A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-di ﬀ erential equation with a nonlocal initial condition. We propose a ﬁ xed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity ﬁ eld of our ﬁ ndings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.


Introduction
Over the last few decades, there has been significant development in the area of ordinary and partial fractional differential equations with boundary integral conditions. This is reflected by various state-of-the-art papers, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Problems with initial integral conditions have many important applications. An example is when a direct measurement quantity is impossible but their mean values are known. The fractional derivatives and integrals appear to be a very efficient tool to model various physical phenomena: kinetic theories, statistical mechanics, dynamics in complex media, control theory, signal processing, bioengineering and biomedical applications, and many others. We refer the reader to [16][17][18].
In an attempt to formulate different problems, distinct definitions of the fractional derivative and fractional integral are available in the literature. We mention some of them.
The relationship between the ψ-Caputo and the ψ-Riemann-Liouville integrals can be written as follows: Lemma 2 [10]. Let f ∈ C n ð½a, bÞ and α > 0. Then, we have In particular, given α ∈ ð0, 1Þ, we have In this paper, we state and prove a weak form of Banach contraction principal. Together with the Schauder's fixed-point theorem, we discuss the existence, uniqueness, Hyers-Ulam-Rassias stability, and regularity of solutions for the following nonlinear fractional mixed Volterra-Fredholm integro-differential equation with nonlocal initial condition: For the sake of simplicity, we denote The main contributions in this paper reside in the following: (i) This work extends, improves, and generalizes several recent state-of-the-art results, including [19][20][21][22][23][24] (ii) Assumptions (A2)-(A6) are not standard; they have been considerably weakened. In fact, they are not supposed to hold on the overall space, but only on a subspace. Example in last section shows the importance of this new form of assumptions (iii) Results in Section 2 are obtained on the basis of a weak form of the Banach contraction principal, which is stated and proved in this paper (iv) We discuss the Hyers-Ulam-Rassias stability of problems (9) and (10)  The remainder of this paper is structured as follows. In Section 2, we briefly recall some basic definitions and some preliminary concepts about fractional calculus and auxiliary results used in the following sections. In Section 3, we discuss the existence and uniqueness of mild solutions for problems (9) and (10), using a weak form of Banach contraction principal and Schauder's fixed-point theorem. Section 4 is devoted to present an Ulam-Hyers-Rassias stability result for problems (9) and (10), and we give a regularity result of solutions for this problem. Finally, we supply in Section 5 a well-suited example to illustrate the application and the validity of our results. In this example, we discuss the existence, uniqueness, Ulam-Hyers-Rassias stability, and regularity of solutions for a problem with nonlocal neutral pantograph equation involving a functional derivative and ψ-Caputo fractional derivative.

Preliminaries
Throughout this paper, we will use the following notations: For the sake of simplicity, we used the same symbol, k·k, for all norms. The closure of a set C will be denoted by C. F , k, h, g are given functions (problems (9) and (10)), and L 2 , N 2 , C 2 , μ denote the constants: ð12Þ For any u ∈ Y, we set: where T n denotes T composed with itself n times. Now, we present some important theorems and lemmas in obtaining the main results.
Theorem 3 (Weak form of the Banach contraction principal). Let ðX, dÞ and ðY, dÞ be two complete metric spaces, C a nonempty subset of X ∩ Y, and T : for all u, v ∈ C. Then, T has a unique fixed point u * in the closure of C. Further, the sequence fT n ug n converges to u * for all u ∈ C.
Proof. For all u, v ∈ C and n ∈ ℕ, using (16) and the triangle inequality, we get on one hand that and on the other that Now, for all u ∈ C and n, m ∈ ℕ, by virtue of inequalities (17) and (18), we obtain This means that fT n ug n is a Cauchy sequence in the complete metric space X. Then, it converges to a point u * of C. Reusing (19), we obtain Letting n, m tend to infinity, taking into account continuity of mapping T, we deduce that u * is a fixed point of 3 Journal of Function Spaces C is a fixed point of T, there exits a sequence fv n g n in C that converges to v * . By (16), we get for all u ∈ C and n ∈ ℕ, Letting n tends to infinity, by the continuity of T in ð C, d C Þ, we deduce that dðTv * , Tu * Þ ≤ λdðv * , u * Þ. But u * , v * are two fixed points of T so v * = u * . This achieves the proof.
Theorem 4 (Schauder's fixed-point theorem). Let F be a closed convex set in a Banach space X and assume that T : F ⟶ F is a continuous mapping such that TðFÞ is a relatively compact subset of F. Then, T has a fixed point.
Theorem 5 (Arzela-Ascoli theorem). Assume that K is a compact set in ℝ n , n ≥ 1. Then, a set S ⊂ CðKÞ is relatively compact in CðKÞ if the functions in S are uniformly bounded and equicontinuous on K.
Lemma 6. Let f : ½a, b → ℝ + be a nondecreasing function and ψ ∈ C 1 ð½a, bÞ an increasing function with ψ ′ ðtÞ ≠ 0 for all t ∈ ½a, b, then (i) Function f is Riemann integrable and bounded (ii) There exist positive constants P 1 and P 2 such that for all t ∈ ½a, b.
Proof. f is a nondecreasing function on ½a, b; then, it is Riemann integrable, and it attains its maximum and minimum at points b and a, respectively. Let t ∈ ½a, b, and clearly, This achieves the proof.
Proof. Let t 1 ∈ ½a, t 0 , and by items (5) and (9), we have Ð t 1 a φ ðξÞdξ ≤ Ð t 0 a φðξÞdξ ≤ P 1 φðt 0 Þ = 0. Then, φðtÞ = 0 a.a. t ∈ ½a, t 1 . So, there exits a sequence ft n g n ⊂ ½a, b converges to t 1 and verifies φðt n Þ = 0, for all n ∈ ℕ. But, by item (10), we deduce that f ðt n Þ ≤ P 2 φðt n Þ. Keeping in mind that f is continuous, it yields f ðt 1 Þ = 0, and this concludes the proof.  (9) and (10) is equivalent to find a fixed point of T, that is, an element u ∈ Y such that (i) If u 0 is a solution of problems (9) and (10), then for all function f ∈ Y, we have Proof. To prove that T is continuous, let u n be a sequence such that u n → u in Y. For each t ∈ I, we have Journal of Function Spaces But F, u, u n , k, h are the continuous functions on the compact I; hence, there exists t 0 ∈ I verifying the following: So, by (26), Therefore, since F : I × Y × X 2 → X is continuous, operator T is continuous. Similarly, we prove that T 0 is continuous.
For point 2, let u be a solution of ψ-fractional integrodifferential problems (9) and (10). Applying the fractional integral operator I α,ψ on both sides of (9), we get On the other hand, keeping in mind initial condition (10), we have by Dirichlet's formula, and it follows that Substituting this latter in (29) and putting Ð λ a gðsÞds = μ, we obtain Conversely, let u in X verifying Tu = u. We have c D α,ψ uð Therefore, Similarly, we get QED. Let u ∈ Y, and the following assumptions are also used: ðA1Þ There exists a positive constant M such that ðA u 2 Þ There exists a positive constant L 1 such that, for all u 1 , u 2 ∈ V u , x 1 , y 1 , x 2 , y 2 ∈ X, and t ∈ I: ðA u 3 Þ There exists a positive constant N 1 such that, for all u 1 , u 2 ∈ V u and t, s ∈ I: ðA u 4 Þ There exists a positive constant C 1 such that, for all u 1 , u 2 ∈ V u and t, s ∈ I: for any t ∈ I, and u 1 , u 2 ∈ V u :Proof. Immediate.

Existence and Uniqueness Results
Our first result is based on the weak form of the Banach contraction principle.
Theorem 10. Let u ∈ Y. If assumptions ðA u 2 Þ − ðA u 5 Þ are satisfied, then the fractional integro-differential problems (9)-(10) have a unique solution u * continuous on I. Furthermore, the sequence fT n ug n converges to u * in Y.
Proof. Let u ∈ Y where assumptions ðA u 2 Þ − ðA u 5 Þ hold. Clearly, TW u ⊂ W u . We shall prove that T is a contraction on V u . For this, let u 1 , u 2 ∈ V u . By (13), we get By means of assumption ðA u 2 Þ, we obtain and by virtue of Lemma 9, it follows that On the other hand, a simple calculation shows that Journal of Function Spaces Substituting estimates (45) and (46) in relation (44), we deduce that or, keeping in mind condition ðA u 5 Þ, operator T is a contraction on V u and consequently continuous on ð W u , d W u Þ. Therefore, by the weak form of the Banach contraction principal, T has a unique fixed point u * = Tu * in the closure of W u , which is a solution of (24) and hence, by item 2 in Lemma 8, a solution of problems (9) and (10). Note that if v * is a solution of (9) and (10), by (9), we get f ða, v * , Kv * ða Þ, Hv * ðaÞÞ = f ða, u * , Ku * ðaÞ, Hu * ðaÞÞ = 0. Assumption (A 6 ) implies that v * ðaÞ = u * ðaÞ; then, and this leads to the uniqueness of solutions. As a recap, (i) Problems (9) and (10) have a unique solution u * (ii) The sequence fT n ug n converges to u * This achieves the proof.
In the next theorem, we shall use Schauder's fixed-point theorem to establish the existence of solutions for problems (9) and (10), with less conditions. Theorem 11. Under assumption (A1), problems (9) and (10) have at least one solution on I.
Proof. Let B r denotes the closed ball in Y of radius r: and defines the operator T on the Banach space Y by Clearly, Tu ∈ B r whenever u ∈ B r , i.e., T : B r → B r , and item 2 in Lemma 8 assures that T is continuous. Now, we shall prove that TB r is an equicontinuous set of Y. Let t 1 , t 2 ∈ I and u ∈ B r , we have Taking into account assumption (A1), we obtain But, Therefore, and consequently, sup u∈B r kTuðt 1 Þ − Tuðt 2 Þk → 0 as t 1 → t 2 , which means that TB r is an equicontinuous set of Y. But T : B r ⊂ CðI, XÞ → B r , so TB r is uniformly bounded, and the Arzela-Ascoli theorem implies that TB r is relatively 7 Journal of Function Spaces compact. Therefore, T : B r → B r is continuous on the closed convex set B r , and TB r is relatively compact. According to the Schauder's fixed-point theorem, mapping T has at least a fixed point. Item 2 in Lemma 8 achieves the proof.

Ulam-Hyers-Rassias Stability
In this section, we discuss the Ulam-Hyers-Rassias stability of problems (9) and (10). First, we introduce a basic definition and some notations and hypotheses for this section.
Definition 12 (Ulam-Hyers-Rassias stability). If for each function u satisfying where ψ is a nonnegative function, there exists a solution u * of the fractional differential problems (9) and (10) and a constant C > 0 independent of u and u * such that for all t; then, we say that problems (9) and (10) are the Hyers-Ulam-Rassias stability.
Let (H) denotes the following hypothesis: We denote by Φ the set of all functions φ ∈ L 1 ðI, ℝ + Þ verifying φ is locally bounded, inf t∈I φðtÞ = 0, and there exist two positive constants P 1 , P 2 such that, for all t ∈ I ð t a φ τ ð Þdτ ≤ P 1 φ t ð Þ, Let φ ∈ Φ, P 3 ∈ ℝ + , and C : I → ℝ + a function. Let u : I → X be a continuously differential function. Assume that assumptions ðA u 2 Þ − ðA u 6 Þ hold together with one of the following: (i) For all t ∈ I we have (ii) For all u 1 , u 2 ∈ V u and t ∈ I we have and we assume that Now, we state and prove the main result of this section. for all t ∈ I. Then, there exists a unique function u * : I → ℝ solution of problems (9) and (10) for all t ∈ I (ii) Using Lemma 6, we can substitute space Φ by the space of nondecreasing functions (iii) The case inf ξ∈I φðξÞ > 0 is obvious, so we have omitted this case, supposing that inf ξ∈I φðξÞ = 0 Proof. For all f 1 , f 2 ∈ Y, we denote by Ω f 1 ,f 2 the set k≤ Cφ t ð Þ, for all t ∈ I f g :