Abstract
In this paper, N-tupled fixed point theorems for two monotone nondecreasing mappings in complete normed linear space are established. The extension of Krasnoseskii fixed point theorem for a version of N-tupled fixed point is given. Our theoretical results are applied to prove the existence of a mild solution of the system of N-nonlinear fractional evolution equations. Finally, an example of a nonlinear fractional dynamical system is given to illustrate the results.
1. Introduction
It is outstanding that the fixed point theorem of Krasnoselskii [1] might be joined with Banach and Schauder’s fixed point theorems. The sum of operators is obviously found in neutral functional equations and delay integral equations, which have been talked about widely in [2, 3], for instance. In 1964, Krasnoselskii studied the existence of fixed point for a sum of two operators, one being a continuous and compact and the other being a strict contraction. A lot of generalizations of Krasnoselskii’s theorem have been done, mostly by relaxing the continuous and compact conditions and sometimes by withdrawing the requirement of the strict contraction or even both.
In 2006, Bhaskar and Lskshmikanthan [4] proposed new approach of the fixed point theorem which was called coupled fixed point. They gave some useful applications for mixed monotone operators.
In 2011, Luong and Thuan [5] extend the coupled fixed point approach to study the unique solution of the integral equations. Recently in 2013, Dhage [6] proposed new a version of the Krasnoselskii fixed point theorem and used it to prove the existence of the solution of nonlinear fractional differential equation.
More recently, in 2017, Yang et al. [7] proved the Krasnoselskii coupled fixed point theorem under some or certain conditions and used their results to study the existence of the solution of a nonlinear coupled system of fractional differential equations. Imdad et al. [8] in 2013, gave a new approach of the fixed point theory which was called N-tupled fixed point. This approach is useful to study a fixed point of mappings , where is a nonempty set. In this paper, we extend the results of Yang et al. [7] to prove a generalized version of N-tupled fixed point theorems. We give an application to the study existence of a solution of a system of nonlinear fractional differential evolution equations.
We organize the paper as follows: In Section 2, some basic definitions are introduced. In Section 3, Krasnoselskii N-tupled fixed point ( KNTFP) is proved. In Section 4, the existence of the solution of a nonlinear N-system of fractional evolution differential equations is considered. In Section 5, we give an example of a nonlinear fractional dynamical system. Finally, in Section 6 the conclusion is presented.
2. Preliminaries
Let and be a mapping. A point is said to be a coupled fixed point of in if and . Throughout this paper, let be a nonempty set and be a partially ordered normed linear space. If is a mapping. Then is said to be monotone nondecreasing if implies , for all . Two elements are said to be comparable if or . If is a nonempty subset of , is said to be a chain if each two elements are comparable. We recall the following definitions which are given in [6, 7].
Definition 1. Let be a mapping. is called partially compact if for each chain subset of , is a relatively compact subset of .
Definition 2. Let be a mapping. Given an element , define the orbit asIf for any sequence such that as implies , for each , then is called orbitally continuous in . Furthermore, is said to be orbitally complete if each sequence converges to an element .
Definition 3. A mapping is said to be function if it is upper semicontinuous and monotone nondecreasing such that .
Definition 4. A mapping is called partially nonlinear contraction in , if for each comparable elements , there exists a function such that the following conditions are satisfied:
(i) ,
(ii) for all .
Definition 5. Let be a orbitally complete linear space. The positive cone is defined as .
The following theorem was given in [7] and will be used as a tool to prove the main results.
Theorem 6. Let be a partially ordered orbitally complete normed linear space and be the positive cone of Let be normal and be a nonempty closed subset of Consider the two monotone nondecreasing mappings such that the following conditions are satisfied:
(1) is orbitally continuous and a partially nonlinear contraction,
(2) is orbitally continuous and partially compact,
(3) there xists an element such that for all ,
(4) every pair of elements in has an upper and lower bound.
Then has a coupled fixed point in
Now, we recall the definition about the fractional derivative; for more detail about the fractional calculus, please see [9, 10]. The fractional derivative is defined via the fractional integral operator. So, we will start by the definition of the fractional integral operator. The fractional integral operator of order of a function such that is defined asAlso, the Riemann-Liouville derivative of order is defined aswhere and .
The Caputo fractional derivative of order is denoted by and is defined aswhere , such that and [11].
3. Krasnoselshii N-Tupled Fixed Point Theorem
In this section we give the main N-tupled fixed point theorem which is used in the proposed applications. Consider and define the two operations (sum and scaler multiplication in ) as follows: if , , and then, we have thatTherefore, the positive cone in can be defined asAlso the norm in can be defined aswhere is the norm defined on . It is easy to prove that is partially ordered normed linear space with order relation induced by . If is a nonempty closed subset of , then is also nonempty closed subset of . On the other hand, if is normal cone, the is normal.
Let . The point is said to be N-tupled fixed point of if and only ifNow, we prove the main fixed point theorem.
Theorem 7. Let be a partially ordered orbitally complete normed linear space and be the positive cone of Let be normal and be a nonempty closed subset of Consider are monotone nondecreasing mappings such that the following are satisfied:
(1) is orbitally continuous and a partially nonlinear contraction,
(2) are orbitally continuous and a partially compact,
(3) there exists an element such that for all ,
(4) every pair of elements in has an upper and lower bounded.
Then has a N-tupled fixed point in .
Proof. According to the fact that is a partially ordered complete normed linear space and is normal, then is also a partially ordered complete normed linear space and is normal. Define , then is closed subset of . Consider such that, for ,Therefore, if we prove that the operator equation has a solution , then we have thatThis implies that is a N-tupled fixed point of in . Our proof of the theorem is divided into 4 steps.
Step 1. is orbitally continuous and , for all comparable Since is orbitally continuous then by the definition of we get that is orbitally continuous.
Let be comparable elements; we have that
Step 2. are orbitally continuous and partially compact.
Since are orbitally continuous, it is easy to prove that are orbitally continuous. Consider is bounded chain. According to the fact that is partially compact in , then we get that is equicontinuous and uniformly bounded in Let , then and is bounded chain in Now, we prove that is equicontinuous and uniformly bounded in . Since is uniformly bounded, there is a constant satisfying for all For any , we can find such that and . Therefore, we have thatThus, is uniformly bounded in . By doing the same steps we get that are uniformly bounded in . According to the fact that is equicontinuous in , for any and , we have that , as Thus, for any Z=, where , we get the following:Thus, is equicontinuous in . Therefore, is relatively compact in Hence, is partially compact. By doing the same steps, we get that are partially compact.
Step 3. There is an element satisfying: for all
Let and . By condition in the theorem, we get that
Step 4. Every pair of elements in has an upper bound and lower bound. For every pair of elements , from condition of the theorem, there exist and such that, for all , we have thatHence, we find thatThus, we get that every pair has an upper and lower bound.
Define as , for all Thus is orbitally continuous and a partially compact. Also there exists an element such that for all . Hence has a solution in Then if the solution is , we get thatTherefore, we have thatHence has a N-tupled fixed point. By changing condition due to the fact that there exists an element such that for all , the theoretical results are also correct.
Theorem 8. Let be a partially ordered orbitally complete normed linear space and be the positive cone of Let be normal and be a nonempty closed subset of Consider are monotone nondecreasing mappings such that the following are satisfied:
(1) is orbitally continuous and a partially nonlinear contraction,
(2) are orbitally continuous and a partially compact,
(3) there exists an element such that for all ,
(4) every pair of elements in has an upper and lower bounded.
Then has a N-tupled fixed point in .
4. Existence of Mild Solution of the System of N- Fractional Evolution Equations
Let be the positive convex cone of orbitally Banach space . Suppose that is normal. It is clear that is partially order by the poset induced by In this section, we prove the existence of mild solution of the system of N- fractional evolution evolution equations:where generates a semigroup of uniformly linear and bounded operators in . Also, are given functions. According to the fact that are bounded, then there exists such that for all .
Define two families of operator and where andAccording to [12, 13], the family of operators and has the following properties:   For any and fixed, we have that  is an equicontinuous semigroup which implies that is equicontinuous in ,  is positive semigroup which implies that and are positive operators.
Define and . It is easy to show is a partially order orbitally complete normed space with the norm: , and a poset induced by . It is easy to show that is normal because is normal.
Let and . Define, operators asThen we can say that an element is a N-tupled mild solution of the system (19) if and only if is the solution of the system of the operator equations:
We will apply Theorem 7 to prove that the system of operator equations (24) has a N-tupled fixed point in under the following important conditions:  The positive semigroups is equicontinuous.  The function is continuous in for all and there exist a constant with and a function and satisfied the fact that for any and for all and   The functions are continuous, nondecreasing, and bounded in   There exists an element which satisfies the fact that
Theorem 9. If conditions , and are satisfied, the system of fractional evolution equations (19) has a N-tupled mild solution on .
Proof. Since condition holds, then are bound for all , and there are constants such that, for all , we get that For all , we have that where . We can define the following open balls in aswhere . Define . It is clear that is bounded and closed subset in Now we apply 3 steps to prove the theorem.
Step 1. The operator is orbitally continues and a partially nonlinear contraction in .
Since condition holds, we get that, for any with , we haveBy taking the sup, we get that for all , such that Therefore, we have the following:Hence, is a function from to . According to the fact that is positive semigroup and the properties , , and , it is provided that is nondecreasing. Let for any such that as Thus by condition , we get that is continuous in for all , and therefore we have the following:Thus is orbitally continuous.
Let be comparable such that . We have the following:Since , we get that Hence, is partially nonlinear contraction operator in .
Step 2. The operators are orbitally continuous, nondecreasing, and partially compact operators in .
First, we prove that are mappings from to . By the condition , for any such that , we get thatSimilarly, we have thatThus, we get the following:Therefore, we get thatHence, are mappings from to . By using the same method as in Step 1, we can prove are orbitally continuous and nondecreasing . Consider is any arbitrary chain subset of For any , there is an element such that , for all For any such that , we get thatwhere Hence, when , we get thatand also,Let and ; the we get that For and where is very small, then we get thatBy using condition , we get that as . Therefore, as Thus, is equicontinuous on . Also, using condition , we get thatHence, is uniformly bounded operator in and is partially compact operator from into itself. Similarly, by doing the same steps we get that are partially compact operators.
Step 3. Prove the existence of the N-tupled fixed point.
By condition , we get that By applying Theorem 7, we get that the system of fractional evolution equations (19) has N-tuple mild solution in .
5. An Application to Fractional Dynamical System
Consider the following fractional dynamical system: such that is defined as and is defined asThus are continuous and nondecreasing, and Hence, condition is satisfied.
Theorem 10. Let the following two conditions hold:  The function is continuous and there exists a constant with and a function and satisfied the fact that for any and for all and such that   There exists an element which satisfies the fact that Then the fractional dynamical system (46) has a N-tupled mild solution.
Proof. It is clear that if , is orbitally complete normed linear space with Let ; the is closed convex positive cone in . Thus is normal. Let , which , be defined asThus we can find A- generates a semigroup which is defined asHence, is an equicontinuous semigroup. Therefore, it is clear that Thus, condition holds. Define,Thus, the fractional dynamical system (46) can be written as the fractional system (19). From Theorem 9 system (46) has a N-tupled mild solution.
6. Conclusions
This paper introduced a new version of Kransnoselskii fixed point theorem. This version is more general because by it we can study the N-tupled fixed point. We applied the abstract proposed fixed point theorem to prove the existence of mild solution of the system of N-fractional evolution equations. We gave an example of the fractional dynamical system to illustrate the abstract fixed point proposed theorem. The results are interesting and can be directly applied to some other interesting problems such as the nonlocal problems for fuzzy implicit fractional differential systems [14] and fuzzy-valued equations [15].
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declared no potential conflicts on interest with respect to the research, authorship, and/or publication of this article.
Acknowledgments
The author extend his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under Grant no. G.R.P-98-39.