step-by-step calculations.

$$\begin{array}{cc}\hfill & \parallel \vee \wedge (t,·)-\vee \tilde{\wedge }{(t,·)\parallel }_{{\mathcal{A}}_{p,\eta }}^p\hfill \\ \hfill & \le 2^{p-1}{\int }_0^x\Lambda \parallel {\int }_0^s\mathcal{D}(t-\kappa )\left(f\right({\wedge }_{\kappa }(·)-f\left({\tilde{\wedge }}_{\kappa }(·)\right)d\kappa {\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dt\hfill \\ \hfill & +2^{p-1}{\int }_0^x\Lambda \parallel {\int }_0^s\mathcal{D}(t-\kappa )(\tau \left({\wedge }_{\kappa }(·)\right)-\tau \left({\tilde{\wedge }}_{\kappa }(·)\right)d\kappa {\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dt\hfill \\ \hfill & :=2^{p-1}({\phi }_4+{\phi }_5).\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\phi }_4& \le d_{\varpi }^p\left(\eta \right)L^p{\int }_0^x\Lambda {\left({\int }_0^s{\parallel {\wedge }_{\kappa }(·)-{\tilde{\wedge }}_{\kappa }(·)\parallel }_{{\mathcal{A}}_1^{\varpi }}d\kappa \right)}^{p}dt\hfill \\ \hfill & \le d(\varpi ,\eta ,p){\int }_0^x{\int }_0^s\Lambda {\left({\int }_{-h}^0{\parallel \wedge (\kappa +\zeta ,·)-\tilde{\wedge }(\kappa +\zeta ,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \right)}^{p/2}d\kappa dt\hfill \\ \hfill & \le d_5(\varpi ,\eta ,p)x^2{\parallel \wedge -\tilde{\wedge }\parallel }_{{\mathcal{A}}_{p,x}}^p.\hfill \end{array}$$

(18)

It follows from (17) that

$$\begin{array}{cc}\hfill {\phi }_5& \le d\left(p\right)a^p\underset{n}{sup}{\parallel e_n\parallel }_{\infty }^p{\int }_0^x\Lambda {\left({\int }_0^s{\parallel [\tau \left({\wedge }_{\kappa }\right)-\tau \left({\tilde{\wedge }}_{\kappa }\right)]\mathcal{D}(t-\kappa )\parallel }_{{\mathcal{A}}_0^{\varpi }}^2\right)}^{p/2}dx\hfill \\ \hfill & \le d_6{\int }_0^x{\int }_0^s\Lambda {\left({\int }_{-h}^0{\parallel \wedge (\kappa +\zeta ,·)-\tilde{\wedge }(\kappa +\zeta ,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \right)}^{p/2}d\kappa dt\hfill \\ \hfill & \le d_6(\varpi ,\eta ,p,h)x^2{\parallel \wedge -\tilde{\wedge }\parallel }_{{\mathcal{A}}_{p,x}}^p.\hfill \end{array}$$

(19)

As a result, for sufficiently small $\tilde{x}$, inequalities (18) and (19) indicate that the mapping ∨ possesses a distinct fixed point within ${\mathcal{A}}_{p,\tilde{x}}$, which corresponds to the solution sought in (11). Moreover, if we consider the problem over intervals $[0,\tilde{x}],[\tilde{x},2\tilde{x}],...$ with $d_6{\tilde{x}}^2<1$, the continuity of the solution, almost surely, in the ${\mathcal{A}}_0^{\varpi }$ norm guarantees both the existence and uniqueness of the solution over the interval $[0,\eta ]$.

The demonstration of estimate (12) is the final remaining task. By referencing Equation (11), we can deduce that, for any $x\in [-h,\eta ]$, the following holds:

$$\begin{array}{cc}\hfill {\Lambda \parallel u(x,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p& \le 3^{p-1}\Lambda {\parallel \mathcal{D}\left(x\right)\lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p+3^{p-1}\Lambda {\left({\int }_0^x{\parallel \mathcal{D}(x-t)f\left(u_t\right)\parallel }_{{\mathcal{A}}_0^{\varpi }}dt\right)}^p\hfill \\ \hfill & +3^{p-1}\Lambda \parallel {\int }_0^x\mathcal{D}(x-t)\tau \left(u_t\right)d\Phi \left(t\right){\parallel }_{{\mathcal{A}}_0^{\varpi }}^p\hfill \\ \hfill & \le 3^{p-1}{d}_{\varpi }{\left(\eta \right)\Lambda \parallel \lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p+3^{p-1}{d}_7{\int }_0^x(1+\Lambda \parallel u_t{\parallel }_{{\mathcal{A}}_1^{\varpi }}^p)dt\hfill \\ \hfill & +3^{p-1}{d}_8\Lambda {\left({\int }_0^x{\parallel \mathcal{D}(x-t)\tau \left(u_t\right)\parallel }_{{\mathcal{L}}^2}^{2}dt\right)}^{p/2}\hfill \\ \hfill & \le d_9\left({\Lambda \parallel \lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p+{\int }_0^x(1+\Lambda \parallel u_t{\parallel }_{{\mathcal{A}}_1^{\varpi }}^p)dt\right).\hfill \end{array}$$

(20)

We consider two distinct scenarios: $x\in [0,h]$ and $x\in [h,\eta ]$. For the case where $x\in [0,h]$, the following inequality (21) holds:

$$\begin{array}{cc}\hfill \Lambda \parallel u_x{\parallel }_{{\mathcal{A}}_1^{\varpi }}^p& =\Lambda {\left({\int }_{-h}^0{\parallel u(x+\zeta ,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \right)}^{p/2}\hfill \\ \hfill & \le 2^{{\displaystyle \frac{p}{2}}-1}\left(\Lambda {\left({\int }_{-h}^{-x}{\parallel u(t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}dt\right)}^{p/2}+\Lambda {\left({\int }_{-x}^0{\parallel u(t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}dt\right)}^{p/2}\right)\hfill \\ \hfill & \le 2^{{\displaystyle \frac{p}{2}}-1}\left({\Lambda \parallel \lambda (x,·)\parallel }_{{\mathcal{A}}_1^{\varpi }}^p+h^{{\displaystyle \frac{p-2}{p}}}{\int }_0^x\Lambda {\parallel u(t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dt\right)\hfill \\ \hfill & \le 2^{{\displaystyle \frac{p}{2}}-1}\left({\Lambda \parallel \lambda (x,·)\parallel }_{{\mathcal{A}}_1^{\varpi }}^p+d_{10}\underset{t\in [0,x]}{sup}\Lambda {\parallel u(t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p\right).\hfill \end{array}$$

(21)

In the case where $x\in [h,\eta ]$, we have the inequality (22) as follows:

$$\Lambda \parallel u_x{\parallel }_{{\mathcal{A}}_1^{\varpi }}^p=\Lambda {\left({\int }_{-h}^0{\parallel u(x+\zeta ,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \right)}^{p/2}\le d_{11}\left(\eta \right)\underset{t\in [0,x]}{sup}\Lambda {\parallel u\left(t\right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p.$$

(22)

Combining (20)–(22), one has that

$$\begin{array}{cc}\hfill & \underset{t\in [0,x]}{sup}\Lambda {\parallel u(t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p\hfill \\ \hfill & \le d_{12}\left(\eta \right)\left({\Lambda \parallel \lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p+{\Lambda \parallel \lambda (x,·)\parallel }_{{\mathcal{A}}_1^{\varpi }}^p+{\int }_0^x\underset{\kappa \in [0,t]}{sup}\Lambda {\parallel u(\kappa ,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{p}dt\right).\hfill \end{array}$$

By treating the final term as a distinct entity, it can be deduced that

$$\underset{t\in [0,x]}{sup}{\Lambda \parallel u(t,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p\le d_{13}{\left(\eta \right)[1+\Lambda \parallel \lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^p+{\Lambda \parallel \lambda (x,·)\parallel }_{{\mathcal{A}}_1^{\varpi }}^p].$$

Combining the aforementioned estimations, it follows that

$$\Lambda \parallel u_x{\parallel }_{{\mathcal{A}}_1^{\varpi }}^p\le d_{14}{\left(\eta \right)(1+\Lambda \parallel z\left(0\right)\parallel }_{{\mathcal{A}}^{\varpi }}^p,$$

thereby concluding the proof.

5. Proofs of Theorem 2

By utilizing the definitions of $z_1$ and $y_2$, the following equation is obtained:

$$\begin{array}{cc}\hfill & \underset{x\in [0,\eta ]}{sup}\Lambda {\parallel z\left(x\right)-z_1\left(x\right)\parallel }_{{\mathcal{A}}^{\varpi }}^2\hfill \\ \hfill & \le \underset{x\in [0,\eta ]}{sup}\Lambda \parallel u(x,\gamma )-u(x,{\gamma }_1){\parallel }_{{\mathcal{A}}_0^{\varpi }}^2+\underset{x\in [0,\eta ]}{sup}\Lambda {\parallel u_x\left(\gamma \right)-u_x\left({\gamma }_1\right)\parallel }_{{\mathcal{A}}_1^{\varpi }}^2.\hfill \end{array}$$

The first term in the above inequality can be estimated as:

$$\underset{x\in [0,\eta ]}{sup}\Lambda \parallel u(x,\gamma )-u(x,{\gamma }_1){\parallel }_{{\mathcal{A}}_0^{\varpi }}^2\le d_{15}\underset{x\in [0,\eta ]}{sup}\Lambda {\parallel \gamma \left(x\right)-{\gamma }_1\left(x\right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^2.$$

Regarding the second term in the original equation, let us investigate two distinct cases: $x\in [h,\eta ]$ and $x\in [0,h]$. Taking into account the above estimation, we can derive the following result:

$$\underset{x\in [0,\eta ]}{sup}\Lambda {\int }_{-h}^0\parallel u(x+\zeta ,\gamma )-u(x+\zeta ,{\gamma }_1){\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \le d_{16}\underset{x\in [0,\eta ]}{sup}\Lambda {\parallel \gamma \left(x\right)-{\gamma }_1\left(x\right)\parallel }_{{\mathcal{A}}^{\varpi }}^2.$$

Hence, the proof is complete.

6. Proofs of Theorem 4

To establish the validity of Theorem 4, it is necessary to utilize several auxiliary lemmas.

Lemma 2.

The operator

$$\Psi {\lambda }_0:=\mathcal{D}({\eta }_0+\zeta ):{\mathcal{A}}_0^{\overline{\varpi }}\to {\mathcal{A}}_1^{\varpi }$$

is a Sobolev–Schmidt operator for every fixed ${\eta }_0>2h$.

Proof. 

By [28], p. 91, an orthonormal basis $\{h_n,n\ge 1\}$ exists in ${\mathcal{A}}_0^{\varpi }$ satisfying ${sup}_n{\parallel h_n\parallel }_{L^{\infty }(\Omega )}<\infty$. It can be easily verified that if $\left\{e_n\right\}$ is an orthonormal basis in $\mathcal{H}=L^2(\Omega )$, where $n\ge 1$, then $\{{\displaystyle \frac{e_n}{{\overline{\varpi }}^{1/2}}}\}$ is an orthonormal basis in ${\mathcal{A}}_0^{\overline{\varpi }}$. Therefore, we have

$$\begin{array}{cc}\hfill & {\parallel \Psi \parallel }_{{\mathcal{L}}^2}^2\hfill \\ \hfill & =\sum _{i=1}^{\infty }{\parallel \Psi {\displaystyle \frac{e_i}{\sqrt{\overline{\varpi }}}}\parallel }_{{\mathcal{A}}_1^{\varpi }}^2\hfill \\ \hfill & =\sum _{i=1}^{\infty }{\parallel \mathcal{D}({\eta }_0+\zeta ){\displaystyle \frac{e_i}{\sqrt{\overline{\varpi }}}}\parallel }_{{\mathcal{A}}_1^{\varpi }}^2\hfill \\ \hfill & =\sum _{i=1}^{\infty }{\int }_{-h}^{0}d\zeta {\int }_{\Omega }|\mathcal{D}({\eta }_0+\zeta ){\displaystyle \frac{e_i}{\sqrt{\overline{\varpi }}}}{|}^2\varpi \left(y\right)dy\hfill \\ \hfill & =\sum _{i=1}^{\infty }{\int }_{-h}^{0}d\zeta {\int }_{\Omega }|{\int }_{\Omega }\mathcal{G}({\eta }_0+\zeta ,y,z){\displaystyle \frac{e_i\left(z\right)}{\sqrt{\overline{\varpi }\left(z\right)}}}dz{|}^2\varpi \left(y\right)dy\hfill \\ \hfill & ={\int }_{-h}^{0}d\zeta {\int }_{\Omega }\varpi \left(y\right){\int }_{\Omega }{\displaystyle \frac{{\mathcal{G}}^2({\eta }_0+\zeta ,y,z)}{\overline{\varpi }\left(z\right)}}dzdy\hfill \\ \hfill & \le {\int }_{-h}^{0}d\zeta {\int }_{\Omega }{\int }_{\Omega }{\displaystyle \frac{\varpi \left(y\right)}{\overline{\varpi }\left(z\right)}}{d}_1\left({\eta }_0\right){({\eta }_0+\zeta )}^{-d}exp\{-2d_2\left({\eta }_0\right){\displaystyle \frac{{|y-z|}^2}{{\eta }_0+\zeta }}\}dzdy\hfill \\ \hfill & \le d_{17}{\int }_{-h}^0{\displaystyle \frac{d\zeta }{{({\eta }_0+\zeta )}^{n/2}}}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1}{{({\eta }_0+\zeta )}^{n/2}}}exp\{-2d_2\left({\eta }_0\right){\displaystyle \frac{{|y-z|}^2}{{\eta }_0+\zeta }}\}{\displaystyle \frac{\varpi \left(y\right)}{\overline{\varpi }\left(z\right)}}dydz.\hfill \end{array}$$

However,

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1}{{({\eta }_0+\zeta )}^{n/2}}}exp\{-2d_2\left({\eta }_0\right){\displaystyle \frac{{|y-z|}^2}{{\eta }_0+\zeta }}\}{\displaystyle \frac{\varpi \left(y\right)}{\varpi \left(z\right)}}dy\varpi \left(z\right)\hfill \\ \hfill & \le d\left(\zeta \right){\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1}{{({\eta }_0+\zeta )}^{n/2}}}exp\left\{-2d_2\left({\eta }_0\right){\displaystyle \frac{{|y-z|}^2}{{\eta }_0+\zeta }}\right\}{(1+|y-z|}^{\zeta })dy\varpi \left(z\right)\hfill \\ \hfill & \le d_{18}(\eta ,\zeta )\varpi \left(z\right).\hfill \end{array}$$

So

$${\parallel \Psi \parallel }_{{\mathcal{L}}^2}^2\le d_{19}(\eta ,\zeta ){\int }_{-h}^0{\displaystyle \frac{d\zeta }{{({\eta }_0+\zeta )}^{n/2}}}{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{1+{\left|z\right|}^{\overline{\zeta }}}{1+{\left|z\right|}^{\zeta }}}dz<\infty .$$

Hence, the proof is complete.  □

Corollary 1.

Based on the proof of Lemma 2, it can be demonstrated that $\mathcal{D}\left(x\right)$ constitutes a compact operator when mapping from ${\mathcal{A}}_0^{\overline{\varpi }}$ to ${\mathcal{A}}_0^{\varpi }$, where $x>0$.

It follows from the method presented in [22], p. 311, that we can express Equations (23) and (24) as follows:

$$\begin{array}{c}\begin{array}{cc} & u\left({\eta }_0\right)\hfill \\ & =\mathcal{D}\left({\eta }_0\right)\lambda (0,·)+{\int }_0^{{\eta }_0}\mathcal{D}({\eta }_0-t)f\left(u_t\right)dt+{\int }_0^{{\eta }_0}\mathcal{D}({\eta }_0-t)\tau \left(u_t\right)d\Phi \left(t\right),\hfill \end{array}\end{array}$$

(23)

$$\begin{array}{c}\begin{array}{cc}\hfill u_{{\eta }_0}& =u({\eta }_0+\zeta )\hfill \\ & =\mathcal{D}({\eta }_0+\zeta )\lambda (0,·)+{\int }_0^{{\eta }_0+\zeta }\mathcal{D}({\eta }_0+\zeta -t)f\left(u_t\right)dt\hfill \\ & +{\int }_0^{{\eta }_0+\zeta }\mathcal{D}({\eta }_0+\zeta -t)\tau \left(u_t\right)d\Phi \left(t\right).\hfill \end{array}\end{array}$$

(24)

Equation (23) represents the solution $u\left({\eta }_0\right)$ at a prescribed time ${\eta }_0$ in terms of the initial condition, the inhomogeneous term $f\left(u_t\right)$, and a perturbation involving $\tau \left(u_t\right)$ against the measure $\Phi \left(t\right)$. The operator $\mathcal{D}(·)$ functions as an evolution operator (or fundamental solution) propagating both the initial data and the cumulative effects of forcing and perturbations. Equation (24) generalizes this representation to the shifted time ${\eta }_0+\zeta$, thereby yielding an iterative form of the variation-of-constants formula. In essence, it reveals how the solution develops beyond ${\eta }_0$, once again expressed through initial data transported by $\mathcal{D}(·)$ and the nonlinear terms accumulated over time. These formulas constitute modifications of the classical variation-of-constants formula (Duhamel’s principle), adapted here to the abstract setting involving delay or measure-driven terms. Their significance lies in offering explicit representations of solutions in terms of initial states and forcing, which form the basis for compactness and fixed-point arguments.

It is possible to immediately apply the justifications in [27], [Theorem 11.29], to Equation (23).

Lemma 3.

Under the condition that $p>2$ and $\beta \ge {\displaystyle \frac{1}{p}}$, the operator $\left({\mathcal{G}}_{\beta }\lambda \right)\left(\zeta \right)$ can be regarded as a compact mapping. Specifically, it maps from $L^p(0,{\eta }_0;{\mathcal{A}}_0^{\overline{\varpi }})$ to $C([-h,0],{\mathcal{A}}_0^{\varpi })$ and is defined as

$$\left({\mathcal{G}}_{\beta }\lambda \right)\left(\zeta \right)={\int }_0^{{\eta }_0+\zeta }{({\eta }_0+\zeta -t)}^{\beta -1}\mathcal{D}({\eta }_0+\zeta -t)\lambda \left(t\right)dt.$$

Remark 2.

The observed compactness in $C([-h,0],{\mathcal{A}}_0^{\varpi })$ suggests a comparable compactness in ${\mathcal{A}}_1^{\varpi }$.

Proof of Lemma 3.

Define

$${\parallel \lambda \parallel }_{L_p}^p:={\int }_0^{{\eta }_0}{\parallel \lambda \parallel }_{{\mathcal{A}}_0^{\overline{\varpi }}}^{p}dx.$$

We will utilize the Phragmén–Lindelöf theorem [15] in its infinite dimensional version. In order to demonstrate this, we must establish the following:

(i)

For any fixed value of $\zeta$ in a range $[-h,0]$, the set $\{{\mathcal{G}}_{\beta }\left(\lambda \right)\left(\zeta \right),\parallel \lambda {\parallel }_{L^p}\le 1\}$ is compact within ${\mathcal{A}}_0^{\varpi }$;

(ii)

There exists a positive constant $\omega$ such that if ${\parallel \lambda \parallel }_{L^p}\le 1$ such that if ${\parallel \lambda \parallel }_{L^p}\le 1$ and for all ${\zeta }_1,{\zeta }_2$ with $|{\zeta }_1-{\zeta }_2|\le \omega$, where one has that

$$\parallel {\mathcal{G}}_{\beta }\left(\lambda \right)\left({\zeta }_1\right)-{\mathcal{G}}_{\beta }\left(\lambda \right)\left({\zeta }_2\right){\parallel }_{{\mathcal{A}}_0^{\varpi }}<\epsilon .$$

To verify (i), assuming a fixed $\zeta$ in the range $[-h,0]$ and $0<\epsilon <{\eta }_0+\zeta$, we introduce

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\beta }^{\epsilon }\lambda & :={\int }_0^{{\eta }_0+\zeta -\epsilon }{({\eta }_0+\zeta -t)}^{\beta -1}\lambda \left(t\right)\mathcal{D}({\eta }_0+\zeta -t)dt\hfill \\ \hfill & =\mathcal{D}\left(\epsilon \right){\int }_0^{{\eta }_0+\zeta -\epsilon }{({\eta }_0+\zeta -t)}^{\beta -1}\lambda \left(t\right)\mathcal{D}({\eta }_0+\zeta -t-\epsilon )dt.\hfill \end{array}$$

It is obvious that ${\int }_0^{{\eta }_0+\zeta -\epsilon }{({\eta }_0+\zeta -t)}^{\beta -1}\mathcal{D}({\eta }_0+\zeta -t-\epsilon )\lambda \left(t\right)dt$ belongs to ${\mathcal{A}}_0^{\overline{\varpi }}$. By employing Corollary 1, we can ascertain that $\mathcal{D}\left(\epsilon \right)$ is a compact operator from ${\mathcal{A}}_0^{\overline{\varpi }}$ to ${\mathcal{A}}_0^{\varpi }$. Based on the approach outlined in [23], p. 137, ${\mathcal{G}}_{\beta }^{\epsilon }$ converges strongly to ${\mathcal{G}}_{\beta }$ as $\epsilon$ approaches 0. Consequently, ${\mathcal{G}}_{\beta }$ is compact, that is to say, (i) is satisfied.

To establish (ii), we fix $\zeta$ and $\zeta$ in such a manner that $-h\le \zeta \le \zeta +\zeta \le 0$, and ${\parallel \lambda \parallel }_{L^p}\le 1$. Then, one has that

$$\begin{array}{cc}\hfill & \parallel \left({\mathcal{G}}_{\beta }\lambda \right)(\zeta +\zeta )-\left({\mathcal{G}}_{\beta }\lambda \right)\left(\zeta \right){\parallel }_{{\mathcal{A}}_0^{\varpi }}\hfill \\ \hfill & =\parallel {\int }_0^{{\eta }_0+\zeta +\zeta }{({\eta }_0+\zeta +\zeta -t)}^{\beta -1}\mathcal{D}({\eta }_0+\zeta +\zeta -t)\lambda \left(t\right)dt\hfill \\ \hfill & -{\int }_0^{{\eta }_0+\zeta }{({\eta }_0+\zeta -t)}^{\beta -1}\mathcal{D}({\eta }_0+\zeta -t)\lambda \left(t\right)dt{\parallel }_{{\mathcal{A}}_0^{\varpi }}\hfill \\ \hfill & \le {\int }_0^{{\eta }_0+\zeta }\parallel {({\eta }_0+\zeta +\zeta -t)}^{(\beta -1)}\mathcal{D}({\eta }_0+\zeta +\zeta -t)\hfill \\ \hfill & -{({\eta }_0+\zeta -t)}^{(\beta -1)}\mathcal{D}({\eta }_0+\zeta -t)\parallel \parallel \lambda \left(t\right)\parallel dt\hfill \\ \hfill & +{\int }_{{\eta }_0+\zeta }^{{\eta }_0+\zeta +\zeta }\parallel {({\eta }_0+\zeta +\zeta -t)}^{(\beta -1)}\mathcal{D}({\eta }_0+\zeta +\zeta -t)\lambda \left(t\right)\parallel dt\hfill \\ \hfill & \le {\left({\int }_0^{{\eta }_0}{\parallel {(\zeta +t)}^{\beta -1}\mathcal{D}(t+\zeta )-t^{\beta -1}\mathcal{D}\left(t\right)\parallel }^{q}dt\right)}^{1/q}{\parallel \lambda \parallel }_{L^p}\hfill \\ \hfill & +d_{20}{\left({\int }_0^{{\eta }_0}{t}^{(\beta -1)q}dt\right)}^{1/q}{\parallel \lambda \parallel }_{L^p}:={\varphi }_1+{\varphi }_2.\hfill \end{array}$$

Through direct calculations, we obtain

$${\varphi }_2=d_{20}{\displaystyle \frac{{\zeta }^{\beta -\frac{1}{p}}}{{((\beta -1)q+1)}^{\frac{1}{q}}}}{\parallel \lambda \parallel }_{L^p}\to 0\mathrm{as}\zeta \to 0.$$

Since $\mathcal{D}\left(x\right)$ is compact, it is also strongly continuous for $x>0$. Therefore, $\parallel \mathcal{D}(t+\zeta )-\mathcal{D}\left(t\right)\parallel \to 0$, where $t>0$. Moreover, the integrand in ${\varphi }_1$ is bounded by $2d_{20}{t}^{(\beta -1)q}$. Thus, by utilizing the dominated convergence theorem, one has that ${\varphi }_1\to 0$ as $\zeta \to 0$, thereby concluding the proof of the lemma.  □

Consider any positive value of $\zeta >0$ and introduce

$$\mathcal{E}\left(\zeta \right):=\{(\mu ,\nu ),\mu \in {\mathcal{A}}_0^{\varpi },\nu \in {\mathcal{A}}_1^{\varpi }\}$$

such that

$$\begin{array}{c}\mu :=\mathcal{D}\left({\eta }_0\right)v+\left({\mathcal{G}}_1\lambda \right)\left(0\right)+\left({\mathcal{G}}_{\beta }h\right)\left(0\right),\\ \nu :=\mathcal{D}({\eta }_0+\zeta )v+\left({\mathcal{G}}_1\lambda \right)\left(0\right)+\left({\mathcal{G}}_{\beta }h\right)\left(0\right)\end{array}$$

with ${\parallel v\parallel }_{{\mathcal{A}}_0^{\varpi }}\le \zeta$, ${\parallel \lambda \parallel }_{L^p(0,{\eta }_0,{\mathcal{A}}_0^{\overline{\varpi }})}\le \zeta$ and ${\parallel h\parallel }_{L^p(0,{\eta }_0,{\mathcal{A}}_0^{\overline{\varpi }})}\le \zeta$. Based on Lemma 2, Corollary 1, and Lemma 3, it can be concluded that $\mathcal{E}\left(\zeta \right)$ is compact in ${\mathcal{A}}^{\varpi }$.

Lemma 4.

According to Theorem 1’s conditions, there exists a positive constant c such that for arbitrary $\zeta >0$ and $y=(y,z)\in {\mathcal{A}}^{\overline{\varpi }}$ with ${\parallel y\parallel }_{{\mathcal{A}}^{\overline{\varpi }}}\le \zeta$, the inequality

$$\mathcal{I}\{(u({\eta }_0,y,z),u_{{\eta }_0}(y,z))\in \mathcal{E}\left(\zeta \right)\}\ge 1-c{\zeta }^{-p}{(1+\parallel y\parallel }_{{\mathcal{A}}^{\overline{\varpi }}}^p\mathit{)}$$

holds true, where $u(0,y,z)=y\in {\mathcal{A}}_0^{\overline{\varpi }}$ and $u_0(y,z)=z\in {\mathcal{A}}_1^{\overline{\varpi }}$.

Proof. 

The factorization formula can be represented as follows:

$$\begin{array}{c}u({\eta }_0,z)=\mathcal{D}\left({\eta }_0\right)x+\left({\mathcal{G}}_{1}f\left(u_t\right)\right)\left(0\right)+{\displaystyle \frac{sin\left(\beta \pi \right)}{\pi }}\left({\mathcal{G}}_{\beta }Y\left(t\right)\right)\left(0\right),\end{array}$$

(25)

$$\begin{array}{c}{u}_{{\eta }_0}\left(z\right)=\mathcal{D}({\eta }_0+\zeta )x+\left({\mathcal{G}}_{1}f\left(u_t\right)\right)\left(\zeta \right)+{\displaystyle \frac{sin\left(\beta \pi \right)}{\pi }}\left({\mathcal{G}}_{\beta }Y\left(t\right)\right)\left(\zeta \right),\end{array}$$

(26)

$$\begin{array}{c}Y\left(t\right)={\int }_0^s{(t-\kappa )}^{-\beta }\mathcal{D}(t-\kappa )\tau \left(u_{\kappa }\right)d\Phi \left(\kappa \right).\end{array}$$

(27)

It follows from Theorem 1’s conditions (25), (26),and (27) that

$$\begin{array}{cc}\hfill & \Lambda {\int }_0^{{\eta }_0}{\parallel Y\left(t\right)\parallel }_{{\mathcal{A}}_0^{\overline{\varpi }}}^{p}dt\hfill \\ \hfill & =\Lambda {\int }_0^{{\eta }_0}{\parallel {\int }_0^s\tau \left(u_{\kappa }\right)\mathcal{D}(t-\kappa )d\Phi \left(\kappa \right)\parallel }_{{\mathcal{A}}_0^{\overline{\varpi }}}^p{(t-\kappa )}^{-\beta }dt\hfill \\ \hfill & \le d_{p,{\eta }_0}\Lambda {\int }_0^{{\eta }_0}{\left({\int }_0^s{(t-\kappa )}^{-2\beta }{\parallel \mathcal{D}(t-\kappa )\tau \left(u_{\kappa }\right)\circ {\mathcal{F}}^{1/2}\parallel }_{{\mathcal{L}}_2(\mathcal{H},{\mathcal{A}}_0^{\overline{\varpi }})}\right)}^{p/2}dt\hfill \\ \hfill & \le d_{21}\Lambda {\int }_0^{{\eta }_0}{\left({\int }_0^s{(t-\kappa )}^{-2\beta }{\parallel \tau \left(u_{\kappa }\right)\parallel }_{{\mathcal{A}}_0^{\overline{\varpi }}}^{2}d\kappa \right)}^{p/2}dt.\hfill \end{array}$$

It follows from (11) and Hausdorff–Young’s inequality that

$$\begin{array}{cc}\hfill \Lambda {\int }_0^{{\eta }_0}{\parallel Y\left(t\right)\parallel }_{{\mathcal{A}}_0^{\overline{\varpi }}}^p& \le d_{21}{\left({\int }_0^{{\eta }_0}{x}^{-2\beta }dx\right)}^{p/2}{\int }_0^{{\eta }_0}\Lambda {\parallel \tau \left(u_x\right)\parallel }_{{\mathcal{A}}_0^{\overline{\varpi }}}^{p}dx\hfill \\ \hfill & \le d_{22}{\int }_0^{{\eta }_0}(1+\Lambda \parallel u_x{\parallel }_{{\mathcal{A}}_1^{\overline{\varpi }}}^p)dx\hfill \\ \hfill & \le d_{23}{(1+\parallel y\parallel }_{{\mathcal{A}}^{\overline{\varpi }}}^p).\hfill \end{array}$$

(28)

Similarly,

$$\Lambda {\int }_0^{{\eta }_0}\parallel f\left(u_t\right){\parallel }_{{\mathcal{A}}_0^{\overline{\varpi }}}^{p}dt\le d_{23}{(1+\parallel y\parallel }_{{\mathcal{A}}^{\overline{\varpi }}}^p).$$

(29)

Hence, if ${\parallel y\parallel }_{{\mathcal{A}}^{\overline{\varpi }}}\le \zeta$, $\parallel f\left(u_t\right){\parallel }_{L^p(0,{\eta }_0,{\mathcal{A}}_0^{\overline{\varpi }})}\le \zeta$, and

$$\parallel \tau \left(u_t\right){\parallel }_{L^p(0,{\eta }_0,{\mathcal{A}}_0^{\overline{\varpi }})}\le {\displaystyle \frac{\pi \zeta }{sin\left(\beta \pi \right)}},$$

then it follows from the definition of $\mathcal{E}\left(\zeta \right)$ that $(u({\eta }_0,z),u_{{\eta }_0}\left(z\right))\in \mathcal{E}\left(\zeta \right)$. Suppose that ${\parallel y\parallel }_{{\mathcal{A}}^{\overline{\varpi }}}\le \zeta$. It follows from (28) and (29) that

$$\begin{array}{cc}\hfill \mathcal{I}\{(u({\eta }_0,z),u_{{\eta }_0}\left(z\right))\notin \mathcal{E}\left(\zeta \right)\}& \le \mathcal{I}\{\parallel f\left(u_t\right){\parallel }_{L^p(0,{\eta }_0;{\mathcal{A}}_0^{\overline{\varpi }})}>r\}+\mathcal{I}\{\parallel Y\left(t\right){\parallel }_{L^p(0,{\eta }_0,{\mathcal{A}}_0^{\overline{\varpi }})}\}\hfill \\ \hfill & \le 2{\zeta }^p{d}_{23}{(1+\parallel y\parallel }_{{\mathcal{A}}^{\overline{\varpi }}}^p).\hfill \end{array}$$

We complete the proof.  □

The remaining part of the proof for Theorem 4 can be derived in a similar manner as demonstrated in [24], [Theorem 11.29].

7. Proofs of Theorem 5 and an Example

The proof of [11], [Theorem 6.1.4], shares many similarities with this proof. However, we need to highlight the differences arising from the inclusion of the delay. Let us denote them as

$${\Lambda \parallel z\left(x\right)\parallel }_{{\mathcal{A}}^{\varpi }}^2=\Lambda {\int }_{{\mathbb{R}}^n}{\left|u(x,y)\right|}^2\varpi \left(y\right)dy+\Lambda {\int }_{-h}^{0}d\zeta {\int }_{{\mathbb{R}}^n}{\left|u(x+\zeta ,y)\right|}^2\varpi \left(y\right)dy.$$

It follows from (11) that

$${\parallel u(x,y)\parallel }_{{\mathcal{A}}_0^{\varpi }}^2\le 3({\phi }_1\left(x\right)+{\phi }_2\left(x\right)+{\phi }_3\left(x\right))$$

where

$$\begin{array}{c}{\phi }_1\left(x\right)={\int }_{{\mathbb{R}}^n}{\left({\int }_{{\mathbb{R}}^n}\mathcal{G}(x,y,z)\lambda (0,z)dz\right)}^2\varpi \left(y\right)dy,\\ {\phi }_2\left(x\right)={\int }_{{\mathbb{R}}^n}{\left({\int }_0^x{\int }_{{\mathbb{R}}^n}\mathcal{G}(x-t,y,z)f\left(u_t\left(z\right)\right)dzdt\right)}^2\varpi \left(y\right)dy,\\ {\phi }_3\left(x\right)={\int }_{{\mathbb{R}}^n}{\left({\int }_{{\mathbb{R}}^n}\mathcal{G}(x-t,y,z)\tau \left(u_t\left(z\right)\right)d\Phi \left(t\right)dz\right)}^2\varpi \left(y\right)dy.\end{array}$$

By (6), one has that for all $x\ge 0$,

$$\begin{array}{cc}\hfill \Lambda {\phi }_1& \le {\int }_{{\mathbb{R}}^n}\left({\int }_{{\mathbb{R}}^n}\mathcal{G}(x,y,z)dz{\int }_{{\mathbb{R}}^n}\mathcal{G}(x,y,z){\lambda }^2(0,z)dz\right)\varpi \left(y\right)dy\hfill \\ \hfill & \le d_{24}\Lambda {\int }_{{\mathbb{R}}^n}\left({\int }_{{\mathbb{R}}^n}\mathcal{E}(x,y-z){\lambda }^2(0,z)dz\right)\varpi \left(y\right)dy\hfill \\ \hfill & \le d_{24}{\parallel \varpi \parallel }_{\infty }\Lambda {\parallel \lambda (0,·)\parallel }_{{\mathcal{A}}_0^{\varpi }}^2<\infty ,\hfill \end{array}$$

where $\mathcal{E}$ represents the heat kernel in ${\mathbb{R}}^n$. The estimations for ${\phi }_2$ and ${\phi }_3$ can be determined following the approach outlined in [24], [Theorem 2.1].

Next, we investigate two distinct cases: $x\in [0,h]$ and $x\ge h$.

If $x\in [0,h]$, then one has that

$$\begin{array}{cc}\hfill \Lambda \parallel u_x{\parallel }_{{\mathcal{A}}_1^{\varpi }}^2& =\Lambda {\int }_{-h}^0{\parallel u(x+\zeta )\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \hfill \\ \hfill & \le \Lambda {\int }_0^h{\parallel u\left(t\right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}dt+\Lambda {\int }_{-h}^0{\parallel u\left(t\right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}dt\hfill \\ \hfill & \le {\Lambda \parallel \lambda (x,·)\parallel }_{{\mathcal{A}}_1^{\varpi }}^2+h\underset{x\ge 0}{sup}\Lambda {\parallel u\left(x\right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^2<\infty .\hfill \end{array}$$

If $x\ge h$, then one has that

$$\Lambda \parallel u_x{\parallel }_{{\mathcal{A}}_1^{\varpi }}^2=\Lambda {\int }_{-h}^0{\parallel u(x+\zeta )\parallel }_{{\mathcal{A}}_0^{\varpi }}^{2}d\zeta \le \underset{x\ge 0}{sup}\Lambda {\parallel u\left(x\right)\parallel }_{{\mathcal{A}}_0^{\varpi }}^2<\infty .$$

This completes the proof.

Example 1.

We introduce the functions

$$f\left[\lambda \right]:=\overline{f}\left({\int }_{-h}^0\lambda \left(\zeta \right)d\zeta \right),\tau \left[\lambda \right]:=\overline{\tau }\left({\int }_{-h}^0\lambda \left(\zeta \right)d\zeta \right),$$

where $\overline{f}$ and $\overline{\tau }$ are Lipschitz functions characterized by σ.

For any two functions ${\lambda }_1,{\lambda }_2\in \mathcal{A}{1}^{\varpi }$, the inequality

$$|f\left[{\lambda }_1\right]-f\left[{\lambda }_2\right]|\le \sigma {\int }_{-h}^0|{\lambda }_1\left(\zeta \right)-{\lambda }_2\left(\zeta \right)|d\zeta$$

holds true. So

$$\parallel f\left[{\lambda }_1\right]-f\left[{\lambda }_2\right]{\parallel }_{{\mathcal{A}}_0^{\varpi }}^2\le {\sigma }^2{\int }_{{\mathbb{R}}^n}{\left({\int }_{-h}^0|{\lambda }_1\left(\zeta \right)-{\lambda }_2\left(\zeta \right)|d\zeta \right)}^2\varpi dy\le {\sigma }^{2}h{\parallel {\lambda }_1-{\lambda }_2\parallel }_{{\mathcal{A}}_1^{\varpi }}^2.$$

In a similar way,

$$\parallel \tau \left[{\lambda }_1\right]-\tau \left[{\lambda }_2\right]{\parallel }_{{\mathcal{A}}_0^{\varpi }}^2\le {\sigma }^{2}h{\parallel {\lambda }_1-{\lambda }_2\parallel }_{{\mathcal{A}}_1^{\varpi }}^2.$$

Hence, f and τ can be regarded as instances of Lipschitz maps originating from ${\mathcal{A}}_1^{\varpi }$ and mapping to ${\mathcal{A}}_0^{\varpi }$, wherein the aforementioned theorems can be confidently applied.

Remark 3.

Example 1 illustrates in detail how the functions f and τ satisfy the required Lipschitz conditions. Such functions naturally occur in diverse applied contexts. For example:

• In population dynamics with memory effects, the present growth rate of a population is modeled in terms of an averaged history of past states. The integral ${\int }_{-h}^0\lambda \left(\zeta \right)d\zeta$ accounts for the cumulative influence over the interval $[-h,0]$, while f and τ represent nonlinear feedback mechanisms subject to Lipschitz regularity.
• In viscoelasticity or hereditary materials, the stress at a given instant depends on the weighted integral of the strain history. Here, f and τ act as nonlinear response functions ensuring stability under perturbations owing to their Lipschitz property.
• In neural field models, neuronal activation is often expressed as a nonlinear functional of time-averaged inputs over a delay window. Functions of the type described in Example 1 serve as canonical components in such formulations.

Thus, the abstract verification of the Lipschitz property in Example 1 has direct implications in practical settings across physical and biological systems, thereby broadening the relevance of the theoretical framework presented here.

8. Proofs of Theorem 6

Considering that

$$\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel \xi \left(x\right)\parallel }_{{\mathcal{A}}_0}^2<\infty ,$$

$$\underset{x\in \mathbb{R}}{sup}{\parallel \xi \left(x\right)\parallel }_{\mathcal{A}}^2\le (1+h)\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel \xi \left(x\right)\parallel }_{{\mathcal{A}}_0}^2,$$

we adopt the methodology presented in [26] to consider the following formulation:

$$d{u}^{(n+1)}=(\Psi u^{(n+1)}+f\left(u_x^{\left(n\right)}\right))dx+\tau \left(u_x^{\left(n\right)}\right)d\Phi \left(x\right).$$

(30)

So

$$\underset{x\in \mathbb{R}}{sup}\Lambda \parallel f\left(u_x^{\left(n\right)}\right){\parallel }_{{\mathcal{A}}_0}^2\le 2\parallel f\left(0\right){\parallel }_{{\mathcal{A}}_0}^2+2L^2{h}^2\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel u^{\left(n\right)}\left(x\right)\parallel }_{{\mathcal{A}}_0}^2<\infty .$$

In a similar way,

$$\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel \tau \left(u_x^{\left(n\right)}\right)\parallel }_{{\mathcal{A}}_0}^2<\infty .$$

Consequently, based on iterative reconstruction algorithm [29], Equation (30) possesses a distinctive solution denoted as $u^{(n+1)}\left(x\right)$, which satisfies the condition

$$\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel u^{(n+1)}\left(x\right)\parallel }_{{\mathcal{A}}_0}^2<\infty ,$$

which yields that

$$\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel u^{(n+1)}\left(x\right)\parallel }_{\mathcal{A}}^2<\infty .$$

So

$$\underset{x\in \mathbb{R}}{sup}\Lambda \parallel u^{\left(n\right)}{\parallel }_{{\mathcal{A}}_0}^2\le (1+h)\underset{x\in \mathbb{R}}{sup}\Lambda \parallel u^{\left(n\right)}{\left(x\right)\parallel }_{{\mathcal{A}}_0}^2\le C+h{\sigma }^2\left({\displaystyle \frac{4}{{\iota }_1^2}}+{\displaystyle \frac{2a}{{\iota }_1}}\right)\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel u^{(n-1)}\parallel }_{{\mathcal{A}}_0}^2.$$

Thus, under the condition

$$h{\sigma }^2\left({\displaystyle \frac{4}{{\iota }_1^2}}+{\displaystyle \frac{2a}{{\iota }_1}}\right)<1,$$

(31)

similar to the approach employed in [10], we have

$$\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel u^{*}\left(x\right)\parallel }_B<\infty$$

and

$$\underset{x\in \mathbb{R}}{sup}\Lambda {\parallel u^n\left(x\right)-u^{*}\left(x\right)\parallel }_{\mathcal{A}}^2\to 0,n\to \infty .$$

Additionally, it can be argued that $u^{*}$ satisfies the following expression:

$$u^{*}\left(x\right)=\mathcal{D}(x-x_0)u^{*}\left(x_0\right)+{\int }_{x_0}^x\mathcal{D}(x-x_0)f\left(u_t^{*}\right)dt+{\int }_{x_0}^x\mathcal{D}(x-t)\tau \left(u_t^{*}\right)d\Phi \left(t\right).$$

(32)

Assume the existence of another solution (32) such that $\varrho \left(x_0\right)$ is ${\mathcal{E}}_{x_0}$-measurable, and $\Lambda |\varrho \left(x_0\right){|}^{2}B<\infty$. Here, $\Lambda \parallel \varrho \left(x_0\right){\parallel }_B^2<\infty$ in $[-h,0]$. Let us demonstrate that $\varrho$ converges exponentially to $u^{*}$. At the same time, since the focus is on the behavior of solutions for a large x, it is assumed that $x>x_0+h$. Consequently, $x+\zeta >x_0$ and $\varrho \left(x\right)$ is determined using Formula (32). So

$$\begin{array}{cc}\hfill & \Lambda \parallel u^{*}{\left(x\right)-\varrho \left(x\right)\parallel }_{{\mathcal{A}}_0}^2\hfill \\ \hfill & \le 3e^{-{\iota }_1(x-x_0)}\Lambda \parallel u^{*}\left(x_0\right)-\varrho \left(x_0\right){\parallel }_{{\mathcal{A}}_0}^2+3{\displaystyle \frac{{\sigma }^2}{{\iota }_1}}{\int }_{x_0}^x\Lambda {\parallel u_t^{*}-{\varrho }_t\parallel }_{{\mathcal{A}}_1}^2{e}^{-{\iota }_1(x-t)}dt\hfill \\ \hfill & +3{\sigma }^{2}a{\int }_{x_0}^x\Lambda {\parallel u_t^{*}-{\varrho }_t\parallel }_{{\mathcal{A}}_1}^2{e}^{-{\iota }_1(x-t)}dt\hfill \\ \hfill & =3e^{-{\iota }_1(x-x_0)}\Lambda {\parallel u^{*}\left(x_0\right)-\varrho \left(x_0\right)\parallel }_{{\mathcal{A}}_0}^2\hfill \\ \hfill & +3\left({\displaystyle \frac{{\sigma }^2}{{\iota }_1}}+{\sigma }^{2}a\right){\int }_{x_0}^x{e}^{-{\iota }_1(x-t)}\Lambda {\parallel u_t^{*}-{\varrho }_t\parallel }_{{\mathcal{A}}_1}^{2}dt.\hfill \end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill \Lambda \parallel u_x^{*}-{\varrho }_x{\parallel }_{{\mathcal{A}}_1}^2& ={\int }_{-h}^0\Lambda {\parallel u^{*}(x+\zeta )-\varrho (x+\zeta )\parallel }_{{\mathcal{A}}_0}^{2}d\zeta \hfill \\ \hfill & +3{\int }_{-h}^0{e}^{-{\iota }_1(x+\zeta -x_0)}\Lambda {\parallel u^{*}\left(x_0\right)-\varrho \left(x_0\right)\parallel }_{{\mathcal{A}}_0}^{2}d\zeta \hfill \\ \hfill & +3{\int }_{-h}^0\left({\displaystyle \frac{{\sigma }^2}{{\iota }_1}}{\int }_{x_0}^{x+\zeta }{e}^{-{\iota }_1(x+\zeta -t)}\Lambda {\parallel u_t^{*}-{\varrho }_t\parallel }_{{\mathcal{A}}_1}^{2}dt\right)d\zeta \hfill \\ \hfill & +3{\int }_{-h}^0\left({\sigma }^{2}a{\int }_{x_0}^{x+\zeta }{e}^{-{\iota }_1(x+\zeta -t)}\Lambda {\parallel u_t^{*}-{\varrho }_t\parallel }_{{\mathcal{A}}_1}^{2}dt\right)d\zeta .\hfill \end{array}$$

But

$$e^{-{\iota }_1(x+\zeta -t)}\le e^{-{\iota }_1(x-t)}·e^{{\iota }_{1}h},$$

so

$$\begin{array}{cc}\hfill \Lambda \parallel u_x^{*}-{\varrho }_x{\parallel }_{{\mathcal{A}}_1}^2& \le 3h{e}^{{\iota }_{1}h}{e}^{-{\iota }_1(x-x_0)}E{\parallel u^{*}\left(x_0\right)-\varrho \left(x_0\right)\parallel }_{{\mathcal{A}}_0}^2\hfill \\ \hfill & +3e^{{\iota }_{1}h}h\left({\displaystyle \frac{{\sigma }^2}{{\iota }_1}}+{\sigma }^{2}a\right){\int }_{x_0}^x{e}^{-{\iota }_1(x-t)}\Lambda {\parallel u_t^{*}-{\varrho }_t\parallel }_{{\mathcal{A}}_1}^{2}dt,\hfill \end{array}$$

which yields that

$$\begin{array}{cc}\hfill \Lambda \parallel u^{*}{\left(x\right)-\varrho \left(x\right)\parallel }_{\mathcal{A}}^2& \le (3e^{{\iota }_{1}h}h+3)e^{-{\iota }_1(x-x_0)}\Lambda {\parallel u^{*}\left(x_0\right)-\varrho \left(x_0\right)\parallel }_{\mathcal{A}}^2\hfill \\ \hfill & +(3+3h{e}^{{\iota }_{1}h})\left({\displaystyle \frac{{\sigma }^2}{{\iota }_1}}+{\sigma }^{2}a\right){\int }_{x_0}^x{e}^{-{\iota }_1(x-t)}\Lambda {\parallel u^{*}\left(t\right)-\varrho \left(t\right)\parallel }_B^{2}dt.\hfill \end{array}$$

In addition, if

$$(3+3h{e}^{{\iota }_{1}h})\left({\displaystyle \frac{{\sigma }^2}{{\iota }_1}}+{\sigma }^{2}a\right):={\vartheta }_0{\sigma }^2<{\iota }_1,$$

then one has that

$$\Lambda \parallel u^{*}{\left(x\right)-\varrho \left(x\right)\parallel }_{\mathcal{A}}^2\le (3e^{{\iota }_{1}h}h+3)e^{({\vartheta }_0{\sigma }^2-{\iota }_1)(x-x_0)}\Lambda {\parallel u^{*}\left(x_0\right)-\varrho \left(x_0\right)\parallel }_{\mathcal{A}}^2.$$

It is possible to prove the existence and uniqueness of an invariant measure by employing a similar approach to that utilized in the previous research conducted by [22], [Theorem 3.4.1].

9. Conclusions

In summary, we have carried out a detailed investigation of nonlinear stochastic functional differential equations in Sobolev spaces. The principal contributions are as follows: we proved the existence and uniqueness of mild solutions together with their continuous dependence on initial conditions; we derived uniform a priori bounds that provide a robust foundation for further qualitative analysis; and we established the existence and uniqueness of invariant measures along with their exponential stability.

Overall, this work deepens the theoretical understanding of nonlinear differential systems in Sobolev spaces, while simultaneously offering fresh insights and a solid platform for future developments in this area.