Extremal solutions for fractional evolution equations of order 1 < γ < 2

: This manuscript considers a class of fractional evolution equations with order 1 < γ < 2 in ordered Banach space. Based on the theory of cosine operators, this paper extends the application of monotonic iterative methods in this type of equation. This method can be applied to some physical problems and phenomena, providing new tools and ideas for academic research and practical applications. Under the assumption that the linear part is an m -accretive operator, the positivity of the operator families of fractional power solutions is obtained by using Mainardi’s Wright-type function. By virtue of the positivity of the family of fractional power solution operators, we establish the monotone iterative technique of the solution of the equation and obtain the existence of extremal mild solutions under the assumption that the upper and lower solutions exist. Moreover, we investigate the positive mild solutions without assuming the existence of upper and lower solutions. In the end, we give an example to illustrate the applied value of our study.


Introduction
Let (X, ∥ · ∥, ≤) be an ordered Banach space, whose positive cone K = {u ∈ X|u ≥ θ} is normal with normal constant N, θ is the zero element of X. In this paper, we discuss the existence and uniqueness of mild solutions for fractional evolution equations t u(t) + Au(t) = f (t, u(t)), t ∈ [0, a], where c D γ t is the Caputo fractional derivative of order γ ∈ (1, 2), −A is the infinitesimal generator of a strongly continuous cosine family {C(t)} t≥0 of uniformly bounded linear operators in X, f : [0, a]×X → X is a given function which will be specified later, x 0 , x 1 ∈ X.
Fractional calculus is an important branch of mathematics, which was born in 1695. It is generally recognized that integral calculus and fractional calculus appear almost simultaneously. They have plenty of similarities. In the fields of turbulence models, Brownian motion, and viscoelastic materials, the researchers found that integer calculus cannot accurately describe the historical memory and spatial correlation of relevant models. However, fractional calculus can comprehensively reflect its behavior. Therefore, fractional calculus has become an indispensable method to deal with practical problems in a number of disciplines. Fractional calculus is widely used in fractional evolution equations. Because differential equations containing fractional derivatives (called fractional differential equations) are abstract formulas for many problems in engineering, flow control system, biological tissue, stochastic process, genetic mechanics, newtonian fluid mechanics, anomalous diffusion, etc, the subject is frequently discussed at home and abroad by scholars. The theory and application of fractional differential equations have yielded several remarkable results (see [14,16,20,28,31,52] and the references therein).
In the natural world, diffusion phenomena occur frequently. For example, the transmission of odor, sound, and light, as well as the transmission of temperature, etc. It has wide applications in many fields. In the realm of physics, it can be harnessed to modulate the resistivity of semiconductors. In industry, diffusion phenomena can optimize the speed and efficiency of chemical reactions. In biology, it allows for the investigation of cellular interactions, thereby enhancing our understanding of life processes. The diffusion phenomena refer to the spontaneous migration of biomass resulting from the thermal motion of particles, including atoms, molecules, and molecular clusters. There exists a correlation between the diffusion coefficient of particles and mean squared displacement. The mean squared displacement depends on time, and the specific relationship is manifested as ⟨x 2 (t)⟩ ∼ const · t γ , γ > 0. When γ = 1, it represents a slow diffusion phenomenon. When γ = 2, it represents a fast diffusion phenomenon, such as the wave equation. When γ ∈ (1, 2) represents the super-diffusion equation, which can be applied to information signal processing. The edge detection based on fractional differential equation can improve the standard of edge detection and the community's standard of noise [35]. It can also be applied to other fields such as viscosity. Additionally, this type of equation and its various variants are applicable to mathematical models of viscoelasticity. Therefore, the study of such equations is very meaningful.
The study of fractional evolution equations has captured the interest of numerous scholars. However, the prevailing research trend involves converting equations into alternative forms, disregarding the constraints of certain equations, and failing to explore iterative sequences. In addition, the majority of the existing literature focuses on integer derivatives or fractional derivatives of 0 < γ < 1. Recently, for instance, Abdollahi et al. [2] analyzed the generalized two-dimensional fractional Volterra integral equations and transformed the proposed equation into an algebraic equation using the two-dimensional Haar wavelet operation matrix method. They derived some sufficient conditions for existence and uniqueness of the equation. Alipour [5] solved for the numerical solution of the two-dimensional time fractional diffusion wave equation using the double interactive boundary element method and provided the iterative format of the time derivative. The author suggested that discretization could convert fractional differential equations into non-homogeneous Helmholtz equations. Avazzadeh et al. [6] studied fractional differential systems for order γ ∈ (0, 1) under the Caputo sense, with initial value conditions. The existence and uniqueness of the system were proven using the GFP model and derivative operation matrix.
In response to the issues mentioned in the above analysis, this paper proposes an effective method, which is the monotonic iteration technique for upper and lower solutions. This method can be briefly described as: if the problem that we consider has a pair of ordered lower and upper solutions, we will construct a simple iterative sequence through these lower and upper solutions, so that it can converges uniformly to the minimum and maximum solutions between lower and upper solutions of problem under certain conditions. This method can obtain both the existence of the solution and the corresponding approximate iteration sequence. This method is comprehensively summarized in document [22]. Many scholars use this method to treat all kinds of fractional evolution equations. In 2009, McRae [36] applied the monotone iteration technique of upper and lower solutions to study the existence of extremal solutions for Riemann-Liouville fractional differential equation. Later, Cabada and Kisela [9] concerned with the existence and uniqueness of solutions for a nonlinear fractional differential equation involving Riemann-Liouville fractional derivative supplied with periodic boundary condition. Abdelouahed and Zakia [1] established the existence of maximal and minimal solutions for two types of fractional nonlinear reaction diffusion problems with periodic conditions or with initial conditions. For more applications of the upper and lower solutions, see [3,38] and the references therein.
In practical operation, utilizing this technology poses certain challenges. If the A is infinitesimal generator of semigroup T (t), we can use T (t) and probability density function to express the operator families of fractional power solutions, which is a very common method. The positivity of the operator families of fractional power solutions is naturally generated by the positivity of the semigroup T (t). A new semigroup T b (t) is generated after T (t) perturbation and T b (t) = e −bt T (t). Thus, we can still obtain the positivity of T b (t) using the positivity of T (t). In this paper, we hope to adopt a similar approach. However, we encountered two very critical issues. On the one hand, we are not sure if the strong continuous cosine family and the strong continuous sine family generated after perturbation are positive; On the other hand, we don't know that the operator families of fractional power solutions described by cosine function theory and Minardi's Wright-type function are positive.
In addition, we focus on the fact that only positive solutions have practical significance in some mathematical and physical models, such as the reaction-diffusion equation, neutron transport equation and heat transfer equation. When 0 < γ < 1, a host of scholars have paid attention to the positive solutions and have achieved many results. Li et al. [30] discussed a class of nonlinear fractional differential equations by constructing the upper and lower control functions of nonlinear terms. They obtained the existence of positive solutions by employing the method of upper and lower solutions and Schauder's fixed point theorem. For more applications, see [9,12,14,25,33,42] and the references therein. When 1 < γ < 2, it is rare to study positive solutions. Li et al. [24] investigated the boundary value problems described by fractional differential equation with Riemann-Liouville fractional derivative. The existence of positive solutions is obtained by using Krasnoselskii's fixed point theorem.
This article attempts to expand Eq (1.1) research from various perspectives. Firstly, the current literature predominantly employs the theory of analytic semigroup to analyze problems while attaching importance to specific fractional evolution equations with order between 0 and 1. Thus, this study classifies and compares these equations, deriving the applicability of upper and lower bound methods to fractional evolution equations with order between 1 and 2. Secondly, this article concentrates on the uniqueness of the cosine family theory of operators, using the translation of cosine function generators and combining it with dissipative operator theory to enhance the understanding of the positivity of fractional power solution operators. This further extends the existing theory of fractional power solution operators. Finally, we overcome the usual constraint that monotonic iterative methods require the existence of upper and lower solutions of the system. Our research methodology proves effective in addressing these limitations, making monotonic iterative methods widely applicable to practical problems. Overall, this paper contributes substantially to the research on Eq (1.1), deepening its understanding while providing new concepts and techniques for further research and application.
The remainder of this paper is organized as follows. In Section 2, the symbols, concepts and lemma required are introduced for this article. By introducing the concept of accretive operator and related properties, the positivity of the strong continuous cosine family and the strong continuous sine family generated after perturbation are obtained. Furthermore, combined with the definition and properties of Mainardi's Wright-type function, the positivity of the operator families of fractional power solutions is obtained. In Section 3, under the condition that the strongly continuous sine family is compact, we use the monotonic iterative technique of upper and lower solutions to demonstrate the existence of mild solutions. In the case that the strongly continuous sine family is noncompact, we explore the existence of the extreme solution of Eq (1.1). By introducing the Kuratowski measure of noncompactness, we establish the uniqueness of solution of Eq (1.1). Finally, the existence of positive solution of Eq (1.1) is obtained without assuming the existence of upper and lower solutions. In Section 4, we present an example to illustrate the applied value of our study.

Preliminaries
In this section, we will list some concepts and definitions used to show our main results. Assume that X is an ordered Banach space with norm ∥ · ∥ and partial order ≤, whose positive cone K = {x ∈ X|x ≥ θ} is normal with normal constant N, θ is the zero element of X. We denote that L b (X, Y) are the spaces of all bounded linear operators from X into Y equipped with the norm We define the domain of −A by D(−A) and the range of −A by R(−A). If −A is a linear operator, then the resolvent set and resolvent of −A are defined by ρ(−A) and R(λ, A significant amount of literature refers to the definition of Caputo fractional derivation, see [13,46] and its references. Now, we propose definitions of the strongly continuous cosine and sine families, recalling a few of their characteristics.
x is a twice continuously differentiable function of t}.
Clearly, −A is a closed dense operator in X.

Lemma 2.2 ([44])
Let {C(t)} t≥0 be a strongly continuous cosine family in X. Then there exist constants M 0 > 0 and ϖ > 0 such that and the convergence is uniform on bounded subsets of t ≥ 0 for any fixed x ∈ X, where b 2 , and b 2 k,n = 0(k > n, n = 1, 2, · · · ). According to [44,Lemma 4.1], note that −(A + L 2 I) can also generate a strongly continuous cosine families C L (t) in X for arbitrary constant L. And and for Reλ > ϖ + |L| and x ∈ X. Let S L (t) denote the strongly continuous sine family associated with C L (t). In view of [43, Proposition 2.1] and (2.6), it is easy to obtain for for Reλ > ϖ + |L| and x ∈ X.
Next, we will introduce solution operators and their related properties. We first introduce the Mainardi's Wright-type function , for z ∈ C, ϱ ∈ (0, 1).
For any t > 0, the Mainardi's Wright-type function has the properties In what follows, we always suppose that −A is the infinitesimal generator of a strongly continuous cosine family of uniformly bounded linear operators For the sake of convenience in writing, we also set q = γ 2 for γ ∈ (1, 2). Thus, we can define three families of operators C q (t)(t ≥ 0), N q (t)(t ≥ 0) and T q (t)(t ≥ 0) in X as following We have the following useful properties for operator families C q (t)(t ≥ 0), N q (t)(t ≥ 0) and T q (t)(t ≥ 0). Lemma 2.6 ( [44]) The operators C q (t), N q (t) and T q (t) admit the following properties: (i) for all t ≥ 0, C q (t), N q (t) and T q (t) are linear operators; (ii) for all t ≥ 0 and for any x ∈ X, To prove the positivity of the proposed operator families, we need to introduce the definition of accretive operator and related lemmas. Definition 2.7 An operator family {C(t)} t≥0 in X is called to be positive if C(t)x ≥ θ for any x ≥ θ and t ≥ 0. (i) There exists P ≥ 0 such that, for every value of λ > 0 and every x ∈ D(A), Proof Since −A is a m-accretive operator, from Definition 2.8 and Definition 2.9, it is easy to obtain that −A is nonnegative. From (2.6), it follows that λ 2 + L 2 ∈ ρ(−A) and for Reλ > ϖ + |L|, one can see which implies that R(λ 2 ; −A − L 2 I) is positive. Thus, according to Lemma 2.5, we can get that From Definition 2.7, we can obtain that the cosine family C L (t) is positive. Similarly, by (2.7), it is easily to obtain that the sine family S L (t) is positive. This completes the proof of Lemma 2.10 .
where x ∈ X, and ξ q (τ) is the Mainardi's Wright-type function. Employing (2.5) and t ∈ [0, a], derive the relation The operators C q (t), N q (t) and T q (t) have properties (i) and (iii) in Lemma 2.6. At the same time, for each t ≥ 0, Taking account to the properties of the Mainardi's Wright-type function, the positivity of {C L (t)} t≥0 , we can deduce that C q (t), N q (t) and T q (t) are positive operators for all t ≥ 0. Lemma 2.11 ([11]) Let −A be a closed operator from D(−A) into X such that R λ (−A) exists and is compact for some λ. Then R λ (−A) is compact for any λ ∈ ρ(−A). Lemma 2.12 Assume that −A is a closed operator. If the strongly continuous sine operator S (t) generated by −A is compact, then the strongly continuous sine operator S L (t) generated by −A − L 2 I is also compact for every t ≥ 0.

14)
If the inequality of (2.14) is inverse, we call it is an upper solution.

Suppose that h(t) is nonnegative and locally integrable on
Then,

Main results
Throughout this part, we always assume that X is an ordered Banach space, whose positive cone K is normal. −A is a linear m-accretive operator and −A generates a strongly continuous cosine family {C(t)} t≥0 in X.
We present and demonstrate several results in this location. First, under the condition that the strongly continuous sine family is compact, we apply monotone iterative method of the lower and upper solutions to consider the existence of the extreme solutions of Eq (1.1). Theorem 3.1.
Let −A be a linear m-accretive operator and −A generate a compact strongly continuous sine family {S (t)} t≥0 in X. Assume that Eq (1.1) has upper and lower solutions w 0 , v 0 with v 0 ≤ w 0 , f : [0, a] × X → X is a continuous function. If the following assumptions are established: (H1) There is a constant number L such that for all t ∈ [0, a] and v 0 (t) ≤ x 1 ≤ x 2 ≤ w 0 (t). Then Eq (1.1) has minimal and maximal mild solutions u, u ∈ [v 0 , w 0 ].
Proof It follows easily from Eq (1.1) where constant L is decided by condition (H1). According to [44,Lemma 4.1], −(A + L 2 I) can generate a strongly continuous cosine families C L (t) in X. And ∥C L (t)∥ L b ≤ M.
Therefore, Q is a continuous monotonic increasing operator. Next, we define two sequences by Since the monotonicity of Q, we get Step 2.

Results (2.1), (3.3) and ∥C L (t)∥ L b ≤ M together imply that
Similarly, by (2.8), we obtain Finally, in view of Lemma 2.3 and the strong continuous of S L (t) for every t ≥ 0, we obtain Using this result, we get Consequently, one can deduce that the set (QΩ 0 )(t) is relatively compact, which means that {v i (t)} is relatively compact on X for each t ∈ [0, a]. Therefore, {v i (t)} is relatively compact on X for every t ∈ [0, a]. Similarly, we can prove that {w i (t)} is relatively compact on X for every t ∈ [0, a].
Combining the normality of the cone and monotonicity, it follows that {v i } themselves is convergent, that is, there exists u ∈ C([0, a], X) such that v i → u, i → ∞. Similarly, there exists u ∈ C([0, a], X) such that w i → u, i → ∞. Taking the limit of (3.5), we can assert that u(t) = Qu(t), u(t) = Qu(t).
Hence, two fixed points of Q, u and u, are mild solutions of Eq (1.1).
Step 3. Minimal and maximal properties of u, u. Assume that u is a fixed point of Q with u ∈ [v 0 , w 0 ], then for every t ∈ [0, a], v 0 (t) ≤ u(t) ≤ w 0 (t) and v It is clear that u ≤ u ≤ u as i → ∞, which means that u and u are minimal and maximal mild solutions of Eq (1.1), and u and u can be obtained by the iterative sequences defined in (3.5) starting from v 0 and w 0 . This completes the proof of Theorem 3.1. □ Furthermore, we delete the compactness of S (t) and investigate the existence of the solution of Eq (1.1) under the Kuratowski measure of noncompactness. Theorem 3.2. Assume that Eq (1.1) has upper and lower solutions w 0 , v 0 with v 0 ≤ w 0 . f : [0, a]× X → X is a continuous function and satisfied condition (H1). If the following assumption is established: for any t ≥ 0 and bounded subset D ⊂ C([0, a], X). Then Eq (1.1) has maximal and minimal solutions u, u between v 0 and w 0 , which can be obtained by monotone iterative sequences starting from v 0 and w 0 , respectively.
Proof We define the operator Q by (3.2). In view of Theorem 3.1, it is easy to derive that Q : [v 0 , w 0 ] → [v 0 , w 0 ] is a continuous monotone increasing operator, v 0 ≤ Qv 0 and Qw 0 ≤ w 0 . Next, we define two sequences {v i } and {w i } by (3.5) satisfy (3.6).
Next, we prove that {v i } and {w i } are convergent on X.
Proof Using the same method as Theorem 3.2, we can obtain According to Lemma 2.18, we can obtain that {v i (t)} is relatively compact in C([0, a], X). This completes the proof of Corrollary 3.3.
Then Eq (1.1) has a unique mild solution between v 0 and w 0 provided that Proof We establish two iterative sequences v i and w i by (3.5), and its satisfy (3.6). By Theorem 3.2, we know that Eq (1.1) has maximal and minimal mild mild solution on interval [v 0 , w 0 ], that is, the operator Q has fixed point defined by (3.2). For any t ∈ [0, a], from (3.2), (3.5) and (H4), one can obtain that From the normality of the cone K, (2.13) and (3.3), we get that for any t ∈ [0, a], that is, Then (H6) There exist nonnegative constants c ∈ (0, ϖ 3 M(1−e a ) ) and d ≥ 0, such that Then Eq (1.1) has minimal positive mild solutions u * .
Proof We define the operator Q by (3.2). The positive mild solution of Eq (1.1) is equivalent to the fixed point of the operator Q. In view of the continuity of f , it is easy to derive that Q : For every u, v ∈ K with u ≤ v, from (H5), (3.2), the positivity of C q (t), N q (t), T q (t), and x 0 , x 1 ≥ θ, it follows that for all t ∈ [0, a], θ ≤ Qu(t) ≤ Qv(t). Thus, i.e., the sequence {v i } is uniformly bounded. By Theorem 3.1, we know that {v i } is relatively compact and equicontinuous in [0, a]. According to the monotonicity of sequence and the normality of cone, {v i } itself is uniformly converge, which means that there is u * ∈ C([0, a], X) such that v i → u * as i → ∞.
Thus, u * is fixed point of Q, which means that u * is a positive mild solution of Eq (1.1).
Obviously, X is an ordered Banach space and K is a normal cone. We define the operator −A by −A = △ and D(−A) = H 1 0 (Ω) ∩ H 2 (Ω). This seems obvious, −A generates a uniformly bounded strongly continuous cosine family C(t) for t ≥ 0. Thus, ∥C(t)∥ L b ≤ M for each t ≥ 0.
As an elliptic operator, Laplace operator has the maximum principle, so we can easily know the operator λ 2 I + A has a positive bounded inverse. By Lemma 2.10, we can obtain that −A − L 2 I generate a positive strongly continuous cosine operator C L (t) for each t ≥ 0. As consequence, the operator families C q (t)(t ≥ 0), N q (t)(t ≥ 0) and T q (t)(t ≥ 0) are positive. By Theorem 3.1, (4.1) has minimal and maximal mild solutions between [0, w]. This completes the proof of Theorem 4.1. □