Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential equations

In this paper, we use some fixed point theorems in Banach space for studying the existence and uniqueness results for Hilfer–Hadamard-type fractional differential equations Dα,βHx(t)+f(t,x(t))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {}_{\mathrm{H}}D^{\alpha ,\beta }x(t)+f\bigl(t,x(t)\bigr)=0 $$\end{document} on the interval (1,e]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1,e]$\end{document} with nonlinear boundary conditions x(1+ϵ)=∑i=1n−2νix(ζi),HD1,1x(e)=∑i=1n−2σiHD1,1x(ζi).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i}),\qquad {}_{\mathrm{H}}D^{1,1}x(e)= \sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}D^{1,1}x( \zeta _{i}). $$\end{document}

The fractional differential equations appear as more appropriate models for describing real world problems. Indeed, these problems cannot be described using the classical integer-order differential equations. In the past years, the theory of fractional differential equations has received much attention from the authors and has become an important field of investigation due to existing applications in engineering, biology, chemistry, economics, and numerous branches of physics [20,27,33,40]. For example, the fractional differential equations are applied to describe the abundant phenomena such as flow in nonlinear electric circuits [15,16,20], properties of viscoelastic and dielectric materials [20,21,32], nonlinear oscillations of an earthquake [28], mechanics [35], aerodynamics, regular variations in thermodynamics [18], etc.
Fractional derivatives can be of several kinds, one of them is the Hadamard fractional derivative innovated by Hadamard in 1892 [17]. It differs from the preceding Riemann-Liouville-and Caputo-type fractional derivatives [33] in the sense that the kernel of the integral contains the logarithmic function of an arbitrary exponent. The properties of Hadamard fractional integral and derivative can be found in [26,27]. Recently, scholars have studied the Hadamard-, Caputo-Hadamard-and Hilfer-Hadamard-type fractional derivatives by using the fixed point theorems with the boundary value problems and have given results of the existence and uniqueness of solutions, see [1-13, 22-25, 30, 31, 34, 36-39, 41, 43-45] and the references mentioned therein.
In this paper, we find a variety of results for the boundary value problem (1.
From this lemma, we notice that if β = 0 then the equation reduces to the equation in Theorem 2.9, and if the β = 1 then the equation reduces to the equation in Theorem 2.10.

Main results
has a unique solution given by Proof In view of Lemma 2.16, the solution of the Hilfer-Hadamard differential equation (3.1) can be written as and The boundary condition where In view of the boundary condition H D 1,1 x(e) = n-2 i=1 σ i H D 1,1 x(ζ i ) and from equations (3.3) and (3.4), we have where By using (3.5) in equation (3.4), we have By substituting the value of c 1 into (3.5), we have Now, substituting the values of c 0 and c 1 in (3.2), we obtain the solution of problem (3.1).
Next, we present the existence and uniqueness of solutions for Hilfer-Hadamard-type fractional differential equation (1.1). For that, suppose that is a Banach space of all continuous functions from [1, e] into R equipped with the norm x = sup t∈J |x(t)|. From Lemma 3.1, we get an operator ρ : K → K defined as It must be noticed that problem ( then the boundary value problem (1.1) has a unique solution on J.
Proof We are using Banach contraction mapping principle to transform the boundary value problem (1.1) into a fixed point problem x = ρx, where the operator ρ is defined by (3.7). We will show that ρ has a fixed point, which is a unique solution of problem (1.1). We put sup |f (τ , 0)| = p < ∞ and choose Now, assume that B r = {x ∈ K : |x| ≤ r}. We will show that ρB r ⊂ B r .
For any x ∈ B r , we have Thus, we have shown that ρB r ⊂ B r . Now, for x, y ∈ K and t ∈ J, we have (ρx)(t) -(ρy)(t) Therefore, it has been shown that (ρx)(t) -(ρy)(t) ≤ C xy , where C < 1. Hence, ρ is a contraction. Thus, by Banach contraction mapping principle, problem (1.1) has a unique solution.

Theorem 3.3 Let f : J × R → R be a continuous function satisfying the assumption
x)f (t, y)| ≤ ϕ(t)(|x -y|/(P * + |x -y|)), t ∈ J, x, y ≥ 0, where ϕ : J → R + is continuous and a constant P * is defined by Then, the boundary value problem (1.1) has a unique solution on J.
Proof We have the operator ρ : K → K defined by (3.7) and by applying Definition 2.11, we can define a continuous nondecreasing function : where the function satisfies (0) = 0 and (φ) < φ for all φ > 0.
For any x, y ∈ K and for each t ∈ J, we have (ρx)(t) -(ρy)(t) 14) which implies that ρxρy ≤ ( xy ). Then, the operator ρ is a nonlinear contraction. Thus, by Lemma 2.12 (Banach contraction mapping principle) the operator ρ has a unique fixed point, which is the unique solution of problem (1.1).
Next, we will give the existence result by using Theorem 2.13 (Krasnoselskii's fixed point theorem). Theorem 3.4 Let f : J × R → R be a continuous function satisfying the assumption (Q 1 ). In addition, assume that then the boundary value problem (1.1) has at least one solution on J.
Proof We put sup t∈J |g(t)| = g and choose a suitable constantr such that r ≥ g , (3.16) where is defined by (3.8). Moreover, we consider the operators F and G on Br = {x ∈ K : x ≤r} defined as For any x, y ∈ Br, we have (3.18) which implies that F x + G x ∈ Br. It follows from assumption (Q 1 ), together with (3.15), that G is a contraction. Furthermore, it is easy to show that the operator F is continuous. Moreover, Hence, F is uniformly bounded on Br.
Next, we prove that the operator F is compact. For that, we put sup (t,x)∈J×Br |f (t, x)| = p < ∞.
Consequently, for t 1 , t 2 ∈ J, we get which is independent of x and tends to zero as t 2 → t 1 . Thus, F is equicontinuous. Hence, F is relatively compact on Br. Therefore, by the Arzelà-Ascoli theorem, F is compact on Br. Thus, by Theorem 2.13, the boundary value problem (1.1) has at least one solution on J.
Now, the final existence result is based on Theorem 2.14 (nonlinear alternative for single-valued maps).
where q ∈ C( [1, e], R + ) is a function; where is defined by (3.8). Then, the boundary value problem (1.1) has at least one solution on J.
Proof We have the operator ρ defined by (3.7). Firstly, we will show that ρ maps bounded sets (balls) into bounded sets in K . For that, letr be a positive number, and Br = {x ∈ K : x ≤r} be a bounded ball in K , where K is defined by (3.6). For t ∈ J, we have which implies that ρx ≤ C 1 . Now, we will show that ρ maps bounded sets into equicontinuous sets of K . For that, let sup (t,x)∈J×Br |f (t, x)| = p < ∞, where ω 1 , ω 2 ∈ J, with ω 1 < ω 2 and x ∈ Br. Hence, we have Clearly, as ω 2 → ω 1 , the right-hand side of the latter inequality tends to zero, which happens independently of x ∈ Br. Thus, by the Arzelà-Ascoli theorem, it follows that ρ : K → K is completely continuous. Finally, let x be a solution. So, for t ∈ J, following similar computations as in the first step, we have Thus, we have In view of (Q 5 ), there exists L such that x = L. Let us set Note that the operator ρ : V → K is continuous and completely continuous. From the choice of V , there is no x ∈ ∂V such that x =λρx for someλ ∈ (0, 1). Thus, by Theorem 2.14, the operator ρ has a fixed point in V , which is a solution of the boundary value problem (1.1).
Hence, L > 1.320578171. Therefore, by Theorem 3.5, the boundary value problem (4.2) has at least one solution on J.