Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses

Abstract

This article deals with some existence, uniqueness, and Ulam-Hyers-Rassias stability results for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and generalized Hilfer Fractional derivative. The results are obtained using the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems.

1. Introduction

Differential equations of fractional order have been recently proved to be a powerful tool to study many phenomena in various fields of science and engineering such as electrochemistry, finance, hydrology, electromagnetics, and viscoelasticity. There are numerous books and articles focused on linear and nonlinear initial and boundary value problems for fractional differential equations involving different kinds of fractional derivatives, see, for example, [1–6]. Impulsive fractional differential equations have been considered by many authors (see, for instance, [7–12]). Recent results involving different fractional derivatives can be found in [13–21] and the references therein.

Ulam was the first who raise the concept of stability of functional equations [22]. In 1941, Hyers [23] provided the first answer to Ulam’s question. Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [24] was able to make a remarkable generalization of Ulam-Hyers stability of mappings by considering variables.

Considerable attention has been given to the study of the Ulam-Hyers and Ulam-Hyers-Rassias stability of all kinds of functional equations; one can see the monograph of Abbas et al. [3] and the paper by Rus [25] who discussed the Ulam-Hyers stability for operational equations (see also [26–29]).

Recently, in [30], Harikrishnan et al. investigated existence theory and different kinds of stability in the sense of Ulam, for the following initial value problem with nonlinear generalized Hilfer-type fractional differential equation and impulses: where and are a generalized Hilfer fractional derivative of order and type and generalized fractional integral of order , respectively; , , and represent the right and left hand limits of at ; ; is a given function; and are given continuous functions.

Motivated by the works mentioned above, in this paper, we establish existence and uniqueness results to the boundary value problem with nonlinear implicit generalized Hilfer-type fractional differential equation and impulses: where are a generalized Hilfer fractional derivative of order and type and generalized fractional integral of order , respectively; are real with ; ; ; and represent the right and left hand limits of at ; is a given function; and , are given continuous functions.

The present paper is organized as follows. In Section 2, some notations are introduced and we recall some preliminaries about generalized Hilfer fractional derivative and auxiliary results. In Section 3, three results for problems (2)–(4) are presented which are based on the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems. In Section 4, we discuss the Ulam-Hyers-Rassias stability for problems (2)–(4). Finally, we give examples to illustrate the applicability of our main results.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let . By , we denote the Banach space of all continuous functions from into with the norm

We consider the weighted spaces of continuous functions with the norms

Consider the Banach space , and there exist and , with },

Also, we consider the weighted space with the norm

Consider the space of those complex-valued Lebesgue measurable functions on for which , where the norm is defined by

In particular, when , the space coincides with the space: .

Definition 1 [31]. Generalized Hilfer fractional integral.
Let , and . A generalized Hilfer fractional integral of order is defined by where is the Euler gamma function defined by .

Definition 2 [31]. Generalized Hilfer fractional derivative.
Let and . A generalized Hilfer fractional derivative of order α is defined by where and .

Theorem 3 [31]. Let , , and . Then, for , the semigroup property is valid, i.e.,

Lemma 4 [31, 32]. Let , and . Then, is bounded from into

Lemma 5 [32]. Let , , , and . If , then is continuous on and

Lemma 6 [33]. Let . Then, for and , we have

Lemma 7 [32]. Let , , and . Then,

Lemma 8 [32]. Let , . If and , then

Definition 9 [32]. Let order α and type β satisfy and , with . A generalized Hilfer fractional derivative to , with of a function , is defined by

In this paper, we consider the case only, because .

Property 10 [32]. The operator can be written as

Property 11 [32]. The fractional derivative is an interpolator of the following fractional derivatives: Hilfer , Hilfer–Hadamard , generalized , Caputo–type , Riemann–Liouville , Hadamard , Caputo , Caputo–Hadamard , Liouville , and Weyl .
Consider the following parameters satisfying

We define the spaces where .

Since , it follows from Lemma 4 that

Lemma 12 [32]. Let , and . If , then

Lemma 13 (Theorem 25 [32]). Let be a function such that , for any . Then, is a solution of the differential equation: if and only if satisfies the following Volterra integral equation: where .

Theorem 14 ([34]) (Banach’s fixed-point theorem). Let be a nonempty closed subset of a Banach space ; then, any contraction mapping of into itself has a unique fixed point.

Theorem 15 ([34]) (Schaefer’s fixed-point theorem). Let be a Banach space and be a completely continuous operator. If the set is bounded, then has a fixed point.

Theorem 16 ([34]) (Krasnoselskii’s fixed-point theorem). Let be a closed, convex, and nonempty subset of a Banach space and the operators such that (1), for all (2) is compact and continuous(3) is a contraction mapping

Then, there exists such that .

Now, we consider the Ulam stability for problems ((2))–((4)) that will be used in Section 4. Let , , , and be a continuous function.

We consider the following inequality:

Definition 17 ([35]). Problems ((2))–((4)) are Ulam-Hyers-Rassias (U-H-R) stable with respect to if there exists a real number such that for each and for each solution of inequality (27), there exists a solution of (1)–(3) with

Remark 18 ([35]). A function is a solution of inequality (27) if and only if there exist and a sequence , such that (1) and (2)(3)

3. Existence of Solutions

We consider the following linear fractional differential equation: where , , and , with the conditions and where and with and

The following theorem shows that problems ((29))–((31)) have a unique solution given by

Theorem 19. Let , where and . If is a function such that , then satisfies problems ((29))–((31)) if and only if it satisfies (34).

Proof. Assume u satisfies (29)–(31). If , then

Lemma 20. implies that

If , then Lemma 13 implies

If , then Lemma 13 implies

Repeating the process in this way, the solution for , , can be written as

Applying on both sides of (39), using Lemma 6 and taking , we obtain

Multiplying both sides of (40) by and using condition (31), we obtain which implies that

Substituting (42) into (39) and (36), we obtain (34).

Reciprocally, applying on both sides of (34) and using Lemma 6 and Theorem 3, we get

Next, taking the limit of (43) and using Lemma 5, with , we obtain

Now, taking in (43), we get

From (44) and (45), we find that which shows that the boundary condition is satisfied. Next, apply operator on both sides of (34), where . Then, from Lemma 6 and Lemma 12, we obtain

Since and by definition of , we have , then (47) implies that

As and from Lemma 4, it follows that

From (48) and (49) and by the definition of the space , we obtain

Applying operator on both sides of (47) and using Lemma 8, Lemma 5, and Property 10, we have that is, (29) holds. Also, we can easily show that

This completes the proof.

As a consequence of Theorem 19, we have the following result.

Lemma 21. Let , where and ; let be a function such that for any . If , then satisfies problems ((2))–((4)) if and only if is the fixed point of the operator defined by where is a function satisfying the functional equation

Assume that the function is continuous and satisfies the following conditions:

(H1). The function be such that

(H2). There exist constants and such that for any and .

(H3). There exists a constant such that for any and

(H4). There exist functions with such that

(H5). The functions are continuous, and there exist constants such that

We are now in a position to state and prove our existence result for problems ((2))–((4)) based on Banach’s fixed point.

Theorem 22. Assume (H1)–(H3) hold. If then problems ((2))–((4)) have a unique solution in .

Proof. The proof will be given in two steps.

Step 1. We show that the operator defined in (53) has a unique fixed point in . Let and ; then, we have where such that

By (H2), we have

Then,

Therefore, for each ,

Thus,

By Lemma 6, we have