Solvability of fractional dynamic systems utilizing measure of noncompactness

Fractional dynamics is a scope of study in science considering the action of systems. These systems are designated by utilizing derivatives of arbitrary orders. In this effort, we discuss the sufficient conditions for the existence of the mild solution (m-solution) of a class of fractional dynamic systems (FDS). We deal with a new family of fractional m-solution in Rn for fractional dynamic systems. To accomplish it, we introduce first the concept of (F, ψ)-contraction based on the measure of noncompactness in some Banach spaces. Consequently, we establish requisite fixed point theorems (FPTs), which extend existing results following the Krasnoselskii FPT and coupled fixed point results as a outcomes of derived one. Finally, we give a numerical example to verify the considered FDS, and we solve it by iterative algorithm constructed by semianalytic method with high accuracy. The solution can be considered as bacterial growth system when the time interval is large. 


Introduction
Fractional calculus includes all fractional concepts, (operators) fractional formulas (equations, inequalities and inclusions) and fractional formal (logic concepts) (see [19,20,22,23,26]) can express the possessions of the history of materials. Practical problems take in classifications of the fractional operators (differential and integral) allowing the procedure of the entity and uniqueness of associations outcome based equity model. For example, fractional diffusion equations (derivatives with respect to time), where elements are more slowly than a traditional diffusion. This concept demonstrated its authority in all sciences.
The main problem in the classes of fractional differentiation arguments (equations, inclusions and systems) is the uniqueness of m-solution. This problem has been discussed by many authors. For recent work, one can see [3, 9-11, 13, 15-17, 28, 32]. Most of these efforts delivered by various types of FPTs in compact sets. Therefore, we shall develop a set of fixed point theorems of measure of noncompactness based on (F, ψ)-contraction functions (see [1-8, 12, 21]).
In our investigation, we establish some basic fixed point results, which generalize some well-known results. Our method is based on the new definition of (F, ψ)-contraction with respect to measures of noncompactness in Banach spaces. Consequently, a set of coupled FPTs is also derived from the main result. Applying our results, we deliver adequate conditions for a constructed mild solution (m-solution) of fractional dynamic systems. Any classical solution is m-solution. In [9], Araya and Lizama provided the idea of α-resolvent sets establishing the entity of m-solutions of equation [9] D α t ν(t) = ∆ν(t) + t n φ(t) α ∈ [1,2], n ∈ Z + , in a Banach space E for automorphism functions φ : R → E. Moreover, the researchers studied the m-solution of D α t ν(t) = ∆ν(t) + φ t, ν(t) , α ∈ [1,2], and D α t ν(t) = ∆ν(t) + φ t, ν(t), ν (t) , α ∈ [1,2]. Many investigators imposed different criteria of m-solution for various classes of FDS (see [13,28]). In [13], Cuevas and Lizama suggested the almost mild solutions for the following class of equation: where ∆ is a linear operator, and φ(t, ν) is Lipschitz in ν. In [3], Agarwal et al. imposed analytic operator establishing the integral formal In [28], Ponce studied the solutions of the following equation: where the linear operator ∆ is closed on a Banach space E, α > 0, ρ ∈ L 1 (R + ) is a kernel of the integral operator, and φ : R × E → E achieves a special type of Lipschitz conditions. In [15], Dhanapalan et al. established m-solution of a class of nonlinear FDS of the form

Measure concept
The following abbreviations are utilized in this manuscript: R is the set of real numbers, R + = [0, +∞); (E, · ) is the Banach space (BC); B(x, r) -the closed ball, B r = B(0, r); M E and N E denote the family of nonempty bounded subsets of E and the subfamily connecting all relatively compact set, respectively; µ denotes the measure of noncompactness (MNC) (see [12]); BCC -the bounded closed convex set.
Definition 1. A mapping µ : M E → R + is called MNC in E if it fulfills the next conditions: (d1) The family ker µ ⊂ N E (the kernel of the MNC) is nonempty set satisfying , and let lim n→∞ µ(Y n ) = 0, then the conclusion setting by Lemma 1. (See [12].) Let C ∈ Λ, and T : C → C be a continuous and µ-set contraction operator such that there is a constant k ∈ [0, 1) with for any nonempty subset Y of C, where µ be the Kuratowski MNC on E. Then T admits a fixed point.
Thereafter, various types of DFPT and their coupled version were considered by utilizing several types of contractive condition in the sense of MNC (for instant, see ). Here, we introduce a new µ-contraction operator know as (F, ψ)-contraction in the sense of MNC, and we prove some new fixed point, Krasnoselskii FPT and coupled FPTs that generalize the outcomes in [5,8,12,27] mainly.
for all Y ⊆ C. Then T admits at least one fixed point in C.
In view of (1), we conclude that
Since C n ⊇ C n+1 and TC n ⊆ C n for all n = 1, 2, . . . , then by (d6) of Definition 1, Y n is nonempty convex closed set, invariant under T and belongs to ker µ. So, Schauder's FPT gives the requested result.
then T admits a unique fixed point in C.
Proof. In view of (d6), it is well know that diam(·) is a MNC, and thus, from Theorem 1 we get the existence of a T-invariant nonempty closed convex subset X ∞ with diam Y ∞ = 0. Consequently, X ∞ is a singleton, and therefore, T has a fixed point in C.
To attain the uniqueness, we assume that there exist two distinct fixed points ζ, ξ ∈ C, then we may define the set Y := {ζ, ξ}. In this case, diam Y = diam(T(Y)) = ξ − ζ > 0. Using (2) and notion of F and ψ, we obtain a contradiction and hence the result. Now we are in position to derive some classical fixed point result from Proposition 1 and Theorem 1.
for all u, v ∈ C. Then T admits a unique fixed point.
which implies that Thus, following Proposition 1, T has an unique fixed point.
Following is the Krasnoselskii FPT: (iii) T 2 is a continuous and compact operator.
Taking the limit in (5) as n → ∞, we get By (iii) hypothesis and (6) we have by the notion of χ that Therefore, from Theorem 1 T has a fixed point u ∈ C.

Coupled fixed point results
In this section, we introduce the result of Theorem 1 for ϕ(t) = 0.
http://www.journals.vu.lt/nonlinear-analysis Theorem 4. Let C ∈ Λ and G : C 2 → C is continuous operator, and let there exist F ∈ F, ψ ∈ Ψ , and F is subadditive such that Then F admits at least one CFP.

Applications
In this section, we construct a m-solution for a class of FDS with delay.

Construction
• Let ν(t) = (ν 1 (t), . . . , ν n (t) T ∈ R n be a vector of variables with continuously differential components in the partition intervals [η 1 , t−τ 1 ], . . . , [η k , t−τ k ], t > τ i , i = 1, . . . , k, and C i = C T i (transpose matrix of C i ) be a constant n × n matrix satisfying the following operational equation: where ∆ : R n → R n . Adding the above relation, we take out the operator D : R n → R n possessing the n × n summation • Let η 1 < t − τ 1 , . . . , η k < t − τ k , η i 0, and M i be a constant n × n matrix such that M i = M T i > 0, i = 1, . . . , k, fulfilling the integral formula Repeating the above construction integral k times, we attain the general summation formal • Let η 1 < t − τ 1 , . . . , η k < t − τ k , η i 0, and R i be a constant n × n matrix satisfying R i = R T i > 0, i = 1, . . . , k, achieving the formal integral equation Adding k times, we bring out Combine (9)-(11) to obtain a new mild solution ν : [0, ∞) → R n : α ∈ (0, 1], ν ∈ R n , F ∈ C(R n ), ℘ α ∈ [0, ∞), Obviously, ν(t) is a continuous and differential function. Our aim is to show that the following dynamic system has a mild solution in the frame of (12): where F : [0, ∞) × R n → R n is continuous, and A n×n and C n×n are constant matrices achieving cij > aij > 0 with the propertŷ

M-Solutions
In this subsection, we establish the m-solution of FDS (13).
Proof. Define an operator Q : R n → R n as follows: Our aim is to show that Q admits at least one fixed point in a BCC set in R n .
Boundedness. By (12) we get Taking the sup norm over t ∈ [0, T ], T < ∞, we have Hence, Q : B r → B r is bounded.
For ν and υ ∈ B r , we have To satisfy the condition of Theorem 1, we follow the same technique in [27]. Consequently, we obtain the desired result.
Equicontinuous. Let t 1 and t 2 ∈ [0, T ] with t 1 > t 2 , then we attain This conclude that the operator Q is equicontinuous in B r . As a consequence, Theorem 1 yields that Q admits at least one fixed point.
Next, we provide the sufficient condition on Q to has a unique fixed point.
Proof. Our aim is to satisfy inequality (3). For ν and υ ∈ B r , we have Thus system (13) admits a unique m-solution ν ∈ BC(R n ).
6 An iterative algorithm to find solution of equation (15) (15) (15) In fact, the above problem is a fractional delay singular integral equations system. We decide to find solution of it by an iterative algorithm. At first, we introduce F (t, ν) function as follows: where By substituting (16) and (17) into (15) and concept of D α 0 ν(t) = DI 1−α ν(t), α = 0.5, we obtain a system of fractional singular delay integral equations of the form where τ 1 = 0.4, η 1 = 0.2 and t ∈ (0.6, 1], also h 1 (t) = −0.4610126329751105 √ t + 1.1605823875518624t 3/2 + 0.75t 2 Now, to solve (18), we use a modified technique constructed an important concept of topology and perturbations theory that named modified homotopy perturbation method. To introduce some applications of the similar method, [29][30][31] hcan be seen. To make the above iterative algorithm, we consider the nonlinear problem in the general form where A is a general differential operator, H is a known function, We distribute the common operator A to N 1 and N 2 nonlinear operators, and correspondingly H function adapts to some functions such as H 1 and H 2 in order to (19) can be represented by N 1 (ν)−H 1 (t)+N 2 (ν)−H 2 (t) = 0. Consequently, we define an adapted homotopy perturbation as tails: and where p is an inserting parameter. According to variations of p = 0 to p = 1, we deliver N 1 (u) = H 1 (t) to A(u) = H(t). Therefore, we can develop a solution of (19) (numerical-solution) for p = 1 and ν(t) lim p→1 u(t). Currently, considering the system of fractional singular delay integral equations (18), we introduce operators N 1 and N 2 and also functions H 1 and H 2 in these forms: Here h 11 (t) and h 21 (t) are simple functions, which are chosen as a prate of functions h 1 (t) and h 2 (t), respectively. Substituting (22) and (21) in (20) concludes that http://www.journals.vu.lt/nonlinear-analysis Rearranging (23) in terms of p powers concludes that By the construction of generalized homotopy perturbation (20) the coefficients of p powers are amounting to zero. Thus, we obtain an iterative process for numerical solution of (18).

Bacterial growth system
From the solution ν(t) in (26) we consider a realistic model of Bacterial growth population as follows: where ν(t) is the next-state function of the growth for two experiences, κ is a positive constant, while is a negative constant. The quadratic term is called a corrected term for the linear term. If the constant is negative, then the growth occurs; otherwise, there is no growth (death). Figure 1 shows the bacterial growth of a population. Moreover, the solution (27) converges to a fixed point of system (15). In fact, that there is an equilibrium state corresponding to a fixed point. The accuracy of the growth is given by the ratio Q := κ/| |. Our example has accuracy = 1.5/0.9 = 1.6, which approximated to the value of the golden ration. The degree of noncompactness of a set is measured by incomes of functions entitled measures of noncompactness. This type of measure can describe the behavior of the growth at infinity.