Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

We study the existence, uniqueness, and various kinds of Ulam–Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.


Introduction
Over the past few decades, differential equations of fractional order have got considerable attention from the researchers due to their significant applications in various disciplines of science and technology.Fractional derivatives introduce amazing instrument for the description of general properties of different materials and processes.This is the primary advantage of fractional derivatives in comparison with classical integer order models, in which such impacts are in fact ignored.The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of properties of gases, liquids, rocks and in many other fields (see [1,2]).Since fractional order differential equations play important roles in modeling real world problems related to biology, viscoelasticity, physics, chemistry, control theory, economics, signal and image processing phenomenon, bioengineering, and so forth (for details, see [3][4][5][6][7]), it is investigated that fractional order differential equations model real world problems more accurately than differential equations of integer order.The area devoted to the study of existence and uniqueness of solutions to initial/boundary value problems for fractional order differential equations has been studied very well and plenty of papers are available on it in the literature.We refer the reader to few of them in [8][9][10][11][12][13][14] and the references therein.To model evolution process and phenomena which are experienced from sudden changes in their states, impulsive differential equations serve as a powerful mathematical tool to model them.In daily life, we observe several physical systems that suffer from impulsive behavior such as the pendulum clock action, heart function, mechanical systems subject to impacts, dynamic of system with automatic regulation, the maintenance of a species through periodic stocking or harvesting, the thrust impulse maneuver of a spacecraft, control of the satellite orbit, disturbances in cellular neural networks, fluctuations of economical systems, vibrations of percussive systems, and relaxational oscillations of the electromechanical systems.For details, see [15][16][17][18][19][20][21][22][23].
In some cases, nonlocal conditions are imposed instead of local conditions.It is sometimes better to impose nonlocal conditions since the measurements needed by a nonlocal condition may be more precise than the measurement given by a local condition in dynamical problems.Also, nonlocal boundary value problems have become an expeditiously 2 Complexity growing area of research.The study of this type of problems is driven not only by a theoretical interest, but also by the fact that several phenomena in physics, engineering, and life sciences can be modeled in this manner, for example, problems with feedback controls such as the steady-states of a thermostat, where a controller at one of its ends adds or removes heat.For more applications, see [24] and references therein.Due to the aforesaid significant behavior, we prefer to take nonlocal boundary conditions.
The definitions of the fractional order derivative are not unique and there exist several definitions, including Grunwald-Letnikov [7], Riemann-Liouville [25], Weyl-Riesz [26], Erdlyi-Kober [27], and the Caputo [28] representation for fractional order derivative.In the Caputo case, the derivative of a constant is zero and we can define, properly, the initial conditions for the fractional differential equations which can be handled by using an analogy with the classical integer case.For these reasons, in this manuscript, we prefer to use the Caputo fractional derivative.
Tian and Bai [29] studied the existence and uniqueness of the following nonlinear impulsive boundary value problem: where R → R are continuous, and with ( +  ),   ( +  ), ( −  ),   ( −  ) are the respective left and right limits of (  ) at  =   .
In 1940, Ulam posed the following problem about the stability of functional equations: "Under what conditions does there exist an additive mapping near an approximately additive mapping?" (see [30]).In the following year, Hyers gave an answer to the problem of Ulam for additive functions defined on Banach spaces [31].Let B 1 , B 2 be two real Banach spaces and  > 0. Then for each mapping  : for all ,  ∈ B 1 , there is a unique additive mapping  : That is why the name of this stability is Ulam-Hyers stability.Later on, Hyers results are extended by many mathematicians; for details, reader may see [32][33][34][35][36][37][38][39] and the reference therein.The mentioned stability analysis is extremely helpful in numerous applications, for example, numerical analysis and optimization, where it is very tough to find the exact solution of a nonlinear problem.We notice that Ulam-Hyers stability concept is quite significant in realistic problems in numerical analysis, biology, and economics.The aforementioned stability has very recently attracted the attention of researchers; we refer the reader to some papers in [40][41][42][43][44][45].Because fractional order system may have additional attractive feature over the integer order system, let us suppose the following example to show which one is more stable in the aforementioned (fractional order and integer order) systems.
The manuscript is structured as follows.In Section 2, we give some definitions, theorems, lemmas, and remarks.In Section 3, we built up some adequate conditions for the existence and uniqueness of solutions to the considered problem (7) through using fixed point theorems of Krasnoselski's and Banach contraction type.In Section 4, we establish applicable results under which the solution of the considered boundary value problem (7) satisfies the conditions of different kinds of Ulam-Hyers stability.The established results are illustrated by an example in Section 5.

Background Materials and Auxiliary Results
This section is devoted to some basic definitions, theorems, lemmas, and remarks which are useful in existence and stability results.
Lemma 4 (see [11]).Let  > 0, then the solution of the differential equation will be in the following form: where  = [] + 1.
Consider U be a convex closed and nonempty subset of Banach space X.Suppose F * , G * be the operators such that (ii) F * is compact and continuous and G * is contraction mapping.
Then there is  ∈ U such that  = F *  + G * .
Theorem 7 ((Banach fixed point theorem) see [47]).Suppose S be a nonempty closed subset of a Banach space B.
Definition 11 (see [48]).The considered problem ( 7) is generalized Ulam-Hyers-Rassias stable with respect to  ∈ (I, R + ), if there is a constant  ,, ∈ R + , such that, for every solution  ∈ X of inequality (19), there is a unique solution  ∈ X of the considered problem (7) with Remark 12.A function  ∈ X is a solution of inequality (17), if there is a function  ∈ X and a sequence   ,  = 1, 2, . . .,  (which depend on  only), such that Remark 13.A function  ∈ X is a solution of inequality (18), if there is a function  ∈ X and a sequence   ,  = 1, 2, . . .,  (which depend on  only), such that (19), if there is a function  ∈ X and a sequence   ,  = 1, 2, . . .,  (which depend on  only), such that
Hence proof is completed.
We define D : X → X by where Suppose that the following hold. ( The following result is based on Banach contraction theorem. Theorem 16.Under the assumptions ( 1 )-( 3 ) and if the considered problem (7) has a unique positive solution.
Proof.Suppose ,  ∈ X and for every  ∈ I, consider where which implies that D is contraction.Hence, the considered problem ( 7) has a unique positive solution.
The following result is based on Krasnoselskii's fixed point theorem.

Theorem 17. In addition to assumptions (𝐴
If then the considered problem (7) has at least one positive solution.

Ulam Stability Results
In this section, we built up some sufficient conditions under which problem (7) satisfies the assumptions of various kinds of Ulam-Hyers stability.

Lemma 18.
If  ∈ X is the solution of inequality ( 17) and 1 <  ≤ 2, then  is the solution of the following inequality: Proof.Since  is the solution of inequality (17), so in view of Remark 13, we have So the solution of (65) will be in the following form: where For convenience, we denote the sum of terms free of  by ](), that is, Therefore, (66) becomes By using (i) of Remark 13, we get Theorem 19.Let assumptions ( 1 )-( 3 ) hold along with the condition Then problem (7) will be Ulam-Hyers stable.
Proof.Suppose  ∈ X be any solution of inequality ( 17) and let  be the unique solution of the considered problem ( 7 where ( 8 ) Suppose a function  ∈ (I, R + ), which is increasing.Then there is   > 0, such that, for every  ∈ I, the following integral inequality holds.

Lemma 21.
Let assumption ( 8 ) hold and suppose  ∈ X is the solution of inequality (18), then  will be the solution of the following integral inequality: Proof.From Lemma 18, we have  (95) Hence, the considered problem ( 7) is Ulam-Hyers-Rassias stable.

Example
Consider the following implicit impulsive fractional differential equations with nonlocal boundary conditions.96) is Ulam-Hyers stable and by Remark 20, it will be generalized Ulam-Hyers stable.Also by demonstrating the conditions of Theorem 22 and Remark 23, it can be easily seen that the considered problem (96) is Ulam-Hyers-Rassias stable and generalized Ulam-Hyers-Rassias stable.

Conclusion
We have successfully built up some proper conditions for existence theory of implicit impulsive fractional order differential equations with nonlocal boundary conditions, by using different kinds of fixed point theorems which are stated in Section 2. The concerned theorems ensure the existence and uniqueness of solution.Further, we did settle some adequate conditions for different kinds of Ulam-Hyers stability (see Theorems 19 and 22 with Remarks 20 and 23, respectively) by using Definitions 8-11.The mentioned stability is rarely investigated for implicit impulsive fractional differential equations and also very important.Finally, we illustrated the main results by giving a suitable example.