Existence of Iterative Cauchy Fractional Differential Equation
Our main aim in this paper is to use the technique of nonexpansive operators in more general iterative and noniterative fractional differential equations (Cauchy type). The noninteger case is taken in sense of the Riemann-Liouville fractional operators. Applications are illustrated.
Fractional calculus and its applications (that is the theory of derivatives and integrals of any arbitrary real or complex order) are important in several widely diverse areas of mathematical, physical, and engineering sciences. It generalized the ideas of integer order differentiation and -fold integration. Fractional derivatives introduce an excellent instrument for the description of general properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. 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.
Our aim in this paper is to consider the existence and uniqueness of nonlinear Cauchy problems of fractional order in sense of Riemann-Liouville operators. Also, two theorems in the analytic continuation of solutions are studied. In the fractional Cauchy problems, we replace the first-order time derivative by a fractional derivative. Fractional Cauchy problems are useful in physics. Recently, the author studied the the fractional Cauchy problems in complex domain .
One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. The Riemann-Liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. Moreover, this operator possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms (see ).
Definition 1. The fractional (arbitrary) order integral of the function of order is defined by When , we write , where denoted the convolution product (see ), and and as where is the delta function.
Definition 2. The fractional (arbitrary) order derivative of the function of order is defined by
Remark 3. From Definitions 1 and 2, we have
Definition 4. The Caputo fractional derivative of order is defined, for a smooth function , by where , (the notation stands for the largest integer not greater than ).
Note that there is a relationship between Riemann-Liouville differential operator and the Caputo operator and they are equivalent in a physical problem (i.e., a problem which specifies the initial conditions, that is, if , then the Riemann-Liouville derivative and the Caputo derivative of order coincide).
We extract here the basic theory of nonexpansive mappings in order to offer the notions and results that will be needed in the next sections of the paper. Let be a metric space. A mapping is said to be an -contraction if there exists such that In the case where , the mapping is said to be nonexpansive. Let be a nonempty subset of a real normed linear space and be a map. In this setting, is non-expansive if
The following result is a fixed point theorem for non expansive mappings, according to Berinde; see for example .
Theorem 5. Let be a nonempty closed convex and bounded subset of a uniformly Banach space . Then any nonexpansive mapping has at least a fixed point.
Definition 6. Let be a convex subset of a normed linear space and let be a self-mapping. Given an and a real number , the sequence defined by the formula is usually called Krasnoselskij iteration or Krasnoselskij-Mann iteration.
Definition 7. Let be a convex subset of a normed linear space and let be a self-mapping. Given an and a real number , the sequence defined by the formula is usually called Mann iteration.
Edelstein  proved that strict convexity of suffices for the Krasnoselskij iteration converge to a fixed point of . While, Egri and Rus  proved that for any subset of , the Mann iteration converge to a fixed point of when is a non-expansive mapping.
We need the following results, which can be found in .
Lemma 8. Let be a convex and compact subset of a Banach space and let be a non-expansive mapping. If the Mann iteration process satisfies the assumptions(a) for all positive integers ,(b) for all positive integers ,(c).
Then converges strongly to a fixed point of .
Lemma 9. Let be a closed bounded convex subset of a real normed space and a non-expansive mapping. If maps closed bounded subset of into closed subset of and is the Mann iteration, with satisfying assumptions (a)–(c) in Lemma 8, then converges strongly to a fixed point of in .
3. Existence Theorems and Approximation of Solutions
For most of the differential and integral equations with deviating arguments that appear in recent literature, the deviation of the argument usually involves only the time itself. However, another case, in which the deviating arguments depend on both the state variable and the time , is of importance in theory and practice. Equations of the form are called iterative differential equations. These equations are important in the study of infection models and are related to the study of the motion of charged particles with retarded interaction (see [3, 8, 9]).
In this section, we establish the existence and uniqueness results for the fractional differential equation with initial condition , where and . For denote It is clear that is a nonempty convex and compact subset of the Banach space , where .
Theorem 10. Assume that the following conditions are satisfied for the initial value problem (11):(A1); (A2), for all ;(A3) if is the Lipschitz constant such that , then (A4) one of the following conditions holds:(a), where ;(b), for all ;(c), for all .
If then there exists at least one solution of problem (11) in which can be approximated by the Krasnoselskij iteration where and are arbitrary.
Proof. Consider the integral operator
Our aim is to show that has a fixed point in . We proceed to apply Schauder fixed point theorem or Banach fixed point theorem.
First we show that is invariant set with respect to , that is, . In virtue of condition (A4(a)) and for all , , we have
Thus . In the similar manner of (A4(a)), we treat the cases (A4(b)) and (A4(c)). Now for every , by (A3), we obtain
Hence whenever . Therefore, (i.e., is a self-mapping of ). Let and , by employing (A2), we have
Now, by taking the supremum in the last assertion, we get
If , then is a contraction mapping and hence in view of Banach fixed point theorem, (11) has a unique solution. Now if
then is non-expansive and, hence, continuous; thus Schauder fixed point theorem implies that (11) has a solution in . Finally, in view of Lemmas 8 and 9, we obtain the second part of the theorem.
Next we establish the solution of (11) in a subset of defined by
It is clear that is nonempty, convex, and compact subset in .
Theorem 11. Assume that the following conditions are satisfied.(A5). (A6) If is the Lipschitz constant such that , then .(A7) There exists a such that and
If (A2), (A4) hold then there exists at least one solution of problem (11) in which can be approximated by the Krasnoselskij iteration where and are arbitrary.
Proof. We assume the Banach space endowed with Bielecki’s norm is given by the formula
Let be defined as in the proof of Theorem 10. By assumptions (A2), (A4), and (A6), it follows that
Now we prove that is an invariant set with respect to the operator . Indeed, if and then in view of (A5) and (A6), we have
that is, .
Let and , we have
where is a continuous function. Then there exists a constant such that
Thus we have
which shows that is Lipschitzian, hence continuous. By Schauder’s fixed point theorem, it follows that has at least one fixed point which is actually a solution of the initial value problem (11).
We proceed to show that is nonexpansive function. The function
is strictly increasing on and ; furthermore,
Similarly for the function
Now the function
is strictly decreasing on ; hence,
For and then by the assumption (A7), there exists a such that
which implies that and hence is strictly increasing on . If we put , we have
But since thus we get
for sufficient , , , and . Moreover, we have
Consequently, we receive
This shows that is non-expansive.
Similar argument holds when , in (38) we have , hence is strictly increasing on . Finally, one can use Lemmas 8 and 9 to obtain the second part of the theorem. This completes the proof.
Example 12. Consider the following initial value problem associated to an fractional iterative differential equation where . We are focused on the solutions belonging to the set
To satisfy (A4(a)), we have , and
Hence (A4(a)) is satisfied. The function is Lipschitzian with the Lipschitz constant . This shows that
Therefore, by Theorem 10, we obtain information on the existence and approximation for the solutions of the initial value problem (44).
If we consider the function in Example 12, then we obtain
Therefore, again by Theorem 10 we pose the existence and approximation of the solutions of the initial value problem (44).
Again, we consider the problem (44) on the interval for , where . We are interested in the solutions belonging to the set Our aim is to satisfy the assumptions of Theorem 11. (A2) and (A4) are valid. Since and , we have hence (A5) is satisfied. Moreover, a computation gives thus (A6) is satisfied. Now we proceed to satisfy (A7), since then for , we impose
Hence in view of Theorem 11, problem (44) has a solution in the set .
We can observe that problem (44) has not a solution on the set over the interval : For , , , , a calculation poses therefore, condition (A5) dose not satisfy.
Finally, problem (44) has not a solution on the set over the interval : (in view of Theorem 11) For , , , , a calculation yields therefore, condition (A5) dose not satisfy.
As such iterative fractional differential equations are used to generalize the model infective disease processes, pattern formation in the plane, and are important in investigations of dynamical systems, future works will be also devoted to them.
The author is thankful to the anonymous referee for his/her helpful suggestions for the improvement of this paper.
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