Next Article in Journal
Variable-Order Time-Fractional Kelvin Peridynamics for Rock Steady Creep
Previous Article in Journal
The 3D Multifractal Characteristics of Urban Morphology in Chinese Old Districts
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications

by
Ghadah Albeladi
1,*,
Mohamed Gamal
2,* and
Youssri Hassan Youssri
3
1
Mathematics Department, College of Sciences & Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
2
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
3
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
*
Authors to whom correspondence should be addressed.
Fractal Fract. 2025, 9(3), 196; https://doi.org/10.3390/fractalfract9030196
Submission received: 14 January 2025 / Revised: 7 March 2025 / Accepted: 17 March 2025 / Published: 20 March 2025
(This article belongs to the Section General Mathematics, Analysis)

Abstract

The primary aim of this manuscript is to establish unique fixed point results for a class of Ψ -contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of Ψ -contraction operators, we introduce a novel function, denoted as ψ , and explore its properties. Our work presents new theoretical results, supported by examples and applications, that enrich the study of G-metric spaces. These results not only generalize and unify a broad range of existing findings in the literature but also expand their use to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. In doing so, we offer a more comprehensive understanding of fixed point theory in the G-metric space framework and broaden its scope in applied mathematics. We also offer a numerical spectral approach for solving fractional initial value problems, utilizing shifted Chebyshev polynomials to construct a semi-analytic solution that inherently satisfies the given homogeneous initial conditions.

1. Mathematical Introduction

Let F be a self-operator on a set X . A fixed point ( FP ) in X is defined as an element q such that F q = q , where F is an operator on X . The FP Theorem argues that under specific conditions (on the operator F and the space X ), an operator F of X into itself admits one or more FP . There are many results on different cases of the FP Theorem. The basic foundations upon which FP theory studies are built include the Banach contraction principle ( BCP ) [1], Brouwer’s FP theorem [2], Schauder’s FP theorem [3], and the contraction operators theorem.
The FP theorem, commonly recognized as the Banach contraction principle, first appeared in definitive form in Banach’s thesis in 1922 [1], where it was used to establish the existence of a solution to an integral equation ( IE ). Since then, its simplicity and usefulness have made it become a widely used tool for solving problems in many mathematical analysis branches. This principle remarks that if ( X , δ ) is a complete metric space and F : X X is a contraction operator (i.e., δ ( F ϖ , F ϱ ) λ δ ( ϖ , ϱ ) for all ϖ , w X , where λ ( 0 , 1 ) is a constant), then F has a FP .
The BCP has been generalized in various ways over the years. Some generalizations, such as those in [4,5] and others, relax the contractive conditions of the operator, while others, including [6,7] and others, weaken the topological assumptions. In [8], Nadler extended the Banach FP theorem from single-valued operators to set-valued contractive operators. Additional FP results for set-valued operators can be found in [9] and the references therein.
Mustafa [10] introduced an extension of metric spaces known as generalized metric spaces (or G-metric spaces) and established FP results for Banach-type contraction operators within this framework. This concept was further explored by Mustafa and Sims [11]. Subsequently, Mustafa et al. [12] established FP results for Lipschitzian-type operators in G-metric spaces, which garnered significant interest from researchers in FP theory. However, Jleli and Samet [13] and Samet et al. [14] later pointed out that many FP results in G-metric spaces are essentially direct consequences of corresponding results in metric spaces. Specifically, Jleli and Samet [13] observed that if a G-metric can be reduced to a quasi-metric, the associated FP results align with known results in quasi-metric spaces. Numerous FP results in such spaces have subsequently appeared (see, for example, [15,16]).
Integral equations ( IE s ) appear in a wide range of applications [17]. Depending on the field, one may encounter Fredholm integro-differential equations ( FIDE s ) [18], Volterra integral equations ( VIE s ) [19], or Fredholm integral equations ( FIE s ) [20]. The latter can be further divided into linear and nonlinear FIE s of the first or second kind. Although they have become a recent research focus, linear FIE s of the first kind are less commonly known due to their frequent ill-posedness. They are typically studied only within a finite interval [21].
Furthermore, most IE s that are closely related to differential equations are FIE s [22]. These equations are typically derived from boundary value problems ( BVP s ) for differential equations and are often solved using various simplified methods. Fredholm’s work on these equations was driven by a strong motivation to explore and understand this class of equations. Consequently, they were named FIE s in his honor. This discovery marked a significant step forward in overcoming key challenges that had hindered the progress of mathematics [23]. It is also worth noting that FIE s can appear in both linear and nonlinear forms, including homogeneous and non-homogeneous types [24].
Fractional calculus has almost three centuries of history. L’Hospital is credited with introducing the idea of the fractional derivative when he posed the query regarding the derivative of order 1 / 2 . Because fractional derivatives have so many uses in biology, economics, science, and engineering, academics have recently focused more on them [25,26,27].
There are now definitions of fractional derivatives that are both local and nonlocal. The history of the function is crucial for many of these applications, hence nonlocal derivatives are more interesting. Singular kernels are used to define some fractional derivatives, including the Riemann–Liouville, Grünwald, and Caputo definitions [28], as well as numerous others found in the literature [29,30]. Caputo–Fabrizio and Atangana–Baleanu fractional derivatives are two examples of more recent formulations of fractional derivatives that are based on nonsingular kernels [31,32].
Because some models of dissipative processes cannot be sufficiently characterized by standard fractional derivatives, fractional derivatives with nonsingular kernels are important. The Hadamard fractional integral and derivative is one of the additional definitions of fractional integrals and derivatives that are available. The Riemann–Liouville or Caputo integral and the Hadamard integral differ primarily in the type of kernel that is employed. Hadamard [33] introduced the Hadamard integral, which incorporates a logarithmic function, whereas the Riemann–Liouville integral utilizes a power function.
Thus, mathematical models based on the Caputo fractional derivative ( CFD ) extend those relying on classical derivatives, offering improved descriptions of certain real-world phenomena. Specifically, BVP s for systems of differential equations with CFD can model the behavior of a damping system in mechanics [34], pollution levels in lakes connected by rivers [35], the spread of COVID-19 [36], global population growth, and tape counter readings at a specific time [37], among others.
The study of G-metric space is considered an extension and generalization of the classical concept of metric space. G-metric space has been presented for analyzing distance and convergence in many mathematical structures. Metric space measures the distance between two points but G-metric space measures the distance between three points that provides us a wide framework for reading and studying geometry of spaces. Also, G-metric space focuses on the notions of compactness, sequence convergence, and completeness. The main motivations for studying G-metric space are its applications in many branches of analysis and applied mathematics such as topology, functional analysis, fixed point theory, differential equations, FDE s , and IE s .
We present a fixed point solution for Ψ -contraction operators in complete G-metric spaces. By generalizing and extending existing fixed point theorems, we establish a new function, denoted as ψ , and investigate its unique features. Our findings not only consolidate and build on prior discoveries, but also offer novel applications to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. Furthermore, we describe a numerical spectral method to fractional initial value problems that employ shifted Chebyshev polynomials, providing an efficient semi-analytic solution that is inherently homogeneous under initial conditions.
Now, we can summarize the content of this manuscript as follows:
Section 1 introduces the definitions and basic concepts that lay the foundation for understanding general metric spaces. Section 2 presents several theorems that generalize the theorems given in [10,38], which prove the existence and uniqueness of FP s. Section 3 explores the unique solutions of BVP s . Section 4 considers applications for boundary value problems. Section 5 focuses on methods for finding solutions to the FIE s in G-metric spaces, and Section 6 discusses the existence and uniqueness of solutions for FDE s in G-metric spaces. Finally, in Section 7, we present a numerical solution for fractional initial value problems using the spectral Chebyshev method, proposing a basis that inherently satisfies the homogeneous initial conditions.

2. Basic Facts

This section introduces the fundamental concepts necessary to achieve our objectives. We present the key definitions and results related to G-metric spaces, which will form the basis for the subsequent sections of the paper.
Definition 1
([11]). Suppose that X is a nonempty set. Let G : X × X × X R + be a function satisfying the following conditions:
(1) 
G ( ϖ , ϱ , κ ) = 0 if and only if ϖ = ϱ = κ ;
(2) 
G ( ϖ , ϖ , ϱ ) G ( ϖ , ϱ , κ ) for all ϖ , ϱ , κ X with ϱ κ ;
(3) 
G ( ϖ , ϱ , κ ) = G ( ϖ , κ , ϱ ) = G ( ϱ , κ , ϖ ) = (symmetry in all three variables);
(4) 
G ( ϖ , ϱ , κ ) G ( ϖ , a , a ) + G ( a , ϱ , κ ) , for all ϖ , ϱ , κ , a X .
Then G is called a G-metric on X and ( X , G ) is called a G-metric space.
Geometric understanding of G ( ϖ , ϱ , κ ) : (See Figure 1) Instead of requesting information on the distance between two points as in a conventional metric space, a G-metric requests information on the ’triangle like interaction’ among three points at once. You can think of this as a measure of how ‘spread out’ three points are in the space.
If G ( ϖ , ϱ , κ ) is small, then ϖ , ϱ , and κ are close.
If G ( ϖ , ϱ , κ ) is large, then at least one of the points is far from the others.
The function G ( ϖ , ϖ , ϱ ) can be seen as a replacement for the usual metric d ( ϖ , ϱ ) , but the value depends on an additional reference point.
Envisioning G-Metric Geometry
Triangle Representation: G ( ϖ , ϱ , κ ) can be viewed as a measurement that establishes the “spread” of a triangle in space since it depends on three points at a time.
Deformation of Distances: Because G-metric spaces depend on a third point, they provide flexibility in measuring distances, unlike Euclidean spaces, where the distances between two points are fixed.
Curvature-Like Behavior: According to some interpretations, the G-metrics can capture curvature-like phenomena, in which the placement of extra points affects distances.
Figure 1. Geometric meaning of G-metric space.
Figure 1. Geometric meaning of G-metric space.
Fractalfract 09 00196 g001
Example 1.
Let X = [ 0 , + ) and G : X × X × X R + such that G ( ϖ , κ , ϱ ) = sup { ϖ , κ , ϱ } for all ϖ , κ , ϱ X . Then, ( X , G ) is a G-metric space.
Example 2.
Let X = [ 0 , + ) and G : X × X × X R + such that G ( ϖ , κ , ϱ ) = sup { d ( ϖ , κ ) , d ( κ , ϱ ) , d ( ϱ , ϖ ) } for all ϖ , κ , ϱ X , where d is a standard metric on X . Then, ( X , G ) is a G-metric space.
Definition 2
([11]). A G-metric space ( X , G ) is said to be symmetric if G ( ϖ , ϱ , ϱ ) = G ( ϱ , ϖ , ϖ ) for all ϖ , ϱ X .
The simplest example of a nonsymmetric G-metric, which also happens to be one that does not originate from any metric in the methods mentioned above, is shown in the example that follows.
Example 3
([11]). Suppose that X = { ϖ , ϱ } . Assume G ( ϖ , ϖ , ϖ ) = G ( ϱ , ϱ , ϱ ) = 0 , G ( ϖ , ϖ , ϱ ) = 1 , and G ( ϖ , ϱ , ϱ ) = 2 and extend G to all of X × X × X using the symmetry in the variables. Then, it is easily verified that G is G-metric, but G ( ϖ , ϱ , ϱ ) G ( ϖ , ϖ , ϱ ) .
Example 4
([39]). Assume that X = [ a , b ] R , where 1 < a < b , and G : X × X × X R + is given by the following:
G ( ϖ , ϱ , κ ) = sup { ϖ ϱ , ϱ κ , κ ϖ } .
Then, ( X , G ) is G-metric space.
Proposition 1
([11]). Suppose that ( X , G ) is a G-metric space; then, for any ϖ , ϱ , κ , a X it follows that
(1) 
if G ( ϖ , ϱ , κ ) = 0 , then ϖ = ϱ = κ ,
(2) 
G ( ϖ , ϱ , κ ) G ( ϖ , ϖ , ϱ ) + G ( ϖ , ϖ , κ ) ,
(3) 
G ( ϖ , ϱ , κ ) G ( ϖ , a , κ ) + G ( a , ϱ , κ ) .
Definition 3
([11]). Suppose that ( X , G ) is a G-metric space, and let { ϖ n } be a sequence of points of X . We say that { ϖ n } is
(1) 
a G-Cauchy sequence if, for any ε > 0 , there is N N such that for all n , m , l N , G ( ϖ n , ϖ m , ϖ l ) < ε ;
(2) 
a G-convergent sequence to ϖ X if, for any ε > 0 , there is N N such that for all n , m N , G ( ϖ , ϖ n , ϖ m ) < ε .
Definition 4
([11]). Suppose that ( X , G ) and ( X * , G * ) are G-metric spaces. Then, a function f : X X * is G-continuous at a point ϖ X if and only if it is G-sequentially continuous at ϖ; that is, whenever a sequence { ϖ n } is G-convergent to ϖ, we have that { f ( ϖ n ) } is G-convergent to f ( ϖ ) .
Definition 5
([11]). A G-metric space ( X , G ) is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X .
Proposition 2
([11]). Let ( X , G ) be a G-metric space. Define on X the metric δ G with δ G ( ϖ , ϱ ) = G ( ϖ , ϱ , ϱ ) + G ( ϱ , ϖ , ϖ ) , for all ϖ , ϱ X . Then, for a sequence { ϖ n } X , the following are equivalent:
(1) 
{ ϖ n } is G-convergent to ϖ;
(2) 
G ( ϖ n , ϖ n , ϖ ) 0 as n + ;
(3) 
δ G ( ϖ n , ϖ ) = 0 ;
(4) 
G ( ϖ n , ϖ , ϖ ) 0 as n + ;
(5) 
G ( ϖ n , ϖ m , ϖ ) 0 as n , m + .
Proposition 3
([11]). Let ( X , G ) be a G-metric space. Then, the following are equivalent:
(1) 
the sequence { ϖ n } is G-Cauchy;
(2) 
G ( ϖ n , ϖ m , ϖ m ) 0 as n , m + .
Proposition 4
([11]). A G-metric space ( X , G ) is G-complete if and only if ( X , δ G ) is complete metric space.
Definition 6.
Assume that Ψ is the family of all functions ψ : [ 0 , + ) [ 0 , + ) fulfilling the conditions below:
(1) 
ψ ( 0 ) = 0 ;
(2) 
lim n + ψ n ( ϖ ) = 0 for ϖ > 0 , where ψ n stands for the n-th iterate of ψ;
(3) 
ψ ( ϖ + ϱ ) ψ ( ϖ ) + ψ ( ϱ ) ;
(4) 
ψ ( ϖ ϱ ) ψ ( ϖ ) ψ ( ϱ ) .
The following example supports Definition 6:
Example 5.
Let ψ : R + R + be defined by ψ ( ϖ ) = ϖ exp ( ϖ ) , which satisfies the four conditions in the family Ψ. Let us check each condition one by one.
(1) 
If ϖ = 0 , then we have ψ ( 0 ) = 0 · exp ( 0 ) = 0 . Thus, ψ ( 0 ) = 0 , so condition (1) is satisfied.
(2) 
lim n + ψ n ( ϖ ) = lim n + ϖ n exp ( n ϖ ) = 0 . Thus, condition (2) is satisfied.
(3) 
For every ϖ , ϱ R + and n N , we have
ψ ( ϖ + ϱ ) = ( ϖ + ϱ ) exp ( ( ϖ + ϱ ) ) = ϖ exp ( ( ϖ + ϱ ) ) + ϱ exp ( ( ϖ + ϱ ) ) ϖ exp ( ϖ ) + ϱ exp ( ϱ ) = ψ ( ϖ ) + ψ ( ϱ ) .
Thus, condition (3) is satisfied.
(4) 
For every ϖ , ϱ R + , we have
ψ ( ϖ ϱ ) = ( ϖ ϱ ) exp ( ( ϖ ϱ ) ) = ϖ exp ( ( ϖ ϱ ) ) ϱ exp ( ( ϖ ϱ ) ) exp ( ϖ ϱ ) ϖ exp ( ϖ ) ϱ exp ( ϱ ) = ψ ( ϖ ) ψ ( ϱ ) .
Thus, condition (4) is satisfied.
Lemma 1.
Let ψ < I , where I is the identity, then ( I ψ ) has an inverse where
( I ψ ) 1 = n = 0 + ψ n .
Proof. 
Suppose that the function ψ : [ 0 , + ) [ 0 , + ) satisfies ψ < I . We want to show that
I ψ 1 = n = 0 + ψ n .
The partial sum S M is as follows:
S M = n = 0 M ψ n .
Using S M , we determine the product
I ψ S M = I ψ n = 0 M ψ n = n = 0 M ψ n n = 0 M ψ n + 1 .
Note that
n = 0 M ψ n n = 0 M ψ n + 1 = ψ 0 ψ M + 1 = I ψ M + 1 .
Condition (2) in Definition 6 states that for each ϖ > 0 , we have
lim M + ψ M + 1 ( ϖ ) = 0 .
Consequently, if we take M + , we obtain
I ψ n = 0 + ψ n = I .
Hence, n = 0 + ψ n operates as a right-inverse of I ψ and converges (in the pointwise sense on [ 0 , + ) ). Similarly, one can confirm that
n = 0 + ψ n ( I ψ ) = I ,
Thus, I ψ is the two-sided inverse of the series. Therefore, ( I ψ ) is invertible and
I ψ 1 = n = 0 + ψ n .
Lemma 2.
For the following BVP
d 2 ϖ d t 2 = λ ( t , ϖ ( t ) ) , t [ 0 , 1 ] , ϖ R + , ϖ ( 0 ) = ϖ ( 1 ) = 0 ,
the solution can be expressed as follows:
ϖ ( t ) = 0 1 Ω ( t , s ) λ ( s , ϖ ( s ) ) d s ,
where λ : [ 0 , 1 ] × R + R + is a continuous function and Ω ( t , s ) is a Green function defined by
Ω ( t , s ) = t ( 1 s ) , if 0 t s 1 , s ( 1 t ) , if 0 s t 1 .
Proof. 
Integrating Equation (1) with respect to t from 0 to t two times yields
ϖ ( r ) ϖ ( 0 ) = 0 r ϖ ( s ) d s = 0 r λ ( s , ϖ ( s ) ) d s ,
and
0 t ϖ ( r ) ϖ ( 0 ) d r = 0 t ϖ ( r ) d r 0 t ϖ ( 0 ) d r = ϖ ( t ) ϖ ( 0 ) t ϖ ( 0 ) ,
Then,
ϖ ( t ) = ϖ ( 0 ) + t ϖ ( 0 ) + 0 t 0 r λ ( s , ϖ ( s ) ) d s d r = 0 + t ϖ ( 0 ) + 0 t λ ( s , ϖ ( s ) ) s t d r d s = t ϖ ( 0 ) + 0 t λ ( s , ϖ ( s ) ) ( t s ) d s = t ϖ ( 0 ) + 0 t ( t s ) λ ( s , ϖ ( s ) ) d s .
To determine ϖ ( 0 ) , we use the condition at t = 1 , ϖ ( 1 ) = 0 ,
ϖ ( 1 ) = 0 = ϖ ( 0 ) + 0 1 ( 1 s ) λ ( s , ϖ ( s ) ) d s .
Therefore,
ϖ ( 0 ) = 0 1 ( 1 s ) λ ( s , ϖ ( s ) ) d s .
Thus, Equation (2) can be written as
ϖ ( t ) = t 0 1 ( 1 s ) λ ( s , ϖ ( s ) ) d s + 0 t ( t s ) λ ( s , ϖ ( s ) ) d s = 0 t ( t s ) t ( 1 s ) λ ( s , ϖ ( s ) ) d s t 1 t ( 1 s ) λ ( s , ϖ ( s ) ) d s = 0 t t s t + t s ) λ ( s , ϖ ( s ) ) d s t 1 t ( 1 s ) λ ( s , ϖ ( s ) ) d s = 0 t t s + s ) λ ( s , ϖ ( s ) ) d s t 1 t ( 1 s ) λ ( s , ϖ ( s ) ) d s = 0 t s 1 t λ ( s , ϖ ( s ) ) d s t 1 t ( 1 s ) λ ( s , ϖ ( s ) ) d s = 0 1 Ω ( t , s ) λ ( s , ϖ ( s ) ) d s ,
in which the kernel is given by
Ω ( t , s ) = t ( 1 s ) , if 0 t s 1 , s ( 1 t ) , if 0 s t 1 .
Definition 7
([30]). Let [ a , b ] ( < a < b < + ) be a finite interval on the real axis R . The Riemann–Liouville fractional integrals I a + α f and I b α of order α C ( R ( α ) > 0 ) are defined by
( I a + α f ) ( s ) : = 1 Γ ( α ) a s f ( t ) d t ( s t ) 1 α s > a ; R ( α ) > 0 ,
and
( I b α f ) ( s ) : = 1 Γ ( α ) s b f ( t ) d t ( t s ) 1 α s < b ; R ( α ) > 0 ,
respectively. Here, Γ ( α ) is the Gamma function. These integrals are called the left-sided and the right-sided fractional integrals. When α = n N , the definitions (3) and (4) coincide with the nth integrals of the form
I a + n f ( s ) = a s d t 1 a t 1 d t 2 a t n 1 f ( t n ) d t n = 1 ( n 1 ) ! a s ( s t ) n 1 f ( t ) d t ( n N ) ,
and
I b n f ( s ) = s b d t 1 t 1 b d t 2 t n 1 b f ( t n ) d t n = 1 ( n 1 ) ! s b ( t s ) n 1 f ( t ) d t ( n N ) .
The Riemann–Liouville fractional derivatives D a + α ϖ and D b α ϖ of order α C ( R ( α ) 0 ) are defined by
D a + α ϖ : = d d s n I a + n α ϖ ( s ) = 1 Γ ( n α ) d d s n a s ϖ ( t ) d t ( s t ) α n + 1 n = [ R ( α ) ] + 1 ; s > a ,
and
D b α ϖ : = d d s n I b n α ϖ ( s ) = 1 Γ ( n α ) d d s n s b ϖ ( t ) d t ( t s ) α n + 1 n = [ R ( α ) ] + 1 ; s < b ,
respectively, where [ R ( α ) ] means the integral part of R ( α ) .
Definition 8
([30]). Suppose that [ a , b ] is a finite interval of the real line R , and let D a + α ϖ ( t ) ( s ) D a + α ϖ ( s ) and D b α ϖ ( t ) ( s ) D b α ϖ ( s ) is the Riemann–Liouville fractional derivatives of order α C ( R ( α ) 0 ) defined by (7) and (8), respectively. The fractional derivatives ( C D a + α ϖ ) ( s ) and ( C D b α ϖ ) ( s ) of order α C ( R ( α ) 0 ) on [ a , b ] are defined via the above Riemann–Liouville fractional derivatives by
D a + α C ϖ ( s ) : = D a + α ϖ ( t ) k = 0 n 1 ϖ ( k ) ( a ) k ! ( t a ) k ( s ) ,
and
D b α C ϖ ( s ) : = D b α ϖ ( t ) k = 0 n 1 ϖ ( k ) ( b ) k ! ( b t ) k ( s ) ,
respectively, where
n = [ R ( α ) ] + 1 f o r α N 0 ; n = α f o r α N 0 .
These derivatives are called the left-sided and right-sided Caputo fractional derivatives of order α.
Lemma 3.
The following nonlinear fractional differential equation ( FDE ) of Caputo type
D 0 + μ C ϖ ( t ) = f ( t , ϖ ( t ) ) , t ( 0 , 1 ) , 1 < μ 2 , ϖ ( 0 ) = 0 , ϖ ( 1 ) = 0 m ϖ ( s ) d s , 0 < m < 1 ,
where D 0 + μ C represents the ( CFD ) of order μ and f : [ 0 , 1 ] × R R is a continuous function. This nonlinear FDE can also be represented as follows:
ϖ ( t ) = 1 Γ ( μ ) 0 t ( t s ) μ 1 f ( s , ϖ ( s ) ) d s 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s .
A function ϖ ( t ) is a solution of the defined nonlinear FDE whenever it is the solution of the FDE (9) and vice-versa.
Proof. 
Applying the R-L fractional integral I 0 + μ to both sides to the equation D 0 + μ C ϖ ( t ) = f ( t , ϖ ( t ) ) , we have
ϖ ( t ) = 1 Γ ( μ ) 0 t ( t s ) μ 1 f ( s , ϖ ( s ) ) d s + C 1 t + C 0 ,
where C 0 , C 1 are arbitrary constants. From the condition ϖ ( 0 ) = 0 we find C 0 = 0 .
Then, in view of the condition ϖ ( 1 ) = 0 μ ϖ ( s ) d s , we have
1 Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s + C 1 = 1 Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s + C 1 0 m s d s C 1 C 1 m 2 2 = 1 Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s 1 Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s C 1 1 m 2 2 = 1 Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s + 1 Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s C 1 = 2 ( 2 m 2 ) [ 1 Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s + 1 Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s ]
substituting the value of C 1 in (10), we have
ϖ ( t ) = 1 Γ ( μ ) 0 t ( t s ) μ 1 f ( s , ϖ ( s ) ) d s 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s .
Example 6.
(numerical example) Let the following be true:
  • f ( t , ϖ ( t ) ) = t ϖ ( t ) + sin ( t ) ,
  • μ = 1.5 (order of the fractional derivative, 1 < μ 2 ),
  • m = 0.5 (satisfying 0 < m < 1 ).
Using the representation in Equation (10), the solution is expressed as follows:
ϖ ( t ) = 1 Γ ( 1.5 ) 0 t ( t s ) 0.5 [ s ϖ ( s ) + sin ( s ) ] d s 2 t ( 2 0 . 5 2 ) Γ ( 1.5 ) 0 1 ( 1 s ) 0.5 [ s ϖ ( s ) + sin ( s ) ] d s + 2 t ( 2 0 . 5 2 ) Γ ( 1.5 ) 0 0.5 0 s ( s τ ) 0.5 [ τ ϖ ( τ ) + sin ( τ ) ] d τ d s .
Here,
Γ ( 1.5 ) = π 2 .
In [10], Mustafa described the well-known Banach contraction operator principle within the framework of G-metric spaces as follows:
Theorem 1
([10]). Suppose that ( X , G ) is a complete G-metric space. Let F : X X be an operator satisfying the following condition for all ϖ , ϱ , κ X :
G ( F ϖ , F ϱ , F κ ) σ G ( ϖ , ϱ , κ ) ,
where σ [ 0 , 1 ) . Then, F has a unique FP .
Theorem 2
([10]). Suppose that ( X , G ) is a G-metric space. Let F : X X be an operator satisfying the following condition for all ϖ , ϱ X :
G ( F ϖ , F ϱ , F ϱ ) σ G ( ϖ , ϱ , ϱ ) ,
where σ [ 0 , 1 ) . Then, F has a unique FP .
Theorem 3
([38]). Suppose that ( X , G ) is a complete G-metric space. Let F : X X be an operator satisfying the following condition for all ϖ , ϱ , κ X :
G ( F ϖ , F ϱ , F ϱ ) σ G ( ϖ , F ϖ , F ϖ ) + G ( ϱ , F ϱ , F ϱ ) + G ( κ , F κ , F κ ) ,
where σ [ 0 , 1 3 ) . Then, F has a unique FP .

3. Main Results

This section consists of an overview of fundamental principles that play a role in achieving our main objectives. Now, we generalize, unify, and extend Mustafa’s [10] results.
Theorem 4.
Suppose that ( X , G ) is a G-complete G-metric space. Let F : X X and G : X × X × X R + be operators satisfying
G ( F ϖ , F ϖ , F ϱ ) ψ G ( ϖ , ϖ , ϱ ) , f o r a l l ϖ , ϱ X ,
where ψ Ψ and ψ < I . Then, F has a unique FP in X .
Proof. 
Let ϖ 0 be an arbitrary element in X . Define a sequence { ϖ n } by F 2 n + 1 ϖ 0 = F ϖ 2 n = ϖ 2 n + 1 , n 1 , then we obtain
G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) = G ( F ϖ 2 n + 1 , F ϖ 2 n + 1 , F ϖ 2 n ) ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) = ψ G ( F ϖ 2 n , F ϖ 2 n , F ϖ 2 n 1 ) ψ 2 G ( ϖ 2 n , ϖ 2 n , ϖ 2 n 1 ) ψ 2 n + 1 G ( ϖ 1 , ϖ 1 , ϖ 0 ) .
Then, for m , n N with n > m , we find that
G ( ϖ n , ϖ n , ϖ m ) G ( ϖ n , ϖ n , ϖ n 1 ) + G ( ϖ n 1 , ϖ n 1 , ϖ n 2 ) + + G ( ϖ m + 1 , ϖ m + 1 , ϖ m ) ψ n 1 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + ψ n 2 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + + ψ m G ( ϖ 1 , ϖ 1 , ϖ 0 ) k = m n 1 ψ k G ( ϖ 1 , ϖ 1 , ϖ 0 ) 0 , for all G ( ϖ 1 , ϖ 1 , ϖ 0 ) > 0 , n , m + .
Hence, { ϖ n } is a G-Cauchyness in X . Based on G-completeness of X , there exists ϖ * X such that lim n + ϖ n = ϖ * . Therefore, one has
0 G ( ϖ * , F ϖ * , F ϖ * ) G ( ϖ * , F ϖ n , F ϖ n ) + G ( F ϖ n , F ϖ * , F ϖ * ) G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) + ψ G ( ϖ n , ϖ * , ϖ * ) .
It follows that
0 G ( ϖ * , ϖ * , F ϖ * ) G ( ϖ * , ϖ * , ϖ * ) + ψ G ( ϖ * , ϖ * , ϖ * ) 0 + ψ ( 0 ) .
Thus, G ( ϖ * , ϖ * , F ϖ * ) 0 , which is a contradiction. Then, G ( ϖ * , ϖ * , F ϖ * ) = 0 , i.e., ϖ * = F ϖ * is a FP of F . Now, if ϱ * ϖ * is another FP of the operator F , then
0 G ( ϖ * , ϖ * , ϱ * ) = G ( F ϖ * , F ϖ * , F ϱ * ) ψ G ( ϖ * , ϖ * , ϱ * ) ,
that is,
( I ψ ) G ( ϖ * , ϖ * , ϱ * ) 0 .
Since ( I ψ ) is invertible, then
( I ψ ) 1 ( I ψ ) G ( ϖ * , ϖ * , ϱ * ) ( I ψ ) 1 ( 0 ) I G ( ϖ * , ϖ * , ϱ * ) n = 0 + ψ n ( 0 ) G ( ϖ * , ϖ * , ϱ * ) 0 .
Again, we obtain a contradiction. Therefore,
G ( ϖ * , ϖ * , ϱ * ) = 0 ϖ * = ϱ * .
This implies that the FP is unique. □
The following example affirms Theorem 4:
Example 7.
Let X = R and the metric G : X × X × X R + be defined by
G ( ϖ , ϱ , κ ) : = sup { ϖ ϱ , ϱ κ , κ ϖ } .
Then, ( X , G ) is a G-complete G-metric space.
To verify that the given function G ( ϖ , ϱ , κ ) is a valid G-metric, we need to check that it satisfies the four conditions required for a G-metric.
(1) 
⇒: If ϖ = ϱ = κ , then | ϖ ϱ | = | ϱ κ | = | κ ϖ | = 0 . Therefore, G ( ϖ , ϱ , κ ) = sup { 0 , 0 , 0 } = 0 .
⇐: If G ( ϖ , ϱ , κ ) = 0 , then sup { | ϖ ϱ | , | ϱ κ | , | κ ϖ | } = 0 , which implies that | ϖ ϱ | = | ϱ κ | = | κ ϖ | = 0 . Hence, ϖ = ϱ = κ .
(2) 
Since G ( ϖ , ϖ , ϱ ) = | ϖ ϱ | and G ( ϖ , ϱ , κ ) = sup { | ϖ ϱ | , | ϱ κ | , | κ ϖ | } .
Clearly, | ϖ ϱ | sup { | ϖ ϱ | , | ϱ κ | , | κ ϖ | } = G ( ϖ , ϱ , κ ) .
(3) 
The function G ( ϖ , ϱ , κ ) = sup { | ϖ ϱ | , | ϱ κ | , | κ ϖ | } is clearly symmetric because the supremum is taken over all the pairwise differences and reordering these differences does not change the result.
(4) 
We need to check if the inequality holds for the given function:
G ( ϖ , ϱ , κ ) = sup { | ϖ ϱ | , | ϱ κ | , | κ ϖ | } ,
and
G ( ϖ , a , a ) = sup { | ϖ a | , | a a | , | a ϖ | } = | ϖ a | ,
G ( a , ϱ , κ ) = sup { | a ϱ | , | ϱ κ | , | κ a | } .
Now,
G ( ϖ , ϱ , κ ) G ( ϖ , a , a ) + G ( a , ϱ , κ ) ,
translates to
sup { | ϖ ϱ | , | ϱ κ | , | κ ϖ | } | ϖ a | + sup { | a ϱ | , | ϱ κ | , | κ a | } .
Since the supremum represents the largest of the distances between the points, and the triangle inequality holds in R , this inequality is valid.
To prove that the G-metric space ( X , G ) is G-complete,we need to show that every G-Cauchy sequence in this space is G-convergent to some point in R .
Let { ϖ n } be a G-Cauchy sequence in the given G-metric space. Then for every ε > 0 there exists an integer N N such that for all n , m , l N ,
G ( ϖ n , ϖ m , ϖ l ) < ε .
That is
G ( ϖ n , ϖ m , ϖ l ) = sup { | ϖ n ϖ m | , | ϖ m ϖ l | , | ϖ l ϖ n | } < ε .
In particular,
| ϖ n ϖ m | < ε and | ϖ m ϖ l | < ε and | ϖ l ϖ n | < ε .
Therefore, for all n , m N , we have | ϖ n ϖ m | < ε . This shows that a G-Cauchy sequence in the G-metric space is also a Cauchy sequence in the usual sense of the real numbers. Since R with the usual metric is a complete metric space, every Cauchy sequence in R converges to some limit in R . Therefore, the sequence { ϖ n } converges to some point ϖ R in the usual metric, i.e., for every ε > 0 , there exists N N such that for all n N ,
| ϖ n ϖ | < ε .
Next, we show that the sequence { ϖ n } also G-converges to ϖ in the G-metric sense. That is, for every ε > 0 , there exists N N such that for all n , m N ,
G ( ϖ , ϖ n , ϖ m ) < ε .
Using the definition of the G-metric, we need to show that
G ( ϖ , ϖ n , ϖ m ) = sup { | ϖ ϖ n | , | ϖ n ϖ m | , | ϖ m ϖ | } < ε .
Since { ϖ n } converges to ϖ in the usual sense, for any ε > 0 , there exists N N such that for all n N , | ϖ n ϖ | < ε 2 . Therefore, for all n , m N ,
| ϖ ϖ n | < ε 2 , | ϖ n ϖ m | < ε 2 , | ϖ m ϖ | < ε 2 .
Thus, for all n , m N ,
G ( ϖ , ϖ n , ϖ m ) = sup { | ϖ ϖ n | , | ϖ n ϖ m | , | ϖ m ϖ | } < sup ε 2 , ε 2 , ε 2 = ε 2 < ε .
Therefore, G ( ϖ , ϖ n , ϖ m ) < ε , proving that { ϖ n } is G-convergent to ϖ in the G-metric space. Since every G-Cauchy sequence in ( X , G ) is also G-convergent to some point in X , the space ( X , G ) is G-complete.
Now, define the operator F : X X by
F ϖ : = ϖ 4 ,
and ψ : R + R + by
ψ ( t ) : = t exp t .
Then, we obtain
G ( F ϖ , F ϱ , F κ ) = sup { F ϖ F ϱ , F ϱ F κ , F κ F ϖ } = 1 4 sup { ϖ ϱ , ϱ κ , κ ϖ } = 1 4 G ( ϖ , ϱ , κ ) G ( ϖ , ϱ , κ ) exp G ( ϖ , ϱ , κ ) = ψ G ( ϖ , ϱ , κ ) .
Therefore, all hypotheses of Theorem 4 are satisfied and 0 is a unique FP of F .
Theorem 5.
Let ( X , G ) be a G-complete G-metric space. Assume that F : X X and G : X × X × X R + are operators satisfying
G ( F ϖ , F ϖ , F ϱ ) ψ G ( ϖ , ϖ , ϱ ) + G ( ϖ , F ϖ , F ϖ ) + G ( ϱ , F ϱ , F ϱ ) , f o r a l l ϖ , ϱ X ,
where ψ Ψ and ψ < I . Then, F has a unique FP in X .
Proof. 
Let ϖ 0 be an arbitrary element in X . Define a sequence { ϖ n } by F 2 n + 2 ϖ 0 = F ϖ 2 n + 1 = ϖ 2 n + 2 , n 1 .
G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) = G ( F ϖ 2 n + 1 , F ϖ 2 n + 1 , F ϖ 2 n ) ψ ( G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + G ( ϖ 2 n + 1 , F ϖ 2 n + 1 , F ϖ 2 n + 1 ) + G ( ϖ 2 n , F ϖ 2 n , F ϖ 2 n ) ) = ψ ( G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + G ( ϖ 2 n + 1 , ϖ 2 n + 2 , ϖ 2 n + 2 ) + G ( ϖ 2 n , ϖ 2 n + 1 , ϖ 2 n + 1 ) ) ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + ψ G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) + ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) .
It follows that
( I ψ ) G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) 2 ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) .
By multiplying ( I ψ ) 1 , we have
G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) 2 ( I ψ ) 1 ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) 2 ( I ψ ) 1 ψ 2 n + 1 G ( ϖ 1 , ϖ 1 , ϖ 0 ) .
Thus, for n , m N with n > m , we obtain
G ( ϖ n , ϖ n , ϖ m ) G ( ϖ n , ϖ n , ϖ n 1 ) + G ( ϖ n 1 , ϖ n 1 , ϖ n 2 ) + + G ( ϖ m + 1 , ϖ m + 1 , ϖ m ) 2 ( I ψ ) 1 ψ n 1 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + 2 ( I ψ ) 1 ψ n 2 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + + 2 ( I ψ ) 1 ψ m G ( ϖ 1 , ϖ 1 , ϖ 0 ) k = m n 1 2 ( I ψ ) 1 ψ k G ( ϖ 1 , ϖ 1 , ϖ 0 ) 0 , for all G ( ϖ 1 , ϖ 1 , ϖ 0 ) > 0 , n , m + .
Hence, { ϖ n } is a G-Cauchyness in X . Based on G-completeness of X , there exists ϖ * X such that lim n + ϖ n = ϖ * . Therefore, we obtain
0 G ( ϖ * , F ϖ * , F ϖ * ) G ( ϖ * , F ϖ n , F ϖ n ) + G ( F ϖ n , F ϖ * , F ϖ * ) = G ( ϖ * , F ϖ n , F ϖ n ) + G ( F ϖ * , F ϖ * , F ϖ n ) G ( ϖ * , F ϖ n , F ϖ n ) + ψ ( G ( ϖ * , ϖ * , ϖ n ) + G ( ϖ * , F ϖ * , F ϖ * ) + G ( ϖ n , F ϖ n , F ϖ n ) ) G ( ϖ * , F ϖ n , F ϖ n ) + ψ G ( ϖ * , ϖ * , ϖ n ) + ψ G ( ϖ * , F ϖ * , F ϖ * ) + ψ G ( ϖ n , F ϖ n , F ϖ n ) = G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) + ψ G ( ϖ * , ϖ * , ϖ n ) + ψ G ( ϖ * , F ϖ * , F ϖ * ) + ψ G ( ϖ n , ϖ n + 1 , ϖ n + 1 ) .
Taking n + , we obtain
G ( ϖ * , F ϖ * , F ϖ * ) G ( ϖ * , ϖ * , ϖ * ) + ψ G ( ϖ * , ϖ * , ϖ * ) + ψ G ( ϖ * , F ϖ * , F ϖ * ) + ψ G ( ϖ * , ϖ * , ϖ * ) 0 + 0 + ψ G ( ϖ * , F ϖ * , F ϖ * ) + 0 ,
that is,
( I ψ ) G ( ϖ * , F ϖ * , F ϖ * ) 0 .
Since ( I ψ ) is invertible, then
( I ψ ) 1 ( I ψ ) G ( ϖ * , F ϖ * , F ϖ * ) ( I ψ ) 1 ( 0 ) .
Thus, G ( ϖ * , F ϖ * , F ϖ * ) 0 , which is a contradiction. Then, G ( ϖ * , F ϖ * , F ϖ * ) = 0 , i.e., ϖ * = F ϖ * is a FP of F . Now, if ϱ * ϖ * is another FP of the operator F , then
0 G ( ϖ * , ϖ * , ϱ * ) = G ( F ϖ * , F ϖ * , F ϱ * ) ψ G ( ϖ * , ϖ * , ϱ * ) + G ( ϖ * , F ϖ * , F ϖ * ) + G ( ϱ * , F ϱ * , F ϱ * ) ψ G ( ϖ * , ϖ * , ϱ * ) + ψ G ( ϖ * , ϖ * , ϖ * ) + ψ G ( ϱ * , ϱ * , ϱ * ) ,
that is,
( I ψ ) G ( ϖ * , ϖ * , ϱ * ) 0 .
Since ψ ( ϖ ) = 0 if and only if ϖ = 0 and I ( ϖ ) ψ ( ϖ ) . Then
G ( ϖ * , ϖ * , ϱ * ) = 0 ϖ * = ϱ * .
This implies that the FP is unique. □
Corollary 1.
Let ( X , G ) be a G-complete G-metric space, F : X X and G : X × X × X R + be operators satisfying
G ( F ϖ , F ϖ , F ϱ ) ψ G ( F ϖ , F ϖ , ϖ ) + G ( F ϱ , F ϱ , ϱ ) , for all ϖ , ϱ X ,
where ψ Ψ and ψ < I . Then, F has a unique FP in X .
The following example affirms Corollary 1:
Example 8.
Consider all hypotheses of Example 7 are true, we conclude that
G ( F ϖ , F ϖ , ϖ ) + G ( F ϱ , F ϱ , ϱ ) = sup { F ϖ F ϖ , F ϖ ϖ , ϖ F ϖ } + sup { F ϱ F ϱ , F ϱ ϱ , ϱ F ϱ } = 3 4 ϖ + 3 4 ϱ = 3 4 ϖ + ϱ 3 4 ϖ ϱ = 3 ϖ 4 ϱ 4 = 3 F ϖ F ϱ = 3 sup { F ϖ F ϖ , F ϖ F ϱ , F ϱ F ϖ } = 3 G ( F ϖ , F ϖ , F ϱ ) ,
which yields,
G ( F ϖ , F ϖ , F ϱ ) 1 3 G ( F ϖ , F ϖ , ϖ ) + G ( F ϱ , F ϱ , ϱ ) G ( F ϖ , F ϖ , ϖ ) + G ( F ϱ , F ϱ , ϱ ) exp G ( F ϖ , F ϖ , ϖ ) + G ( F ϱ , F ϱ , ϱ ) .
It follows that
G ( F ϖ , F ϖ , F ϱ ) ψ G ( F ϖ , F ϖ , ϖ ) + G ( F ϱ , F ϱ , ϱ ) .
Therefore, all conditions of Corollary 1 are verified and 0 is a unique FP of F .
The example below supports Theorem 5:
Example 9.
Considering that all hypotheses of Example 7 are true, we conclude that
G ( F ϖ , F ϖ , F ϱ ) = sup { F ϖ F ϖ , F ϖ F ϱ , F ϱ F ϖ } = 1 4 ϖ ϱ = 1 4 ϖ F ϖ + F ϖ ϱ 1 4 ϖ F ϖ + F ϖ ϱ = 1 4 ϖ F ϖ + 1 4 F ϖ ϱ = 1 4 ϖ F ϖ + 1 4 F ϖ F ϱ + F ϱ ϱ 1 4 ϖ F ϖ + 1 4 F ϖ F ϱ + 1 4 F ϱ ϱ = 1 4 sup { ϖ F ϖ , F ϖ F ϖ , F ϖ ϖ } + 1 4 sup { F ϖ F ϖ , F ϖ F ϱ , F ϱ F ϖ } + 1 4 sup { ϱ F ϱ , F ϱ F ϱ , F ϱ ϱ } = 1 4 G ( ϖ , F ϖ , F ϖ ) + G ( F ϖ , F ϖ , F ϱ ) + G ( ϱ , F ϱ , F ϱ ) G ( ϖ , ϖ , ϱ ) + G ( ϖ , F ϖ , F ϖ ) + G ( ϱ , F ϱ , F ϱ ) exp G ( ϖ , ϖ , ϱ ) + G ( ϖ , F ϖ , F ϖ ) + G ( ϱ , F ϱ , F ϱ ) .
Then,
G ( F ϖ , F ϖ , F ϱ ) ψ G ( ϖ , ϖ , ϱ ) + G ( ϖ , F ϖ , F ϖ ) + G ( ϱ , F ϱ , F ϱ ) .
Therefore, all conditions of Theorem 5 are verified and 0 is a unique FP of F .
Theorem 6.
Let ( X , G ) be a G-complete G-metric space. Assume that F : X X and G : X × X × X R + are operators verifying
G ( F ϖ , F ϖ , F ϱ ) ψ G ( ϖ , ϖ , ϱ ) + G ( ϖ , F ϖ , F ϖ ) . G ( ϱ , F ϱ , F ϱ ) 1 + G ( ϖ , ϖ , ϱ ) , for all ϖ , ϱ X ,
where ψ Ψ and ψ < I . Then, F has a unique FP in X .
Proof. 
Let ϖ 0 be an arbitrary element in X . Define a sequence { ϖ n } by
F 2 n + 2 ϖ 0 = F ϖ 2 n + 1 = ϖ 2 n + 2 , n 1 ;
then, we have
G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) = G ( F ϖ 2 n + 1 , F ϖ 2 n + 1 , F ϖ 2 n ) ψ ( G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + G ( ϖ 2 n + 1 , F ϖ 2 n + 1 , F ϖ 2 n + 1 ) . G ( ϖ 2 n , F ϖ 2 n , F ϖ 2 n ) 1 + G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) ) = ψ ( G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) 1 + G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) ) ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + ψ G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) .
Then,
( I ψ ) G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) .
By multiplying ( I ψ ) 1 , we have
G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) ( I ψ ) 1 ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) ( I ψ ) 1 ψ n G ( ϖ 1 , ϖ 1 , ϖ 0 ) .
Thus, for n , m N with n > m , we obtain
G ( ϖ n , ϖ n , ϖ m ) G ( ϖ n , ϖ n , ϖ n 1 ) + G ( ϖ n 1 , ϖ n 1 , ϖ n 2 ) + + G ( ϖ m + 1 , ϖ m + 1 , ϖ m ) ( I ψ ) 1 ψ n 1 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + ( I ψ ) 1 ψ n 2 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + + ( I ψ ) 1 ψ m G ( ϖ 1 , ϖ 1 ϖ 0 ) k = m n 1 ( I ψ ) 1 ψ k G ( ϖ 1 , ϖ 1 , ϖ 0 ) 0 , G ( ϖ 1 , ϖ 1 , ϖ 0 ) > 0 , n , m + .
Hence, { ϖ n } is a G-Cauchyness in X . Based on G-completeness of X , there exists ϖ * X such that lim n + ϖ n = ϖ * . Therefore, one has
0 G ( ϖ * , F ϖ * , F ϖ * ) G ( ϖ * , F ϖ n , F ϖ n ) + G ( F ϖ n , F ϖ * , F ϖ * ) G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) + ψ ( G ( ϖ * , ϖ * , ϖ n ) + G ( ϖ * , F ϖ * , F ϖ * ) . G ( ϖ n , F ϖ n , F ϖ n ) 1 + G ( ϖ * , ϖ * , ϖ n ) ) = G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) + ψ ( G ( ϖ * , ϖ * , ϖ n ) + G ( ϖ * , F ϖ * , F ϖ * ) . G ( ϖ n , ϖ n + 1 , ϖ n + 1 ) 1 + G ( ϖ * , ϖ * , ϖ n ) ) .
Taking n + , we obtain
G ( ϖ * , F ϖ * , F ϖ * ) G ( ϖ * , ϖ * , ϖ * ) + ψ G ( ϖ * , ϖ * , ϖ * ) + G ( ϖ * , F ϖ * , F ϖ * ) . G ( ϖ * , ϖ * , ϖ * ) 1 + G ( ϖ * , ϖ * , ϖ * ) .
Thus, G ( ϖ * , F ϖ * , F ϖ * ) 0 , which is a contradiction. Then, G ( ϖ * , F ϖ * , F ϖ * ) = 0 , i.e., ϖ * = F ϖ * is an FP of F . Now, if ϱ * ϖ * is another FP of the operator F , then
0 G ( ϖ * , ϖ * , ϱ * ) = G ( F ϖ * , F ϖ * , F ϱ * ) ψ G ( ϖ * , ϖ * , ϱ * ) + G ( ϖ * , F ϖ * , F ϖ * ) . G ( ϱ * , F ϱ * , F ϱ * ) 1 + G ( ϖ * , ϖ * , ϱ * ) = ψ G ( ϖ * , ϖ * , ϱ * ) ,
that is,
( I ψ ) G ( ϖ * , ϖ * , ϱ * ) 0 .
Since ( I ψ ) is invertible, then
( I ψ ) 1 ( I ψ ) G ( ϖ * , ϖ * , ϱ * ) ( I ψ ) 1 ( 0 ) .
Then,
G ( ϖ * , ϖ * , ϱ * ) = 0 ϖ * = ϱ * .
This implies that the FP is unique. □
In Theorem 6, if ψ = λ I then we obtain the result below:
Corollary 2.
Let ( X , G ) be a G-complete G-metric space. Suppose that F : X X and G : X × X × X R + are operators satisfying
G ( F ϖ , F ϖ , F ϱ ) λ G ( ϖ , ϖ , ϱ ) + G ( ϖ , F ϖ , F ϖ ) . G ( ϱ , F ϱ , F ϱ ) 1 + G ( ϖ , ϖ , ϱ ) , f o r a l l ϖ , ϱ X ,
where 0 λ < 1 is a constant. Then, F has a unique FP in X .
Theorem 7.
Let ( X , G ) be a G-complete G-metric space. Assume that F : X X and G : X × X × X R + are operators satisfying
G ( F ϖ , F ϖ , F ϱ ) ψ G ( ϖ , ϖ , ϱ ) + G ( w , F ϖ , F ϖ ) . G ( ϖ , F ϱ , F ϱ ) 1 + G ( ϖ , ϱ , ϱ ) , f o r a l l ϖ , ϱ X ,
where ψ Ψ and ψ < I . Then, F has a unique FP in X .
Proof. 
Let ϖ 0 be an arbitrary element in X . Define a sequence { ϖ n } by F 2 n + 2 ϖ 0 = F ϖ 2 n + 1 = ϖ 2 n + 2 , n 1 then, we have
G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) = G ( F ϖ 2 n + 1 , F ϖ 2 n + 1 , F ϖ 2 n ) ψ ( G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + G ( ϖ 2 n , F ϖ 2 n + 1 , F ϖ 2 n + 1 ) . G ( ϖ 2 n + 1 , F ϖ 2 n , F ϖ 2 n ) 1 + G ( ϖ 2 n + 1 , ϖ 2 n , ϖ 2 n ) ) = ψ ( G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) + G ( ϖ 2 n , ϖ 2 n + 2 , ϖ 2 n + 2 ) . G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n + 1 ) 1 + G ( ϖ 2 n + 1 , ϖ 2 n , ϖ 2 n ) ) ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) .
Thus,
G ( ϖ 2 n + 2 , ϖ 2 n + 2 , ϖ 2 n + 1 ) ψ G ( ϖ 2 n + 1 , ϖ 2 n + 1 , ϖ 2 n ) ψ 2 n + 1 G ( ϖ 1 , ϖ 1 , ϖ 0 ) .
Thus, for m , n N with n > m , we find
G ( ϖ n , ϖ n , ϖ m ) G ( ϖ n , ϖ n , ϖ n 1 ) + G ( ϖ n 1 , ϖ n 1 , ϖ n 2 ) + + G ( ϖ m + 1 , ϖ m + 1 , ϖ m ) ψ n 1 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + ψ n 2 G ( ϖ 1 , ϖ 1 , ϖ 0 ) + + ψ m G ( ϖ 1 , ϖ 1 , ϖ 0 ) k = m n 1 ψ k G ( ϖ 1 , ϖ 1 , ϖ 0 ) 0 , for all G ( ϖ 1 , ϖ 1 , ϖ 0 ) > 0 , n , m + .
Hence, { ϖ n } is a G-Cauchyness in X . Based on G-completeness of X , there exists ϖ * X such that lim n + ϖ n = ϖ * . Therefore, one has
0 G ( ϖ * , F ϖ * , F ϖ * ) G ( ϖ * , F ϖ n , F ϖ n ) + G ( F ϖ * , F ϖ * , F ϖ n ) G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) + ψ ( G ( ϖ * , ϖ * , ϖ n ) + G ( ϖ n , F ϖ * , F ϖ * ) . G ( ϖ * , F ϖ n , F ϖ n ) 1 + G ( ϖ * , ϖ n , ϖ n ) ) G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) + ψ ( G ( ϖ * , ϖ * , ϖ n ) + G ( ϖ n , F ϖ * , F ϖ * ) . G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) 1 + G ( ϖ * , ϖ n , ϖ n ) ) G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) + ψ G ( ϖ * , ϖ * , ϖ n ) + ψ G ( ϖ n , F ϖ * , F ϖ * ) . G ( ϖ * , ϖ n + 1 , ϖ n + 1 ) 1 + G ( ϖ * , ϖ n , ϖ n ) .
Taking n + , we obtain
G ( ϖ * , F ϖ * , F ϖ * ) G ( ϖ * , ϖ * , ϖ * ) + ψ G ( ϖ * , ϖ * , ϖ * ) + ψ G ( ϖ * , F ϖ * , F ϖ * ) . G ( ϖ * , ϖ * , ϖ * ) 1 + G ( ϖ * , ϖ * , ϖ * ) = 0 + ψ ( 0 ) + ψ ( 0 ) = 0 .
Thus, G ( ϖ * , F ϖ * , F ϖ * ) 0 , that is a contradiction. Then, G ( ϖ * , F ϖ * , F ϖ * ) = 0 , i.e., ϖ * = F ϖ * is a FP of F . Now, if ϱ * ϖ * is another FP of the operator F , then
0 G ( ϖ * , ϖ * , ϱ * ) = G ( F ϖ * , F ϖ * , F ϱ * ) ψ G ( ϖ * , ϖ * , ϱ * ) + G ( ϱ * , F ϖ * , F ϖ * ) . G ( ϖ * , F ϱ * , F ϱ * ) 1 + G ( ϖ * , ϱ * , ϱ * ) ψ G ( ϖ * , ϖ * , ϱ * ) + G ( ϱ * , ϖ * , ϖ * ) . G ( ϖ * , ϱ * , ϱ * ) 1 + G ( ϖ * , ϱ * , ϱ * ) 2 ψ G ( ϖ * , ϖ * , ϱ * ) ,
that is,
( I 2 ψ ) G ( ϖ * , ϖ * , ϱ * ) 0 .
Since ψ ( ϖ ) = 0 if ϖ = 0 and I ( ϖ ) 2 ψ ( ϖ ) . Then,
G ( ϖ * , ϖ * , ϱ * ) = 0 ϖ * = ϱ * .
This implies that the FP is unique. □
In Theorem 7, if ψ = λ I then we have the result below:
Corollary 3.
Let ( X , G ) be a G-complete G-metric space. Assume that F : X X and G : X × X × X R + are operators satisfying
G ( F ϖ , F ϖ , F ϱ ) λ G ( ϖ , ϖ , ϱ ) + G ( w , F ϖ , F ϖ ) . G ( ϖ , F ϱ , F ϱ ) 1 + G ( ϖ , ϱ , ϱ ) , f o r a l l ϖ , ϱ X ,
where 0 λ < 1 is a constant. Then, F has a unique FP in X .

4. Applications to Boundary Value Problems

In this section, we use Theorem 4 to establish the conditions under which the solution to BVP can be expressed
d 2 ϖ d t 2 = λ ( t , ϖ ( t ) ) , t [ 0 , 1 ] , ϖ R + , ϖ ( 0 ) = ϖ ( 1 ) = 0 ,
where λ : [ 0 , 1 ] × R + R + is a continuous function. The ideas presented in this section are inspired by [40,41,42,43].
Note that (11) corresponds to the IE
ϖ ( t ) = 0 1 Ω ( t , s ) λ ( s , ϖ ( s ) ) d s , t [ 0 , 1 ] ,
where Ω ( t , s ) is called Green function, defined by
Ω ( t , s ) = t ( 1 s ) if 0 t < s 1 , s ( 1 t ) if 0 s < t 1 .
Let X = C ( [ 0 , 1 ] , R + ) denote the set of all continuous real-valued functions defined on the interval [ 0 , 1 ] . We equip X with an operator defined for all ϖ , ϱ , κ X as follows:
G ( ϖ , ϱ , κ ) = sup t [ 0 , 1 ] ϖ ( t ) ϱ ( t ) + ϖ ( t ) κ ( t ) + ϱ ( t ) κ ( t ) .
Then, ( X , G ) is a G-complete G-metric space. Consider the self-operator F : X X be defined as
F ϖ ( t ) = 0 1 Ω ( t , s ) λ ( s , ϖ ( s ) ) d s , t [ 0 , 1 ] .
Then, any solution ϖ * to (11) is a FP of F and conversely.
The physical meaning of boundary value problem (11) is as follows:
A vibrating beam with fixed ends ϖ ( 0 ) = ϖ ( 1 ) = 0 , that mean the ends are fixed at zero.
The displacement of an elastic beam under force, where
(a)
ϖ ( t ) indicates the displacement at position t,
(b)
λ ( t , ϖ ( t ) ) indicates the applied force, which depend on both displacement and position,
(c)
the boundary conditions ϖ ( 0 ) = ϖ ( 1 ) = 0 indicate the beam is fixed at both ends.
Now, we examine the conditions for the existence of solutions to the BVP (11) under the following assumptions.
Theorem 8.
Assume that ψ Ψ and F : X X is a self-operator on X . Assume that the conditions below are satisfied:
  • (C1) there exists t [ 0 , 1 ] and for all ϖ , ϱ , κ R , we have
    λ ( t , ϖ ) λ ( t , ϱ ) + λ ( t , ϖ ) λ ( t , κ ) + λ ( t , ϱ ) λ ( t , κ ) ψ ϖ ϱ + ϖ κ + ϱ κ ;
  • (C2) 0 1 Ω ( t , s ) d s 1 .
Then, the boundary value problem (11) has a solution in X .
Proof. 
Taking (12) and (13) into account, and for all ϖ , ϱ X with t [ 0 , 1 ] , we have
G ( F ϖ , F ϖ , F ϱ ) = sup t [ 0 , 1 ] F ϖ ( t ) F ϖ ( t ) + F ϖ ( t ) F ϱ ( t ) + F ϖ ( t ) F ϱ ( t ) sup t [ 0 , 1 ] 0 1 Ω ( t , s ) [ λ ( s , ϖ ( s ) ) λ ( s , ϖ ( s ) ) + λ ( s , ϖ ( s ) ) λ ( s , ϱ ( s ) ) + λ ( s , ϖ ( s ) ) λ ( s , ϱ ( s ) ) ] d s sup t [ 0 , 1 ] 0 1 Ω ( t , s ) ψ ϖ ( s ) ϖ ( s ) + ϖ ( s ) ϱ ( s ) + ϖ ( s ) ϱ ( s ) d s 0 1 Ω ( t , s ) ψ sup t [ 0 , 1 ] ϖ ( s ) ϖ ( s ) + ϖ ( s ) ϱ ( s ) + ϖ ( s ) ϱ ( s ) d s ψ G ( ϖ , ϖ , ϱ ) 0 1 Ω ( t , s ) d s ψ G ( ϖ , ϖ , ϱ ) .
We see that F verifies all hypotheses of Theorem 4. It follows that F has a FP ϖ * in X , which corresponds to solution of the BVP . □

5. Application to Fredholm-Type Integral Equations

In this part, we let X = C ( [ t , r ] , R ) , represent the set of all continuous real-valued functions defined on the interval [ t , r ] , where t , r R . We use Theorem 4 to derive a result guaranteeing a unique solution for nonlinear Fredholm-type integral equations:
ϖ ( α ) = γ t r K α , η , ϖ ( η ) d η ,
where K : [ t , r ] × [ t , r ] × R R is a given continuous function, γ is constant and ϖ C ( [ t , r ] , R ) .
The physical meaning of each operator in the Fredholm-type integral equation is: please confirm
(a)
ϖ ( α ) represents the temperature distribution at point α in a heat conduction problem,
(b)
γ indicates the thermal conductivity coefficient of the material,
(c)
the kernel function K ( α , η , ϖ ( η ) ) represents the heat transfer kernel that describes how heat at point η affects the temperature at point α ,
(d)
the integral from t to r indicates the total accumulation of heat effects over the entire physical domain from point t to r,
(e)
the interval [ t , r ] represents the physical length of the conducting material.
Theorem 9.
Assume that the kernel function K satisfies the following condition:
| K ( α , η , ϖ ( η ) ) K ( α , η , ϱ ( η ) ) | + | K ( α , η , ϱ ( η ) ) K ( α , η , κ ( η ) | + | ( K ( α , η , ϖ ( η ) ) K ( α , η , κ ( η ) ) | ψ | ϖ ( η ) ϱ ( η ) | + | ϱ ( η ) κ ( η ) | + | ϖ ( η ) κ ( η ) | ,
and consider the nonlinear IE problem (14). Then, for some constant 0 < γ 1 r t the Equation (14) has a unique solution ϖ C ( [ t , r ] , R ) .
Proof. 
Let F : X X be defined as
F ϖ ( α ) = γ t r K α , η , ϖ ( η ) d η , for all ϖ X ,
where X = C ( [ t , r ] , R ) . Assume for all ϖ , ϱ , κ X
G ( ϖ , ϱ , κ ) = sup η [ t , r ] | ϖ ( η ) ϱ ( η ) | + | ϖ ( η ) κ ( η ) | + | ϱ ( η ) κ ( η ) | .
It is clear that ( X , G ) form a G-complete G-metric space.
G F ϖ , F ϖ , F ϱ = sup η [ t , r ] F ϖ ( η ) F ϖ ( η ) + F ϖ ( η ) F ϱ ( η ) + F ϖ ( η ) F ϱ ( η ) sup η [ t , r ] γ [ | t r K ( α , η , ϖ ( η ) ) K ( α , η , ϖ ( η ) ) d η | + | t r K ( α , η , ϖ ( η ) ) K ( α , η , ϱ ( η ) ) d η | + | t r K ( α , η , ϖ ( η ) ) K ( α , η , ϱ ( η ) ) d η | ] sup η [ t , r ] γ [ t r | K ( α , η , ϖ ( η ) ) K ( α , η , ϖ ( η ) ) | d η + t r | K ( α , η , ϖ ( η ) ) K ( α , η , ϱ ( η ) ) | d η + t r | K ( α , η , ϖ ( η ) ) K ( α , η , ϱ ( η ) ) | d η ]
sup η [ t , r ] γ t r [ | K ( α , η , ϖ ( η ) ) K ( α , η , ϖ ( η ) ) | + | K ( α , η , ϖ ( η ) ) K ( α , η , ϱ ( η ) ) | + | K ( α , η , ϖ ( η ) ) K ( α , η , ϱ ( η ) ) | ] d η sup η [ t , r ] γ t r ψ | ϖ ( η ) ϖ ( η ) | + | ϖ ( η ) ϱ ( η ) | + | ϖ ( η ) ϱ ( η ) | d η γ ψ sup η [ t , r ] | ϖ ( η ) ϖ ( η ) | + | ϖ ( η ) ϱ ( η ) | + | ϖ ( η ) ϱ ( η ) | t r d η ψ G ( ϖ , ϖ , ϱ ) γ ( r t ) ψ G ( ϖ , ϖ , ϱ ) .
Therefore, the nonlinear FIE (14) has a unique solution. □

6. Application of Nonlinear Fractional Differential Equation

The aim of this part is to introduce an application of Theorem 4 to obtain a common solution of the nonlinear fractional differential equation ( FDE ) of Caputo type for a pair of generalized Ψ -contraction of the operator in G-metric space. The Caputo nonlinear fractional differential equation proposed by Kilbas [30] is given as follows:
D 0 + μ C ϖ ( t ) = f ( t , ϖ ( t ) ) , t ( 0 , 1 ) , 1 < μ 2 , ϖ ( 0 ) = 0 , ϖ ( 1 ) = 0 m ϖ ( s ) d s , 0 < m < 1 ,
where D 0 + μ C represents the ( CFD ) of order μ and f : [ 0 , 1 ] × R R is a continuous function. This nonlinear FDE can also be represented in the form as follows:
ϖ ( t ) = 1 Γ ( μ ) 0 t ( t s ) μ 1 f ( s , ϖ ( s ) ) d s 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s .
A function ϖ ( t ) is a solution of the defined nonlinear FDE whenever it is the solution of the FDE (15) and vice versa.
The physical meaning of function f is:
It indicates the forcing function in the system and describes how the system responds to both time t and the current state ϖ ( t ) .
Theorem 10.
Consider the space of all continuous functions X = C ( [ 0 , 1 ] , R ) constructed on closed interval [ 0 , 1 ] , equipped with a binary relation B = C [ 0 , 1 ] × C [ 0 , 1 ] . Let C [ 0 , 1 ] , R be the Banach space of all continuous functions from [ 0 , 1 ] into R with norm ϖ = sup t [ 0 , 1 ] ϖ ( t ) ; for all ϖ ( t ) C [ 0 , 1 ] . This space defines the metric as follows:
G ( ϖ , ϱ , κ ) = sup t [ 0 , 1 ] ϖ ( t ) ϱ ( t ) + ϖ ( t ) κ ( t ) + ϱ ( t ) κ ( t ) , f o r a l l ϖ , ϱ , s X .
Now, if we construct self-operator F : X X as
F ϖ ( t ) = 1 Γ ( μ ) 0 t ( t s ) μ 1 f ( s , ϖ ( s ) ) d s 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 f ( s , ϖ ( s ) ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m 0 s ( s τ ) μ 1 f ( τ , ϖ ( τ ) ) d τ d s .
Then, F has a unique FP if we have the following assumptions:
(1) 
there exists a continuous function f : [ 0 , 1 ] × R R satisfying that
| f ( s , ϖ ( s ) ) f ( s , ϖ ( s ) ) | + | f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) | + | f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) | ψ | ϖ ( s ) ϖ ( s ) | + | ϖ ( s ) ϱ ( s ) | + | ϖ ( s ) ϱ ( s ) | ,
(2) 
t μ ( 2 m 2 ) ( μ + 1 ) + 2 t ( m μ + 1 μ 1 ) ( 2 m 2 ) ( μ + 1 ) 1 .
Proof. 
Obviously, C [ 0 , 1 ] , R is a complete metric space.
(a)
Since F : X X is a self operator, and B = { X × X } , then for any ϖ 0 B we have F ϖ 0 B . Thus, ϖ 0 , F ϖ 0 B , and hence B is nonempty.
(b)
For any arbitrary element ϖ 0 X , the sequence { F n ϖ 0 } n N X , where F : X X is a self-operator. Therefore, { ( F n 1 ( ϖ 0 ) , F n ( ϖ 0 ) } n N B , where F 0 = I , and hence B is F -closed.
(c)
It is only required to prove that F is a generalized Ψ -contractive operator.
G ( F ϖ , F ϖ , F ϱ ) = sup t [ 0 , 1 ] F ϖ ( t ) F ϖ ( t ) + F ϖ ( t ) F ϱ ( t ) + F ϱ ( t ) F ϖ ( t ) = sup t [ 0 , 1 ] | 1 Γ ( μ ) 0 t ( t s ) μ 1 ( f ( s , ϖ ( s ) ) f ( s , ϖ ( s ) ) + f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) + f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) ) 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 ( f ( s , ϖ ( s ) ) f ( s , ϖ ( s ) ) + f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) + f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m [ 0 s ( s τ ) μ 1 ( f ( τ , ϖ ( τ ) ) f ( τ , ϖ ( τ ) ) + f ( τ , ϖ ( τ ) ) f ( τ , ϱ ( τ ) ) + f ( τ , ϖ ( τ ) ) f ( τ , ϱ ( τ ) ) ) d τ ] d s | = sup t [ 0 , 1 ] [ 1 Γ ( μ ) 0 t ( t s ) μ 1 ( | f ( s , ϖ ( s ) ) f ( s , ϖ ( s ) ) | + | f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) | + | f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) | ) d s 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 ( | f ( s , ϖ ( s ) ) f ( s , ϖ ( s ) ) | + | f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) | + | f ( s , ϖ ( s ) ) f ( s , ϱ ( s ) ) | ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m [ 0 s ( s τ ) μ 1 ( | f ( τ , ϖ ( τ ) ) f ( τ , ϖ ( τ ) ) | + | f ( τ , ϖ ( τ ) ) f ( τ , ϱ ( τ ) ) | + | f ( τ , ϖ ( τ ) ) f ( τ , ϱ ( τ ) ) | ) d τ ] d s ] sup t [ 0 , 1 ] [ 1 Γ ( μ ) 0 t ( t s ) μ 1 ψ ( | ϖ ( s ) ϖ ( s ) | + | ϖ ( s ) ϱ ( s ) | + | ϖ ( s ) ϱ ( s ) | ) d s 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 ψ ( | ϖ ( s ) ϖ ( s ) | + | ϖ ( s ) ϱ ( s ) | + | ϖ ( s ) ϱ ( s ) | ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m [ 0 s ( s τ ) μ 1 ψ ( | ϖ ( τ ) ϖ ( τ ) | + | ϖ ( τ ) ϱ ( τ ) | + | ϖ ( τ ) ϱ ( τ ) | ) d τ ] ] d s
= 1 Γ ( μ ) 0 t ( t s ) μ 1 ψ sup t [ 0 , 1 ] | ϖ ( s ) ϖ ( s ) | + | ϖ ( s ) ϱ ( s ) | + | ϖ ( s ) ϱ ( s ) | d s 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 ψ ( sup t [ 0 , 1 ] ( | ϖ ( s ) ϖ ( s ) | + | ϖ ( s ) ϱ ( s ) | + | ϖ ( s ) ϱ ( s ) | ) ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m [ 0 s ( s τ ) μ 1 ψ ( sup t [ 0 , 1 ] ( | ϖ ( τ ) ϖ ( τ ) | + | ϖ ( τ ) ϱ ( τ ) | + | ϖ ( τ ) ϱ ( τ ) | ) ) d τ ] d s = 1 Γ ( μ ) 0 t ( t s ) μ 1 ψ G ( ϖ , ϖ , ϱ ) 2 t ( 2 m 2 ) Γ ( μ ) 0 1 ( 1 s ) μ 1 ψ G ( ϖ , ϖ , ϱ ) d s + 2 t ( 2 m 2 ) Γ ( μ ) 0 m 0 s ( s τ ) μ 1 ψ G ( ϖ , ϖ , ϱ ) d τ d s .
Therefore,
G ( F ϖ , F ϖ , F ϱ ) ψ G ( ϖ , ϖ , ϱ ) Γ ( μ ) [ 0 t ( t s ) μ 1 d s 2 t ( 2 m 2 ) 0 1 ( 1 s ) μ 1 d s + 2 t ( 2 m 2 ) 0 m 0 s ( s τ ) μ 1 d τ d s ] = ψ G ( ϖ , ϖ , ϱ ) μ Γ ( μ ) t μ 2 t ( 2 m 2 ) + 2 t ( 2 m 2 ) × m μ + 1 ( μ + 1 ) = ψ G ( ϖ , ϖ , ϱ ) μ Γ ( μ ) t μ ( 2 m 2 ) ( μ + 1 ) + 2 t ( m μ + 1 μ 1 ) ( 2 m 2 ) ( μ + 1 ) .
By using given condition ( 3 ) , we obtain
G ( F ϖ , F ϖ , F ϱ ) ψ G ( ϖ , ϖ , ϱ ) .
Therefore, all the hypothesis of Theorem 4 are satisfied; hence, F has a unique FP . □

Numerical Solution of Fractional Initial Value Problems

In this section, we present a numerical spectral approach to obtain a semi-analytic solution for the following fractional initial value problem:
D 0 + μ C ϖ ( t ) = f t , ϖ ( t ) , t ( 0 , 1 ) , 1 < μ 2 ,
subject to the following homogeneous initial conditions:
ϖ ( 0 ) = ϖ ( 0 ) = 0 .
We assume the following approximate solution:
ϖ ( t ) ϖ n ( t ) = i = 0 n u i t 2 C i ( t ) ,
where C i ( t ) = cos ( i cos 1 ( 2 t 1 ) ) represents the shifted Chebyshev polynomials of the first kind [44]. This choice ensures that the approximate solution inherently satisfies the homogeneous initial conditions in (17).
Note 1.
If the initial value problem (16) satisfies nonhomogeneous initial conditions, such as the following:
y ( 0 ) = y 0 , y ( 0 ) = y 1 ,
we can apply the following transformation:
y ( t ) = u ( t ) + y 0 + y 1 t .
With this substitution, u ( t ) satisfies the homogeneous initial conditions given in (17).
Lemma 1.
The following power series representation of the shifted Chebyshev polynomials holds [45]:
C ( t ) = k = 0 ( 1 ) k 2 2 k ( + k 1 ) ! ( k ) ! ( 2 k ) ! t k , > 0 .
Theorem 11.
The following fractional derivative formula holds:
D 0 + μ C t 2 C i ( t ) = 2 ( 1 ) i Γ ( 3 μ ) t 2 μ F 2 3 3 , i , i ; 1 2 , 3 μ ; t .
Proof. 
Using Lemma 1, we express the following:
t 2 C i ( t ) = i k = 0 i ( 1 ) i k 2 2 k ( i + k 1 ) ! ( i k ) ! ( 2 k ) ! t k + 2 .
Applying the fractional derivative D 0 + μ C , we obtain the following:
D 0 + μ C t 2 C i ( t ) = k = 0 i i 4 k ( 1 ) i k Γ ( k + 3 ) Γ ( i + k ) ( 2 k ) ! ( i k ) ! Γ ( k μ + 3 ) t k + 2 μ ,
and simplifying the right-hand side yields the desired result. □
The residual of (16) is given by the following:
R ( t ) = C D 0 + μ ϖ n ( t ) f t , ϖ n ( t ) .
Using Theorem 1, we express the simplified residual as follows:
R ˜ ( t ) = t μ R ( t ) = t 2 i = 0 n u i 2 ( 1 ) i Γ ( 3 μ ) F 2 3 3 , i , i ; 1 2 , 3 μ ; t t μ f t , t 2 i = 0 n u i C i ( t ) .
Applying the collocation method at the first roots of C n + 1 ( t ) , denoted by t i for i = 1 ( 1 ) n + 1 , we obtain the following:
R ˜ ( t i ) = 0 , i = 1 , 2 , . . . , n + 1 .
Newton’s iterative method with a vanishing initial guess is employed to solve the resulting system of ( n + 1 ) nonlinear algebraic equations, yielding the unknown expansion coefficients u i .
Example 10.
Consider the following fractional initial value problem [46]:
D 0 + α C y ( x ) = 40320 Γ ( 9 α ) x 8 α 3 Γ ( 5 + α / 2 ) Γ ( 5 α / 2 ) x 4 α / 2 + 9 4 Γ ( α + 1 ) + 3 2 x α / 2 x 4 3 [ y ( x ) ] 3 / 2 .
The initial conditions are homogeneous: y ( 0 ) = 0 , y ( 0 ) = 0 . The exact solution is as follows:
y ( x ) = x 8 3 x 4 + α / 2 + 9 4 x α .
We apply the proposed Chebyshev collocation method. Table 1 lists the absolute errors for different values of n, while Table 2 compares our best error with that reported in [46].
Example 11.
Consider the fractional oscillator problem arising in physics [47]:
D μ u ( t ) + u ( t ) = 0 , u ( 0 ) = 1 , u ( 0 ) = 0 , t ( 0 , L ) ,
where 1 < μ 2 . The exact solution is given by the following:
u ( t ) = k = 0 + ( t μ ) k Γ ( k μ + 1 ) = E μ ( t μ ) ,
where E μ ( t ) denotes the well-known Mittag–Leffler function. Table 3 compares our method with the fifth-kind Chebyshev method proposed in [47], using the following truncated series:
u ( t ) = k = 0 10 ( t μ ) k Γ ( k μ + 1 ) = E μ ( t μ ) .
In Table 3, we compare our method and the fifth-kind Chebyshev method offered in [47], we compare the numerical solutions with the following solution,
u ( t ) = k = 0 10 t μ k Γ ( k μ + 1 ) = E μ t μ .
Example 12.
Consider the following fractional nonlinear Bratu’s initial value problem appears in physics [48]:
D 0 + α C y ( x ) = 2 e y ( x ) .
The initial conditions are homogeneous: y ( 0 ) = 0 , y ( 0 ) = 0 . The exact solution is available only when α = 2 :
y ( x ) = 2 ln ( cos ( x ) ) .
We apply the proposed Chebyshev collocation method. Table 4 lists the maximum residual errors for different values of n, where the residual error is defined by
r e s n ( x ) = | C D 0 + α y n ( x ) 2 e y n ( x ) | .
In Figure 2, we depict the residual errors for different values of α, when n = 15 , these results show the exponential convergence.
From the results presented in Table 1, Table 2, Table 3 and Table 4, we confirm that our proposed method demonstrates superior accuracy compared to the approach presented in recent literature. This highlights the effectiveness and superiority of our method over other contemporary techniques published in reputable journals.

7. Conclusions and Future Work

This manuscript contributes significantly to the field of mathematical analysis by introducing new results within the framework of G-metric spaces. The research expands upon existing findings, providing a deeper understanding of FP theorems for single and paired operators in G-metric spaces. The application of these theorems to establish the existence of unique solutions for boundary value problems, Fredholm-type integral equations, and Caputo fractional differential equations in G-metric spaces not only confirms the theoretical outcomes but also underscores their practical importance. Looking forward, several promising directions for future research in C * -metric spaces present themselves. Extending the results obtained in G-metric spaces to modular metric space and modular G-metric space and investigating the application of these FP theorems in solving more complex differential and integral equations within such spaces could be particularly valuable. We also discuss the numerical solution of fractional initial value problems using the spectral collocation method to provide a comprehensive study.

Author Contributions

Conceptualization, writing—original draft: G.A.; methodology, data curation: M.G. and Y.H.Y.; formal analysis, writing—review and editing: M.G. and Y.H.Y.; project administration: G.A.; supervision: M.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. Banach, S. Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fund. Math. 1922, 3, 133–181. [Google Scholar] [CrossRef]
  2. Brouwer, L.E.J. Uber Abbildungen von Mannigfaltigkeiten. Math. Ann. 1912, 71, 97–115. [Google Scholar] [CrossRef]
  3. Schauder, J. Der Fixpunktsatz in Funktionalraumen. Stud. Math. 1930, 2, 171–180. [Google Scholar] [CrossRef]
  4. Kirk, W.A. Fixed points of asymptotic contractions. J. Math. Anal. Appl. 2003, 277, 645–650. [Google Scholar] [CrossRef]
  5. Suzuki, T. Fixed point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces. Nonlinear Anal. 2006, 64, 971–978. [Google Scholar] [CrossRef]
  6. Vetro, C. On Branciari’s theorem for weakly compatible mappings. Appl. Math. Lett. 2010, 23, 700–705. [Google Scholar] [CrossRef]
  7. Turinici, M. Functional contractions in local Branciari metric spaces. arXiv 2012, arXiv:1208.4610. [Google Scholar]
  8. Nadler, S.B. Multi-valued contraction operators. Pac. J. Math. 1969, 30, 475–488. [Google Scholar] [CrossRef]
  9. Berinde, M.; Berinde, V. On a general class of multi-valued weakly Picard mappings. J. Math. Anal. Appl. 2007, 326, 772–782. [Google Scholar] [CrossRef]
  10. Mustafa, Z. A New Structure for Generalized Metric Spaces—With Applications to Fixed Point Theory. Ph.D. Thesis, The University of Newcastle, Callaghan, Australia, 2005. [Google Scholar]
  11. Mustafa, Z.; Sims, B. A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7, 289–297. [Google Scholar]
  12. Mustafa, Z.; Obiedat, H.; Awawdeh, F. Some fixed point theorem for mapping on complete G-metric spaces. Fixed Point Theory Appl. 2008, 2008, 189870. [Google Scholar] [CrossRef]
  13. Jleli, M.; Samet, B. Remarks on G-metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012, 2012, 210. [Google Scholar] [CrossRef]
  14. Samet, B.; Vetro, C.; Vetro, F. Remarks on G-metric spaces. Int. J. Anal. 2013, 2013, 917158. [Google Scholar] [CrossRef]
  15. Shatanawi, W. Fixed point theory for contractive mappings satisfying ϕ-maps in G-metric spaces. Fixed Point Theory Appl. 2010, 2010, 181650. [Google Scholar] [CrossRef]
  16. Shatanawi, W. Some fixed point theorems in ordered G-metric spaces and applications. Abstr. Appl. Anal. 2011, 2011, 126205. [Google Scholar] [CrossRef]
  17. Lonseth, A.T. Sources and applications of integral equations. SIAM Rev. 1977, 2, 241–278. [Google Scholar] [CrossRef]
  18. Akyuz-Dascioglu, A.; Sezer, M. A Taylor polynomial approach for solving high–order linear Fredholm integro–differential equations in the most general form. Int. J. Comput. Math. 2007, 84, 527–539. [Google Scholar] [CrossRef]
  19. Li, X.F.; Fang, M. Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion. Int. J. Comput. Math. 2006, 83, 637–649. [Google Scholar] [CrossRef]
  20. Ghasemi, M.; Babolian, E.; Kajani, M.T. Numerical solution of linear Fredholm integral equations using sine–cosine wavelets. Int. J. Comput. Math. 2007, 84, 979–987. [Google Scholar] [CrossRef]
  21. Groetsch, C.W. Integral equations of the first kind, inverse problems and regularization: A crash course. J. Phys. Conf. Ser. 2007, 73, 012001. [Google Scholar] [CrossRef]
  22. Jerri, A.J. Introduction to Integral Equations with Applications; John Wiley Sons: Hoboken, NJ, USA, 1999. [Google Scholar]
  23. Delves, L.M.; Mohamed, J.L. Computational Methods for Integral Equations; Cambridge University Press: Cambridge, UK, 1985; pp. 2–4. [Google Scholar]
  24. Wazwaz, A.M. Linear and Nonlinear Integral Equations Methods and Applications; Higher Education Press: Beijing, China; Springer: Berlin/Heidelberg, Germany; London, UK; New York, NY, USA,, 2011; pp. 469–510. [Google Scholar]
  25. Atangana, A.; Alkahtani, B.S. Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel. Adv. Mech. Eng. 2015, 7, 1687814015591937. [Google Scholar] [CrossRef]
  26. Saad, K.M.; Atangana, A.; Baleanu, D. New fractional derivatives with non-singular kernel applied to the Burgers equation. Chaos 2018, 28, 63–109. [Google Scholar] [CrossRef] [PubMed]
  27. Baleanu, D.; Sadat Sajjadi, S.; Jajarmi, A.; Asad, J.H. New features of the fractional Euler-Lagrange equations for a physical system within non-singular derivative operator. Eur. Phys. J. Plus 2019, 134, 181. [Google Scholar] [CrossRef]
  28. Caputo, M. Linear model of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astr. Soc. 1967, 13, 529–539. [Google Scholar] [CrossRef]
  29. Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Elsevier: Amsterdam, The Netherlands, 1998. [Google Scholar]
  30. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Math. Studies; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
  31. Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 2015, 1, 73–85. [Google Scholar]
  32. Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Threm. Sci. 2016, 20, 763–769. [Google Scholar] [CrossRef]
  33. Hadamard, J. Essai sur l’étude des fonctions, données par leur développement de Taylor. J. Math. Pures Appl. 1892, 8, 101–186. [Google Scholar]
  34. Diethelm, K. The analysis of fractional differential equations: An application-oriented exposition using differential operators of caputo type. In Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
  35. Shiri, B.; Baleanu, D. A general fractional pollution model for lakes. Commun. Appl. Math. Comput. 2022, 4, 1–26. [Google Scholar] [CrossRef]
  36. Li, X.P.; Alrihieli, H.F.; Algehyne, E.A.; Khan, M.A.; Alshahrani, M.Y.; Alraey, Y.; Riaz, M.B. Application of piecewise fractional differential equation to COVID-19 infection dynamics. Results Phys. 2022, 39, 105685. [Google Scholar] [CrossRef]
  37. Almeida, R.; Bastos, N.R.O.; Monteiro, M.T.T. Modeling some real phenomena by fractional differential equations. Math. Meth. Appl. Sci. 2016, 39, 4846–4855. [Google Scholar] [CrossRef]
  38. Mustafa, Z.; Obiedat, B. A fixed point theorem of Reich in G-metric spaces. CUBO 2010, 12, 83–93. [Google Scholar] [CrossRef]
  39. Gaba, Y.U. Fixed point theorems in G-metric spaces. J. Math. Anal. Appl. 2017, 455, 528–537. [Google Scholar] [CrossRef]
  40. Mustafa, Z.; Arshad, M.; Khan, S.U.; Ahmad, J.; Jaradat, M.M.M. Common Fixed Points for Multivalued operators in G-metric spaces with Applications. J. Nonlinear Sci. Appl. 2017, 10, 2550–2564. [Google Scholar] [CrossRef]
  41. Karapınar, E.; Abdeljawad, T.; Jarad, F. Applying new fixed point theorems on fractional and ordinary differential equations. Adv. Diff. Equat. 2019, 2019, 421. [Google Scholar] [CrossRef]
  42. Younis, M.; Singh, D.; Radenović, S.; Imdad, M. Convergence theorems for generalized contractions and applications. Filomat 2020, 34, 945–964. [Google Scholar] [CrossRef]
  43. Jiddah, J.A.; Shagari, M.S.; Imam, A.T. On fixed points of a general class of hybrid contractions with ulam-Type stability. Sahand Commun. Math. Anal. 2023, 20, 39–64. [Google Scholar]
  44. Youssri, Y.H.; Atta, A.G. Chebyshev Petrov–Galerkin method for nonlinear time-fractional integro-differential equations with a mildly singular kernel. J. Appl. Math. Comput. 2025, 1–21. [Google Scholar] [CrossRef]
  45. Youssri, Y.H.; Atta, A.G. Adopted Chebyshev Collocation Algorithm for Modeling Human Corneal Shape via the Caputo Fractional Derivative. Contemp. Math. 2025, 6, 1223–1238. [Google Scholar] [CrossRef]
  46. Diethelm, K.; Ford, N.J.; Freed, A.D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam. 2002, 29, 3–22. [Google Scholar] [CrossRef]
  47. Abd-Elhameed, W.M.; Youssri, Y.H. Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations. J. Comput. Appl. Math. 2018, 37, 2897–2921. [Google Scholar] [CrossRef]
  48. Atta, A.G.; Soliman, J.F.; Elsaeed, E.W.; Elsaeed, M.W.; Youssri, Y.H. Spectral collocation algorithm for the fractional Bratu equation via Hexic shifted Chebyshev polynomials. In Computational Methods for Differential Equations; University of Tabriz: Tabriz, Iran, 2024. [Google Scholar]
Figure 2. Residual errors of Example 12, (left) ( α = 1.7 ), (middle) ( α = 1.8 ), (right) ( α = 1.9 ).
Figure 2. Residual errors of Example 12, (left) ( α = 1.7 ), (middle) ( α = 1.8 ), (right) ( α = 1.9 ).
Fractalfract 09 00196 g002
Table 1. Absolute Errors of Example 10 for α = 5 4 .
Table 1. Absolute Errors of Example 10 for α = 5 4 .
n46810
Absolute Error 2.3 × 10 5 3.7 × 10 10 5.2 × 10 16 2.2 × 10 16
Table 2. Comparison between best errors of Example 10 for α = 5 4 .
Table 2. Comparison between best errors of Example 10 for α = 5 4 .
Present Method n = 10 Predictor-Corrector Method [46]
2.2 × 10 16 3.25 × 10 11
Table 3. Maximum pointwise error of Example 11 ( n = 16 ).
Table 3. Maximum pointwise error of Example 11 ( n = 16 ).
μ 1.951.91.851.81.75
[47] 3.47 × 10 14 2.52 × 10 13 5.78 × 10 12 2.71 × 10 11 3.25 × 10 11
Our Method 5.72 × 10 15 2.78 × 10 15 2.51 × 10 14 7.64 × 10 14 2.22 × 10 14
Table 4. Maximum residual errors of Example 12 for different values of α .
Table 4. Maximum residual errors of Example 12 for different values of α .
n371115
α = 1.9 3.4 × 10 4 7.2 × 10 9 2.6 × 10 14 2.2 × 10 17
α = 1.8 5.3 × 10 4 8.1 × 10 9 3.8 × 10 14 2.2 × 10 17
α = 1.7 2.7 × 10 4 6.3 × 10 9 4.7 × 10 14 2.2 × 10 17
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Albeladi, G.; Gamal, M.; Youssri, Y.H. G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications. Fractal Fract. 2025, 9, 196. https://doi.org/10.3390/fractalfract9030196

AMA Style

Albeladi G, Gamal M, Youssri YH. G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications. Fractal and Fractional. 2025; 9(3):196. https://doi.org/10.3390/fractalfract9030196

Chicago/Turabian Style

Albeladi, Ghadah, Mohamed Gamal, and Youssri Hassan Youssri. 2025. "G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications" Fractal and Fractional 9, no. 3: 196. https://doi.org/10.3390/fractalfract9030196

APA Style

Albeladi, G., Gamal, M., & Youssri, Y. H. (2025). G-Metric Spaces via Fixed Point Techniques for Ψ-Contraction with Applications. Fractal and Fractional, 9(3), 196. https://doi.org/10.3390/fractalfract9030196

Article Metrics

Back to TopTop