Discontinuous solutions of delay fractional integral equation via measures of noncompactness

1.   Introduction

This article investigates and examines the presence and then the uniqueness of a.e. nondecreasing solutions to the problem

in weighted Lebesgue space $L_1^N({\mathbb{R}}^+)$, which is a larger space than the classical Lebesgue space $L_1({\mathbb{R}}^+)$. This permits us to concentrate on our aims under more general conditions. To attain these aims, we prove a modified version of Darbo's fixed point principle ^[1] combined with a suitable measure of noncompactness (M.N.C.) in $L_1^N({\mathbb{R}}^+)$. We use the notion of sets are compact in measure to prove that our (M.N.C.) is equivalent to the Hausdorff (M.N.C.). The technique used in this article differs from the ones used in ^[2,3,4], where we dispense the compactness assumptions.

Moreover, we focus on nondecreasing solutions, which don't belong to $L_1({\mathbb{R}}^+)$, so we consider our solutions in the space $L_1^N({\mathbb{R}}^+)$ to bypass these difficulties.

Equation (1.1) represents a generalization and extension of the classical, convolution, and fractional integral equations discussed in the former literature ^{[5,6,7,8,9,10,11]}.

The authors in ^[12] had examined the existence and the uniqueness of a.e. nonincreasing results of some delay-Volterra Hammerstein integral problems

in both $L_1({\mathbb{R}}^+)$ and $L_1^{loc}({\mathbb{R}}^+)$.

Models involving delay integral or differential equations arise in mathematical biology, physics, medicine, and in models of machine operations (see e.g. ^[13,14]).

In ^[15], Cooke and Kaplan created the following model to describe the noticed periodic epidemics of several infectious diseases with periodic contact levels that vary seasonally

which has been also examined in ^[16,17,18].

In ^[19,20] the authors studied equations of the type

using contraction mappings and combining Lyapunov's direct method and Krasnoselskii-type fixed point theorem. Many physical and biological models such as electric, pneumatic, and hydraulic networks (see ^[21,22]) are described by delayed differential or integral equations with discontinuous functions. For example, in ^[23] the authors considered the discontinuity solutions for the delay differential equation

The following Abel integral equation reconstructs the potential $V(z)$ for measurements of the duration of oscillations $T$ of a pendulum,

where $m$ and $E$ denote the particle mass and energy, respectively (cf. ^[5]).

This article is motivated by inspecting and studying the existence and the uniqueness of discontinuous monotonic solutions for a general fractional integral equation in $L_1^N(\mathbb{R^+})$. We give an example to demonstrate the applicability and significance of our theorems.

2.   Preliminaries

Let $\mathbb{R} = (-\infty, \infty), \; \mathbb{R^+} = [0, \infty)$ and the symbols M.N.C. (M.W.N.C.) refer to the measure of noncompactness (weak noncompactness), respectively. Denote by $L_1^N = L_1^N(\mathbb{R^+})$ the weighted Lebesgue space, which is the Banach space of all Lebesgue integrable functions $z$ on $\mathbb{R^+}$ related to the norm

If N = 0 we have classical Lebesgue space $L_1$ with the standard norm.

Now, we need to recall some operators with their properties on $L_1^N$, which will be needed in the sequel.

Definition 2.1 ^[24] Suppose that the function $f(\theta, z) = f:\mathbb{R^+}\times \mathbb{R}\rightarrow \mathbb{R}$ fulfills the Carathéodory conditions, i.e. it is measurable in $\theta$ for any $z\in \mathbb{R}$ and continuous in $z$ for almost all $\theta\in \mathbb{R^+}$. Then, we denote the Nemytskii (Superposition) operator by

Lemma 2.2. ^[4] Suppose that the function $f$ fulfills the Carathéodory conditions and

where $a \in L_1^N$ and $b \geq 0$ for all $\theta \in {\mathbb{R}}^+$ and $z \in {\mathbb{R}}$. Then $F_f: L_1^N \to L_1^N$ is continuous.

Definition 2.3. ^[25,26] Let $z\in L_1$, $\alpha\in \mathbb{R^+}$. The Riemann-Liouville (R-L) fractional integral of function $z$ of order $\alpha$ is defined as:

where $\Gamma(\alpha) = \int_0^\theta e^{-\theta} \theta^{\alpha-1} d\theta$.

Lemma 2.4. ^[4,25] For $\alpha > 0$, we have

(a) the operator $I^\alpha:\; L_1^N \; \to\; L_1^N$ continuously;

(b) $\| I^\alpha z\|_{L_1^N}\leq \frac{1}{N^\alpha} \|z\|_{L_1^N};$

(c) the operator $I^\alpha$ takes a.e. nondecreasing and nonnegative functions into functions have the same properties.

Let $J = [a, b]$ and the symbol $B_r$ points to the closed ball has radius $r$ and center at zero element $\kappa$. Denote by $S = S(J)$ the set of all measurable functions (in Lebesgue sense) on $J$. The functions equal a.e. in the set $S$ corresponding to the metric

construct a complete metric space. Furthermore, the convergence with respect to the metric $d$ is the same as the convergence in measure on $J$ (Proposition 2.14 in ^[27]).

Remark 2.5. Concerning the case of ${\mathbb{R}}^+$, as the measure is $\sigma$-finite, a notion of convergence in finite measure is used and it means, that $(z_n)$ is convergent to $z$ in finite measure iff it converges to $z$ on every set $T \subset {\mathbb{R}}^+$ of finite measure. We will call the compactness in these spaces "compactness in finite measure" ("compactness in measure").

Remark 2.6. Let $Z\subset L_1^N(J)$ be a bounded set. Suppose that there is a family $(\Omega_c)_{0\leq c\leq b-a} \subset J,$ such that meas$\Omega_c = c$ for every $c \in [0, b-a]$, and for every $z \in Z$, $z(t_1) \geq z(t_2),$ $(t_1 \in \Omega_c, t_2 \not \in \Omega_c)$. That family is equimeasurable and the set $Z$ is compact in measure in $L_1^N(J)$. Obviously, by taking $\Omega_c = [0, c) \cup W$ or $\Omega_c = [0, c) \setminus W$, where $W$ is a set of measure zero, such family consists of nondecreasing functions (possibly except for a set $W$). The functions from this family are called "a.e. nondecreasing" functions. It is clear that the same is true for ${\mathbb{R}}^+$.

Remark 2.7. Since $\theta \to e^{-N\theta}$ is nonincreasing on ${\mathbb{R}}^+$ (for $N > 0$), then the pointwise product of this function with monotonic (nondecreasing or nonincreasing) integrable functions do not change their monotonicity properties. Immediately, as in the case of $L_1$ ^[6], we get:

Theorem 2.8. Let $Z\subset L_1^N(J)$ be a bounded set containing functions that are a.e. nonincreasing (or a.e. nondecreasing) on the interval $J$. Then the set $Z$ represents a compact in measure set in $L_1^N(J)$.

Next, we will extend these results from bounded domain $J$ to $\mathbb{R}^+$.

Corollary 2.9. Let $Z\subset L_1^N$ be a bounded set containing functions that are a.e. nondecreasing (or a.e. nonincreasing) on ${\mathbb{R}}^+$. Then the set $Z$ represents a compact in measure set in $L_1^N$.

Proof. Let $L_1^N(T)$ be the space for $\sigma$-finite measure space $T$ and then there exists some equivalent finite measure $\nu$ ($\nu({\mathbb{R}}^+) < \infty$) [27,Corollary 2.20 and Proposition 2.1.].

Therefore, the convergence of sequences in $S$ is equivalent to the metric $d$ and $d_{\nu}(z, y) = \inf_{\rho > 0}[\rho + \nu \{\theta : |z(\theta) - y(\theta)| \geq \rho \}]$ [28,Proposition 2.2]. Let $(z_n) \subset Z$ be an arbitrary bounded sequence.

As a subset of a metric space $Z$ = ($L_1^N({\mathbb{R}}^+), d_{\nu}$) that sequence is compact in this metric space (Theorem 2.8). Then there exists a subsequence $(z_{n_k})$ of $(z_n)$ that is convergent in the space $Z$ to some $z$, i.e.

As said before these two metrics have the same convergent sequences, then

Then, the set $Z$ is compact in finite measure in $L_1^N$. □

Remark 2.10. Let $Q_r$ be the set of all functions $z \in L_1^N$ that is a.e. nondecreasing and a.e. positive on $\mathbb{R^+}$. Then $Q_r$ is closed, nonempty, convex and bounded subset of $L_1^N$, such that $\|z\|_{L_1^N} < r, \; r > 0$. Moreover, the set $Q_r$ is compact in measure (cf. ^[6] and [29,Lemma 4.10]).

Definition 2.11. ^[1] Let $Z \neq \emptyset$ be a bounded subset of a Banach space $E$. The Hausdorff M.N.C. $\chi(Z)$ is given by

Moreover, the De Blasi M.W.N.C. $\beta$ is given by ^[30]:

Theorem 2.12. ^[2,3] Let $\emptyset \neq Z \subset L_1^N$ be a bounded set and $\varepsilon > 0$, we have

and

Then

forms a M.W.N.C. on the space $L_1^N$.

Next, we will demonstrate that M.W.N.C. $\gamma$ and M.N.C. $\chi$ are equivalent, which is important for establishing our main findings.

Theorem 2.13. If $\emptyset \neq Z \subset L_1^N$ is a bounded and compact in measure set, then

Proof. Let $\chi(Z) = r$ and $\varepsilon > 0$ be arbitrary. Then we can obtain a finite set $Y\subset L_1^N$, such that $Z\subset Y+(r+\varepsilon)B_1$. By the properties of $\gamma$, we have

and since $\varepsilon$ is arbitrary, we get

Moreover, let $\emptyset \neq Z \subset L_1^N$ be compact in finite measure. Suppose that $\chi(Z) = r$ and $c(Z) = r_1, \; d(Z) = r_2$, where $r_1 + r_2 = r$. Fix an arbitrary $\eta > 0$. Then for any measurable subset $D \subset [0, T]$, such that $meas\; D < \varepsilon$,

for any $z \in Z$ there exist $T > 0$ and $\varepsilon > 0$, such that

Now, for $z \in Z$ and an arbitrary $h \geq 0$ be arbitrary, we symbolize

Since $Z$ is bounded, we deduce

By this consideration, we can select $h_0\geq 0$, such that

for any $z\in Z$. Then, by using (2.4), we have

for an arbitrary $z \in Z$.

Next, for any $z \in Z$ we denote by $z_{h_0}$ the function

Since $Z$ is compact in finite measure, which indicates that $Z_{h_0} = \{z_{h_0} : z \in Z\}$ is also compact in finite measure. It is clear that $c(Z_{h_0}) = d(Z_{h_0}) = 0$ which indicates that $\gamma (Z_{h_0}) = 0$. Then the set $Z_{h_0}$ is compact in $L_1^N$.

Consequently

Now, applying (2.5) we infer

Moreover, by (2.6) we get

Thus, considering (2.7) we infer

and consequently

Thus

and since $\eta$ is arbitrary, we have

This inequality in conjunction with (2.3) fulfills the proof. □

It allows us to prove the next modified version of the Darbo-type fixed point hypothesis.

Corollary 2.14. Let $\emptyset \neq Q \subset L_1^N$ be a convex, bounded, and closed set. Also, assume $Q$ consists of functions which are a.e. positive and a.e. nondecreasing (or a.e. nonincreasing) on $\mathbb{R}^+$. Suppose $H:Q \to Q$ is a continuous operator and takes a.e. positive and a.e. nondecreasing (or a.e. nonincreasing) functions on $\mathbb{R}^+$ into functions of the same type. Finally, suppose there exists $0\leq k < \frac{1}{2}$ with

for any set $\emptyset \neq Z \subset Q$. Then $H$ has at least one fixed point in $Q$.

Proof. Let $Z$ be a subset of $Q$. Note from Remark 2.10 that $Z$ and $H(Z)$ are compact in measure in $L_1^N$. Then from Theorem 2.13, we have

The above estimation with $0 \leq k < \frac{1}{2}$ completes the proof. □

3.   Main results

In what follows, we will examine the presence and the uniqueness of the solutions for Eq (1.1). Allow us to rewrite (1.1) in the operator form

where $z_\tau(\theta) = z(\theta-\tau), \; \tau < \theta, \; \; I^\alpha$ is defined in Definition 2.3 and $F_{f_i}, i = 1, 2, 3$ are superposition operators as in Definition 2.1. Note that, for any integrable function $z$, a function $z_\tau$ is integrable too.

3.1.   Presence of the solutions

The next presented assumptions are more general than the ones considered earlier, for example, all growth and bound conditions are expressed in terms of functions from $L_1^N$.

(i) Suppose that, $l, h:\mathbb{R^{+}}\rightarrow \mathbb{R}^+$ are a.e. nondecreasing functions and $l$ is a bounded function, such that $\sup_{\theta\in \mathbb{R^{+}}}|l(\theta)|\leq M$ and $h\in L_1^N$.

(ii) Assume that the functions $f_i:\mathbb{R^{+}}\times \mathbb{R}\rightarrow \mathbb{R}, \; i = 1, 2, 3$ fulfill Carathéodory conditions. Moreover, each $f_i (\theta, z)\geq 0$ for a.e. $(\theta, z) \in \mathbb{R^{+}}\times \mathbb{R}$ and $f_i, i = 1, 2, 3$ are supposed to be nondecreasing concerning the two variables $\theta$ and $z$, independently.

(iii) There exist positive functions $a_i\in L_1^N$ and constants $b_i\geq0$, such that

for almost all $\theta\in \mathbb{R^{+}}$ and all $z\in \mathbb{R}.$

(iv) There exists a constant $N > 0$, such that

Theorem 3.1. Suppose assumptions (i)–(iv) are satisfied. Then (1.1) has at least one solution $z \in L_1^N$ that is a.e. nondecreasing on $\mathbb{R^{+}}$.

Proof. From assumptions (ii), (iii), and Lemma 2.2, we have that $F_{f_i}, i = 1, 2, 3$ map $L_1^N$ into itself continuously. Since $I^\alpha$ maps $L_1^N$ into itself and is continuous, then by utilizing assumption (i), we indicate that the operator $H: L_1^N \; \to L_1^N$ and it is continuous. Using (3.1) with assumptions (i)–(iii) and Lemma 2.4, we have for $z \in L_1^N$ that

Therefore,

Put $\theta-\tau = u$ and since $e^{-N(u+\tau)}\leq e^{-Nu}$, we obtain

Thus if $z\in B_r = \{ m\in L_1^N:\, \, \|m\|_{L_1^N} \leq r\}$ ($r$ is given below) we have

where

Thus $H: B_r \to B_r$ is continuous.

Let us denote by $Q_r \subset B_r$ the set of all positive and a.e. nondecreasing functions on $\mathbb{R^+}$. The set $Q_r$ is bounded, nonempty, closed, convex, and compact in measure in regards to Remark 2.10.

Now, we shall demonstrate that $H$ preserves the positivity and the monotonicity of functions. Choose $z\in Q_r$. Then $z(\theta)$ is positive and a.e. nondecreasing on $\mathbb{R^+}$ and thus each $f_i$ is of the same type according to assumption (ii). In addition, $I^\alpha$ is positive and a.e. nondecreasing on $\mathbb{R^+}$. Thus by assumption (i) we infer that $(Hz)$ is positive and a.e. nondecreasing on $\mathbb{R^+}$. Then $H: Q_r\rightarrow Q_r$ is continuous.

In what follows, let us fix a nonempty subset $Z$ of $Q_r$. For $z \in Z$ and fix arbitrary $\varepsilon > 0,$ such that for any $D\subset \mathbb{R^+}$ with $meas(D)\leq\varepsilon$, we have

where the notation $\|\cdot\|_{L_1^N(D)}$ refers to the operator norm which maps the space $L_1^N(D) \to L_1^N(D)$. Since $h, a_i \in L_1^N, \; i = 1, 2, 3$, we have

Thus, by using Definition (2.1), we have

For $T > 0$ and $z\in Z$, we have the following estimate

where the notation $\|\cdot\|_{L_1^N(T)}$ refers to the operator norm which maps the space $L_1^N[T, \infty) \to L_1^N[T, \infty)$. As $T \to \infty$ we get

Joining (3.3) and (3.4), and by recalling Definition (2.2), we have

From assumption (iv) (and the properties of $H$ on $Q_r$) we may apply Corollary 2.14 which fulfills the proof. □

3.2.   Uniqueness of the solutions

Next, we examine the uniqueness of solution for Eq (1.1).

Theorem 3.2. Let assumptions of Theorem 3.1 be fulfilled, but replace (3.2) by the following one:

(v) There exist constants $b_i\geq 0$ and positive functions $a_i\in L_1^N,$ such that

where $Q_r$ is given in Theorem 3.1.

Then (1.1) has a unique integrable solution in the set $Q_r$.

Proof. By using the above suppositions, we get

Then all assumptions of Theorem 3.1 are fulfilled and therefore Eq (1.1) has at least one integrable solution $z\in L_1^N$.

Next, let $z$ and $y$ be any two distinct solutions of Eq (1.1), we have

Put $\theta-\tau = u$ with $e^{-N(u+\tau)}\leq e^{-Nu}$ and since $z(u) = y(u) = z_0$ on $[-\tau, 0)$, we have

From the above inequality with $\left(b_1+\frac{Mb_2b_3}{N^\alpha}\right) < \frac{1}{2}$, we deduce that $z = y$, which completes the proof. □

4.   Example

Next, we give an example to demonstrate the applicability and significance of our theorems.

Example 4.1. Consider the following integral equation

Equation (4.1) is a special case of Eq (1.1), where

and

such that

(1) $a_1(\theta) = \frac{1+\theta^2}{2}, \; a_2(\theta) = e^{\theta}, \; a_3(\theta) = e^{2\theta}, \; b_1 = b_2 = b_3 = \frac{1}{20}$,

(2) $\sup_{\theta\in \mathbb{R^+}}|l(\theta)| = \sup_{\theta\in \mathbb{R^+}}|\sin \theta|\leq1 = M,$

(3) for $N > 0$, we have

By recalling Theorem 3.1, we can indicate that (4.1) has at least one integrable solution a.e. nondecreasing on $\mathbb{\mathbb{R^+}}$.

Moreover, we have

$(1)$ $| f_1(\theta, 0)| = \frac{1+\theta^2}{2}, \; | f_2(\theta, 0)| = e^{\theta}$ and $| f_3(\theta, 0)| = e^{2\theta}$.

$(2)$ $| f_i(\theta, z)-f_i(\theta, y) |\leq\frac{1}{20}|z-y|, \; i = 1, 2, 3$.

By recalling Theorem 3.2, Eq (4.1) has a unique integrable solution $z \in L^N_1$.

5.   Conclusions

We examine the existence and the uniqueness of monotonic solutions of a delay integral equation of fractional order in the weighted Lebesgue space $L_1^N({\mathbb{R}}^+)$, which is a larger space than the classical Lebesgue space $L_1({\mathbb{R}}^+)$. Our analysis uses a suitable measure of noncompactness (M.N.C.), a modified version of Darbo's fixed point theorem, and fractional calculus in the mentioned space. To show the applicability and significance of our outcomes, we present an illustrated example in that space.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.