Discontinuous solutions of delay fractional integral equation via measures of noncompactness

: This article considers the existence and the uniqueness of monotonic solutions of a delay functional integral equation of fractional order in the weighted Lebesgue space L N 1 ( R + ). Our analysis uses a suitable measure of noncompactness, a modiﬁed version of Darbo’s ﬁxed point theorem, and fractional calculus in the mentioned space. An illustrated example to show the applicability and signiﬁcance of our outcomes is included.


Introduction
This article investigates and examines the presence and then the uniqueness of a.e. nondecreasing solutions to the problem Γ(α) f 2 (s, z(s − τ))ds , θ ∈ R + z(θ) = z 0 on [−τ, 0), 0 < α < 1 (1.1) in weighted Lebesgue space L N 1 (R + ), which is a larger space than the classical Lebesgue space L 1 (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 N 1 (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 (R + ), so we consider our solutions in the space L N 1 (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].
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 N 1 (R + ). We give an example to demonstrate the applicability and significance of our theorems.

Preliminaries
Let R = (−∞, ∞), R + = [0, ∞) and the symbols M.N.C. (M.W.N.C.) refer to the measure of noncompactness (weak noncompactness), respectively. Denote by L N 1 = L N 1 (R + ) the weighted Lebesgue space, which is the Banach space of all Lebesgue integrable functions z on 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 N 1 , which will be needed in the sequel.
Definition 2.1. [24] Suppose that the function f (θ, z) = f : R + × R → R fulfills the Carathéodory conditions, i.e. it is measurable in θ for any z ∈ R and continuous in z for almost all θ ∈ 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 ∈ L N 1 and b ≥ 0 for all θ ∈ R + and z ∈ R. Then F f : L N 1 → L N 1 is continuous. Definition 2.3. [25,26] Let z ∈ L 1 , α ∈ R + . The Riemann-Liouville (R-L) fractional integral of function z of order α is defined as: where Γ(α) = θ 0 e −θ θ α−1 dθ. Lemma 2.4. [4,25] For α > 0, we have (a) the operator I α : (c) the operator I α 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 κ. 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 R + , as the measure is σ-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 ⊂ R + of finite measure. We will call the compactness in these spaces "compactness in finite measure" ("compactness in measure").
, and for every z ∈ Z, z(t 1 ) ≥ z(t 2 ), (t 1 ∈ Ω c , t 2 Ω c ). That family is equimeasurable and the set Z is compact in measure in L N 1 (J). Obviously, by taking 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 R + .
Remark 2.7. Since θ → e −Nθ is nonincreasing on 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: 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 N 1 (J). Next, we will extend these results from bounded domain J to R + . Therefore, the convergence of sequences in S is equivalent to the metric d and d ν (z, y) = inf ρ>0 [ρ + ν{θ : |z(θ) − y(θ)| ≥ ρ}] [28, Proposition 2.2]. Let (z n ) ⊂ Z be an arbitrary bounded sequence.
As a subset of a metric space Z = (L N 1 (R + ), d ν ) 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. d ν (z n k , z) k→∞ −→ 0.
As said before these two metrics have the same convergent sequences, then Then, the set Z is compact in finite measure in L N 1 . Remark 2.10. Let Q r be the set of all functions z ∈ L N 1 that is a.e. nondecreasing and a.e. positive on R + . Then Q r is closed, nonempty, convex and bounded subset of L N 1 , such that z L N 1 < r, r > 0. Moreover, the set Q r is compact in measure (cf. [6] and [29,Lemma 4.10]).
Moreover, the De Blasi M.W.N.C. β is given by [30]: Then forms a M.W.N.C. on the space L N 1 . Next, we will demonstrate that M.W.N.C. γ and M.N.C. χ are equivalent, which is important for establishing our main findings.
1 is a bounded and compact in measure set, then Proof. Let χ(Z) = r and ε > 0 be arbitrary. Then we can obtain a finite set Y ⊂ L N 1 , such that Z ⊂ Y + (r + ε)B 1 . By the properties of γ, we have and since ε is arbitrary, we get γ(Z) ≤ 2χ(Z).
Moreover, let ∅ Z ⊂ L N 1 be compact in finite measure. Suppose that χ(Z) = r and c(Z) = r 1 , d(Z) = r 2 , where r 1 + r 2 = r. Fix an arbitrary η > 0. Then for any measurable subset D ⊂ [0, T ], such that meas D < ε, for any z ∈ Z there exist T > 0 and ε > 0, such that Now, for z ∈ Z and an arbitrary h ≥ 0 be arbitrary, we symbolize Since Z is bounded, we deduce By this consideration, we can select h 0 ≥ 0, such that measΩ(z, h 0 ) ≤ ε for any z ∈ Z. Then, by using (2.4), we have for an arbitrary z ∈ Z.
Next, for any z ∈ Z we denote by z h 0 the function for θ ∈ Ω(z, h 0 ).
Since Z is compact in finite measure, which indicates that Moreover, by (2.6) we get Thus, considering (2.7) we infer Thus and since η 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 ∅ Q ⊂ L N 1 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 R + . Suppose H : Q → Q is a continuous operator and takes a.e. positive and a.e. nondecreasing (or a.e. nonincreasing) functions on R + into functions of the same type. Finally, suppose there exists 0 ≤ k < 1 2 with γ(H(Z)) ≤ 2kγ(Z) for any set ∅ Z ⊂ 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 N 1 . Then from Theorem 2.13, we have The above estimation with 0 ≤ k < 1 2 completes the proof.

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 τ (θ) = z(θ − τ), τ < θ, I α 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 τ is integrable too.

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 N 1 . (i) Suppose that, l, h : R + → R + are a.e. nondecreasing functions and l is a bounded function, such that sup θ∈R + |l(θ)| ≤ M and h ∈ L N 1 . (ii) Assume that the functions f i : R + × R → R, i = 1, 2, 3 fulfill Carathéodory conditions. Moreover, each f i (θ, z) ≥ 0 for a.e. (θ, z) ∈ R + × R and f i , i = 1, 2, 3 are supposed to be nondecreasing concerning the two variables θ and z, independently. (iii) There exist positive functions a i ∈ L N 1 and constants b i ≥ 0, such that for almost all θ ∈ R + and all z ∈ 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 ∈ L N 1 that is a.e. nondecreasing on R + .
Proof. From assumptions (ii), (iii), and Lemma 2.2, we have that F f i , i = 1, 2, 3 map L N 1 into itself continuously. Since I α maps L N 1 into itself and is continuous, then by utilizing assumption (i), we indicate that the operator H : L N 1 → L N 1 and it is continuous. Using (3.1) with assumptions (i)-(iii) and Lemma 2.4, we have for z ∈ L N 1 that Therefore, Put θ − τ = u and since e −N(u+τ) ≤ e −Nu , we obtain Thus if z ∈ B r = {m ∈ L N 1 : m L N 1 ≤ r} (r is given below) we have Thus H : B r → B r is continuous. Let us denote by Q r ⊂ B r the set of all positive and a.e. nondecreasing functions on 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 ∈ Q r . Then z(θ) is positive and a.e. nondecreasing on R + and thus each f i is of the same type according to assumption (ii). In addition, I α is positive and a.e. nondecreasing on R + . Thus by assumption (i) we infer that (Hz) is positive and a.e. nondecreasing on R + . Then H : Q r → Q r is continuous.
In what follows, let us fix a nonempty subset Z of Q r . For z ∈ Z and fix arbitrary ε > 0, such that for any D ⊂ R + with meas(D) ≤ ε, we have where the notation · L N 1 (D) refers to the operator norm which maps the space Thus, by using Definition (2.1), we have For T > 0 and z ∈ Z, we have the following estimate where the notation · L N 1 (T ) refers to the operator norm which maps the space From assumption (iv) (and the properties of H on Q r ) we may apply Corollary 2.14 which fulfills the proof.

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 ≥ 0 and positive functions a i ∈ L N 1 , 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 ∈ L N 1 . Next, let z and y be any two distinct solutions of Eq (1.1), we have From the above inequality with b 1 + Mb 2 b 3 N α < 1 2 , we deduce that z = y, which completes the proof.

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