Study of a boundary value problem for fractional order ψ -Hilfer fractional derivative

This manuscript is devoted to the existence theory of a class of random fractional differential equations (RFDEs) involving boundary condition (BCs). Here we take the corresponding derivative of arbitrary order in ψ -Hilfer sense. By utilizing classical ﬁxed point theory and nonlinear analysis we establish some basic results of the qualitative theory such as existence, uniqueness and stability of solutions to the considered boundary value problem of RFDEs. Further, for the justiﬁcation of our analysis we provide two examples.

viscoplasticity), (bio-)chemistry (modelling of polymers and proteins), electrical engineering (transmission of ultrasound waves), medicine (modelling of human tissue under mechanical loads), etc. Since it is clear that dealing Riemann-Liouville derivative in various applied problems is very difficult, therefore certain modifications were introduced to avoid the above-mentioned difficulties. In this regard some new type fractional order derivative operators were introduced in literature like Caputo, Hidamard, etc. Recently, Hilfer [12] initiated an extended Riemann-Liouville fractional derivative, named Hilfer fractional derivative, which interpolates Caputo fractional derivative and Riemann-Liouville fractional derivative. This said operator arose in the theoretical simulation of dielectric relaxation in glass-forming materials. Further Hilfer et al. [13] initially presented linear differential equations with the new Hilfer fractional derivative and applied operational calculus to solve such generalized fractional differential equations.
In last few years, many researchers have studied FDEs using other definitions of fractional derivative like Hilfer, Hadammard, etc, see [1,12]. The mentioned FDEs were studied by many authors in last few years, see [9,11] and the references therein. The authors have mainly considered initial value problems corresponding to Hilfer derivative. But to the best of our knowledge investigation of BVPs corresponding to RFDEs is very rarely considered. The concerned FDEs involving Hilfer derivative have many applications, we refer [26] and the references cited therein. There are actual world occurrences with uncharacteristic dynamics such as atmospheric diffusion of pollution, signals transmissions through strong magnetic fields, the effect of theory on the profitability of stocks in economic markets, network traffic, and so on. The area devoted to RFDEs is a natural extension of such deterministic situations, occurring in several applications and has been examined by many mathematicians, see [1,19,34] and the references therein. The mentioned study has addressed existence and stability of solutions to RFDEs.
On the other hand stability analysis has been establish for various kinds of FDEs. Since the mentioned aspect is very important from numerical and optimization point of view. Different kinds of stability like exponential, Lyapanove and Mittag-Leffler type have been studied very well for functional, integral and differential equations, see [18,29,30]. Another kind of stability which has been given much attention in last few years for FDEs is known as Ulam-Hyers (HU), generalized Ulam-Hyers (g-UH), Ulam-Hyers-Rassias (UHR) and generalized Ulam-Hyers-Rassias (g-UHR) stability, see [22,31,32]. The mentioned stability has been very well studied for FDEs involving Caputo and Riemann-Liouville fractional derivative, see [6,20,35] and the references therein. UH stability concept is relatively important in practical problems in mathematical analysis in different fields such as biology and economics. Thus, the generalized results of the considered stability have been discussed in many books and papers, see [7,14,23,36]. For detailed definitions of the said stability and its generalization, we suggest [10,28,33] about HFDEs. In this work we establish different kinds of the aforesaid stability for the given BVP of RFDEs as: Here D ν,β;ψ represents ψ-HFD and I 1−μ;ψ is ψ-integral of orders 1 − μ(μ = ν + β − νβ). Let R be a Banach space, y : J × × R → R is a given continuous function and where a, b and c are some constants. The detailed study and development of ψ-HFD can be seen in [27]. The equivalent integral equation of Eq.
(1) is set ash where We list some hypotheses to prove our required results.

Preliminaries
Definitions and results to obtain the solution are established in this section, see [2]. Let C be the Banach space of all continuous functionsh : J × → R endowed with Consider weighted space

Definition 2.1 [27] The ψ-fractional integral of a functionh(t) is defined by
Definition 2.2 [27] The ψ-HFD in Riemann-Liouville sense of a functionh with respect to ψ of order ν is defined by where n = [ν] + 1.

Definition 2.3 [27]
The left ψ-Caputo derivative of order ν is given by Definition 2.4 [27] The ψ-HFD of functionh is given by The ψ-HFD can be written in another form as Next, we shall give the definitions g-UHR stable for the problem D ν,β;ψh (t, ρ) = y(t, ρ,h(t, ρ)), t ∈ J.

Theorem 3.2 Under the hypotheses [H2] and if
We can easily prove that the solutions fulfill the conditions of various stabilities like UH, g-UH, UHR and g-UHR stability.

Conclusion
In this research work we have considered a class of RFODEs by taking ψ-HFD under boundary conditions. We have investigated existence theory as well as various kinds of Ulam stability results for the solutions of the considered problem. We claim that such type of BVPs of the aforesaid differential equations have been very rarely studied earlier. The obtained results have been investigated via an example.