Existence of Solution for Variable-order FDEs with IBCs

Existence of Solution for Variable-order FDEs with IBCs

Existence of Solution for Variable-order FDEs with IBCs

Osama Ahmad and Adnan Ali

vilniaus universitetas

Abstract

This paper presents a comprehensive analytical approach to establish the exis-tence of solutions for variable-order fractional dierential equations (FDEs) with integral boundary conditions (IBCs) in the Caputo derivative sense. The investi-gation delves into the scrutiny of existence and uniqueness of solutions, guided by the specied assumptions (A1)(A4). The theoretical foundation of this investi-gation is rmly rooted in the Banach contraction principle (BCP), the Schauder xed point theorem (SFPT), and the preconditions of the Arzel`a-Ascoli theorem. To substantiate the theoretical ndings, the paper incorporates numerical exam-ples that not only serve as empirical validation but also arm the reliability and robustness of the obtained solutions. Through these examples, the ecacy of the developed methods is vividly demonstrated, further enhancing the credibility of the analytical approach. The utilization of established assumptions and numerical validations ensures a consistent and rigorous investigation into the variable-order FDEs considered in this paper.

Keywords: Fractional dierential equations, Integral boundary conditions, Banach contraction principle, Schauder xed point theorem, Arzela`-Ascoli theorem.

1. Introduction

   The investigation into the emerging discipline of fractional calculus (FC) is framed within the historical development of calculus and its evolutionary path. The study of FC finds its roots in the pioneering works of LHospital and Leibniz, who made significant contributions to the understanding of fractional derivatives [1, 2]. Tradi-tional differential equations of integer-order (IO) have served as fundamental tools for modeling dynamic systems across various disciplines, including physics, biology,

   engineering, and economics [3]. These equations trace their origins to the seminal contributions of luminaries such as Newton and Leibniz.

   Fractional dierential equations have been widely used for modeling a wide range of physical processes and phenomena. They nd applications in areas such as structural probability theory, cell growth, quantum mechanics, linear and nonlinear system dynamics, astrophysics, and electrodynamics [46]. To analyze fractional derivatives, several fractional dierential operators have been developed, including Caputo, Febrizio, Atangana-Baleonu-Caputo, and Riemann-Liouville [79]. FO dier-ential equations are investigated from numerous perspectives, encompassing existence theory and numerical approximation [1620]. Researchers have utilized xed-point theory, particularly the BCP and SFPT, to explore the existence and stability analy-sis of solutions for the dierential equations [21, 22]. These xed-point theorems play a crucial role in examining the existence of solutions [2325].

   Recent research has placed notable emphasis on visualizing solutions of integral fractional boundary value problems (FBVPs) to gain insights into their behavior in physical settings. This visualization aspect is crucial for understanding complex phe-nomena like heat conduction and uid ow. The integration of qualitative analysis, leveraging tools such as fractional Greens functions and topological degree theory, with visualization techniques oers a comprehensive approach to studying integral FBVPs and their applications in real-world scenarios [1012]. Of particular interest are IBCs, which signicantly inuence phenomena such as heat conduction, uid ow, and viscoelasticity. IBCs impose restrictions on physical processes over the entire interval of consideration, providing a more holistic perspective than localized boundary conditions [1315].

   Building on the aforementioned results, the authors investigated a class of panto-graph implicit FO dierential equations with anti-periodic boundary conditions [26]. Specically, the authors focused on studying the equation:

   0

   0

   (1)

   (C D() = f (, (), (), CD()), [0,T ], 2 < 3,

   0

   0

   0

   0

   (0) = (T ), CD(0) = C D(T ), CD(0) = C D(T )

   ×

   where, 0 < < 1, 0 < < 1, 1 < < 2, and f : [0, T ] R3 R is a continuous function. The author investigates the qualitative theory of problem (1) using xed-

   point theory and focusing on Hyers-Ulam types of stability. The study explores the signicance of variable-order operators in FC, an area that has garnered considerable attention [16]. Previous studies have examined boundary value problems (BVPs) related to various forms of FDEs. For instance, in [27], the author established a solution for a nonlinear FDE with order (2, 3] and the author in [28] obtain a solution for implicit fractional-order dierential equations. The study presented in [ 29] addressed implicit FO dierential e quations a nd d iscussed t he e xistence of s olutions f or two-point BVPs involving singular FDEs of variable order. Furthermore, [30] conducted a

   qualitative analysis of problem (2) within the context of variable-order derivatives.

   0+

   (C D()() = f1

   0+

   (, (), I()()), J

   (2)

   (0) = 0, (T ) = 0,

   where, J = [0,T ], 0 < T < , () : J (1, 2] represents the variable order of the fractional derivatives, f1 : J × R × R R is a continuous function.

   The primary contribution in [30] lies in examining the existence of a solution to the variable-order BVP. This is achieved through the transformation of the variable-order BVP into a comparable standard Caputo BVP, which maintains a fractional constant order. The investigation centers on the existence of solutions to the adjusted BVP by utilizing Darbos xed-point theorem and adopting the denition of Ulam-Hyers type stability. In [31], a study on a BVP for Hadamard fractional dierential equations of variable order, focusing on existence criteria and stability, is conducted. The existence-uniqueness solution for a Caputo type variable order fractional dierential equation, presenting new UlamHyers stability results [32]. Furthermore, a nonlinear variable order fractional dierential system incorporating a p-Laplacian operator is formulated in [33]. The investigation concentrates on establishing the existence of solutions and analyzing HyersUlam stability, with an application in a waterborne disease model.

   Motivated by the above studies,the primary aim of this paper is to explore the impact of the variable-order function () on the solutions of FDEs and the impor-tance of solution existence and uniqueness. To address this objective, the problem is formulated as follows:

   0

   (cD()() = f (),

   0

   0

   ()

   (0) = + / 1 (1)1 g(, ())d, 0 < < 1 (3)

   0

   where and

   f () = f (, (),c D()()), g() = g(, ()).

   0

   D

   In equation (3), the functions f (), g() are known functions, while the variable-order fractional derivative operator, denoted as c (), plays a signicant role. The order of the derivative, represented by the function (), is not constant but varies. We have described this function as a mapping: : [0, 1) (0, 1), such that 0 < () < 1 for all [0, 1).

   The paper further endeavors to address a lacuna in existing literature by examin-ing innovative mathematcal methodologies from xed p oint t heory to a scertain the presence of solutions to the BVP outlined in equation (3). The investigation will draw inspiration from the research conducted by [26], which lays the groundwork for explor-ing the existence and uniqueness of solutions for variable-order FDEs characterized by

   IBCs. Mathematical tools including the BCP, SFPT, and Arzela`-Ascoli theorem will be utilized throughout the study.

2. Preliminaries

   Some useful denitions and lemmas will be introduced in this section.

   Denition 1. ([34]) The variable-order RiemannLiouville integral of function ()

   is dened as:

   I

   RL ()

   0,

   () =

   1 r

   ( s)()1(s) ds, > 0, () > 0. (4)

   (()) 0

   Denition 2. ([34]) The variable-order RiemannLiouville derivative of function

   () is dened as:

   RL ()

   1 dn r

   n()1

   D0,

   () = (n ()) dn

   ( s)

   0

   (s) ds, (5)

   where n N such that n> (), and , () > 0.

   Denition 3. ([34]) The variable-order Caputo derivative of () is given as:

   D

   C () 0,

   1

   r

   () =

   (n ()) 0

   ( s)

   n1()

   I(s) ds, > 0, () > 0. (6)

   Denition 4. ([34]) The denitions of variable-order derivatives (5) and (6) are not often equivalent; however, they can be linked by the following relationship [35]:

   n1

   (k)

   k()

   r

   0,

   (1 + k ())

   0,

   RLD()() = X (0) + CD()(), (7)

   k=0

   D

   C () 0,

   1

   () =

   (n ()) 0

   ( s)

   n1()

   I(s) ds, > 0, () > 0. (8)

   Lemma 1. ([19]) Let B(0, 1), then the solution of fractional dierential equation

   c

   D () = 0,

   with order n 1 < n, is given by

   () = c0 + c1 + · · · + cn1n1, i = 0, 1, · · · , n 1.

   Lemma 2. ([19]) Suppose that C(0, 1), with derivative of fractional order , then

   c n1

   I D () = () + c0 + c1 + · · · + cn1 , i = 0, 1, · · · , n 1.

   2.1 Nonlinear analysis

   Example 1. Consider the Banach space CI(P, Q) dened as follows:

   CI(P, Q) = { C(P, Q) : cD1 C(P, Q)},

   where P and Q are suitable intervals, and represents a continuous function from P

   to Q. The norm in this space is dened as:

   c 1

   /I/ICI = max{/I/I, /I D /I}.

   Consider two specic functions 1 and 2 dened within this Banach space as follows:

   1(x) = x2, for x P,

   2(x) = sin(x), for x P.

   It can be veried that both 1 and 2 belong to CI(P, Q), as they are continuous functions and their fractional derivatives exist and are continuous as well.

   Next, lets compute the norms of these functions:

   c 1

   /I1/ICII = max{/I1/I, /I D 1/I} = max{1, 2} = 2,

   c

   1

   /I2/ICII = max{/I2/I, /I D 2/I} = max{1, } = .

   Certainly, 1 and 2 are elements of CI(P, Q), with their norms being nite. Addi-tionally, these norms serve as measures of the maximum absolute value of the functions and their fractional derivatives within this Banach space.

3. Existence and uniqueness

   Lemma 3. Let f : J × R2 R is continuous, then the BVP (3) i-e

   ()

   c

   D0 () = f (), 0 < () < 1, [0, 1),

   0

   (0) = 0 +

   g()d, 0 < < 1,

   ()

   r 1 (1 )1

   0

   (9)

   where and

   has a solution

   f () = f (, (),c D()()), g() = g(, ()),

   () = 0 + r 1 (1 )1 g()d + 1 r ( )()1f ()d. (10)

   0 () (()) 0

   r

   Proof. With the help of the Lemma 1,

   1

   () = c0 +

   ( )()1f ()d, (11)

   (()) 0

   by using the IBCs

   1 r 0 ()1

   (0) = c0 + ( ) f ()d, (12)

   (()) 0

   (0) = 0 +

   1 (1 )1

   r

   g()d, 0 < < 1. (13)

   0 ()

   (0) = c0

   Corollary 1. On the basis of Lemma 3, solution of (3) is given as:

   () = 0 + r 1 (1 )1 g()d + 1 r ( )()1f ()d, (14)

   0 () (()) 0

   Now for the main result, we considered the following assumptions:

   (A1). f : J R2 R is continuous and constant L m > 0 and 0 < L n < 1 with

   | f (, u, v) f (, u¯, v¯) | L m | u u¯ | + L n | v v¯ | J & u, u¯, v, v¯ R.

   (A2). g : J R R is continuous and constant L g > 0 with

   | g(, u) g(, u¯) | L g | u u¯ |, J & u, u¯ R.

   (A3). There exist a non decreasing and continuous function : R+ R+

   and h C(J , R), such that | f (, u, v) |< h()(| V |) for J and u, v Rj

   K

   +

   and a constant > 0 such that {| Y0 h = Sup{h(), J )} and | | .

   (A4). | g(, ()) | K | () | .

   | +(+1)

   h() ((1)+1)

   } where

   0

   Consider the set B = { X | cD() X} and the operator T :

   dene as:

   T () = 0 +

   T () = (),

   r

   1 (1 )1

   g(, ())d

   0

   r

   0 ()

   1

   +

   ( )()1f (, (),c D()())d,

   (()) 0

   c ()

   c ()

   r 1 (1 )1

   D0 T () = D0 {0 +

   0

   g(, ())d

   ()

   + I()f (, (),c D()())d},

   c ()

   0 0

   c ()

   and

   D0 T () = f (, (),

   D0 ()).

   c ()

   c ()

   r 1 (1 )1

   D0 () = D0 {0 +

   1 r

   g(, ())d

   0

   ()

   ()

   +

   (())

   ()

   ( )()1f (, (),c D0 ())d}, (15)

   0

   = cD0 {0 +

   0

   ()

   g(, ())d}

   r 1 (1 )1

   0

   + cD(){

   1 r

   0

   ( )()1f (, (),c D()())d},

   (()) 0

   = 0 +c D()I()f (, (), cD()()),

   c ()

   0 0 0

   c ()

   D0 () = f (, (), D0 ()), (16)

   Now by using (A1) and equation (16), we have

   | f (, (), cD()()) f (, ¯(), cD()¯()) | L m | () ¯()

   0 0

   0

   0

   +L n |c D()() cD()¯() |,

   | f (, (), cD()()) f (¯, ¯(), cD()¯()) | L n |c D()()

   0 0 0

   0

   cD()¯() |

   L m | () ¯() |,

   | f (, (), cD()()) f (¯, ¯(), cD()¯()) | L n | f (, ())cD()()

   0 0 0

   0

   f (¯, ¯()cD()¯() |

   L m | () ¯() |,

   0

   0

   | f (, (), cD()()) f (¯, ¯(), cD()¯()) | (1 L n) L m | () ¯() |,

   0 0

   | f (, (), cD()()) f (¯, ¯(), cD()¯()) | L c | () ¯() |, (17)

   where

   L c =

   L m

   .

   1 L n

   Theorem 1. If the assumptions (A1) and (A2) hold and L b < 1 , then problem (3)

   has a unique solution.

   0

   Proof. Since f is continuous on J = [0, 1] and cD()

   C(J , R). For this let

   (), ¯() B, we have

   /IT () T ¯()/I = max | {0 +

   r 1 (1 )1

   g(, ())d

   J 0

   +

   ( )()1f (, (),c D0 ())d}

   1 r

   ()

   ()

   (())

   0

   {0 +

   g(, ())d

   ()

   r 1 (1 )1 ¯

   0

   +

   1 r

   ( )()1f (, (),c D0

   ())d} |,

   ¯ () ¯

   (())

   0

   =

   0

   () g(, ())d

   0

   g(, ())d

   ()

   r 1 (1 )1 r 1 (1 )1 ¯

   r

   1

   +

   ( )()1f (, (),c D()())d

   (()) 0

   0

   1 r 1

   ()1

   ¯ c () ¯

   (())

   ( )

   0

   f (, (), D0

   ())d,

   /IT () T ()/I = max{

   ()

   | g(, ()) g(, ())) | d

   ¯ r 1 (1 )1 ¯

   J

   0

   0

   r

   1 1

   +

   ( )()1 | f (, (),c D()())

   () 0

   0

   f (, ¯(),c D()¯() | d},

   Using assumption (A1), (A2) and inequality (17), we have

   /IT () T ¯()/I max[

   r 1 (1 )1

   L- g | () ¯() | d

   r

   J 0 ()

   1

   +

   {( )()1L- c | () ¯() | d}],

   (()) 0

   ¯

   r 1 (1 )1

   {L- g | () () |

   +

   L- c | () () |

   1

   (())

   d

   0 ()

   ( )()1d},

   ¯ r

   0

   ¯ (1 )1 1

   {L- g | () () | (

   1 ¯

   () |0)

   ( )()1()

   + (()) L- c | () () | ( () |0)},

   L- g

   () ¯() ( 1 )

   | |

   ()

   (()) J

   1

   + L- c | () ¯() | max(

   ()()

   ()

   ),

   L- g

   () ¯() ( 1 )

   | |

   ( + 1)

   1

   + L- c

   ((1) + 1)

   | () ¯() |,

   | |

   L- g () ¯() ( + 1)

   1

   + L- c

   ((1) + 1)

   | () ¯() |,

   | |

   L- g () ¯() ( + 1)

   1

   + L- c

   ((1) + 1)

   | () ¯() |,

   L- g 1

   ( + 1)

   ((1) + 1)

   c

   ( + L-

   ) | () ¯() |,

   L- g 1

   ( + 1)

   ((1) + 1)

   c

   ( + L-

   ) | () ¯() |, (18)

   1L- n

   (+1)

   (()+1)

   where L- c = L- m and L- b = L- g + L- c . Which proves that T is a contraction

   mapping. So by BCP, problem (3) has a unique solution.

   Theorem 2. Consider (A1)(A1) hold, then BVP (3) has at least one solution.

   Proof. To show that T satises the assumption of SFPT.

   Let us consider a closed convex subset E = { C(J , R) , /I/I }of C(J , R)

   Step 1:

   To show that T is continuous.

   r

   Let us consider a sequence {n}, such that n in C(J , R) and let P > 0 such that for each J , we have

   1 (1 )1

   /IT n() T () = max{

   | g(, n()) g(, ()) | d

   r

   J 0 ()

   1

   +

   ( )()1 | f (, n(),c D()

   n())

   (()) 0

   0

   c ()

   f (, (), D0 () | d}. (19)

   By assumptions (A1), (A2) and inequality (17), we have

   /IT n() T ()/I max{

   ()

   L- g | n() () | d

   r 1 (1 )1

   0

   r

   1

   +

   ( )()1(L- c | n() () |)d},

   (()) 0

   /IT n() T ()/I {L- g | n() () |

   +

   1

   (())

   1 (1 )1

   r

   d

   0 ()

   L- c | n() () |

   r 1

   0

   ( )()1d},

   max{L- g | n() () | (

   (1 )1

   ()

   |0)

   1

   1 ( )()

   + (()) L- c | n() () | ( () |0)},

   1

   L- g | n() () | ( ())

   1

   + (()+ 1) L- c | n() () |,

   L- g 1

   ( ( + 1) + (()+ 1) L- c) | n() () |, (20)

   /IT n T /I 0, this implies that n ,therefore T is a continuous linear

   operator.

   Step 2:

   The image of a bounded set under T is bounded.

   Let E, we have to show that T () E, for each J ,

   J

   r 1 (1 )1

   /IT ()/I max {| 0 +

   0

   1 r

   ()

   g(, ())d

   ()

   +

   0

   (())

   ( )()1f (, (),c D0 ())d | }, (21)

   r 1 (1 )1

   J

   max {| 0 | +

   +

   ( )()1 | f (, (),c D0 ()) | d }, (22)

   1 r

   (())

   0

   0

   ()

   | g(, ()) | d

   ()

   0

   r

   by assumption (A3), let K = / 1 | g(, ()) | d, then we have

   max {| 0 | + K

   1 (1 )1

   d

   J

   +

   1 r

   (())

   0

   0 ()

   ( )()1h() |c D0 ()) | d },

   ()

   1 1 1

   | 0 | + K ( + 1) + ((1)) h /I/I (1) ,

   Where h = Sup{(h(), J )}.

   K

   h ()

   which show that

   Step 3:

   /IT ()/I ( + 1) + ((1) + 1) ,

   /IT ()/I , (23)

   T (E) E.

   To Show that T is equi-continuous on C[J , R].

   Let 1, 2 J , 1 < 2 and E, then

   r

   1 (1 )1

   /IT (2) T (1)/I max {0 +

   | g(, ()) | d

   r

   J

   0 ()

   1 2

   +

   (2 )()1 | f (, (),c D()()) | d

   r

   ((2)) 0

   0

   0

   1 (1 )1

   0

   () | g(, ()) | d

   1 r 1

   ()1

   c ()

   ((1)) 0

   (1 )

   | f (, (), D0 ()) | d },

   (24)

   r

   max

   J

   1 (1 )1

   0 ()

   | g(, ()) | d

   {

   1 (1 )1

   0

   () | g(, ()) | d

   +

   ((2))

   (2 )()1 | f (, (),c D0 ()) | d

   1 r 2

   r

   ()

   0

   1 r 1

   ()1

   c ()

   ((1)) 0

   (1 )

   | f (, (), D0 ()) | d },

   ()

   max { r ( )()1 | f (, (), D ()) | d

   c

   1 2

   2 0

   J (2) 0

   1 r 1

   ()1

   c ()

   (1) 0

   (1 )

   | f (, (), D0 ()) | d },

   1 r 1

   ()1

   c (1)

   { (2) 0

   (2 )()1 | f (, (),c D0 ()) | d

   r 2

   +

   (2 )

   | f (, (), D0 ()) | d

   (1)

   1

   1 r 1

   ()1

   c (1)

   (1)

   (1 )

   0

   2

   0

   c (1)

   | f (, (),

   1 r 1

   D0 ()) | d },

   ()1

   | f (, (),

   D0 () | { ( )

   (2 ) d

   1 r 1

   ()1

   r

   (1)

   (1 ) d

   0

   0

   1 2

   +

   (2 )()1 | f (, (),c D(1)()) | d },

   (2) 1

   c

   | f (, (),

   (1) 1

   2

   2

   D0 () | [ ( )+ 1 {(2 )

   (2) 2

   } |

   0

   1

   (1)

   (1

   )+ 1 {(1 )

   (1)

   } |0

   1

   1

   +

   (2

   )+ 1 {(2 )

   (2)

   } |1 ],

   c

   | f (, (),

   D0 () |

   1

   ((2

   )+ 1)(2)

   (2)

   1(0)

   (2 1)(1 )

   ((1

   )+ 1)

   c

   +

   (1)

   ((2

   ,

   )+ 1)

   (2)(2)

   /IT (2) T (1)/I | f (, (),

   (1)(1)

   D0 () | { ((

   2

   (2 1)(1 )

   ,

   )+ 1)

   ((1

   +

   )+ 1)

   ((2

   )+ 1) }, (25)

   As 2 1, then /IT (2) T (1)/I 0

   Step 1-3 implies that T satisfy all the condition of ArzelaIr Aseoli, Theorem. Hence T is completely continuous.

   So by SFPT T has a xed point E, which is a solution of BVP (3).

4. Examples

   In this section, the validity of the proposed approach in Section 3 is veried via two numerical examples.

   Example 2. Consider a Caputo fractional dierentil equation with boundary conditions as

   c

   2

   0

   D 2 () =

   2( () +

   r (1 )

   13 + |()|

   2

   D ()

   c 2

   0

   2

   0

   17 + |cD 2 ()|

   ), [0, 1)

   (26)

   (0) =0 +

   1 1

   2

   0

   2

   2

   ( 1 )

   ()

   .

   11 + |()|

   2 1

   c 2 ()

   2

   0

   )

   0

   cD 2 ()

   Clearly, () = 2 , = 2

   and f (, (),

   D 2 ()) =

   ( 13+|()| + 2

   11+|()|

   and g(, ()) = () .

   17+|cD02 ()|

   Let , ¯ [J , R], one has

   0

   0

   2 2

   I ()

   2

   c 2

   2

   17 + |cD 2 ()|

   D ()

   13 + |()|

   |f (, (),c D 2 ()) f (, ¯(),c D 2 ¯())| |2|I( + 0

   0

   2

   D

   ¯ c 2 ¯

   ()

   13 + |¯()|

   0 ()

   2

   0

   17 + |cD 2 ¯|

   )I (27)

   2 () ¯()

   2

   c 2

   D0 ()

   2

   0

   cD 2 ¯()

   | |(| 13 + |()| 13 + |¯()| | + |

   2

   0

   D ¯()

   17 + |cD 2 ()|

   2

   0

   17 + |cD 2 ¯()|

   |),

   ()

   | 13

   ¯() c 2

   1

   0

   13 | + | 17

   1

   c 2

   D ()

   0

   17 |,

   1 1 c 1 c 1

   0

   0

   17

   13 |() ¯()| + | D 2 () D 2 ¯()|,

   and

   | | |

   g(, () g(, ¯() ()

   11 + |()|

   ¯()

   11 + |¯()| |,

   ()

   | 11

   1

   ¯()

   11 |,

   ¯

   13

   Here L m = 1 ,

   L n = 1

   11 |() ()|, (28)

   17

   11

   and L g = 1 . Hence,

   2

   L b 11 +

   for any value of [0, 1).

   17

   2

   208 (1 + 2 ) < 1,

   Now, if we choose = 0.4, 0.6, and 0.8, the values of L b are 0.187741, 0.191059, and

   0.193936 respectively. So by Theorem (1), problem (26) has unique solution.

   Example 3. Consider a Caputo fractional dierential equation with boundary conditions as

   c 3

   2 ()

   3

   c

   D0 ()

   0

   r (1 )

   D0 () = ( 15 + |()| + 17 + |cD3 ()|), [0, 1)

   (29)

   (0) = 0 +

   1 1

   2

   0

   3

   ( 1 )

   ()

   .

   13 + |()|

   3 ()

   cD3 ()

   3

   Clearly, () = 3 , = 1

   11+|()|

   and g(, ()) = () .

   Let , ¯ [J , R], then

   and f (, (),c D0

   ()) = 2(

   15+|()| +

   0

   0

   17+|cD3

   )

   ()|

   c 3 ¯ c 3 ¯ 2 ()

   0

   0

   15 + |()|

   |f (, (), D ()) f (, (), D ())| | | I( +

   3

   17 + |cD3 ()|

   c

   D0 ()

   0

   0

   ¯()

   15 + |¯()|

   17 + |cD3 ¯|

   cD3 ¯()

   0

   )I,

   ()

   ¯()

   3

   c

   0

   D ()

   cD3 ¯()

   2

   0

   | |(| 15 + |()| 15 + |¯()| | + | 17 + |cD3 ()| 17 + |cD3 ¯()| |),

   D ()

   D

   ()

   | 15

   ¯() c 1

   0

   15 | + | 17

   0 0

   0

   c 1¯()

   17 |,

   1 ¯ 1 c 1

   c 1 ¯

   15 |() ()| + 17 | D0() D0()|,

   and

   | | |

   g(, () g(, ¯() ()

   13 + |()|

   ¯()

   13 + |¯()| |,

   ()

   | 13

   1

   ¯()

   13 |,

   ¯

   13 |() ()|, (30)

   15

   17

   13

   Here L m = 1 , L n = 1 and L g = 1 . Hence

   1

   3

   L b 13 ( 4 ) +

   17

   240 (1 + 3) < 1,

   for any value of [0, 1).

   Now, if we choose = 0.4, 0.6, and 0.8, the value ofLb are 0.159401, 0.163633, and 0.166029 respectively, which are less than 1. Hence by Theorem (1), problem (26) has unique solution.

5. Conclusion

In this paper, the existence of solutions for variable-order FDEs with IBCs in the Caputo derivative sense is established. Through rigorous analysis guided by specied assumptions (A1A4) and leveraging foundational theorems such as the BCP and the

SFPT, the reliability and robustness of the proposed analytical approach are demon-strated. Numerical examples further validate the ndings and underscore the ecacy of the proposed methods. The work contributes to the advancement of understand-ing in variable-order FDEs and provides insights for both theoretical developments and practical applications. Future research avenues can explore extensions to more complex systems, investigate alternative numerical methods, and explore applications across diverse scientic and engineering disciplines.

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