Abstract
The main intention of this article is that new techniques of existence theory are used to derive some required criteria pertinent to a given fractional multi-term problem and its inclusion version. In such an approach, we do our research on a fractional integral equation corresponding to the mentioned BVPs. In more precise words, by virtue of this integral equation, we construct new operators which belong to a special category of functions named α-admissible and α-ψ-contraction maps coupled with operators having (AEP)-property. Next, by considering some new properties on the existing Banach space having properties (B) and (C_{alpha }), our argument for ensuring the existence of solutions is completed. In addition, we also add two simulative examples to review our findings by a numerical view.
Highlights
1 Introduction As is well known, fractional calculus (FC), for the sake of its higher accuracy than that of the integer one, is an essential topic that is considered as a strong tool in description of natural laws in many branches of science including electrical networks, rheology, biology, dynamical systems, biophysics coupled with many mathematical modelings formulated by a vast diversity of fractional operators; review for details [1,2,3,4,5,6,7,8]
Most of them have focused on applying Caputo, Riemann–Liouville (RL), Hadamard, and many other derivation operators to illustrate the underlying fractional differential equations
We observe different published articles recently in which the existence of solutions is derived for interesting categories of fractional local or nonlocal, multi-term, multi-point, multi-strip, multi-order fractional differential equations; see [9,10,11,12,13,14,15,16,17,18,19,20,21]
Summary
Fractional calculus (FC), for the sake of its higher accuracy than that of the integer one, is an essential topic that is considered as a strong tool in description of natural laws in many branches of science including electrical networks, rheology, biology, dynamical systems, biophysics coupled with many mathematical modelings formulated by a vast diversity of fractional operators; review for details [1,2,3,4,5,6,7,8].Among theoretical concepts and methods, the theory of the existence of solution on the large domain of different fractional constructions including differential equations and inclusions has gained the attention of many mathematicians and relevant researchers. By studying a wide range of published articles pertinent to the existence and uniqueness notions in the context of fractional boundary value problems, we see that many authors usually utilize some standard methods based on the famous fixed point techniques to derive desired results in relation to the existence of solutions. By using these functions on a space having properties (B) and (Cα), we derive the existence results for both suggested BVPs (1) and (2).
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