# Performance optimizations on deep noise suppression models

###### Abstract

We study the role of magnitude structured pruning as an architecture search to
speed up the inference time of a deep noise suppression (DNS) model. While
deep learning approaches have been remarkably successful in enhancing audio
quality, their increased complexity inhibits their deployment in real-time
applications. We achieve up to a 7.25X inference speedup over the baseline,
with a smooth model performance degradation. Ablation studies indicate that
our proposed network re-parameterization (i.e., size per layer) is the major
driver of the speedup, and that magnitude structured pruning does comparably
to directly training a model in the smaller size. We report inference speed
because a parameter reduction does not necessitate speedup, and we measure
model quality using an accurate non-intrusive objective speech quality metric.

Index Terms—  speech enhancement, noise reduction, real-time, inference
speedup, structured pruning

##  1 Introduction

There has been much work in compressing deep learning methods so that they can
efficiently operate within the real-time and hardware constraints of many
audio enhancement applications [1, 2, 3, 4]. This interest stems from the fact
that deep learning methods – while typically providing superior audio
enhancement – come with a greater computational complexity than classical
signal processing methods [1]. In real-time applications, the computational
complexity becomes the primary constraint. The available memory per device
varies, but the available time per computation does not. Thus we measure and
present our compression results in terms of the inference speed. Calculating
the memory or parameter reduction is not an accurate proxy - see Section 5.3.

We investigate the application of structured pruning and fine-tuning to speed
up our baseline CRUSE model [1]. Structured pruning aims to find a dense sub-
network that well approximates the original. This type of model compression
immediately transfers to an inference speedup and reduced storage costs
because we are performing dense and smaller matrix multiplications. In
addition, we propose a new scalable per-layer parameter configuration for the
CRUSE architecture class to specify the pruned network sizes.

###  1.1 Contributions

Using the CRUSE [1] architecture we demonstrate up to a 7.25X speedup over the
baseline model, with a smooth degradation of model quality. Ablation studies
indicate that the proposed network parameter configuration is in fact
responsible for the successful scalability. Our structured pruning method does
no better than directly training a model in a given size. The value then of
structured pruning is in the architecture search: discovering which network
parameterizations can reduce model complexity with minimal model degradation.

##  2 Related work

Tan and Wang [3, 4] use sparse regularization, iterative pruning, and
clustering-based quantization to compress DNN speech enhancement models.
However they use STOI and PESQ [5] to evaluate the quality after compression,
which has been shown to have low correlation to subjective quality [6, 7]. In
addition, no run-time benchmarks are given to show real-world improvements,
the noise suppression model used is relatively simple and not state of the
art, and the training and test set are simplistic. Therefore it isn’t clear
from this study what optimizations work well on a best-in-class noise
suppressor on a much more challenging test set such as [7].

Kim et. al. [2] use a combination of unstructured pruning, quantization, and
knowledge distillation to compress keyword spotting models. The authors
motivate their work via edge computing, but do not provide any complexity
measurements to indicate real-world improvements. In addition, no comparisons
are made to any other compression methods.

###  2.1 Deep noise suppression

Braun et. al. [1] developed the CRUSE class of models for real-time deep noise
suppression. It is based on the U-Net architecture [8], another real-time
model for DNS. Unlike earlier network architectures primarily based on
recurrent neural networks whose models have hit a performance saturation [9,
10, 11], CRUSE is in the class of convolutional recurrent networks [8, 12, 13,
14]. These models have increased performance, albeit at a computational cost
that prohibits their real-time deployment on consumer devices. We investigate
two versions of the CRUSE model, a more complex model called CRUSE32 and a
less complex model called CRUSE16.

###  2.2 Model compression

Pruning aims to remove parts of a neural network while maintaining its
accuracy. It can either remove individual parameters resulting in sparse
matrices (unstructured) [15], or it can remove groups of parameters such as
channels or neurons (structured) [16]. There are many strategies to prune a
neural network, but simple magnitude pruning has been shown to be superior to
more complicated methods on ImageNet [17]. Other methods of model compression
include quantization, matrix factorization, and knowledge distillation [18].
Frankle and Carbin [15] present the Lottery Ticket Hypothesis: dense randomly
initialized neural networks contain sparse subnetworks (winning tickets) that
can be trained to comparable accuracy of the original network, in a comparable
number of epochs. The work of Liu et. al. [16] instead focuses on structured
pruning, and presents a different message: fine-tuning a pruned model is
comparable or worse to training a model directly in that size.

Operator Type | Inference Time Proportion
---|---
GRU | 75%
(De)Convolution | 18%
Activations, etc | 7%
Table 1: Profiling CRUSE32 to identify main targets for
inference speedup.

##  3 Experimental methodology

We employ two simple but important experimental methodologies which enable a
more accurate assessment of performance in real-world scenarios. First, we
provide timing results in the ONNX Runtime inference engine [19]. Timing
results are crucial because of the strict real-time requirements for
background noise suppression. It is not necessary that reduced parameter
counts give a faster inference time. For example, we measured no meaningful
speedup for sparse pruned models in ONNX Runtime (see Section 5.3). Benchmark
results are conducted on an Intel Core i7-10610U CPU. Second, the model
quality is evaluated using the INTERSPEECH 2021 and ICASSP 2021 DNS Challenge
test set [7, 20]. We use a new non-intrusive objective speech quality metric
called DNSMOS P.835 [21] employing the ITU-T P.835 [22] standard which
provides separate scores for speech (SIG), background (BAK), and overall
(OVLR) quality. DNSMOS P.835 has a Pearson Correlation Coefficient of 0.94 for
speech, 0.98 for background, and 0.98 for overall compared to subjective
quality, which gives sufficient accuracy to do fast pruning.

Fig. 1: Speedup via structured pruning for the CRUSE class of models. DNSMOS
P.835 reported for the 2021 DNS challenge test sets [7, 20]. Degree 2
polynomial trend line with $R^{2}$ values.

##  4 Performance optimizations
via pruning architecture search

We aim to improve the inference time of the CRUSE architecture class while
maintaining minimal model degradation. The CRUSE architecture we study
consists of 4 convolution-encoder and deconvolution-decoder layer pairs, with
a central parallel GRU layer. Full details of the architecture can be found in
Braun et. al. [1]. Because the DNS models operate in a real-time environment,
we will focus on reducing the inference time. We first profile the inference
time components of CRUSE, with results in Table 1. The GRU operations dominate
the computation, and so we will focus on compressing them.

Config | Network | Mem | Benchmark
---|---|---|---
Name | Param | (MB) | (ms)
CRUSE32 | 32 | 64 | 128 | 256 | 33.6 | 3.41 ($\pm$ .02)
CRUSE16 | 16 | 32 | 64 | 128 | 8.41 | 1.19 ($\pm$ .02)
P.125 | 32 | 64 | 128 | 224 | 26.0 | 2.61 ($\pm$ .01)
P.250 | 32 | 64 | 128 | 192 | 19.5 | 2.10 ($\pm$ .01)
P.500 | 32 | 64 | 128 | 128 | 9.22 | 1.39 ($\pm$ .02)
P.5625 | 32 | 64 | 112 | 112 | 7.16 | 1.14 ($\pm$ .01)
P.625 | 32 | 64 | 96 | 96 | 5.34 | 0.96 ($\pm$ .04)
P.6875 | 32 | 64 | 80 | 80 | 3.79 | 0.79 ($\pm$ .01)
P.750 | 32 | 64 | 64 | 64 | 2.52 | 0.65 ($\pm$ .003)
P.875 | 32 | 32 | 32 | 32 | 0.68 | 0.47 ($\pm$ .01)
Table 2: Structured pruning configurations for CRUSE architecture. Model
memory is of the ONNX format. Benchmarks include 95% confidence intervals.
CRUSE{32,16} are baseline models from Braun et. al. [1].

Structured pruning requires that the number of parameters in each layer be
adjusted. Our DNS model can be parameterized by a length 4 vector
$[c_{1},c_{2},c_{3},c_{4}]$. Because of the skip connections between encoder-
decoder layer pairs, CRUSE is symmetric about the central GRU layer. For
example, $c_{1}$ specifies the output channels of the first convolution layer
and the input channels of the second convolution layer. And $c_{4}$ specifies
the output channels of the last convolution layer and the hidden state of the
GRU layer. Table 2 specifies the baseline CRUSE32 network parameterizations,
as well as the 8 configurations we consider.

We develop a heuristic framework for resizing the CRUSE architecture. First,
reduce the dimensions of the GRU layer via a pruning configuration parameter.
This reduction corresponds to modifying $c_{4}$ in the network
parameterization. For example, with the P.625 configuration the pruning
parameter is $0.625$. Thus, we change $c_{4}\leftarrow(1-0.625)*256=96$. Note
that we must also change the output channels in the last convolution layer.
Then, we enforce what we call “network monotonicity”, that $c_{1}\leq
c_{2}\leq c_{3}\leq c_{4}$. This monotonicity condition is commonly seen in
neural network architecture design where the number of channels increase
through the network [23, 24]. Recall that $c_{3}=128$, and thus $c_{3}>c_{4}$.
We then set $c_{3}\leftarrow c_{4}$ to satisfy the condition. Our proposed
configuration scheme can be applied to other U-Net [8] style architectures, as
it only requires a symmetric encoder-decoder structure.

We employ structured magnitude pruning to construct a dense sub-network to the
size specified by our network parameterization. Consider a generic tensor
$W\in\mathbb{R}^{d_{1}\times\dots\times d_{p}}$. To construct a sub-tensor
$\widehat{W}\in\mathbb{R}^{d_{1}\times\dots\times
d_{i-1}\times\hat{d}_{i}\times d_{i+1}\times\dots\times d_{p}}$ with reduced
dimension $\hat{d}_{i}<d_{i}$, choose a set of coordinates in dimension $i$
satisfying:

$\widehat{\mathcal{J}}=\operatorname*{argmax}_{\mathcal{J}\subseteq\\{1,\dots,d_{i}\\}^{|\hat{d}_{i}|}}\sum_{j\in\mathcal{J}}\|W_{i}^{j}\|$

where $W_{i}^{j}$ represents the tensor $W$ indexed at coordinate $j$ in
dimension $i$. We apply this generic structured magnitude pruning method to
reduce the number of input or output channels in (de)convolution layers, as
well as the input and hidden dimensions of the GRU layer. We prune the CRUSE32
model, our baseline. Fine-tuning is critical to recover model accuracy after
pruning. We use the same optimization hyper-parameters used to train the
baseline CRUSE32 model [1].

##  5 Results

Figure 1 plots the speedup achieved via structured magnitude pruning to the
configurations specified in Table 2, against the resulting signal, background,
and overall DNSMOS P.835. Importantly, structured pruning achieves a smooth
trade-off between complexity and model quality. This trade-off is quite flat;
we can decrease complexity without incurring much model degradation. For
example, a 3.64X memory reduction and a 2.46X inference speedup over the
baseline CRUSE32 model only incurs a 0.01 degradation in overall DNSMOS P.835.
The P.125 configuration achieves higher DNSMOS P.835 than the baseline CRUSE32
because we are doing additional retraining on top of the baseline model. At
the extreme, a 7.25X inference speedup incurs a 0.2 overall DNSMOS P.835
degradation. Overall, we have shown the CRUSE class of models is viable at
complexity levels previously thought untenable.

###  5.1 Ablation studies

| DNSMOS P.835
---|---
Config | (SIG, BAK, OVRL)
Name | Struct Prune | Direct Train
P.125 | 3.86 | 4.34 | 3.60 | 3.85 | 4.33 | 3.59
P.250 | 3.85 | 4.33 | 3.59 | 3.85 | 4.34 | 3.59
P.500 | 3.83 | 4.31 | 3.56 | 3.82 | 4.32 | 3.56
P.750 | 3.76 | 4.25 | 3.47 | 3.77 | 4.26 | 3.48
P.875 | 3.70 | 4.15 | 3.37 | 3.68 | 4.17 | 3.36
Table 3: Prune vs direct training a configuration.

We have introduced two new variables in compressing the CRUSE class of models:
the network re-parameterization, and the structured magnitude pruning method
itself. The CRUSE16 model follows the previous network parameterization [1],
but with half the number of parameters as CRUSE32. In Table 2 we see that
CRUSE16 lies between the P.500 and P.5625 prune configurations in terms of
memory and benchmark speed. However Figure 1 shows CRUSE16 has a worse signal
and overall DNSMOS P.835, with equivalent background. Holding complexity
constant, our new network parameterization in Table 2 achieves superior audio
quality.

Table 3 shows the results of directly training a model in a given
configuration, using the same total number of training epochs as the pruned
models. The results are effectively indistinguishable, which indicates that
structured magnitude pruning is not providing any added value. Additional
tuning of the fine-tuning procedure for structured pruning did not improve
results, see Table 4.

Init | DNSMOS P.835
---|---
LR | (SIG, BAK, OVRL)
1e-3* | 3.76 | 4.25 | 3.47
1e-4 | 3.70 | 4.18 | 3.39
1e-5 | 3.61 | 4.08 | 3.24
1e-6 | 3.46 | 3.80 | 3.04
Table 4: Tuning LR for structured pruning fine-tuning. (*) is the selected hyper-parameter, used for fine-tuning, direct training, and is the original CRUSE32 training parameter.  | DNSMOS P.835
---|---
Baseline | (SIG, BAK, OVR
CRUSE32 | 3.80 | 4.35 | 3.57
CRUSE16 | 3.73 | 4.30 | 3.49
Table 5:  DNSMOS P.835 for baseline CRUSE32, CRUSE16 models.

###  5.2 The value of fine-tuning

| DNSMOS P.835
---|---
| (SIG, BAK, OVRL)
Prune | Before | After
Name | Fine-Tuning | Fine-Tuning
P.125 | 2.22 | 2.87 | 2.11 | 3.86 | 4.34 | 3.60
P.250 | 2.23 | 2.87 | 2.15 | 3.85 | 4.33 | 3.59
P.500 | 2.03 | 2.83 | 1.96 | 3.83 | 4.31 | 3.56
P.5625 | 2.04 | 2.77 | 2.04 | 3.80 | 4.27 | 3.52
P.625 | 2.11 | 2.74 | 2.02 | 3.79 | 4.28 | 3.52
P.6875 | 2.21 | 2.81 | 2.00 | 3.79 | 4.26 | 3.50
P.750 | 2.02 | 2.75 | 1.89 | 3.76 | 4.25 | 3.47
P.875 | 2.66 | 2.74 | 2.22 | 3.70 | 4.15 | 3.37
Table 6: DNSMOS P.835 of structure pruned CRUSE32 models before and after fine-tuning. | DNSMOS P.835
---|---
| (SIG, BAK, OVRL)
Frac | Before | After
GRU | Fine-Tuning | Fine-Tuning
0.25 | 3.80 | 4.34 | 3.56 | 3.86 | 4.32 | 3.59
0.50 | 3.78 | 4.32 | 3.54 | 3.86 | 4.33 | 3.60
0.75 | 3.68 | 4.22 | 3.40 | 3.86 | 4.33 | 3.59
Table 7:  DNSMOS P.835 of unstructured (sparse) pruned CRUSE32 models before
and after fine-tuning.

In Table 6 the DNSMOS P.835 is reported after structured magnitude pruning,
and again after fine-tuning. We see that fine-tuning is crucial to recover the
model quality. The MOS scores are increased by approximately 1, a large
margin. Table 7 reports the DNSMOS P.835 after unstructured (sparse) magnitude
pruning, and again after fine-tuning. Remarkably, we see that unstructured
pruning does not degrade the model quality nearly as much. Note, the “Frac GRU
0.25” unstructured setting does not modify any convolution layers and is not
equivalent to the “P.250” configuration. Subsequently, fine-tuning has less to
recover.

###  5.3 Sparsity does not imply inference speedup

Frac | Benchmark
---|---
GRU | (ms)
0.00 | 4.91 ($\pm$ .07)
0.25 | 4.92 ($\pm$ .04)
0.50 | 4.92 ($\pm$ .08)
0.75 | 4.90 ($\pm$ .03)
Table 8:  Benchmark results of unstructured (sparse) magnitude pruning the GRU
layers in CRUSE32.

Despite the promising results in Table 7, reducing the parameter count via
sparsity does not easily translate to an inference speedup. Table 8 shows the
benchmark results for sparse magnitude pruning of the GRU layer in CRUSE32. A
certain fraction of the GRU weights were set to zero. The times are not
distinguishable with their confidence intervals. This means that ONNX Runtime
provides no speedup from sparse linear algebra operations. Specialized sparse
inference support is needed to realize speedup from sparsity. The Neural Magic
inference engine [25] is one such option, but in our experimentation we found
that support for RNN layers was still in development.

##  6 Conclusions

We have achieved up to a 7.25X inference speedup over the baseline CRUSE32
model. Our experiments indicate that the proposed network parameterization
(size of each layer) is the primary driver of our speedup results, not
structured magnitude pruning. These conclusions support Liu et. al. [16] in
that the value of structured pruning is in conducting an architecture search.
Additionally, our methodological choice of measuring the inference speed –
rather than using parameter count as a proxy – revealed the difficulty of
realizing practical benefits from sparse pruning methods. Our proposed network
parameterization only requires a symmetric encoder-decoder structure, and thus
can be applied to other U-Net style architectures.

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# Stability of additive-quadratic functional equation in modular space

Abderrahman Baza1 1Departement of Mathematics, Ibn Tofail University, Kenitra,
Morocco<EMAIL_ADDRESS>,  Mohamed Rossafi2 2Departement of
Mathematics, Ibn Tofail University, Kenitra, Morocco<EMAIL_ADDRESS>and  Choonkil Park3∗ 3Research Institute for Natural Sciences, Hanyang
University, Seoul 04763, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

Using the direct method, we prove the generalised Hyers-Ulam stability of the
following functional equation

$\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w)=0$ (0.1)

in modular space satisfying the Fatou property or $\Delta_{2}$-condition.

###### Key words and phrases:

Hyers-Ulam stability, radical cubic functional inequalities, modular sapaces,
fuzzy Banach, $\Delta_{2}$-condition.

###### 2010 Mathematics Subject Classification:

Primary 39B82, Secondary 39B52

∗Corresponding author

##  1\. Introduction and preliminaries

Nakano [13] looked into the concept of modular linear spaces in 1950. Many
authors, including Luxemburg [9], Amemiya [1], Musielak [12], Koshi [8], Mazur
[11], Turpin [17], and Orlicz [14], have now thoroughly proved these
hypotheses. There are numerous uses for Orlicz spaces [14] and the idea of
interpolation [14] in the context of modular spaces. As suggested by Khamsi
[7], several scholars used a fixed point approach of quasi- contractions to
test stability in modular spaces without using the 2-condition. Recent results
on the stability of various functional equations combining the
$\Delta_{2}$-condition and the Fatou property were reported by Sadeghi [16].
Firstly, we review some vocabulary, notations, and common characteristics of
the theory of Modulars and modular spaces.

######  Definition 1.1.

Let $Y$ be an arbitrary vector space. A functional
$\rho:Y\rightarrow[0,\infty)$ is called a modular if for arbitrary $u,v\in Y$;

  1. (1)

$\rho(u)=0$ if and only if $u=0$.

  2. (2)

$\rho(\alpha u)=\rho(u)$ for every scalar $\alpha$ with $|\alpha|=1$.

  3. (3)

$\rho(\alpha u+\beta v)\leq\rho(u)+\rho(v)$ if and only if $\alpha+\beta=1$
and $\alpha,\beta\geq 0$.
If (1) is replaced by:

  4. (4)

$\rho(\alpha u+\beta v)\leq\alpha\rho(u)+\beta\rho(v)$ if and only if
$\alpha+\beta=1$ and $\alpha,\beta\geq 0$, then we say that $\rho$ is a convex
modular.

  5. (5)

It is said that the modular $\rho$ has the Fatou property if and only if
$\rho(x)\leq\liminf_{n\to\infty}\rho(x_{n})$ whenever
$x=\rho-\lim_{n\to\infty}x_{n}$.

A modular $\rho$ defines a corresponding modular space, i.e., the vector space
$Y_{\rho}$ given by:

$Y_{\rho}=\\{u\in Y:\rho(\lambda u)\rightarrow 0\text{ as }\lambda\rightarrow
0\\}.$

A function modular is said to satisfy the $\Delta_{2}$-condition if there
exist $\tau>0$ such that $\rho(2u)\leq\tau\rho(u)$ for all $u\in Y_{\rho}$.

######  Definition 1.2.

Let $\left\\{u_{n}\right\\}$ and $u$ be in $Y_{\rho}$. Then:

  1. (1)

The sequence $\left\\{u_{n}\right\\}$, with $u_{n}\in Y_{\rho}$, is
$\rho$-convergent to $u$ and write: $u_{n}\to u$ if
$\rho\left(u_{n}-u\right)\to 0$ as $n\rightarrow\infty$.

  2. (2)

The sequence $\\{u_{n}\\}$, with $u_{n}\in Y_{\rho}$, is called $\rho$-Cauchy
if $\rho\left(u_{n}-u_{m}\right)\to 0$ as $n,m\to\infty$.

  3. (3)

$Y_{\rho}$ is called $\rho$-complete if every $\rho$-Cauchy sequence in
$Y_{\rho}$ is $\rho$-convergent.

######  Proposition 1.3.

In modular space,

  * •

If $u_{n}\overset{\rho}{\to}u$ and a is a constant vector, then
$u_{n}+a\overset{\rho}{\to}u+a$.

  * •

If $u_{n}\overset{\rho}{\to}u$ and $v_{n}\overset{\rho}{\to}v$ then $\alpha
u_{n}+\beta v_{n}\overset{\rho}{\to}\alpha u+\beta v$, where $\alpha+\beta\leq
1$ and $\alpha,\beta\geq 1$.

######  Remark 1.4.

Note that $\rho(u)$ is an increasing function, for all $u\in X$. Suppose
$0<a<b$, then property $(4)$ of Definition 1.1 with $v=0$ shows that
$\rho(au)=\rho\left(\dfrac{a}{b}bu\right)\leq\rho(bu)$ for all $u\in Y$.
Morever, if $\rho$ is a convexe modular on $Y$ and $|\alpha|\leq 1$, then
$\rho(\alpha u)\leq\alpha\rho(u)$.

In general, if $\lambda_{i}\geq 0$, $i=1,\dots,n$ and
$\lambda_{1},\lambda_{2},\dots,\lambda_{n}\leq 1$ then
$\rho(\lambda_{1}u_{1}+\lambda_{2}u_{2}+\dots+\lambda_{n}u_{n})\leq\lambda_{1}\rho(u_{1})+\lambda_{2}\rho(u_{2})+\dots+\lambda_{n}\rho(u_{n})$.

If $\\{u_{n}\\}$ is $\rho$-convergent to $u$, then $\\{cu_{n}\\}$ is
$\rho$-convergent to $cu$, where $|c|\leq 1$. But the $\rho$-convergent of a
sequence $\\{u_{n}\\}$ to $u$ does not imply that $\\{\alpha u_{n}\\}$ is
$\rho$-convergent to $\alpha u_{n}$ for scalars $\alpha$ with $|\alpha|>1$.

If $\rho$ is a convex modular satisfying $\Delta_{2}$ condition with
$0<\tau<2$, then
$\rho(u)\leq\tau\rho(\dfrac{1}{2}u)\leq\dfrac{\tau}{2}\rho(u)$ for all $u$.
Hence $\rho=0$. Consequently, we must have $\tau\geq 2$ if $\rho$ is convex
modular.

The functional equations are crucial for the research of stability issues in a
variety of contexts. Ulam [18] was the first to question the stability of
group homomorphisms. We refer to an equation as being stable if it only
permits one distinct solution. Ulam [18] developed the next Equation of Cauchy
function. Hyers [6] addressed Cauchy’s functional equation in the setting of a
Banach space to overcome this issue. Aoki [2] improved the work of Hyers by
taking an unbounded Cauchy difference. In his study, Rassias [15] examined
additive mappings, and Gavruta [4] has already provided the same conclusions.
See [3, 6] for further information regarding the stability results.
in this paper wich is constituted of 3 sections, we study the Hyers Ulam
stability of the following functional equation:

$\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w)=0\text{ for all }x,y,z,w\in
X.$ (1.1)

in modular space satisfying the Fatou property or with $\Delta_{2}$-condition.

##  2\. Hyers Ulam stability of (0.1) in modular space satisfying the Fatou
property

######  Lemma 2.1 ([5]).

Let $X$ and $Y$ be vector spaces and $\phi:X^{2}\rightarrow Y$ be a mapping
satisfies: $\phi(0,z)=\phi(x,0)=0$ and

$\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w)=0\text{ for all }x,y,z,w\in
X.$

Then $\phi$ is additive in the first variable and quadratic in the second
variable. The mapping $\phi$ is called an additive-quadratic mapping.

######  Theorem 2.2.

Let $X$ be a vector space, and $Y_{\rho}$ be a $\rho$-complete convex modular
space satisfying the Fatou property. Let
$\alpha:X^{2}\longrightarrow[0,\infty)$ be a function such that

$\varphi(x,y)=\sum_{j=1}^{\infty}\frac{1}{2^{j}}\alpha\left(2^{j-1}x,2^{j-1}y\right)<\infty$
(2.1)

and

$\psi(x,y)=\sum_{j=1}^{\infty}\frac{1}{4^{j}}\alpha\left(2^{j-1}x,2^{j-1}y\right)<\infty$
(2.2)

for all $x,y\in X$. Let $\phi:X^{2}\rightarrow Y_{\rho}$ be a mapping that
satisfies

$\phi(x,0)=\phi(0,z)=0,$

and

$\rho(\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w))\leq\alpha(x,y)\alpha(z,w)$
(2.3)

for all $x,y,z,w\in X$. Then there exists a unique additive quadratic mapping
$H:X^{2}\rightarrow Y_{\rho}$ such that

$\rho(\phi(x,z)-H(x,z))\leq\min\\{\varphi(x,x)\alpha(z,0),\alpha(x,0)\psi(z,z)\\}\text{
for all }x,z\in X.$ (2.4)

###### Proof.

Letting $y=x$ and $w=0$ in (2.3), we get

$\rho\left(\phi(2x,z)-2\phi(x,z)\right)\leq\alpha(x,x)\alpha(z,0)$

Then by convexity of $\rho$, we have

$\rho\left(\frac{1}{2}\phi(2x,z)-\phi(x,z)\right)\leq\frac{1}{2}\alpha(x,x)\alpha(z,0)$
(2.5)

and by induction, we have

$\rho\left(\frac{1}{2^{k}}\phi\left(2^{k}x,z\right)-\phi(x,z)\right)\leq\sum_{j=1}^{k}\frac{1}{2^{j}}\alpha\left(2^{j-1}x,2^{j-1}x\right)\alpha(z,0)$
(2.6)

for all $x,z\in X$ and all positive integer $k$. For $k=1$, we obtain (2.5).
Suppose that (2.6) holds for a fixed $k\in\mathbb{N}$. We have

$\displaystyle\rho\left(\frac{1}{2^{k+1}}\phi\left(2^{k+1}x,z\right)-\phi(x,z)\right)$
$\displaystyle=\rho\left(\frac{1}{2}\left(\frac{1}{2^{k}}\phi\left(2^{k}\cdot
2x,z\right)-\phi(2x,z)\right)+\frac{1}{2}\phi(2x,z)-\phi(x,z)\right)$
$\displaystyle\leq\frac{1}{2}\sum_{j=1}^{k}\frac{1}{2^{j}}\alpha\left(2^{j}x,2^{j}x\right)\alpha(z,0)+\frac{1}{2}\alpha(x,x)\alpha(z,0)$
$\displaystyle=\sum_{j=1}^{k+1}\frac{1}{2^{j}}\alpha\left(2^{j-1}x,2^{j-1}x\right)\alpha(z,0)$

Hence, (2.6) holds for every $k\in\mathbb{N}$.

Let $m,n$ be positive integers such that $n>m$. We write

$\displaystyle\rho\left(\frac{1}{2^{n}}\phi\left(2^{n}x,z\right)-\frac{1}{2^{m}}\phi\left(2^{m}x,z\right)\right)$
$\displaystyle=\rho\left(\frac{1}{2^{m}}\left(\frac{\phi\left(2^{n-m}\cdot
2^{m}x,z\right)}{2^{n-m}}-\phi\left(2^{m}x,z\right)\right)\right)$
$\displaystyle\leq\frac{1}{2^{m}}\sum_{j=1}^{n-m}\frac{1}{2^{j}}\alpha\left(2^{m+j-1}x,2^{m+j-1}x\right)\alpha(z,0)$
$\displaystyle=\sum_{k=m+1}^{n}\frac{1}{2^{k}}\alpha\left(2^{k-1}x,2^{k-1}x\right)\alpha(z,0)$
(2.7)

it follows from (2.7) and (2.1) that
$\left\\{\frac{\phi\left(2^{n}x,z\right)}{2^{n}}\right\\}$ is a Cauchy
sequence in $Y_{\rho}$ which is complete, and this guarantees the existence a
mapping $A:X^{2}\rightarrow Y_{\rho}$ such that

$A(x,z)=\rho-\operatorname{limit}\frac{\phi\left(2^{n}x,z\right)}{2^{n}},\qquad
x,z\in X.$ (2.8)

Now, by application the Fatou property, we have

$\displaystyle\rho(A(x,z)-\phi(x,z))$
$\displaystyle\leq\liminf_{n\rightarrow\infty}\rho\left(\frac{1}{2^{n}}\phi(2^{n}x,z)-\phi(x,z)\right)$
$\displaystyle\leq\sum_{k=1}^{\infty}\frac{1}{2^{k}}\alpha\left(2^{k-1}x,2^{k-1}x\right)\alpha(z,0)$

Hence

$\rho(A(x,z)-\phi(x,z))\leq\varphi(x,x)\alpha(z,0)$ (2.9)

Now, we prove that $A$ is an additive-quadratic mapping. In the first, we have

$\rho\left(\frac{1}{2^{n}}\phi\left(2^{n}(x+y),z+w\right)+\frac{1}{2^{n}}\phi\left(2^{n}(x-y),z-w\right)-\frac{2}{2^{n}}\phi\left(2^{n}x,z\right)-\frac{2}{2^{n}}\phi\left(2^{n}x,w\right)\right)\\\
\leq\frac{1}{2^{n}}\alpha\left(2^{n}x,2^{n}y\right)\alpha(z,w)\longrightarrow
0\text{ as }n\rightarrow\infty$

for all $x,y,z,w\in X$. And we have

$\rho\left(\frac{1}{12}(A(x+y,z+w)+A(x-y,z-w)-2A(x,z)-2A(x,w))\right)\\\
\leq\frac{1}{6}\left[\frac{1}{2}\rho\left(A(x+y,z+w)-\frac{\phi\left(2^{n}(x+y),z+w\right)}{2^{n}}\right)\right.\\\
\left.+\frac{1}{2}\rho\left(A(x-y,z-w)-\frac{\phi\left(2^{n}(x-y),z-w\right)}{2^{n}}\right)\right.\\\
\left.+\rho\left(A(x,z)-\frac{\phi\left(2^{n}x,z\right)}{2^{n}}\right)+\rho\left(A(x,w)-\frac{\phi\left(2^{n}x,w\right)}{2^{n}}\right)\right]\\\
+\frac{1}{12}\rho\left(\frac{\phi\left(2^{n}(x+y),z+w\right)}{2^{n}}+\frac{\phi\left(2^{n}(x-y),z-w\right)}{2^{n}}-\frac{2}{2^{n}}\phi\left(2^{n}x,z\right)-\frac{2}{2^{n}}\phi\left(2^{n}x,w\right)\right)\\\
\longrightarrow 0\text{ as }n\longrightarrow\infty.$

Then we get

$A(x+y,z+w)+A(x-y,z-w)-2A(x;z)-2A(x,w)=0$

and by Lemma 2.1, we conclude that $A$ is an additive-quadratic mapping. Now,
letting $B:X^{2}\rightarrow Y_{\rho}$ be another additive-quadratic mapping
satisfying (2.9). We have

$\displaystyle\rho\left(\frac{A(x,z)-B(x,z)}{2}\right)$
$\displaystyle=\rho\left(\frac{1}{2}\left(\frac{A\left(2^{k}x,z\right)}{2^{k}}-\frac{\phi\left(2^{k}x,z\right)}{2^{k}}\right)+\frac{1}{2}\left(\frac{\phi\left(2^{k}x,z\right)}{2^{k}}-\frac{B\left(2^{k}x,z\right)}{2^{k}}\right)\right)$
$\displaystyle\leq\frac{1}{2}\rho\left(\frac{A\left(2^{k}x,z\right)}{2^{k}}-\frac{\phi(2^{k}x,z)}{2^{k}}\right)+\frac{1}{2}\rho\left(\frac{\phi(2^{k}x,z)}{2^{k}}-\frac{B(2^{k}x,z)}{2^{k}}\right)$
$\displaystyle\leq\frac{1}{2}\cdot\frac{1}{2^{k}}\left\\{\rho\left(A\left(2^{k}x,z\right)-\phi\left(2^{k}x,z\right)\right)+\rho\left(\phi\left(2^{k}x,z\right)-B\left(2^{k}x,z\right)\right)\right\\}$
$\displaystyle\leq\frac{1}{2^{k}}\varphi\left(2^{k}x,2^{k}x\right)\alpha(z,0)$
$\displaystyle=\sum_{j=1}^{\infty}\frac{1}{2^{k+j}}\alpha\left(2^{k+j-1}x,2^{k+j-1}x\right)\alpha(z,0)$
$\displaystyle=\sum_{l=k+1}^{\infty}\frac{1}{2^{l}}\alpha\left(2^{l-1}x,2^{l-1}x\right)\alpha(z,0)\to
0\text{ as }k\to\infty.$

Then $A(x,z)=B(x,z)$ for all $x,z\in X$.

On the other hand, letting $y=0$ and $w=z$ in (2.3), we get

$\rho(\phi(x,2z)-4\phi(x,z))\leq\alpha(z,z)\alpha(x,0)$

Then

$\rho\left(\frac{1}{4}\phi(x,2z)-\phi(x,z)\right)\leq\frac{1}{4}\alpha(z,z)\alpha(x,0)$

by simple induction, we have

$\rho\left(\frac{1}{4^{k}}\phi\left(x,2^{k}z\right)-\phi(x,z)\right)\leq\sum_{j=1}^{k}\frac{1}{4^{j}}\alpha\left(2^{j-1}z,2^{j-1}z\right)\alpha(x,0).$

Let $m,n$ be positive integers with $n>m$. We have

$\displaystyle\rho\left(\frac{1}{4^{n}}\phi\left(x,2^{n}z\right)-\frac{1}{4^{m}}\phi\left(x,2^{m}z\right)\right)$
$\displaystyle=\rho\left(\frac{1}{4^{m}}\left(\frac{\phi\left(x,2^{n-m}\cdot
2^{m}z\right)}{4^{n-m}}-\phi\left(x,2^{m}z\right)\right)\right)$
$\displaystyle\leq\frac{1}{4^{m}}\sum_{j=1}^{n-m}\frac{1}{4^{j}}\alpha\left(2^{m+j-1}z,2^{m+j-1}z\right)\alpha(x,0)$
$\displaystyle=\sum_{k=m+1}^{n}\frac{1}{4^{k}}\alpha\left(2^{k-1}z,2^{k-1}z\right)\alpha(x,0)$
(2.10)

by (2.10) and (2.2), we conclude that
$\left\\{\dfrac{\phi(x,2z)}{4^{n}}\right\\}$ is a Cauchy sequence in
$Y_{\rho}$ wich is $\rho$-complete.

Then there exists a mapping $C:X^{2}\rightarrow Y_{\rho}$ such that

$C(x,z)=\rho\operatorname{limit}\frac{\phi\left(x,2^{n}z\right)}{4^{n}},\qquad
x,z\in X.$

Moreover

$\displaystyle\rho(C(x,z)-\phi(x,z))$
$\displaystyle\leq\liminf_{n\rightarrow\infty}\rho\left(\frac{1}{4^{n}}\phi\left(x,2^{n}z\right)-\phi(x,z)\right)$
$\displaystyle\leq\sum_{k=1}^{\infty}\frac{1}{4^{k}}\alpha\left(2^{k-1}z,2^{k-1}z\right)\alpha(x,0)$

Hence

$\rho(C(x,z)-\phi(x,z))\leq\psi(z,z)\alpha(x,0)$ (2.11)

Now,we prove that $C$ is an additive-quadratic mapping.

We have

$\rho\left(\frac{1}{4^{n}}\phi\left(x+y,2^{n}(z+w)\right)+\frac{1}{4^{n}}\phi\left(x-y,2^{n}(z-w)\right)-\frac{2}{4^{n}}\phi\left(x,2^{n}z\right)-\frac{2}{4^{n}}\phi\left(x,2^{n}w\right)\right)\\\
\leq\frac{1}{4^{n}}\alpha(x,y)\alpha\left(2^{n}z,2^{n}w\right)\rightarrow
0\text{ as }n\rightarrow\infty$

for all $x,y,z,w\in X$. And we have

$\rho\left(\frac{1}{12}(C(x+y,z+w)+C(x-y,z-w)-2C(x,z)-2C(x,w))\right)\\\
\leq\frac{1}{6}\left[\frac{1}{2}\rho\left(C(x+y,z+w)-\frac{\phi\left(x+y,2^{n}(z+w)\right)}{4^{n}}\right)\right.\\\
\left.+\frac{1}{2}\rho\left(C(x-y,z-w)-\frac{\phi\left(x-y,2^{n}(z-w)\right)}{4^{n}}\right)\right.\\\
\left.+\rho\left(C(x,z)-\frac{\phi\left(x,2^{n}z\right)}{4^{n}}\right)+\rho\left(C(x,w)-\frac{\phi\left(x,2^{n}w\right)}{4^{n}}\right)\right]\\\
+\frac{1}{12}\rho\left(\frac{\phi\left(x+y,2^{n}(z+w)\right)}{4^{n}}+\frac{\phi\left(x-y,2^{n}(z-w)\right)}{4^{n}}-\frac{2}{4^{n}}\phi\left(x,2^{n}z\right)-\frac{2}{4^{n}}\phi\left(x,2^{n}w\right)\right)$
$\longrightarrow 0\text{ as }n\rightarrow\infty$

Hence, we get

$C(x+y,z+w)+C(x-y,z-w)-2C(x,z)-2C(x,w)=0,$

by Lemma 2.1, we deduce that $C$ is an additive-quadratic mapping.

To shows the uniqueness of $C$, letting $D:X^{2}\rightarrow Y_{\rho}$ be an
other mapping satisfying (2.11). We have

$\displaystyle\rho\left(\dfrac{C(x,z)-D(x,z)}{2}\right)$
$\displaystyle=\rho\left(\frac{1}{2}\left(\dfrac{C\left(x,2^{k}z\right)}{4^{k}}-\dfrac{\phi\left(x,2^{k}z\right)}{4^{k}}\right)+\dfrac{1}{2}\left(\phi\dfrac{\left(x,2^{k}z\right)}{4^{k}}-\dfrac{D\left(x,2^{k}z\right)}{4^{k}}\right)\right)$
$\displaystyle\leq\frac{1}{2}\cdot\frac{1}{4^{k}}\left\\{\rho(C(x,2^{k}z)-\phi(x,2^{k}z)+\rho(\phi(x,2^{k}z)-D(x,2^{k}z))\right\\}$
$\displaystyle\leq\frac{1}{4^{k}}\psi\left(2^{k}z,2^{k}z\right)\alpha(x,0)$
$\displaystyle=\sum_{j=1}^{\infty}\frac{1}{4^{k+j}}\alpha\left(2^{k+j-1}z,2^{k+j-1}z\right)\alpha(x,0)$
$\displaystyle=\sum_{l=k+1}^{\infty}\frac{1}{4^{l}}\alpha\left(2^{l-1}z,2^{l-1}z\right)\alpha(x,0)\longrightarrow
0\text{ as }k\rightarrow\infty.$

Then $C(x,z)=D(x,z)$ for all $x,z\in X$.

It follows from (2.9) and (2.11) that

$\displaystyle\rho\left(\frac{C(x,z)-A(x,z)}{2}\right)$
$\displaystyle=\rho\left(\frac{1}{2}\left(\frac{C\left(2^{n}x,z\right)}{2^{n}}-\frac{\phi\left(2^{n}x,z\right)}{2^{n}}\right)+\frac{1}{2}\left(\frac{\phi\left(2^{n}x,z\right)}{2^{n}}-\frac{A(2x,z))}{2^{n}}\right)\right)$
$\displaystyle\leq\frac{1}{2}\cdot\frac{1}{2^{n}}\left\\{\rho(C(2^{n}x,z)-\phi(2^{n}x,z))+\rho\left(\phi(2^{n}x,z)-A(2^{n}x,z)\right)\right\\}$
$\displaystyle\leq\frac{1}{2}\cdot\frac{1}{2^{n}}\alpha\left(2^{n}x,0\right)\psi(z,z)+\frac{1}{2}\cdot\frac{1}{2^{n}}\varphi\left(2^{n},2^{n}x\right)\alpha(z,0)\longrightarrow
0\text{ as }n\rightarrow\infty$ $\displaystyle\text{ for all }x,z\in X.$

This implies that $C(x,z)=A(x,z)=H(x,z)$ for all $x,z\in X$. Then there exists
a unique additive-quadratic mapping $H:X^{2}\rightarrow Y_{\rho}$ such that

$\rho(\phi(x,z)-H(x,z))\leq\min\\{\varphi(x,x)\alpha(z,0),\alpha(x,0)\psi(z,z)\\}$

for all $x,z\in X$. ∎

######  Corollary 2.3.

Let $X$ be a vector space and $Y_{\rho}$ be a $\rho$-complete convex modular
space. Let $0<r<1$ and $\theta$ be positives real numbers and
$\phi:X^{2}\longrightarrow Y_{\rho}$ be a mapping satisfying

$\phi(x,0)=\phi(0,z)=0$

and

$\rho(\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w))\leq\theta\left(\|x\|\left\|{}^{r}+\right\|y\|^{r}\right)\left(\|z\|^{r}+\|w\|^{r}\right)$
(2.12)

for all $x,y,z,w\in X$. Then there exists a unique additive-quadratic mapping
$H:X^{2}\longrightarrow Y_{\rho}$ such that

$\rho(\phi(x,z)-H(x,z))\leq\frac{2\theta\|z\|^{r}\|x\|^{r}}{4-2^{r}},\qquad
x,z\in X.$

###### Proof.

The proof is a result of Theorem 2.2 by taking
$\alpha(x,y)=\sqrt{\theta}(\|x\|^{r}+\|y\|^{r})$ for all $x,y\in X$ and
remarking that

$\min\left\\{\frac{2\theta}{2-2^{r}}\|x\|^{r}\|z\|^{r},\frac{2\theta\|x\|^{r}\|z\|^{r}}{4-2^{r}}\right\\}=\frac{2\theta}{4-2^{r}}\|x\|^{r}\|z\|^{r}$

for all $x,z\in X$. Now, we obtain a classical Ulam stability, by putting
$\alpha=\varepsilon>0$. ∎

######  Corollary 2.4.

Let $X$ be a vector space, $Y_{\rho}$ be a $\rho$-complete modular space with
$\rho$ is a convex modular. Let $\phi:X^{2}\rightarrow Y_{\rho}$ be a mapping
such that $\phi(x,0)=\phi(0,z)=0$ and

$\rho(\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w))\leq\varepsilon^{2}$

for all $x,y,z,w\in X$. Then there exists a unique additive quadratic mapping
$H:X^{2}\longrightarrow Y_{\rho}$ such that
$\rho(\phi(x,z)-H(x,z))\leq\frac{\varepsilon^{2}}{3}$, for all $x,z\in X$.

##  3\. Stability of (0.1) modular space satisfying $\Delta_{2}$-condition

######  Theorem 3.1.

Let $X$ be a vector space, $Y_{\rho}$ be a $\rho$-complete convex modular
space, and satisfying the $\Delta_{2}$-condition. Let
$\alpha:X^{2}\longrightarrow[0,\infty)$ be a function such that

  1. (i)

$\varphi(x,y)=\sum_{j=1}^{\infty}\left(\frac{\tau^{2}}{2}\right)^{j}\alpha\left(\frac{x}{2^{j}},\frac{y}{2^{j}}\right)<\infty\text{
and
}\lim_{n\rightarrow\infty}\tau^{n}\alpha\left(\frac{x}{2^{n}},\frac{y}{2^{n}}\right)=0$
(3.1)

  2. (ii)

$\psi(x,y)=\sum_{j=1}^{\infty}\left(\frac{\tau^{3}}{2}\right)^{j}\alpha\left(\frac{x}{2^{j}},\frac{y}{2^{j}}\right)<\infty\text{
and
}\lim_{n\rightarrow\infty}\tau^{2n}\alpha\left(\frac{x}{2^{n}},\frac{y}{2^{n}}\right)=0.$
(3.2)

Let $\phi:X^{2}\longrightarrow Y_{\rho}$ be a mapping such that
$\phi(x,0)=\phi(0,z)=0$ and

$\rho(\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w))\leq\alpha(x,y)\alpha(z,w)$
(3.3)

for all $x,y,z,w\in X$. Then there exists a unique additive-quadratic mapping
$H:X^{2}\longrightarrow Y_{\rho}$ such that

$\rho(\phi(x,z)-H(x,z))\leq\min\left\\{\frac{1}{2}\varphi(x,x)\alpha(z,0),\frac{1}{2\tau}\psi(z,z)\alpha(x,0)\right\\}$

for all $x,z\in X$.

###### Proof.

Letting $y=x$ and $w=0$ in (3.3), we get

$\rho(\phi(2x,z)-2\phi(x,z))\leq\alpha(x,x)\alpha(z,0)$

Hence

$\rho\left(\phi(x,z)-2\phi\left(\frac{x}{2^{n}},z\right)\right)\leq\alpha\left(\frac{x}{2},\frac{x}{2}\right)\alpha(z,0)$

Then, by using the $\Delta_{2}$-condition and the convexity of $\rho$, we have

$\displaystyle\rho\left(\phi(x,z)-2^{n}\phi\left(\frac{x}{2^{n}},z\right)\right)$
$\displaystyle=\rho\left(\sum_{j=1}^{n}\frac{1}{2^{j}}\left(2^{2j-1}\phi\left(\frac{x}{2^{j-1}},z\right)-2^{2j}\phi\left(\frac{x}{2^{j}},z\right)\right)\right)$
$\displaystyle\leq\frac{1}{\tau}\sum_{j=1}^{n}\left(\frac{\tau^{2}}{2}\right)^{j}\alpha\left(\frac{x}{2^{j}},\frac{x}{2^{j}}\right)\alpha(z,0)$

for all $x,z\in X$. So, for all positifs integers $m$ and $n$ with $n>m$, we
have

$\displaystyle\rho\left(2^{n}\phi\left(\frac{x}{2^{n}},z\right)-2^{m}\phi\left(\frac{x}{2^{m}},z\right)\right)$
$\displaystyle\leq\tau^{m}\rho\left(2^{n-m}\phi\left(\frac{x}{2^{n}},z\right)-\phi\left(\frac{x}{2^{m}},z\right)\right)$
$\displaystyle\leq\tau^{m-1}\sum_{j=1}^{n-m}\left(\frac{\tau^{2}}{2}\right)^{j}\alpha\left(\frac{x}{2^{m+j}},\frac{x}{2^{m+j}}\right)\alpha(z,0)$
$\displaystyle=\frac{1}{\tau}\left(\frac{2}{\tau}\right)^{m}\sum_{l=m+1}^{n}\left(\frac{\tau^{2}}{2}\right)^{l}\alpha\left(\frac{x}{2^{l}},\frac{x}{2^{l}}\right)\alpha(z,0)$
(3.4)

it follows from (3.4) and (3.1) that
$\left\\{2^{n}\phi\left(\frac{x}{2^{n}},z\right)\right\\}$ is a Cauchy
sequence in $Y_{\rho}$ wich is complete. Hence, we define a mapping
$A:X^{2}\rightarrow Y_{\rho}$ as

$A(x,z)=\rho-\lim_{n\rightarrow\infty}2^{n}\phi\left(\frac{x}{2^{n}},z\right);x,z\in
X.$

Now, we have

$\displaystyle\rho(\phi(x,z)-A(x,z))$
$\displaystyle\leq\frac{1}{2}\rho\left(2\phi(x,z)-2^{n+1}\phi\left(\frac{x}{2^{n}},z\right)\right)+\frac{1}{2}\rho\left(2^{n+1}\phi\left(\frac{x}{2^{n}},z\right)-2A(x,z)\right)$
$\displaystyle\leq\frac{\tau}{2}\rho\left(\phi(x,z)-2^{n}\phi\left(\frac{x}{2^{n}},z\right)\right)+\frac{\tau}{2}\rho\left(2^{n}\phi\left(\frac{x}{2^{n}},z\right)-A(x,z)\right)$
$\displaystyle\leq\frac{1}{2}\sum_{j=1}^{n}\left(\frac{\tau^{2}}{2}\right)^{j}\alpha\left(\frac{x}{2^{j}},\frac{x}{2^{j}}\right)\alpha(z,0)+\frac{\tau}{2}\rho\left(2^{n}\phi\left(\frac{x}{2^{n}},z\right)-A(x,z)\right)$

for all $x,z\in X$. Passing to limit $n\rightarrow\infty$, we obtain

$\rho(\phi(x,z)-A(x,z))\leq\frac{1}{2}\varphi(x,x)\alpha(z,0)$ (3.5)

Now, we prove that $A$ is an additive-quadratic mapping.

In the first, we have:
$\displaystyle\rho\left(2^{n}\phi\left(\frac{x+y}{2^{n}},z+w\right)+2^{n}\phi\left(\frac{x-y}{2^{n}},z-w\right)-2\cdot
2^{n}\phi\left(\frac{x}{2^{n}},z\right)-2\cdot
2^{n}\phi\left(\frac{x}{2^{n}},w\right)\right)$
$\displaystyle\leq\tau^{n}\alpha\left(\frac{x}{2^{n}},\frac{y}{2^{n}}\right)\alpha(z,w)\longrightarrow\text{
o as }n\rightarrow\infty.$

for all $x,y,z,w\in X$. And we have

$\rho(A(x+y,z+w)+A(x-y,z-w)-2A(x,z)-2A(x,w))\\\
\leq\frac{1}{8}\rho\left(8\left(A(x+y,z+w)-2^{n}\phi\left(\frac{x+y}{2^{n}},z+w\right)\right)\right)+\\\
\frac{1}{8}\rho\left(8\left(A(x-y,z-w)-2^{n}\phi\left(\frac{x-y}{2^{n}},z-w\right)\right)\right)$
$+\frac{1}{8}\rho\left(16\left(A(x,z)-2^{n}\phi\left(\frac{x}{2^{n}},z\right)\right)\right)+\frac{1}{8}\rho\left(16\left(A(x,w)-2^{n}\phi\left(\frac{x}{2^{n}},w\right)\right)\right)\\\
+\frac{1}{8}\rho\left(8\left(2^{n}\phi\left(\frac{x+y}{2^{n}},z+w\right)+2^{n}\phi\left(\frac{x-y}{2^{n}},z-w\right)-2\cdot
2^{n}\phi\left(\frac{x}{2^{n}},z\right)-2\cdot
2^{n}\phi\left(\frac{x}{2^{n}},w\right)\right)\right)$

$\leq\frac{\tau^{3}}{8}\rho\left(A(x+y,z+w)-2^{n}\phi\left(\frac{x+y}{2^{n}},z+w\right)\right)+\\\
\frac{\tau^{3}}{8}\rho\left(A(x-y,z-w)-2^{n}\phi\left(\frac{x-y}{2^{n}},z-w\right)\right)\\\
+\frac{\tau^{4}}{8}\rho\left(A(x,z)-2^{n}\phi\left(\frac{x}{2^{n}},z\right)\right)+\frac{\tau^{4}}{8}\rho\left(A(x,w)-2^{n}\phi\left(\frac{x}{2^{n}},w\right)\right)\\\
\quad+\frac{\tau^{3}}{8}\rho\left(2^{n}\phi\left(\frac{x+y}{2^{n}},z+w\right)+2^{n}\phi\left(\frac{x-y}{2^{n}},z-w\right)\right.\\\
\left.-2\cdot 2^{n}\phi\left(\frac{x}{2^{n}},z\right)-2\cdot
2^{n}\phi\left(\frac{x}{2^{n}},w\right)\right)\quad\longrightarrow 0\text{ as
}n\rightarrow\infty$

for all $x,y,z,w\in X$. Then we conclude that

$A(x+y,z+w)+A(x-y,z-w)-2A(x,z)-2A(x,w)=0$

and by Lemma 2.1, we deduce that $A$ is an additive-quadratique mapping. Now,
let $B:X^{2}\rightarrow Y_{\rho}$ another additive-quadratic mapping
satisfying (3.5), we have

$\displaystyle\rho(A(x,z)-B(x,z))\leq$
$\displaystyle\frac{1}{2}\rho\left(2^{n+1}A\left(\frac{x}{2^{n}},z\right)-2^{n+1}\phi\left(\frac{x}{2^{n}},z\right)\right)$
$\displaystyle+\frac{1}{2}\rho\left(2^{n+1}\phi\left(\frac{x}{2^{n}},z\right)-2^{n+1}B\left(\frac{x}{2^{n}},z\right)\right)$
$\displaystyle\leq$
$\displaystyle\frac{\tau^{n+1}}{2}\rho\left(A\left(\frac{x}{2^{n}},z\right)-\phi\left(\frac{x}{2^{n}},z\right)\right)$
$\displaystyle+\frac{\tau^{n+1}}{2}\rho\left(\phi\left(\frac{x}{2^{n}},z\right)-B\left(\frac{x}{2^{n}},z\right)\right)$
$\displaystyle\leq$
$\displaystyle\frac{\tau^{n+1}}{2}\varphi\left(\frac{x}{2^{n}},\frac{x}{2^{n}}\right)\alpha(z,0)$
$\displaystyle=$
$\displaystyle\left(\frac{2}{\tau}\right)^{n-1}\sum_{l=n+1}^{\infty}\left(\frac{\tau^{2}}{2}\right)^{l}\alpha\left(\frac{x}{2^{l}},\frac{x}{2^{l}}\right)\alpha(z,0)\longrightarrow
0\text{ as }n\rightarrow\infty.$

for all $x\in X$, and all positive integer $n$. Then, we have

$A(x,z)=B(x,z)\text{ for all }x,z\in X.$

On the other hand, letting $y=0$ and $w=z$ in (3.3), we get

$\rho(\phi(x,2z)-4\phi(x,z))\leq\alpha(z,z)\alpha(x,0).$

Hence

$\rho\left(\phi(x,z)-4\phi\left(x,\frac{z}{2}\right)\right)\leq\alpha\left(\frac{z}{2},\frac{z}{2}\right)\alpha(x,0)$

Then, by using the $\Delta_{2}$-condition and the convexity of $\rho$, we have
(remarking that $\sum_{j=1}^{k}\dfrac{1}{2^{k}}\leq 1$)

$\displaystyle\rho\left(\phi(x,z)-4^{n}\phi\left(x,\frac{z}{2^{n}}\right)\right)$
$\displaystyle=\rho\left(\sum_{j=1}^{n}\frac{1}{2^{j}}\left(2^{3j-2}\phi\left(x,\frac{z}{2^{j}}\right)-2^{3j}\phi\left(x,\frac{z}{2^{j}}\right)\right)\right)$
$\displaystyle\leq\frac{1}{\tau^{2}}\sum_{j=1}^{n}\left(\frac{\tau^{3}}{2}\right)^{j}\alpha\left(\frac{z}{2^{j}},\frac{z}{2^{j}}\right)\alpha(x,0)$

for all $x,z\in X$. Now, we have

$\displaystyle\rho\left(4^{m}\phi\left(x,\frac{z}{2^{m}}\right)-4^{n+m}\phi\left(x,\frac{z}{2^{n+m}}\right)\right)$
$\displaystyle\leq\tau^{2m}\rho\left(\phi\left(x,\frac{z}{2^{m}}\right)-4^{n}\phi\left(x,\frac{z}{2^{n+m}}\right)\right)$
$\displaystyle\leq\tau^{2m-2}\sum_{j=1}^{n}\left(\frac{\tau^{3}}{2}\right)^{j}\alpha\left(\frac{z}{2^{m+j}},\frac{z}{2^{m+j}}\right)\alpha(x,0)$
$\displaystyle\leq\frac{2^{m}}{\tau^{m+2}}\sum_{l=m+1}^{n+m}\left(\frac{\tau^{3}}{2}\right)^{l}\alpha\left(\frac{z}{2^{l}},\frac{z}{2^{l}}\right)\alpha(x,0)$
$\displaystyle\rightarrow\text{ as }m\rightarrow\infty(\text{ because
}\dfrac{2}{\tau}\leq 1)$

for all $x,z\in X$. Hence, the sequence
$\left\\{4^{n}\phi\left(x,\frac{z}{2^{n}}\right)\right\\}$ is a $\rho$-Cauchy
sequence in $Y_{\rho}$ wish is $\rho$ complete. Then we have a mapping
$C:X^{2}\rightarrow Y_{\rho}$ such that
$C(x,z)=\rho-\lim_{n\rightarrow\infty}4^{n}\phi\left(x,\frac{z}{2^{n}}\right);x\in
X$. By using the $\Delta_{2}$-condition, we write

$\displaystyle\rho\left(\phi(x,z)-C\left(x,z\right)\right)$
$\displaystyle\leq\frac{1}{2}\rho\left(2\phi(x,z)-2\cdot
4^{n}\phi\left(x,\frac{z}{2^{n}}\right)\right)+\frac{1}{2}\rho\left(2\cdot
4^{n}\phi\left(x,\frac{z}{2^{n}}\right)-2C(x,z)\right)$
$\displaystyle\leq\frac{\tau}{2}\rho\left(\phi(x,z)-4^{n}\phi\left(x,\frac{z}{2^{n}}\right)\right)+\frac{\tau}{2}\rho\left(4^{n}\phi\left(x,\frac{z}{2^{n}}\right)-C(x,z)\right)$
$\displaystyle\leq\frac{1}{2\tau}\sum_{j=1}^{n}\left(\frac{\tau^{3}}{2}\right)^{j}\alpha\left(\frac{z}{2^{j}},\frac{z}{2^{j}}\right)\alpha(x,0)+\frac{\tau}{2}\rho\left(4^{n}\phi\left(x,\frac{z}{2^{n}}\right)-C(x,z)\right)$

for all $x,z\in X$. Passing to limit $n\rightarrow\infty$, we obtain.

$\rho(\phi(x,z)-C(x,z))\leq\frac{1}{2\tau}\psi(z,z)\alpha(x,0).$ (3.6)

For proving that $C$ is an additive-quadratic mapping, we will start with

$\rho\left(4^{n}\phi\left(x+y,\frac{z+w}{2^{n}}\right)+4^{n}\phi\left(x-y,\frac{z-w}{2^{n}}\right)-2\cdot
4^{n}\phi\left(x,\frac{z}{2^{n}}\right)-2\cdot
4^{n}\phi\left(x,\frac{w}{2^{n}}\right)\right)\\\
\leq\tau^{2n}\alpha(x,y)\alpha\left(\frac{z}{2^{n}},\frac{w}{2^{n}}\right)\longrightarrow\infty\operatorname{as}n\rightarrow\infty.$

for all $x,y,z,w\in X$. And we have:

$\rho\left(\frac{1}{7}C(x+y,z+w)+\frac{1}{7}C(x-y,z-w)-\frac{2}{7}C(x,z)-\frac{2}{7}C(x,w)\right)\\\
\leq\frac{1}{7}\rho\left(C(x+y,z+w)-4^{n}\phi\left(x+y,\frac{z+w}{2^{n}}\right)\right)\\\
+\frac{1}{7}\rho\left(C(x-y,z-w)-4^{n}\phi\left(x-y,\frac{z-w}{2^{n}}\right)\right)\\\
+\frac{2}{7}\rho\left(C(x,z)-4^{n}\phi\left(x,\frac{z}{2^{n}}\right)\right)+\frac{2}{7}\rho\left(C(x,w)-4^{n}\phi\left(x,\frac{w}{2^{n}}\right)\right)\\\
+\frac{1}{7}\rho\left(4^{n}\phi\left(x+y,\frac{z+w}{2^{n}}\right)+4^{n}\phi\left(x-y,\frac{z-w}{2^{n}}\right)-2\cdot
4^{n}\phi\left(x,\frac{z}{2^{n}}\right)-2\cdot
4^{n}\phi\left(x,\frac{w}{2^{n}}\right)\right),\\\ \longrightarrow 0\text{ as
}n\rightarrow\infty$

for all $x,y,z,w\in X$. Hence, we have

$C(x+y,z+w)+C(x-y,z-w)-2C(x,z)-2C(x,w)=0$

and by Lemma 2.1, we deduce that $C$ is an additive-quadratic mapping. To
shows that $C$ is unique, letting $D$ be another mapping satisfying (3.6). We
have

$\rho(C(x,z)-D(x,z))\leq\frac{1}{2}\rho\left(2\cdot
4^{n}C\left(x,\frac{z}{2^{n}}\right)-2\cdot
4^{n}\phi\left(x,\frac{z}{2^{n}}\right)\right)\\\ +\frac{1}{2}\rho\left(2\cdot
4^{n}\phi\left(x,\frac{z}{2^{n}}\right)-2\cdot
4^{n}D\left(x,\frac{z}{2^{n}}\right)\right)\\\
\leq\frac{\tau^{2n+1}}{2}\rho\left(C\left(x,\frac{z}{2^{n}}\right)-\phi\left(x,\frac{z}{2^{n}}\right)\right)+\frac{\tau^{2n+1}}{2}\rho\left(\phi\left(x,\frac{z}{2}\right)-D\left(x,\frac{z}{2^{n}}\right)\right)\\\
\leq\frac{\tau^{2n}}{2}\psi\left(\frac{z}{2^{n}},\frac{z}{2^{n}}\right)\alpha(x,0)\\\
=\frac{2^{n-1}}{\tau^{n}}\sum_{l=n+1}^{\infty}\left(\frac{\tau^{3}}{2}\right)^{l}\alpha\left(\frac{z}{2^{l}},\frac{z}{2^{l}}\right)\alpha(x,0)\\\
\longrightarrow 0\text{ as }n\rightarrow\infty.$

This implies that: $C(x,z)=D(x,z)$ for all $x,z\in X$. It follows from (3.6)
that

$\displaystyle\rho\left(\frac{2^{n}\phi\left(\dfrac{x}{2^{n}},z\right)-2^{n}C\left(\dfrac{x}{2^{n}},z\right)}{2}\right)$
$\displaystyle\leq\frac{\tau^{n}}{2}\rho\left(\phi\left(\frac{x}{2^{n}},z\right)-C\left(\frac{x}{2^{n}},z\right)\right)$
$\displaystyle\leq\frac{\tau^{n}}{4\tau}\alpha\left(\frac{x}{2^{n}},0\right)\psi(z,z)\longrightarrow
0\text{ as }n\rightarrow\infty.$

Since $C:X^{2}\rightarrow Y_{\rho}$ is additive in the first variable, we get

$\rho\left(\frac{1}{2}A(x,z)-\frac{1}{2}C(x,z)\right)\leq 0,$

thus we conclude that

$A(x,z)=C(x,z)=H(x,z)\text{ for all }x,z\in X.$

Hence, there exists a unique additive-quadratique mapping

$H:X^{2}\to Y_{\rho}$

such that

$\rho(\phi(x,z)-H(x,z))\leq\min\left\\{\frac{1}{2\tau}\psi(z,z)\alpha(x,0),\frac{1}{2}\varphi(x,x)\alpha(z,0)\right\\}$

for all $x,z\in X$, and hat complete the proof.

∎

######  Corollary 3.2.

Let $X$ be a vector space, $Y_{\rho}$ be a $\rho$-complete convex modular
sapace satisfying $\Delta_{2}$-condition. Let
$r>\log_{2}\left(\dfrac{\tau^{2}}{2}\right)$ and $\theta$ be positive real
numbers, and $\phi:X^{2}\rightarrow Y_{\rho}$ be mapping satisfying
$\phi(x,0)=\phi(0,z)=0$ and

$\rho(\phi(x+y,z+w)+\phi(x-y,z-w)-2\phi(x,z)-2\phi(x,w))\leq\theta(\|x\|^{r}+\|y\|^{r})(\|z\|^{r}+\|w\|^{r})$

for all $x,y,z,w\in X$. Then there exists a unique additive-quadratic mapping
$H:X^{2}\rightarrow Y_{\rho}$ such that:

$\rho(\phi(x,z)-H(x,z))\leq\frac{\theta\tau^{2}\|x\|^{r}\|z\|^{r}}{2^{r+1}-\tau^{2}};\qquad
x,z\in X.$ (3.7)

###### Proof.

The proof is a result of Theorem 3.1 by taking

$\alpha(x,y)=\sqrt{\theta}(\|x\|^{r}+\|y\|^{r})$

for all $x,y\in X$ and remarking that

$\min\left\\{\frac{\theta\tau^{2}\|x\|^{r}\|z\|^{r}}{2^{r+1}-\tau^{2}},\frac{\theta\tau^{2}\|x\|^{r}\|z\|^{r}}{2^{r+1}-\tau^{3}}\right\\}=\frac{\theta
k^{2}\|x\|^{r}\|z\|^{r}}{2^{r+1}-\tau^{2}},$

for all $x,z\in X$. ∎

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