Extended Existence Results For FDEs with Nonlocal Conditions

: This paper discusses the existence of solutions for fractional di ff erential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point 0 ≤ c < d and fixed points at the end of the interval [0 , d ]. For i = 0 , 1, the boundary conditions are as follow a i , b i > 0, a 0 p ( c ) = − b 0 p ( d ) , a 1 p ′ ( c ) = − b 1 p ′ ( d ). Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional di ff erential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we’ve achieved.


Introduction
In this paper, we study the existence of solutions for the following NFDE: where a i , b i ∈ R + for i = 0, 1, c D ζ represents the Caputo derivative of order ζ, and a continuous function q : [0, d] × R −→ R, by using the Krasnoselskii fixed-point theorem and the contraction principle.
Fractional differential equations (FDEs) have loomed as a dominant and masterful mathematical framework.They are a generalization of an integer-order of differential equations, which have comprehensive applications over diverse scientific disciplines.Unlike traditional differential equations, FDEs provide a unified framework to address phenomena characterized by fractional-order dynamics due to their incorporate on of information in a wider range of points in the domain [1,2].
The Riemann-Liouville fractional derivative (RLFD) and Caputo fractional derivatives (CFD) are tools in the realm of fractional calculus.They are used to deal with initial and nonlocal conditions involving information, insuring the description of particular phenomena in a diverse applications [1].
Over the years, several methodologies have been used to solve FDEs, such as Laplace transformation, asymptotic methods, and numerical methods.The investigation of solutions under essential conditions has been explored widely concerning various intriguing topics.For some interesting topics, see [5,6,7].In [5], authors achieved an accurate solution for the analyzed model, a methodology has been proposed that combines the Adomian Decomposition Method with the Laplace Transform, while in [6], researchers solved a fractional order partial differential equation (PDE) using the Laplace residual power series method, which was proven to be effective for numerical solutions.These methods been used widely in single and multi-point initial conditions of fractional differential equations.To the best of our knowledge, very few research papers have delved into the exploration of a nonlocal boundary condition, which presents a promising avenue for further investigation.In [7], Zuo and Wang employed Krasnosel'skii fixed point theorem with Green's function transformation to demonstrate the existence of positive solutions in a fractional differential equation with periodic boundary conditions.
In [12], existence results are presented using Leray-Schauder theory for the following FDE with order 1 For the same order, Bashir and Espinar [13] consider the following FD inclusion: where and obtained the existence results for a given function q : [0, d] × R −→ R using standard fixed-point theorems.
Nonlocal fractional differential equations (NFDEs) serve enchanting and multifaceted applications in physics and mathematical models.These applications and models for characterizing real-world phenomena have been highly influential in seizing the behavior of various systems in engineering, physics, and biology.For some results related to these applications, see [8,9].Researchers are delving deeper into mathematical models with non-integer orders to solve practical problems, fostering the advancement of solutions for NFDEs in an engineering context and promising profound insights into the complex and multifaceted dynamics that underpin engineered systems.Such equations are particularly valuable in scenarios where memory effects or correlations beyond neighboring points significantly impact the system's dynamics, such as in modeling anomalous diffusion or complex transport phenomena, interesting results can be found in [10,11].
The structure of the paper is outlined as follows.Section 2 delves into the materials and methods employed in this study, particularly the Krasnoselskii's fixed point theorem and the necessary assumptions to investigate the existence of the solution.In section 3 , significant theorems that prove the existence of solutions to NFDE (1.1) are stated and demonstrated.Notably, section 4 features examples that serve to illustrate the results.Finally, the conclusion is presented in section 5 .

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Remark.For i = 0, 1, a i = b i = 1, and c = 0 problem (1.3) represents the classical FDE of Lemma 2.5 in [12] .Comparing the solution in [12] and (1.4), we see additional terms are added in (1.4).On the other hand, for i = 0, 1, a i = b i = 1, the solution (1.4) is the same as that of Lemma 2.1 in [14] .

Materials and methods
Numerous approaches are used to prove existence of a solution for FDEs, such as Leray-Schauder's, Krasnosel'skii's, Schaefer's, fixed-point theorems.In this research, we use Krasnoseliskii's fixed-point theorem and the contraction principle.
Additionally, let us assume the following: Let B be the Banach space of all continuous functions, B = C([0, d], R), and define an operator q(r, p(r))dr . (2.1) The solution exists for (1.
Then there exists p in M such that O 1 p + O 2 p = p.

Results
In this sections we will state and prove important theorems that discuss the existing of solutions to NFDEs (1.1).
Proof If p λT p, then for 0 ∈ B r and the unite operator I, we have Hence, for at least one p ∈ B r , g 1 (s) = p − λT p = 0.
For p 1 , p 2 ∈ B r we have, , As s 2 approaches s 1 , the norm will tend to zero.Therefore, we proved that the operator is uniformly bounded on B r and is relatively compact on B r .Theorem 2.1 implies the compactness of T 1 .Finally, T 2 is a contraction mapping for Thus, Theorem 2.2 is satisfied, which implies that FBVP (1.

Conclusions
This paper explored a novel type of FDE.For a point c ∈ [0, d) in problem (1.1), Theorem 3.1 is reduced to Theorem 3.1 in [14].While when c = 0 Theorem 3.2 is an extension of Theorem 3.1 in
Theorem 2.1.[4, Theorem 1.2]If a family Q{q(s)} in C(J, R) is uniformly bounded and equicontinuous on J, and if for any s * ∈ J, {q(s * )} is relatively compact, then, Q has uniformly convergent subsequence {q n } ∞ n=1 .Theorem 2.2.[3, Theorem 1.2.2,Contraction Mapping Theorem] Any contraction mapping of a complete nonempty metric space space Ω into Ω has a unique fixed point in Ω. Banach space B. Suppose that O 1 and O 2 map M into B and, 1) iff T p = p, for p ∈ [0, d].