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Dongming Nie, Azmat Ullah Khan Niazi, Bilal Ahmed, "On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation", Mathematical Problems in Engineering, vol. 2020, Article ID 3276873, 12 pages, 2020. https://doi.org/10.1155/2020/3276873
On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation
We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results.
Fractional differential equations (FDEs) have boosted considerably due to their application in various fields of sciences, such as engineering, chemistry, mechanics, and physics. Recently, fractional ordinary differential equations (ODEs) and partial differential equations (PDEs) have been developed significantly. Indeed, we can also find applications in control, electromagnetism, and electrochemistry (see [1, 2]). For more details in this area, one can also see the monographs of Kilbas et al. , Li et al. [4, 5], Liu et al. [6–9], Miller and Ross , Podulbny , Rehman et al. , and the references therein.
Positive solution of fractional ODEs have already discussed in [13–15]. However, until now research on existence of positive solution for fractional functional differential equations (FFDEs) is rare. Research in different fields including engineering, physics, and biosciences have proved that numerous system structures explain more exactly with the help of FDEs [16–21] and similarly FDEs with delay [22–27] are more accurate to illustrate the real world problems compared to FDEs without delay. So, in , Miller and Ross mentioned that this field has been touched by almost all fields of engineering and science.
Idea of Ulam stability started in 1940, when during the talk at Wisconsin University Ulam posed a problem that “When can we assert that an approximate solution of a functional equation can be approximated by a solution of the corresponding equation?” (for more details, see ). After one year, Hyers first gave the answer to the Ulam’s question  in case of Banach spaces. After that, stability of this type is called the Ulam–Hyers stability. Rassias  in 1978 provided an outstanding generalization of the Ulam–Hyers stability of mappings by considering variables.
Usually, the discussion of the existence and uniqueness of a solution for nonlinear FDEs normally fixed point theory has been used [16, 32, 33]. Motivated by these and [34–36], in this work, we have discussed existence and uniqueness of solution also after applying some sufficient conditions, obtained positive solution, and at the end established local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability and presented stability results graphically for the class of nonlinear FDEs with delay given bywhere is the Caputo derivative and , while and where is space of continuous functions, is a nonnegative real number also are continuous functions, and is defined as .
Remaining paper is arranged as follows. Section 2 includes some basic definitions and lemmas used throughout this paper. In Section 3, we have the main results. In Section 4, we establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for problem (1), and in Section 5, simulation of an example is given to show the applicability of our results.
Let be a cone in a real Banach space and partial order introduced by in is as
Definition 1. Let be the order interval defined as
Definition 2. The functional is said to be nondecreasing w.r.t. on such thatimplies thatfor any .
Definition 3. (see ). The fractional integral of order for a function with lower limit 0 can be defined aswhere is the gamma function and right-hand side of upper equality is defined pointwise on .
Definition 4. (see ). The left Caputo fractional derivative of order is given bywhere ( stands for the bracket function of ).
Lemma 1. (Leray Schauder fixed point theorem, see ). Let be a nonempty, closed, bounded, and convex subset of Banach space and is a compact and continuous map; then, has a fixed point in .
Theorem 1. If , where is a cone of space of partial order , be nondecreasing. If there exists such that , , and is lower solution and is an upper solution of , then has minimum solution and maximum solution in such that , and if anyone of these conditions holds(i) is reflexive, is weak continuous or continuous, and is normal(ii) is completely continuous and is normal(iii) is continuous and is regular
3. Main Results
In this section, we have discussed existence and uniqueness of solution and some conditions for positive solution of equation (1). Consider is a Banach space endowed with the supremum norm and cone is defined as . From equation (1), we haveand we can easily get that
Let be the function defined by
So, and for every with , and we defined as
Let satisfy equation (9), and we can decompose as so that , for every , where is such that and the function satisfies
Set , where is a cone and and is a Banach space with Sup norm . Let the operator be defined by
Before proving main results, we introduce the following conditions:(C1)Let us take with , where function is nondecreasing and and satisfy Lipchitz condition that is, and there exists nonnegative constants such that(C2)If be bounded, is nondecreasing andwhere . Let . Define the operator by
Theorem 2. Assume that and are nonnegative and continuous; then, operator is completely continuous.
Proof. As we know that and are nonnegative and continuous, so operator is continuous. Let is bounded, i.e., there exists a constant such that . Here, we have to keep in mind the condition that . If , and for all , we get thatHence, is bounded. Now, we show that is equicontinuous, and the proof is divided into three possible steps. Step 1: for every and such that , thenAs , the right-hand side of above inequality tends to zero. Step 2: for every , as and while , there exists such that when , .If , then for , we get that Step 3: for every , , and where . As is continuous, we haveHence, is equicontinuous. So, is compact by Arzela–Ascoli Theorem.
Theorem 3. Suppose that C1 and C2 hold. Let and . Then, there exists a unique solution for IVP (1) on the interval .
Proof. Consider the operator defined asWe will show that is a contraction. Let , then we obtain the following sequence of inequalities:Therefore,So, is a contraction, and hence by Banach’s contraction principle has a unique fixed point.
Definition 5. The function is said to be a lower solution of equation (1) (operator ) ifand the function is said to be an upper solution of equation (1) ifIf inequalities are strict, then and are strict lower and upper solutions.
Theorem 4. Let(H1) and from to are continuous, and and are nondecreasing in for each (H2) is lower and is an upper solution of equation (1), satisfying condition , and Then, equation (1) has at least a positive solution.
Proof. We only take fixed point operator . By Theorem 2, is completely continuous and obviously by equation (16), and are lower and upper solutions of , respectively. By (H1), , we haveSo, the operator is nondecreasing. Also, we haveAs is a lower solution, therefore we can say that , similarly from the definition of upper solution of .
Hence, is a completely continuous operator. Since is a normal cone, Theorem 1 implies that has a fixed point .(H3)Here, we suppose that there exists such that .
Theorem 5. Assume conditions (H1) and (H3) holds; then, equation (1) has a positive solution.
Proof. Consider the equationObviously, equation (28) has a solution , which is a lower solution of equation (1). Similarly, consider the equationWe know that , is an upper solution of equation (1) and , so the above definition proves that equation (1) has minimum one positive solution.
For getting another result for a positive solution of equation (1), now we take the more general case of (1) and find the existence of a positive solution for the equations:
Theorem 6. Equation (30) has a least positive solution.
Proof. Equation (30) is equivalent toLet be an operator defined as is completely continuous by Theorem 2.
Let us take a bounded, closed, and convex subset of the Banach space , where and . The possible two cases are Case 1: for all , we obtainSince , henceHere, . Case 2: if and , thenHence, . The Leray–Schauder fixed point theorem guarantees that operator has at least one fixed point and so equation (30) has at least one positive solution.
In this section, we will discuss local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for a class of fractional neutral differential equations. For the case with are continuous functions on a closed interval or more generally compact sets, then they are bounded so we can replace the supremum by the maximum. In this case, norm is also called the maximum norm. Let be a Banach space with , where is a nonnegative real number and . We consider the following differential equation:where is a Caputo derivative with and . We focus on the following inequalities:
Definition 6. (see ). Equation (36) is local generalized Ulam–Hyers stable if there exists a positive real number such that, for each positive and for every solution of (37), there exists a solution of (36) with .
Remark 1. A solution of differential equation is stable (asymptotically stable) if it attracts all other solutions with sufficiently close initial values. On the contrary, in Hyers–Ulam stability, we compare solution of given differential equation with the solution of differential inequality. We say solution of differential equation is stable if it stays close to solution of differential inequality. Hyers–Ulam stability may not imply the asymptotic stability.
Remark 3. Let and be a solution of inequality (37); then, is a solution of the following inequality:From Remark 2, we havethenTherefore,By using the same process as in Section 3, i.e., let be the function defined byTherefore, and for every with . We defined asLet satisfies equation (42). We can decompose as so that , for every , where is such that and the function satisfiesBy following the conditions in Section 3, we get from (42) thatHence,If is a solution of inequality (38) then is a solution of the following inequality:Before stating stability results, let us take condition (H4) as(H4)
Theorem 7. Suppose that is true and other two conditions(a), , and , .(b) and satisfy Lipchitz conditions:where and are nonnegative real numbers holds, then(i)Equation (36) has a unique solution(ii)Equation (36) is local generalized Ulam–Hyers stable
Proof. Let be a unique solution of equation (37). Denote as the unique solution of the equationThen, we haveWe can see that , for . For , we obtainFor , by usingThus,