Introduction

A multi-agent system (MAS) is a self-possessed system of multiple networking, intelligent agents contained by an environment (cloud system CS). Agent environments can be allocated into: discrete environment; virtual environment; and continuous environment. These environments provide backward compatibility to guarantee easy mobility allowance. More data and information are required to cope with the increasing need for service delivery. This need is particularly obvious when considering high-bitrate multimedia applications that demand high quality of service (QoS) levels. These systems can be employed to resolve problems that are demanding or difficult for a separate agent or a uniform system to solve. Aptitude may involve some well-designed, functional method. Moreover, there is significant overlap, and MAS is not continuously the same as an agent-based system. Subjects where MAS investigation may carry a fitting method to solve scientific and social problems. MAS can be designed in many ways. Discrete methods (whether the total of potential actions in the environment is finite); dynamic methods (how many agents guidance the environment at the moment); periodicity (whether agent travels in confident time periods); and dimensional spaces (whether features are significant factors of the environment, such that each agent in MAS reflects space in its decision making).

During the last five decades, a huge amount of investigators has been worked in almost all fields of sciences and engineering in fractional calculus (fractional derivative and integral) of arbitrary calculus. This field is a major concept in the mathematical analysis to describe the non-linear. This indicates the importance of fractional calculus as an exciting mathematical method for solving various problems in science and engineering. Nowadays, the fractional calculus is considered as a key for opining the generalizations, modifications, and extensions in all sciences. The most well known of these operators that have been promoted in the world of fractional calculus are the Riemann–Liouville, Caputo (continuous and discrete fractional differential and integral operators), and Grunwald–Letnikov (discrete fractional differential operator) (see [1]). There are many types of fractional operators in real and complex planes, which are named by their finders.

Recently, the author suggested different classes of fractional multi-agent computing systems (see [2,3,4,5,6,7]). The first author showed the advantages of utilizing fractional calculus. Specifically, when it was detected that the explanation of some complex systems is more accurate when the fractional derivative is utilized. In addition, this subject deals with integro-differential equations, where the integrals are of convolution category and display kernels of power-law form. Hence, the fractional calculus discoveries usage in many categories of science and engineering, containing fluid flow, diffusive transport theory, electrical networks, electromagnetic theory, probability and statistics, theory of chaos and fractals, viscoelasticity, image processing, signal processing, food processing, chemical processing, and information theory.

In this study, we suggest a new fractional multi-agent cloud computing system. We introduce a new criteria to minimize the cost with high accuracy of product description, contingent upon a class of fractional differential equations. By employing the fractional difference method on fractional Poisson’s equation, we demonstrate that the solution is bounded for some domain. Simulation results are illustrated in the sequel. Outcomes obviously imply that the proposed method exhibits the highest performance compared with other methods.

The paper is devoted as follows: “Formulation” deals with the preliminary and mathematical formulation; “Proposed algorithm” contains the proposed algorithm; “Applications” involves applications modeling systems; “Stability” studies the stability of the proposed algorithm; and “Conclusion” concludes our work.

Formulation

We aim to improve the dynamic of the computing model change between the choice of the traditional on-site computing model and the cloud computing model. We investigate a large economy in a continuous time situation; therefore, there are an appropriately big number of companies, businesses, and firms permitting to be close to importance. The planning pressures on the effect of the general trend of choices of technology transition times. In such a state, a continuum of agents having non-homogeneous references pay a cost to transfer from one point to another point in the state space. Using the usual financial derivatives is caused many disadvantageous. The most renowned disadvantage that financial derivatives could imply some financial difficulties. These difficulties can be recognized when the MAS may frustrate about some operations on derivative tools, thus losing morale for attempting another innovative financial tool. Therefore, here, we avoid using the usual derivative to keep our system in the stable case.

In general, for integer systems, the stability properties of any dynamic system are completely calculated from the location of the Laplacian eigenvalues of the network [8]. In this work, we suggest another method, based on the fractional Gâteaux derivative (FGD) [9]. This type of derivative connects two agents \(\chi _i\) and \(\chi _j\) in the formula as follows:

Definition 2.1

Let (\(\Xi\), \(\Vert .\Vert\)) be a real Banach space (the agreement space) with the dual space \(\Xi ^*.\) A function \(\psi : \Xi \rightarrow \mathbb {R}\) has a fractional Gâteaux derivative, of the order \(0< \wp <1\) at \(\chi _i \in \Xi\) if \(\psi ^{(\wp )}(\chi _i) \in \Xi ^*\) exists, such that for a constant \(\hbar >0,\) the forward operator \({\rm FW}_{\chi _j}(\hbar ), \, \chi _j \in \Xi\) is introduced by the equality

$${\rm FW}_{\chi _j}(\hbar )\psi (\chi _i):= \psi (\chi _i+\hbar \, \chi _j),$$

with the fractional difference on the right

$$\bigtriangleup ^{\wp }_{\chi _j}\psi (\chi _i):= \Big ({\rm FW}_{\chi _j}(\hbar ) -1\Big ) ^\wp \psi (\chi _i),$$

and its fractional derivative on the right

$$\lim _{\hbar \rightarrow 0^+ } \frac{\bigtriangleup ^{\wp }_{\chi _j}[\psi (\chi _i)-\psi (0)]}{\hbar ^\wp }= \psi ^{(\wp )}(\chi _i)(\chi _j), \quad \forall \, \chi _j \in \Xi .$$

The Sobolev spaces involve a natural norm:

$$\begin{aligned} \Vert \phi \Vert _{\kappa ,\wp } = \left( \sum _{n=0}^\kappa \left\| \phi ^{(n)} \right\| _{L^\wp }^\wp \right) ^{\frac{1}{\wp }} = \left( \sum _{n=0}^\kappa \int \left| \phi ^{(n)}(\chi ) \right| ^\wp \,{\rm d} \chi \right) ^{\frac{1}{\wp }}. \end{aligned}$$

Subjected to the norm \(||. ||_{\kappa ,\wp },\) \(W_{\kappa ,\wp }\) yields a Banach space. A generalization of the space is imposed for an open set \(\Omega \in \mathbb {R}^n\), \(\kappa \in \mathbb {N}\), and \(1\le p< \infty .\) The Sobolev space \(W^{\kappa ,p}(\Omega )\) is presented to be the set of all functions \(\phi\) defined on \(\Omega\), such that for every multi-index \(\imath\) with \(|\imath | = \kappa ,\) the mixed partial derivative

$$\begin{aligned} \phi ^{(\wp )} = \frac{\partial ^{| \wp |} \phi }{\partial \chi _{1}^{\wp _{1}} \dots \partial \chi _{n}^{\wp _{n}}} \end{aligned}$$

is both locally integrable and in \(L_p(\Omega ),\) that is

$$\begin{aligned} \left\| \phi ^{(\wp )} \right\| _{L^{p}} < \infty . \end{aligned}$$

That is, the Sobolev space \(W^{\kappa ,p}(\Omega )\) is introduced as

$$\begin{aligned} W^{\kappa ,p}(\Omega ) = \left\{ \nu \in L^p(\Omega ) : \nu {\text {\,is\,absolutely\,continuos}} \, {\text {and}} \quad D^{\wp }\nu \in L^p(\Omega ) \,\, \forall |\wp | \le \kappa \right\} . \end{aligned}$$

The advantages of suggesting the FGD are included the distribution, stabilizing, and increasing the number of agents in the system and the purchase cost is fixed in contractual format.

The agent i has the cost function \(F_{i}\big (\chi _i,u_i,\mu _i,u^{(\wp )}(\chi _i)(\chi _j),u^{(2\wp )}(\chi _i)(\chi _j) \big )\) representing what an agent pays to have the characteristics \(\chi _i\) (i.e., the level of cloud computing at any time t) under the controller input \(u_i\) and the density \(\mu _i\) of the population for a given level of \(\chi _i.\) Thus, we may present the fractional equation:

$$\begin{aligned} F_{i}\big (\chi _i,u_i,\mu _i, u_i^{(\wp )}(\chi _i)(\chi _j),u_i^{(2\wp )}(\chi _i)(\chi _j) \big )= & {} \vartheta _i(\chi ), \quad i=1,\ldots ,n \nonumber \\= & {} \sum _{j \in \mathbb {N}} \alpha _{ij} \big (\chi _j - \chi _i \big )+ u_i +\mu _i, \end{aligned}$$
(1)
$$\begin{aligned} \Big (D_i=\chi _i\Big ), \end{aligned}$$

where \(D_i\) is the outcome at any time t,  \(\chi _i^{(\wp )}\) is the fractional \(\wp\)-order derivative, such that \(\wp \in (0,1),\) \(\alpha _{ij} >0\) is the (i, j) (agent i can receive information from agent j; otherwise, \(\alpha _{ij} =0\)) element of the adjacency matrix. In matrix form, we can extend Eq. (1) as follows:

$$\begin{aligned} F\big (\chi ,u,\mu , \nabla ^{(\wp )}u(\chi ),\nabla ^{(2\wp )}u(\chi ) \big )&= \vartheta (\chi ), \end{aligned}$$
(2)
$$\begin{aligned} \big (u(\chi )=D(\chi )\big ), \end{aligned}$$

where \(\chi =(\chi _1,\ldots ,\chi _n)^\top , \, u=(u_1,\ldots ,u_n)^\top , \, \mu =(\mu _1,\ldots ,\mu _n)^\top , \, \vartheta (t)=(\vartheta _1,\ldots ,\vartheta _n)^\top\) and \(\nabla ^{(\wp )}\) is the fractional gradient. It is clear that the \(u(\chi )\) is a function of independent variable \(\chi \in \mathbb {R}^n.\) In operating form, we have the system:

$$\begin{aligned} \mathcal {L}u(\chi )= \vartheta (\chi ), \quad \chi \in \Omega \subset \mathbb {R}^n, \end{aligned}$$
(3)
$$\begin{aligned} \big (\Theta u(\chi )= \phi (\chi )\, \text {on} \, \partial \Omega \big ), \end{aligned}$$

where \(\mathcal {L}\) and \(\Theta\) are differential operators in \(\mathbb {R}^n\) and \(\Omega\) is bounded in \(\mathbb {R}^n.\) Our aim is to minimize the problem:

$$\begin{aligned} \int _\Omega \Big (\mathcal {L}u(\chi )- \vartheta (\chi )\Big )^2{\rm d} \chi , \quad \chi \in \Omega \subset \mathbb {R}^n, \end{aligned}$$
(4)

such that

$$\begin{aligned} \int _{\partial \Omega } \Big (\Theta u(\chi )- \phi (\chi )\Big )^2 {\rm d}\chi =0, \end{aligned}$$

which the objectives and constraint are functional with \(W^{\kappa ,2}(\Omega ).\) We shall solve the problem (4) approximately.

Proposed algorithm

To solve problem (4), we need the following facts:

Objective function Our chief objective is to investigate a company’s optimal flat of usage of cloud computing. With the cost proposed in “Formulation”, all rational companies make the decision on the flat of switch to the cloud computing pattern to minimize its estimated discount cost with respect to the effort cost. The objective function provides how much each variable contributes to the value to be optimized in the problem. Moreover, the negative of this function is called the utility function, in which to be maximized. Therefore, it is a function that maps all variables onto a real number intuitively representing some evolution with the minimization of the cost. For this purpose, we define a suitable objective function as follows:

$$\begin{aligned} \widehat{F}(\chi )= F(\chi )+ \sum ^m_{i=1}f (\alpha _i,\rho _i(\chi ))+ \sum ^n_{i=1}[f (\beta _i,\sigma _i(\chi ))+ f(\beta _i,-\sigma _i(\chi ))] \end{aligned}$$
(5)

satisfying the conditions

$$\begin{aligned} \Big ( \rho _i(\chi )\le 0, \, i=1,\ldots ,m , \,\, \sigma _j(\chi )= 0, \, j=1,\ldots ,n\Big ), \end{aligned}$$

where \(\alpha _i\) and \(\beta _i\) are positive constants, f is the penalty function, which modifies the original objective function, and \(\rho _i\) and \(\sigma _i\) are the constraints. Our method is based on the finite-difference techniques. The penalty strategy makes the use of of finite-difference techniques on any boundary conditions on a boundary as fitness function in the cloud.

Fitness function (FF) The fitness function of the cloud, between agent \(\chi _i\) and \(\chi _j,\) is employed to control and summarize how close a given design solution to achieve the set aims of the process during the evolution of the system. Equation (4) can be reduced to minimize the problem:

$$\begin{aligned} \widehat{E}_\Omega (\widehat{u}(\chi ))=\sum _{\chi _i\in \Omega }\big ( \mathcal {L}\widehat{u}(\chi _i)- \vartheta (\chi _i)\big )^2 \end{aligned}$$
(6)

satisfying the conditions:

$$\begin{aligned} \widehat{E}_{\partial \Omega }(\widehat{u}(\chi ))=\sum _{\chi _i\in \partial \Omega }\big ( \Theta \widehat{u}(\chi _i)- \phi (\chi _i)\big )^2=0, \end{aligned}$$
(7)

where \(\widehat{u}\) is the estimated value of u. Problems (6)–(7) can be solved by assuming the fitness function:

$$\begin{aligned} \Phi (\chi )=\widehat{E}_\Omega (\widehat{u}(\chi ))+ \tau \widehat{E}_{\partial \Omega }(\widehat{u}(\chi )), \end{aligned}$$
(8)

where \(\tau\) is the penalty parameter.

Fractional Poisson’s equation (FPE) It is a generalization of Laplace’s equation, which is used widely in the cloud computing systems [8]. Poisson’s equation arises to describe the potential connection caused by a given charge distribution of the agent i, with the agent j. Moreover, Poisson’s equation is employed to reconstruct a smooth 3D system based on a large number of cloud agents \(\chi _i,\, i=1,\ldots ,n,\) where each agent carries an estimate of the local cost. By applying the fractional derivative in “Formulation”, we may generalize the Poisson’s equation on two-dimensional rectangular domains:

$$\begin{aligned} u^{(2\wp )}_{\chi _i}+ u^{(2\wp )}_{\chi _j}= \vartheta (\chi _i,\chi _j), \quad (a<\chi _i<b, \, c<\chi _j<d, \, 0< \wp \le 1), \end{aligned}$$
(9)

satisfying

$$\begin{aligned} u(a,\chi _j)=u_1(\chi _j), \, u(b,\chi _j)=u_2(\chi _j), \, u(\chi _i,c)=u_3(\chi _i),\, u(\chi _i,d)=u_4(\chi _i), \end{aligned}$$

where \(u_{ij}\) is the approximated value of \(u(\chi _i,\chi _j).\) By applying the finite-difference method for fractional order, then Poisson’s equation implies

$$\begin{aligned} \frac{u_{i+1,j}-2\wp u_{i,j}+ \wp (2\wp -1)u_{i-1,j}}{h^{2\wp }}+ \frac{u_{i,j+1}-2\wp u_{i,j}+ \wp (2\wp -1)u_{i,j-1}}{k^{2\wp }}=\vartheta _{i,j}. \end{aligned}$$
(10)

Note that, when \(\wp =1,\) Eq. (10) becomes in normal form. This method is admitted consistency, stability, accuracy \((u\approx \widehat{u})\), and convergence for fractional differential equations. These properties of the method help us to understand how well a numerical approximation can be schemed for various classes of differential equations. We introduce a formal relation of the consistency that can be utilized for any partial differential equation defined on any domain. Stability deals with the behavior of solution \(|u_{i,j}-u(ih,jk)|\) as numerical calculation progresses for fixed discrete steps.

Based on Eq. (10), the fitness function can be calculated as follows:

$$\begin{aligned} \Phi (\chi _i,\chi _j)= & {} \widehat{E}_\Omega (\widehat{u}(\chi ))+ \tau \widehat{E}_{\partial \Omega }(\widehat{u}(\chi ))\nonumber \\= & {} \sum _{\Omega } \big [ 2\wp (1+\kappa ) u_{ij}- (u_{i-1,j}+u_{i+1,j})-(u_{i,j-1}+u_{i,j+1})- h^{2\wp }\vartheta _{ij}\big ]^2\nonumber \\&\quad +\tau \sum _{\partial \Omega }\big [ (u_{1,j}-u_1(\chi _j))^2+ (u_{m,j}-u_2(\chi _j)) + (u_{i,1}-u_3(\chi _i))^2 +(u_{i,n}-u_4(\chi _i))^2\big ], \end{aligned}$$
(11)

where \(i=1,\ldots ,m,\, j=1,\ldots ,n, \, \kappa = h^{2\wp }/k^{2\wp }.\) The function is in fact an evaluation of good and bad connecting results in the cloud system.

Examples

We proceed to illustrate two examples to describe the proposed algorithm.

Fig. 1
figure 1

Convergence of solutions of systems \(\nabla ^\wp =4\) and \(\nabla ^{\wp }u= (\chi _i-2)e^{-\chi _i}+\chi _ie^{-\chi _j}\)

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The above examples are shown the domain of the MAS. The first example is described a convex domain, while the second is suggested in a concave domain.

Table 1 Fractional multi-agent system
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Table 2 Utility distribution
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Table 1 shows two examples of the proposed method. Figure 1 shows the solution for the two systems for different \(\wp \in (0,1]\) values. The approximate solution of the first problem is convex type. The error is calculated utilizing the minimum of the fitness function of these outcomes given across domains of various measures \((h=k= 0.1, 0.2).\) The comparison is imposed by well-known techniques, such as the Genetic Algorithm (GA), LMI, and PSOA. For example, the minimum error for the first problem for \(\wp =0.75,\) is equal to \(E_{\rm m}= 2.35e-005,\) while \(E_{\rm GA}= 2.12\) and \(E_{\rm PSOA}= 8.1e-004.\) The second problem has a concave solution. It is evident from the above fractional systems that the proposed technique is a very powerful to solve not only initial value problem, but boundary value problems too.

Table 2 shows five-agent system dynamic of its utility in a convex domain (example 1) and concave domain (example 2) comparing with the utility function which is given in [10] as follows:

$$\begin{aligned} U_{ij}= \sum ^n_{j=1} \alpha _{ij} \chi _i. \end{aligned}$$

Convexity has provided a good utility rather than the concavity domain. The distribution is stable and gives best-connected on satisfying a diversity of radio access technologies that collaborate to construct an integrated cloud computing system that reunites the requests of agents. Thus, users are provided the optimum service delivery through selecting the most appropriate network among different available wireless networks. The MAS utility is utilized to analytically process [11]. A hybrid system can be realized depending on the weight and quality levels for scoring [12]. Numerous investigations which used cloud computing systems to solve various problems are summarized in [13].

Applications

To faithfully model customer forms agreeing to their preferences, we utilize the above multi-agent systems, each with a characteristic form of the utility function (convex and concave forms). Our setting closely looks like a public game, where customers acquire costs for providing incomes to the public, and in this case, it is often presumed that self-interest is the most related parameters that illustrate the behavior of customers. Moreover, the utility function is capable to seizure various procedures of utility functions, such as the convex utility functions, which makes it very flexible. Therefore, all customers will choose their separate optimal value subject to their level of self-sacrifice and the convexity of their preferences (see Fig. 2). In this system model, agents request resources service by service. A service is composed of multiple tasks, such as data downloading and computing. Agent i wants to complete the jobs involved in the service as soon as possible. To process jobs, agents request to utilize their resources. The set of the sizes of jobs that agent i has to use is introduced by \(\mu _i= \{ \mu ^\ell , \ell \, \text {\,type\,of\,the\,job} \}.\) The proposed method which allows two different jobs may require the same type of resources in which case the specific type of resource has to be connected between them. In addition, agent i requests extra resources and agent j shares its resources with agent i. A job of \(\mu ^\ell\) can be divided into smaller parts \((\mu ^\ell )\), and when agent j shares resources with agent i,  agent i outsources some parts of the job to agent j. Therefore, the mechanism of the process is as follows:

Demand \((\chi _{i,j})\)  \(\Longrightarrow\) Controller of utility (u) \(\Longrightarrow\) outcomes (mini of the cost function).

Hence, we have

$${\rm Cost} \, \, = \alpha _{ij} |i-j| * |\chi _i- \chi _j|^{\wp } *u_{i,j}* \mu ^\ell .$$
(12)

Thus, the total cost of the agent i,  connecting with n agents, is calculated by

$${\rm Total\,Cost} \, \, = \sum ^{n}_{j=1}\alpha _{ij} |i-j| * |\chi _i- \chi _j|^{\wp } *u_{i,j}* \mu ^\ell .$$
(13)

Note that the distance between agents i and j takes the maximal value in the rectangle, which is determined by the diameter of the rectangle, i.e, \(|i-j|=5.83\) (for the first example) and \(|i-j|=1\) (for the second example).

Fig. 2
figure 2

Cost function with respect to utility function \(u_i\)

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Figure 2 shows the cost for the convex and concave systems. The convex system approaches to cloudSim at the value \(\wp =0.75,\) while the concave system converges to the exact solution \(\wp =1.\)

Stability

The mathematical term well-posed problem (Hadamard well-posed) is defined that mathematical platforms of physical phenomena should have the following properties:

  1. (i)

    A solution exists.

  2. (ii)

    The solution is unique.

  3. (iii)

    The solution’s behavior changes continuously with the initial conditions.

Otherwise, it is called ill-posed. It is well known that if the problem is well-posed, then it can be viewed as a good algorithm of outcome on a computer employing a stable algorithm. Since the system (1) is formulated for multi-agent users, it is enough to show that (1) has multi-solutions. In this case, we satisfy part (i) of the well-posed problem.

The discrete-time collective dynamics of the network under this algorithm can be read as follows:

$$\begin{aligned} \bigtriangleup ^{\wp }_{\chi _j} \chi _i(k-1)+ \aleph (k) \chi _i(k)= \Pi _i(k,\chi _i), \quad k \in [1, N],\, i=1,\ldots ,n, \end{aligned}$$
(14)
$$\begin{aligned} \chi _i(0)= \chi _i(N+1)=0, \end{aligned}$$

where \(\Pi _i: [1, N] \rightarrow \mathbb {R}\) is a continuous function, \(\aleph : = \sum \alpha _{ij}\ge 0.\)

Introduce the Banach space as [14]

$$\begin{aligned} \mathcal {B}:= \{ \chi _i:[0,N+1] \rightarrow \mathbb {R} : \chi _i(0)=\chi _{i}(N+1)=0\} \end{aligned}$$

endowed with a discrete norm:

$$\begin{aligned} \Vert \chi _i\Vert := \left( \sum ^{N+1}_{k=1} | \bigtriangleup ^{\wp }_{\chi _j} \chi _i(k-1)|^p+\aleph (k)|\chi _i|^p\right) ^{1/p}, \end{aligned}$$

such that

$$\begin{aligned} \max _{k \in [1,N]} |\chi _i| \le \frac{(N+1)^{(p-1)/p} }{2}\Vert \chi _i\Vert , \quad \forall \, \chi _i \in \mathcal {B}, \, i=1,\ldots ,n. \end{aligned}$$

Let

$$\begin{aligned} \Phi (\chi _i) := \frac{\Vert \chi _i\Vert ^p}{p}, \quad \Psi (\chi _i):= \sum ^N_{k=1} \Pi _i(k), \quad \Theta (\chi _i):= \Phi (\chi _i) -\Psi (\chi _i), \quad \forall \, \chi _i \in \mathcal {B}, \end{aligned}$$
$$\begin{aligned} \overline{\Pi }_i(k,u):= \int ^u_0 \Pi _i(k,w)dw. \end{aligned}$$

Note that \(\Theta \in C^1(\mathcal {B}, \mathbb {R}), \, \Theta (0)=0\) and that all the critical points of \(\Theta\) are the solutions of (12).

Definition 5.1

[15] A function \(\theta : \Xi \rightarrow \mathbb {R}\) is standard as a Gâteaux differentiable and verifies the Palais–Smale (PS) condition if any bounded sequence \(\{\chi _n\}\) with \(\lim _{n\rightarrow \infty } \Vert \theta '(\chi _n)\Vert _{\Xi ^*}=0\) has a convergent subsequence.

We need the following result in the sequel [15].

Lemma 5.1

Consider three positive constants \(C, C_1\) and \(C_2\), such that

$$\begin{aligned} C_1< \frac{(2+\overline{\sigma }) ^{1/p} (N+1)^{(p-1)/p} C}{2}< C_2, \end{aligned}$$
(15)

where \(\overline{\sigma }:= \sum ^N_{k=1} \sigma _k.\) If

$$\begin{aligned} \beta _1:= \frac{\sum ^N_{k=1} \max _t H(k,t) -\sum ^N_{k=1} H(k,C)}{(2C_1)^p- (2+\overline{\sigma } )(N+1) ^{p-1}C^p} \le \frac{1}{p(N+1)^{p-1}} \end{aligned}$$

and

$$\begin{aligned} \beta _2 := \frac{\sum ^N_{k=1} \max _t H(k,t) -\sum ^N_{k=1} H(k,C)}{(2C_2)^p- (2+\overline{\sigma } )(N+1) ^{p-1}C^p}\le \frac{1}{p(N+1)^{p-1}}, \end{aligned}$$

where \((2C_i)^p\ne (2+\overline{\sigma } )(N+1) ^{p-1}C^p, \, i=1,2,\) then (12) has at least one non-trivial solution \(\mu ^*\), such that

$$\begin{aligned} \frac{2C_1}{(N+1)^{(p-1)/p}}< \Vert \mu ^*\Vert < \frac{2C_2}{(N+1)^{(p-1)/p}}. \end{aligned}$$

Lemma 5.2

[16] Let the \(\Phi\) satisfied (PS)-condition. Then there is a sequence \(u_n\) of pairwise distinct critical points (local minima) with \(lim_n\Phi (u_n)=inf \Phi\) which weakly converges to a global minimum of \(\Phi\).

In view of Lemmas 5.1 and 5.2 , we conclude the following result.

Theorem 5.1

Assume that there exist two real positive sequences \(\xi _n\) and \(\zeta _n\) with \(\lim _{n\rightarrow \infty } \zeta _n=0\), such that

$$\begin{aligned} \xi _n < \frac{2}{(\aleph +2)^{1/p} (N+1)^{(p-1)/p}}\zeta _n, \quad n \in \mathbb {N}, \end{aligned}$$
$$\begin{aligned} \ell := \frac{\sum ^N_{k=1} \max _u \overline{\Pi }_i(k,u) -\sum ^N_{k=1} \overline{\Pi }_i(k,K)}{(2\zeta _n)^p- (2+\aleph )(N+1) ^{p-1}\xi _n^p} \le \frac{\beta }{(\aleph +2)(N+1)^{p-1}} \end{aligned}$$

and

$$\begin{aligned} \beta := \lim \sup _{u\rightarrow 0} \frac{\sum ^N_{k=1} \overline{\Pi }_i(k,u)}{u^p}. \end{aligned}$$

Then, (12) admits a sequence of non-zero solutions which converges to zero.

Conclusion

We discussed the question as how to optimize resources, scheduling based on an algorithm in cloud computing. The technique was additional capable and surpasses those of mathematical software design and reproducing the definite benefit of saving with the overall cost as well as task distribution. In addition, the objective function is taken in the sense of the fractional Poisson’s equation. Utilizing this equation, we imposed a modification of the fitness function. This algorithm is proposed to evaluate and solve well-posed problems, such as initial and boundary value problems on finite domain. The method admitted two advantages: transforms the problem of constrained optimization into unconstrained, and with a suitable choice of the fractional order, the method is a good approximation. We suggested special type of fractional calculus named the fractional Gâteaux derivative. This class allows us to connect two agents in a multi-agent system. Moreover, one may suggests multi-connection, utilizing the above method. We studied the stability in view of Hadamard well-posed strategy. We proved that for some special class of fractional differential equations, the problem is stable, by approximating the solution to a convergent sequence. Finally, the method can be extended to higher dimension when the number of agents in the multi-agent system becomes large.