Qualitative study of linear and nonlinear relaxation equations with $ \psi $-Riemann-Liouville fractional derivatives

1.   Introduction

As an extension of the current development and generalizations in the field of fractional calculus ^[1,2,3], the investigation of solution behaviors and qualitative properties of the solution in classical or fractional differential equations has become a matter of intense interest for researchers. This reflects the extent of its uses in several applied and engineering aspects. It draws amazing applications in nonlinear oscillations of seismic tremors, the detection of energy transport rate, and energy generation rate. The importance of fractional equations has been recognized in many physical phenomena, in addition to its importance in the mathematical modeling of diseases and viruses to limit and reduce their spread.

The relaxation differential equation gives as $ u^{\prime }(\varkappa)+u(\varkappa) = f(t); \; u(0^{+}) = u_{0}, $ whose solution is

Some recent contributions to the theory of FDEs can be seen in ^[4]. In ^[5], the authors studied the following problem

where $ 0 < \kappa < 1, \; f:[0, 1]\times \mathbb{R} ^{+}\rightarrow \mathbb{R} ^{+} $ is continuous and $ f(\varkappa, \cdot) $ is non-decreasing for $ \varkappa \in \lbrack 0, 1] $, by lower and upper (LU) solution method. The existence and uniqueness of solutions of the FDE

were obtained in ^[1,2,6], by using the fixed point theorem (FPT) of Banach.

In ^[7], the authors discussed the existence and uniqueness of solutions of the following FDE

by using the LU solution method and its associated monotone iterative (MI) method. The problem (1.3) with non-monotone term has been studied by Bai et al. ^[8].

In ^[9], a new appraoch of the maximum principle was presented by using the completely monotonicity of the Mittag-Leffler (ML) function.

On the other hand, there are several definitions and generalizations of the fractional operators (FOs) that contributed a lot to the development of this field. The generalization of RL's FOs based on a local kernel containing a differentiable function was first introduced by Osler ^[3]. Next, Kilbas et al. ^[1] dealt with some of the properties of this operator. Then, the interesting properties for this operator have been discussed by Agarwal ^[10]. Recently, Jarad and Abdeljawad ^[11] achieved some properties in accordance with the generalized Laplace transform with respect to another function.

In this regard, most of the results similar to our current work are covered under the generalized FOs of Caputo ^[12] and Hilfer ^[13], for instance, see ^{[14,15,16,17,18,19]}, whereas, very few considered results related to the dependence on generalized RL's definition. The authors in ^[20,21], investigated the existence and uniqueness of positive solutions of the fractional Cauchy problem in the frame of generalized RL and Caputo, respectively.

For this end, as an additional contribution and enrichment to this active field, we consider the following linear and nonlinear relaxation equations with non-monotone term under $ \psi $-RL fractional derivatives ($ \psi $ -RLFD):

where $ 0 < h < +\infty $, $ 0 < \kappa, \delta < 1, \; \lambda \geq 0, \; f\in C([0, h], \mathbb{R}) $ and

where $ f\in C([0, h]\times \mathbb{R}, \mathbb{\ \mathbb{R} }) $, $ D_{0+}^{\kappa; \psi } $and $ D_{0+}^{\delta; \psi } $ are RL fractional derivatives of order $ \kappa $ and $ \delta, $ respectively, with respect to another function $ \psi \in C^{1}([0, h], \mathbb{ \mathbb{R} }) $, which is increasing, and $ \psi ^{\prime }(\varkappa)\neq 0 $ for all $ [0, h] $. The main contributions of this work stand out as follows:

i) With a new version of Laplace transform, we obtain Hyers-Ulam (HU) and generalized Hyers-Ulam (GHU) stabilities on the finite time interval to check whether the approximate solution is near the exact solution for a $ \psi $-RL linear FDEs (1.4) and (1.5).

ii) We establish a condition to derive the existence and uniqueness of solutions for $ \psi $-RL nonlinear FDEs (1.6) and (1.7), by using LU solution method along with the Banach contraction map (this generalizes the results in ^[7]).

iii) We formulate a new condition on the nonlinear term to ensure the equivalence between the solution of the proposed problem and the corresponding fixed point. Then in light of that, we discuss the maximal and minimal solutions for (1.6) and (1.7).

Remark 1.1.

(1) Our results remain valid if $ \lambda = 0 $ on problems (1.4)–(1.7), which reduce to

and

(2) If $ \psi (\varkappa) = \varkappa, $ the problems (1.10) and (1.11) reduces to problem (1.3) considered in ^[7].

(3) The linear versions (1.4) and (1.5) generalizes that given in Theorem 5.1 by Jarad et al. ^[11].

Observe that in the preceding works, the nonlinear term needs to fulfill the monotone or other control conditions. Indeed, the nonlinear FDE with a non-monotone term can respond better to generic regulation, so it is vital to debilitate the control states of the nonlinear term.

This work is coordinated as follows. Section 2 provides some consepts of $ \psi $ -fractional calculus. Section 3 studies the stability results for the $ \psi $ -RL linear FDEs (1.4) and (1.5). In Section 4, we investigate of the the existence and uniqueness results for the $ \psi $-RL nonlinear FDEs (1.6) and (1.7). Moreover, the existence of maximal and minimal solutions is also obtaind. At the end, we provide some examples in the last section.

2.   Preliminaries

Given $ 0\leq a < b < +\infty $ and $ s > 0 $, and let $ \psi _{s}(\varkappa, a): = (\psi (\varkappa)-\psi (a))^{s} $. Define a set

Clearly, $ C_{s; \psi }[a, b] $ is a Banach space with the norm

Definition 2.1. ^[1] Let $ \theta > 0, $ and $ f:[a, b] \rightarrow \mathbb{R} $ be an integrable function. Then the generalized RL fractional integral and derivative with respect to $ \psi $ is given by

and

respectively, where $ n = [\kappa ]+1 $, and $ \psi :[a, b]\rightarrow \mathbb{ \mathbb{R} } $ is an increasing with $ \psi ^{\prime }(\varkappa)\neq 0, $ for all $ t\in \lbrack a, b]. $

Lemma 2.1. (^[11], Theorem 5.1) Let $ 0 < \kappa < 1, \; \lambda \in \mathbb{R} $ is a constant, and $ \phi \in L(0, h) $. Then the linear version

has the following solution

Lemma 2.2. ^[1] For $ 0 < \kappa \leq 1 $, the ML function $ E_{\kappa, \kappa }(-\lambda \left(\psi (\varkappa)-\psi (0)\right) ^{\kappa }) $ satisfies

Lemma 2.3. (see ^[11], Lemma 4.2) For $ Re(s) > \left\vert \lambda \right\vert ^{\frac{1}{\kappa -\delta }}, $ we have

Lemma 2.4. ^[22] Assume that $ U $ is an ordered Banach space, $ u_{0}, v_{0}\in U $, $ u_{0}\leq v_{0} $, $ D = [u_{0}, v_{0}] $, $ Q:D\rightarrow U $ is an increasing completely continuous map and $ u_{0}\leq Qu_{0}, \; v_{0}\geq Qv_{0} $. Then, $ Q $ has $ u^{\ast } $ and $ v^{\ast } $ are minimal and maximal fixed point, respectively. If we set

then

Definition 2.2. We say that $ v(\varkappa)\in C_{1-\kappa; \psi }[0, h] $ is a lower solution of (1.6) and (1.7), if it satisfies

Definition 2.3. We say that $ w(\varkappa)\in C_{1-\kappa; \psi }[0, h] $ is an upper solution of (1.6) and (1.7), if it satisfies

Theorem 2.4. ^[11] Let $ 0 < \kappa < 1 $. Then, the generalized Laplace transform of $ \psi $-RL fractional derivative is given by

where

3.   Stability analysis for a linear problem

Here, we discuss the HU and GHU stability of $ \psi $-RL linear problems (1.4) and (1.5), by using the $ \psi $-Laplace transform. Before proceeding to prove the results, we will provide the following auxiliary lemmas:

Lemma 3.1. Let $ 0 < \kappa < 1, $ and $ u\in C_{1-\kappa; \psi }[0, h] $. If

then

Proof. The proof is obtained by the same technique presented in Lemma 3.2, see ^[1], taking into account the properties of the $ \psi $ function.

Lemma 3.2. Let $ 0 < \kappa, \delta < 1, \; \lambda \geq 0 $ is a constant, and $ f : [0, h]\rightarrow \mathbb{R} $ is a continuous function. Then the linear problems (1.4) and (1.5) has the following solution

Proof. Taking the generalized Laplace transform of (1.4) as

Via Theorem 2.4, we have

From (1.5) and Lemma 3.1, we have $ \left. I_{0+}^{1-\kappa; \psi }u(\varkappa)\right\vert _{\varkappa = 0} = \Gamma (\kappa)u_{0}. $ It follows that

One has,

Taking $ \mathscr{L}_{\psi }^{-1} $ to both sides of (3.2), it follow from Lemma 2.3 that

which is (3.1).

Theorem 3.1. Let $ 0 < \kappa, \delta < 1 $, $ \lambda \geq 0 $, and $ f:[0, h]\rightarrow \mathbb{R} $ is a continuous function. If $ u\in C_{1-\kappa; \psi }[0, h] $ satisfies the inequality

for each $ \varkappa \in (0, h] $ and $ \epsilon > 0 $, then there exists a solution $ u_{a}\in C_{1-\kappa; \psi }[0, h] $ of (1.4) such that

Proof. Let

As per (3.3), $ \left\vert \Upsilon (\varkappa)\right\vert \leq \epsilon. $ Taking the $ \psi $-Laplace transform of (3.4) via Theorem 2.4, we have

From (1.5) and Lemma 3.1, $ \left. I_{0+}^{1-\kappa; \psi }u(\varkappa)\right\vert _{\varkappa = 0} = \Gamma (\kappa)u_{0}. $ It follows that

One has,

Set

Taking the Laplace transform of (3.6), It follow from Lemma 2.3 that

Note that

Substituting (3.7) into (3.8), we get

which implies that $ u_{a}(\varkappa) $ is a solution of (1.4) and (1.5) due to $ \mathscr{L}_{\psi } $ is one-to-one. It follow from (3.5) and (3.7) that

which implies

Thus, from (Definition 2.6, ^[21]) and Lemma 2.2, we obtain

Remark 3.1. If $ h < \infty $, then (1.4) is HU stable with the constant $ K: = \frac{ \psi _{\kappa }(h, 0)}{\Gamma (\kappa +1)}. $

Corollary 3.1. On Theorem 3.1, let $ \varphi :[0, \infty)\rightarrow \lbrack 0, \infty) $ is continuous function$. $ If we set $ \varphi (\epsilon) = \frac{\psi _{\kappa }(h, 0)}{\Gamma (\kappa +1)}\epsilon $, which satisfies $ \varphi (0) = 0 $, then (1.4) is GHU stable.

4.   Existence results for a nonlinear problem

In this section, we prove the existence and uniqueness results for $ \psi $ -RL nonlinear FDEs (1.6) and (1.7), by using the LU solution method and the Banach contraction mapping. Moreover, we discuss the maximal and minimal solutions for the problem at hand. The following hypotheses will be used in our forthcoming analysis:

(A1) There exist constants $ A, B\geq 0 $ and $ 0 < s_{1}\leq 1 < s_{2} < 1/(1-\kappa) $ such that for $ \varkappa \in \lbrack 0, h] $,

(A2) $ f:[0, h]\times \mathbb{ \mathbb{R} }\rightarrow \mathbb{ \mathbb{R} } $ satisfies

where $ \lambda \geq 0 $ is a constant and $ \hat{u}, \tilde{u} $ are lower and upper solutions of problems (1.6) and (1.7) respectively.

(A3) There exist constant $ \aleph > 0 $ such that

Remark 4.1. Suppose that $ f(\varkappa, u) = a(\varkappa)g(u) $ with $ g $ is a Hölder continuous and $ a(\varkappa) $ is bounded, then (4.1) holds.

Theorem 4.1. Suppose (A1) holds. The function $ u $ solves problems (1.6) and (1.7) iff it is a fixed-point of the operator $ Q:C_{1-\kappa; \psi }[0, h]\rightarrow C_{1-\kappa; \psi }[0, h] $ defined by

Proof. At first, we show that the operator $ Q $ is well defined. Indeed, for every $ u\in C_{1-\kappa; \psi }[0, h] $ and $ \varkappa > 0 $, the integral

belongs to $ C_{1-\kappa; \psi }[0, h] $, due to

bearing in mind that

is continuous on $ [0, h] $.

By the condition (4.1), we have

where $ C = \max_{\varkappa \in \lbrack 0, h]}f(\varkappa, 0) $.

By Lemma 2.2, for $ u(\varkappa)\in C_{1-\kappa; \psi }[0, h] $, we have

Thus, the integral exists and belongs to $ C_{1-\kappa; \psi }[0, h] $.

The previous inequality and the hypothesis $ 0 < s_{1}\leq 1 < s_{2} < 1/(1-\kappa) $ imply that

Since $ \lim_{\varkappa \rightarrow 0+}E_{\kappa, \kappa }(-\lambda \psi _{\kappa }(\varkappa, 0)) = E_{\kappa, \kappa }(0) = 1/\Gamma (\kappa) $ it follows that

.

The above arguments concerted along with Lemma 2.1 yields that the fixed-point of $ Q $ solves (1.6) and (1.7). And the vice versa. The proof is complete.

Next, we consider the compactness of $ C_{s; \psi }[0, h] $. Let $ F\subset C_{s; \psi }[0, h] $ and $ X = \{g(\varkappa) = \psi _{s}(\varkappa, 0)h(\varkappa)\mid h(\varkappa)\in F\} $, then $ X\subset C[0, h] $. It is obvious that $ F $ is a bounded set of $ C_{s; \psi }[0, h] $ iff $ X $ is a bounded set of $ C[0, h] $.

Thus, to prove that $ F\subset C_{s; \psi }[0, h] $ is a compact set, it is sufficient to show that $ X\subset C[0, h] $ is a bounded and equicontinuous set.

Theorem 4.2. Let $ f:[0, h]\times \mathbb{ \mathbb{R} }\rightarrow \mathbb{ \mathbb{R} } $ is a continuous and (A1) holds. Then $ Q $ is a completely continuous.

Proof. Given $ u_{n}\rightarrow u\in C_{1-\kappa; \psi }[0, h] $, with the definition of $ Q $ and condition (A1), we get

Thus, $ Q $ is continuous.

Assume that $ F\subset C_{1-\kappa; \psi }[0, h] $ is a bounded set. Theorem 4.1 shows that $ Q(F)\subset C_{1-\kappa; \psi }[0, h] $ is bounded.

Finally, we show the equicontinuity of $ Q(F) $. Given $ \epsilon > 0 $, for every $ u\in F $ and $ \varkappa _{1}, \varkappa _{2}\in \lbrack 0, h], \varkappa _{1}\leq \varkappa _{2} $,

As $ E_{\kappa, \kappa }(-\lambda \psi _{\kappa }(\varkappa, 0)) $ is uniformly continuous on $ [0, h] $. Thus

where $ p_{1} = (\kappa -1)s_{1}+1 $ and $ p_{2} = (\kappa -1)s_{2}+1, $

where

as $ \varkappa _{2}\rightarrow \varkappa _{1} $ along with the continuity of $ \psi. $ To summarise,

Thus, $ Q(F) $ is equicontinuous. The proof is complete.

Theorem 4.3. Let $ f:[0, h]\times \mathbb{ \mathbb{R} }\rightarrow \mathbb{ \mathbb{R} } $ is a continuous, (A1) and (A2) hold, and $ v, w\in C_{1-\kappa; \psi }[0, h] $ are lower and upper solutions of (1.6) and (1.7), respectively, such that

Then, the problems (1.6) and (1.7) has $ x^{\ast } $ and $ y^{\ast } $ as minimal and maximal solution, respectively, such that

Proof. Obviously, if functions $ v, w $ are lower and upper solutions of problems (1.6) and (1.7), then there are $ v\leq Qv, $ and $ w\geq Qw $. Indeed, by the definition of the lower solution, there exist $ \underline{q} (\varkappa)\geq 0 $ and $ \epsilon \geq 0 $ such that

Using Theorem 4.1 and Lemma 2.2, we obtain

By the definition of the upper solution, there exist $ \overline{q}(\varkappa)\geq 0 $ such that

Similarly, there is $ w\geq Qw $.

By Theorem 4.2, $ Q:C_{1-\kappa; \psi }[0, h]\rightarrow C_{1-\kappa; \psi }[0, h] $ is increasing and completely continuous. Setting $ D: = [v, w] $, by the use of Lemma 2.4, the existence of $ x^{\ast }, y^{\ast } $ is gotten. The proof is complete.

Theorem 4.4. Let $ f:[0, h]\times \mathbb{ \mathbb{R} }\rightarrow \mathbb{ \mathbb{R} } $ is a continuous and (A3) hold. Then problems (1.6) and (1.7) has a unique solution $ \widetilde{u} $ in the sector $ \lbrack v_{0}, w_{0}] $ on $ [0, h] $, provided

where $ v_{0}, w_{0} $ are lower and upper solutions, respectively, of (1.6) and (1.7) , and $ v_{0}(\varkappa)\leq w_{0}(\varkappa). $

Proof. Let $ \widetilde{u} $ is a solution of (1.6) and (1.7). Then $ v_{0}\leq \widetilde{u}\leq w_{0} $. Consider the operator $ Q:C_{1-\kappa; \psi }[0, h]\rightarrow C_{1-\kappa; \psi }[0, h] $ defined by (4.2). For any $ u_{1}, u_{2}\in C_{1-\kappa; \psi }[0, h] $, we have

From (4.5), we obtain

According to Banach's contraction mapping ^[23], $ Q $ has a unique fixed point, which is unique solution.

5.   Examples

In this part, we provide two examples to illustrate main results.

Example 5.1. Consider the following problem

Here $ \varkappa \in (0, 1] $, $ \lambda = \frac{1}{10}, \; D_{0+}^{\kappa; \psi } $ is the $ \psi $-RL fractional derivative of order $ 0 < \kappa < 1. $ Obviously,

and for $ \varkappa \in \lbrack 0, 1], \; u, \widetilde{u}\in \lbrack 0, \infty), $ we have

for $ 0 < s_{1} = 0.5 < s_{2} = 1.5 < 1/(1-\kappa) = 2, $ for $ \kappa = \frac{1}{2}, $ here $ A = B = \frac{1}{\Gamma (1-\kappa)\sqrt{\pi }}.\ $Moreover, we have

where $ C = \max_{\varkappa \in \lbrack 0, 1]}f(\varkappa, 0) = \frac{1}{\Gamma (1-\kappa)\sqrt{\pi }} $. Also, for $ \varkappa \in \lbrack 0, 1], \; u, \widetilde{u}\in \lbrack 0, \infty ], $ we have

where $ \lambda = \frac{1}{\Gamma (1-\kappa)\sqrt{\pi }} > 0. $ From the foregoing, we conclude that (A1), (A2) and (4.3) are satisfied. Hence, problem (3.4) has a solution on $ [0, 1]. $

Example 5.2. Consider the following problem

Here $ \varkappa \in (0, 1] $, $ \lambda = \frac{1}{10}, \; D_{0+}^{\kappa; \psi } $ is the $ \psi $-RL fractional derivative of order $ 0 < \kappa < 1. $ Obviously,

and for $ \varkappa \in \lbrack 0, 1], \; u, \widetilde{u}\in \lbrack 0, \infty), $we have

for $ s_{2} = 1.5 < 1/(1-\kappa) = 2, $ for $ \kappa = \frac{1}{2}, $ here $ A = 0, \; B = \frac{3}{2}.\ $Take $ v_{0}(\varkappa) \; = 0, \ w_{0}(\varkappa) = \psi _{2}(\varkappa, 0) = \left[ \psi (\varkappa)-\psi (0)\right] ^{2}, $ it is not difficult to verify that $ v_{0}(\varkappa) $, $ w_{0}(\varkappa) $ be lower and upper solutions, respectively, of (5.1), and $ v_{0}(\varkappa)\leq w_{0}(\varkappa) $. Then for $ \varkappa \in \lbrack 0, 1], $

where we used $ \psi (\varkappa) = \varkappa. $ In addition, let $ \epsilon > 0 $, $\underline{ q } (\varkappa) = \frac{\varkappa }{2}, $ and $ \overline{q} (\varkappa) = \varkappa ^{2}, $ and consider

and

By Lemma 4.1, we have

and

Thus, all assumptions of Theorem 4.2 are fulfilled. As per Theorem 4.3, problem (5.1) has minimal and maximal solutions $ u^{\ast }\in \lbrack v_{0}, w_{0}] $, $ \widetilde{u}^{\ast }\in \lbrack v_{0}, w_{0}] $, which can be obtained by

where

and

6.   Conclusions

In the current work, we have investigated two classes of fractional relaxation equations. Our results were based on generalized Laplace transform, fixed point theorem due to lower and upper solutions method, and functional analysis approaches. The $ psi $- RL fractional operator, which is connected with numerous well-known fractional operators, has been used in our study. Ulam-Hyer's stability of solutions for the linear version has been shown by the generalized Laplace transform approach. Then by establishing the method of lower and upper solutions along with Banach's fixed point technique, we have investigated the existence and uniqueness of iterative solutions for the nonlinear version with the non-monotone term $ f(\varkappa, u(\varkappa)) $, which permits the nonlinearity $ f $ to manage the condition (A1) to $ |f(\varkappa, u)|\leq A|u|^{s_{1}}+B|u|^{s_{2}}+C $. Besides, we have also discussed the maximal and minimal solutions to the nonlinear version. Then, some known results in the literature have been extended. Finally, two examples to illustrate the obtained results have been provided.

Acknowledgments

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No. 757].

Conflict of interest

No conflicts of interest are related to this work.