On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

Etemad, Sina; Ntouyas, Sotiris K.; Stamova, Ivanka; Tariboon, Jessada

doi:10.3390/fractalfract8040236

Open AccessArticle

On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

¹

Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 3751-71379, Iran

²

Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

³

Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA

⁴

Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

^*

Author to whom correspondence should be addressed.

Fractal Fract. 2024, 8(4), 236; https://doi.org/10.3390/fractalfract8040236

Submission received: 17 February 2024 / Revised: 13 April 2024 / Accepted: 15 April 2024 / Published: 17 April 2024

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

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Abstract

:

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned

(p; q)

-difference Navier problems in two single-valued and set-valued versions. We use the main properties of the

(p; q)

-operators in this regard. Application of the fixed points of the

ρ

-

θ

-contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential

(p; q)

-difference Navier problems.

Keywords:

endpoint; existence theorem; fixed point; navier problem; post-quantum derivative; set-valued function

MSC:

05A30; 26A33; 26E25; 34A12; 39A13

1. Introduction

An important part of mathematics, which focuses on operators of arbitrary real orders and is known as fractional calculus, has brought significant theoretical and practical achievements and results in various fields of engineering and modeling. The main reason for emphasizing the importance of the results of this field can be found in various studies that have been conducted in recent years on topics such as the solutions of fractional differential equations (continuous and discrete) and their solution techniques and related algorithms. For example, fractional order techniques and models are abundantly seen in mathematical structures defined in economics, medical simulations, physics, image processing, clinical disciplines, etc., [1,2,3]. If we want to discuss examples, we can point out to the exact results of fractional algorithms and methods that clearly show their power in finding approximate numerical solutions and even the existence of analytical solutions of fractional equations. That is why fractional order modeling replaces integer order modeling and simulate solutions with the least error. We recommend some sources in this regard for finding more information [4,5,6,7,8,9,10].

In 1910, a mathematician named Jackson [11,12] began a systematic and classified study of quantum calculus, which was later abbreviated as

q

-calculus. In this new calculus, most of the basic operators are defined without using the concept of limit, an idea that opens a new path in mathematics. In the first step,

q

-calculus established its place in physics. For example, Fock [13] defined a kind of quantum difference equations and proved that hydrogen atoms are symmetric. Mathematicians then investigated various properties of

q

-series and

q

-operators, which led to the development of new concepts in the theory of quantum mechanics, orthogonal polynomials and hypergeometric functions and combinatorics [14,15]. Moreover, see other papers in this regard such as [16,17,18,19,20].

As the quantum calculus continued to develop and expand, a new calculus emerged that depends on two parameters

0 < q < p \leq 1

and is known as post-quantum calculus or

(p; q)

-calculus. Of course, this new calculus is not a real generalization of

q

-calculus and it cannot be obtained by placing

\frac{q}{p}

instead of

q

, but its rules and concepts are defined in such a way that assuming

p = 1

, we can achieve the concepts of

q

-calculus.

The early and fundamental studies of

(p; q)

-calculus began with an article by Chakrabarti and Jagannathan [21]. Hypergeometric series [22], approximation methods [23,24], Bézier surfaces and curves [25], physical models [26], Lie groups are some of those theories in which

(p; q)

-operators play an important role. In the following, Sadjang [27] proved the fundamental theorems of

(p; q)

-calculus and introduced the Taylor’s

(p; q)

-formula. In 2018 and 2019, Cheng et al. [28] and Milovanovic et al. [29] extended the

(p; q)

-Gamma and

(p; q)

-Beta functions, respectively. Soontharanon et al. combined

(p; q)

-calculus with fractional calculus and established fractional

(p; q)

-operators and their properties [30]. In the last three or four years, mathematicians have expanded their studies to the areas of existence theory, and with the help of the important tool of fixed point, they have proved the uniqueness of the solution for various types of post-quantum boundary value problems (BVPs) in the form of fractional

(p; q)

-difference equations.

Soontharanon et al. [31], in 2020, continued their studies on solutions of an

r

-th order

(p; q)

-integro-difference problem of the Riemann-Liouville-like type with the

(p; q)

-Robin conditions given by

\{\begin{matrix} {}^{R}D_{(p; q)}^{r_{0}} y (t) = E (t, y (t), {}^{R}I_{(p; q)}^{r_{1}} (f y) (t), {}^{R}D_{(p; q)}^{r_{2}} y (t)), r_{1}, r_{2} \in (0, 1], t \in I_{(p; q)}^{T}, r_{0} \in (1, 2], \\ α_{1} y (a) + α_{2} {}^{R}D_{(p; q)}^{r_{3}} y (a) = h_{1} (y (t)), r_{3} \in (0, 1], α_{1}, α_{2} \in R^{+}, \\ β_{1} y (\frac{T}{p}) + β_{2} {}^{R}D_{(p; q)}^{r_{3}} y (\frac{T}{p}) = h_{2} (y (t)), β_{1}, β_{2} \in R^{+}, 0 < q < p \leq 1, \end{matrix}

with

a \in I_{(p; q)}^{T} - \{0, \frac{T}{p}\}

so that

I_{(p; q)}^{T} : = \{\frac{q^{j}}{p^{j + 1}} T : j \in N_{0}\} \cup {0}

. Also, by assuming

A = R^{3}

, they consider the nonlinear continuous function

E : I_{(p; q)}^{T} \times A \to R

and continuous functions

h_{1}, h_{2} : C (I_{(p; q)}^{T}, R) \to R

and

f : I_{(p; q)}^{T} \times I_{(p; q)}^{T} \to [0, \infty)

so that

{}^{R}I_{(p; q)}^{r_{1}} (f y) (t) = \frac{1}{Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} - 1)} f (t, v) y (\frac{v}{p^{r_{1} - 1}}) d_{(p; q)} v .

Moreover,

{}^{R}D_{(p; q)}^{r *}

is the

r^{*}

-th order

(p; q)

-derivative of the Riemann-Liouville-like type with

r^{*} = r_{i} (i = 0, 2, 3)

.

Neang et al. [32] conducted another analysis on the existence results for a nonlinear

r

-th order

(p; q)

-difference problem of the Caputo-like type with the separated boundary conditions, formulated by

\{\begin{matrix} {}^{c}D_{(p; q)}^{r} y (t) = E (t, y (p^{r} t)), r \in (1, 2], 0 \leq t \leq \frac{T}{p^{r}}, \\ α_{1} y (0) + β_{1} D_{(p; q)} y (0) = γ_{1} y (w_{1}), α_{1}, β_{1}, γ_{1} \in R, \\ α_{2} y (T) + β_{2} D_{(p; q)} y (\frac{T}{p}) = γ_{2} y (w_{2}), α_{2}, β_{2}, γ_{2} \in R, 0 < q < p \leq 1, \end{matrix}

so that

D_{(p; q)}

and

{}^{c}D_{(p; q)}^{r}

are the 1st order and

r

-th order

(p; q)

-difference and

(p; q)

-derivative of the Caputo-like type, respectively, and

E \in C ([0, \frac{T}{p^{r}}] \times R, R)

.

Once again in 2022, the same authors [33] defined a new function

E \in C ([0, T] \times R, R)

to simplify the domain of it, and to study the existence theorems, modeled the

r

-th order

(p; q)

-difference problem of the Caputo-like type as

\{\begin{matrix} {}^{c}D_{(p; q)}^{r} y (t) = E (p^{r} t, y (p^{r} t)), r \in (1, 2], 0 \leq t \leq T, \\ α_{1} y (0) + α_{2} D_{(p; q)} y (0) = α_{3}, α_{1}, α_{2}, α_{3} \in R, \\ α_{4} y (T) + α_{5} D_{(p; q)} y (p T) = α_{6}, α_{4}, α_{5}, α_{6} \in R, 0 < q < p \leq 1, \end{matrix}

with the above notations and

(p; q)

-operators. For more information on newly conducted studies in the context of

(p; q)

-calculus, refer to [34,35,36,37,38,39].

In order to fully understand the nature and dynamics of phenomena in the real world, modern science needs mathematical models of these phenomena and processes. Based on such exact mathematical models that are defined by operators and other mathematical concepts, we can examine the dynamical and behavioral structure of various engineering, mechanical and physical systems in laboratory environments and generalize the results to real dimensions in the surrounding world [40,41,42].

One of the important mathematical models that has attracted the attention of engineers in recent years and is used for technical simulations in advanced and complex structures is the mathematical dynamical model based on elastic beam. In huge engineering structures such as building structures, towers, bridges, aviation industry, giant ocean-going ships and spaceships, the application of elastic beam technology is considered one of the basic necessities. Based on this, from a mathematical point of view, the boundary value problem including a fourth-order Navier differential equation and two-point boundary conditions was modeled by Reiss et al. [43] in 1976, who studied the dynamic behavior of elastic beam changes, and its form is as follows

\{\begin{matrix} y^{(4)} (t) = E (t, y (t), y^{″} (t)), t \in I : = [0, 1], \\ y (0) = y (1) = 0 = y^{″} (0) = y^{″} (1), \end{matrix}

(1)

provided that the source function

E : I \times R^{2} \to R

is continuous. In 1986, Aftabizadeh [44] transformed (1) into a 2-

nd

order integro–differential equation by assuming E to be a bounded nonlinear function, and proved the existence theorems under the Schauder’s fixed point theorem. In 1997, Ma et al. [45] studied the Navier problem (1) and analyzed it based on the lower and upper solutions. Bai et al. [46], in 2004, adopted a monotone technique for the lower and upper solutions of the given beam elastic problem (1). Bachar et al. [47] extended the fractional version of the beam elastic problem (1) of the Riemann-Liouville type and completed their theoretical analysis on the existence and positivity of the unique solutions of the fractional Navier system given by

\{\begin{matrix} {}^{R}D^{r_{1}} ({}^{R}D^{r_{2}} y) (t) = E (t, y (t), {}^{R}D^{r_{2}} y (t)), \\ y (0) = y (1) = 0 = {}^{R}D^{r_{2}} y (0) = {}^{R}D^{r_{2}} y (1), \end{matrix}

(2)

provided

r_{1}, r_{2} \in (1, 2]

,

{}^{R}D^{r_{1}}

and

{}^{R}D^{r_{2}}

specify the fractional derivatives of the Riemann-Liouville type and

E \in C (I \times R^{2}, R)

. If

r_{1} = r_{2} = 2

, then (2) reduces to integer order problem (1).

In 2021, Etemad et al. [48] discussd a new sequential generalized fractional

q

-Navier problem given by

\{\begin{matrix} {}^{c}D_{(q)}^{r_{1}} ({}^{c}D_{(q)}^{r_{2}} y) (t) = E (t, y (t), {}^{c}D_{(q)}^{r_{2}} y (t)), t \in I : = [0, 1], q \in (0, 1), \\ γ y (0) = δ y (1) = λ {}^{c}D_{(q)}^{r_{2}} y (0) = β {}^{c}D_{(q)}^{r_{2}} y (1) = 0, \end{matrix}

(3)

so that

r_{1} \in (1, 2]

,

r_{2} \in (1, 2]

and

γ, δ, λ, β \in R^{+}

. Moreover, here

{}^{c}D_{(q)}^{(\cdot)}

is the fractional

q

-derivative of the Caputo-like type.

Based on the integer order Navier Equation (1) and other aforementioned models, we inspired to study a new sequential

(p; q)

-model of the elastic beam with

(p; q)

-Navier difference equation of the Caputo-like type

\{\begin{matrix} {}^{c}D_{(p; q)}^{r_{1}} ({}^{c}D_{(p; q)}^{r_{2}} y) (t) = E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)), (t \in I_{(p; q)}^{T} = [0, \frac{T}{p}], 0 < q < p \leq 1), \\ β y (t) |_{t = 0} = λ y (t) |_{t = \frac{T}{p}} = δ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = 0} = γ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = \frac{T}{p}} = 0, \end{matrix}

(4)

and the

(p; q)

-Navier difference inclusion of the Caputo-like type

\{\begin{matrix} {}^{c}D_{(p; q)}^{r_{1}} ({}^{c}D_{(p; q)}^{r_{2}} y) (t) \in E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)), (t \in I_{(p; q)}^{T} = [0, \frac{T}{p}], 0 < q < p \leq 1), \\ β y (t) |_{t = 0} = λ y (t) |_{t = \frac{T}{p}} = δ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = 0} = γ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = \frac{T}{p}} = 0, \end{matrix}

(5)

where

T > 0

,

r_{1}, r_{2} \in (1, 2]

,

β, λ, δ, γ \in R^{+}

, and the

(p; q)

-derivative of the Caputo-like type is denoted by

{}^{c}D_{(p; q)}^{(\cdot)}

. We consider

E : [0, \frac{T}{p}] \times R^{2} \to R

and

E : [0, \frac{T}{p}] \times R^{2} \to P (R)

as the continuous single-valued arbitrary function and multivalued arbitrary function, respectively, with some conditions.

The main contribution of this paper can be expressed as follows: here, we have extended the standard model of the elastic beam equation to two fractional structures of Navier

(p; q)

-difference equation and

(p; q)

-difference inclusion for the first time, and aslo, we have presented our numerical and theoretical results to study the existence of solutions for new sequential

(p; q)

-models (4) and (5) of the elastic beam. It is clear that if

r_{1} = r_{2} \to 2

,

p = 1

,

q \to 1

and

β = λ = δ = γ = 1

, we obtain the standard Navier model (1) of the elastic beam. Unlike many similar papers in the field of existence theory, we use a set of non-decreasing mappings and special contractions and endpoints to prove the existence of the solutions of the above two systems. The presented model of elastic beam based on the

(p; q)

-operators helps us to study some behaviors of fractional difference systems without the limit notion, and this can be an important advantage for the fast and easy simulation of the real phenomena in the context of discrete-type

(p; q)

-calculus.

We have prepared the next sections of the paper as follows: Section 2 aims to recall some basic notions of fractional

q

-calculus. Section 3 begins with a proposition for computing some

(p; q)

-integrals, and then, continues with a lemma which formulates the solution of the generalized sequential

(p; q)

-Navier problems (4) and (5) in the form of an

(p; q)

-integral equation. In the following, this section uses the well known fixed point theorem attributed to Krasnoselskii [49] and a special set of the operators proposed by Samet et al. [50] to study the existence theory for the solutions of single-valued operators. Section 4 considers the generalized sequential

(p; q)

-difference Navier inclusion problem (5) and examines the same existence theory for the solutions of (5), but this time, by applying the methods proposed by Mohammadi et al. [51] and also, by the approximate endpoint property. In Section 5, we assign our focus to validate the theorems proved in the previous Section 3 and Section 4. The last section concludes our study by giving some remarks and future ideas.

2. Preliminaries

In the next three subsections, we are going to state some definitions, lemmas and theorems on the context of

q

-calculus,

(p; q)

-calculus, and fixed point theory as a reminder. Throughout this paper, let

q \in (0, 1)

and

0 < q < p \leq 1

.

2.1. `q`-Calculus

The

q

-power function is the

q

-analoge of

{(s_{1} - s_{2})}^{m}

which is defined as

{(s_{1} - s_{2})}_{(q)}^{(0)} = 1, {(s_{1} - s_{2})}_{(q)}^{(m)} = \prod_{a = 0}^{m - 1} (s_{1} - s_{2} q^{a}), (s_{1}, s_{2} \in R, m \in N_{0}),

Ref. [52]. If

m = r \in R

, then generally we have

{(s_{1} - s_{2})}_{(q)}^{(r)} = s_{1}^{r} \prod_{a = 0}^{\infty} \frac{1 - (\frac{s_{2}}{s_{1}}) q^{a}}{1 - (\frac{s_{2}}{s_{1}}) q^{r + a}},

(6)

for

s_{1} \neq 0

. In a special case,

{(s_{1})}_{(q)}^{(r)} = s_{1}^{r}

if

s_{2} = 0

[52]. The

q

-number

{[s]}_{(q)}

and

q

-Gamma function

Γ_{(q)} (\cdot)

, for

s \in R

and

r \in R ∖ Z^{\leq 0}

, are given by

{[0]}_{(q)} = 0, {[s]}_{(q)} = q^{s - 1} + \dots + q + 1 = \frac{1 - q^{s}}{1 - q} (s \neq 0), Γ_{(q)} (r) = \frac{{(1 - q)}_{(q)}^{(r - 1)}}{{(1 - q)}^{r - 1}} .

(7)

Moreover,

Γ_{(q)} (r + 1) = {[r]}_{(q)} Γ_{(q)} (r)

[52].

Definition 1

([53]). The

q

-derivative of the function

y

is defined as

D_{(q)} y (t) = {[\frac{d}{d t}]}_{(q)} y (t) = \frac{y (t) - y (qt)}{(1 - q) t} .

(8)

For the highr orders, we define

D_{(q)}^{m} y (t) = D_{(q)} (D_{(q)}^{m - 1} y (t))

for each

m \in N

and also, define

D_{(q)}^{0} y (t) = y (t)

[53].

Definition 2

([54,55]). The

r

-th order

q

-integral of the Riemann-Liouville-like type for the function

y \in C ([0, + \infty), R)

is given by

{}^{R}I_{(q)}^{r} y (t) = \{\begin{matrix} \frac{1}{Γ_{(q)} (r)} \int_{0}^{t} {(t - qv)}_{(q)}^{(r - 1)} y (v) d_{(q)} v, & r > 0, \\ y (t), & r = 0, \end{matrix}

(9)

provided the integral converges.

Definition 3

([54,55]). Let

λ = [r] + 1

. The

r

-th order

q

-derivative of the Caputo-like type for

y \in C^{(λ)} ([0, + \infty), R)

is defined by

{}^{c}D_{(q)}^{r} y (t) = \frac{1}{Γ_{(q)} (λ - r)} \int_{0}^{t} {(t - qv)}_{(q)}^{(λ - r - 1)} D_{(q)}^{λ} y (v) d_{(q)} v,

(10)

provided the integral converges.

2.2. (`p`; `q`)-Calculus

All definitions of this subsection can be transformed into the aforementioned definitions in the previous subsections if

p = 1

.

The

(p; q)

-power function is the

(p; q)

-analogue of

{(s_{1} - s_{2})}^{m}

which is given by

{(s_{1} - s_{2})}_{(p; q)}^{(0)} = 1, {(s_{1} - s_{2})}_{(p; q)}^{(m)} = \prod_{a = 0}^{m - 1} (s_{1} p^{a} - s_{2} q^{a}), (s_{1}, s_{2} \in R, m \in N_{0}),

Ref. [30]. If

m = r \in R

, then generally, we have

{(s_{1} - s_{2})}_{(p; q)}^{(r)} = s_{1}^{r} \prod_{a = 0}^{\infty} \frac{1}{p^{r}} (\frac{1 - (\frac{s_{2}}{s_{1}}) {(\frac{q}{p})}^{a}}{1 - (\frac{s_{2}}{s_{1}}) {(\frac{q}{p})}^{r + a}}),

(11)

for

s_{1} \neq 0

. Also, if

s_{2} = 0

, then

{(s_{1})}_{(p; q)}^{(r)} = \frac{1}{p^{r}} s_{1}^{r}

[30]. The

(p; q)

-number

{[s]}_{(p; q)}

and

(p; q)

-Gamma function

Γ_{(p; q)} (\cdot)

, for

s \in R

and

r \in R ∖ Z^{\leq 0}

, are given by

{[0]}_{(p; q)} = 0, {[s]}_{(p; q)} = p^{s - 1} {[s]}_{\frac{q}{p}} = \frac{p^{s} - q^{s}}{p - q}, Γ_{(p; q)} (r) = \frac{{(p - q)}_{(p; q)}^{(r - 1)}}{{(p - q)}^{r - 1}},

(12)

and also,

Γ_{(p; q)} (r + 1) = {[r]}_{(p; q)} Γ_{(p; q)} (r)

[30]. Furthermore, the

(p; q)

-Beta function

B_{(p; q)} (\cdot, \cdot)

is defined as

B_{(p; q)} (r, \tilde{r}) = \int_{0}^{1} v^{r - 1} {(1 - qv)}_{(p; q)}^{(\tilde{r} - 1)} d_{(p; q)} v = \frac{Γ_{(p; q)} (r) Γ_{(p; q)} (\tilde{r})}{Γ_{(p; q)} (r + \tilde{r})} p^{\frac{1}{2} (\tilde{r} - 1) (2 r + \tilde{r} - 2)}, r, \tilde{r} > 0 .

(13)

Definition 4

([30]). The

(p; q)

-derivative of the function

y

is given by

D_{(p; q)} y (t) = \frac{y (pt) - y (qt)}{(p - q) t} .

(14)

If

p = 1

, then

D_{(1; q)} y (t) = D_{(q)} y (t)

, and if

q \to 1

, then

D_{(1; q \to 1)} y (t) = y^{'} (t)

.

Definition 5

([30]). The

(p; q)

-integral of

y \in C ([0, T], R)

is defined by

I_{(p; q)} y (t) = \int_{0}^{t} y (v) d_{(p; q)} v = (p - q) t \sum_{a = 0}^{\infty} \frac{q^{a}}{p^{a + 1}} y [\frac{q^{a}}{p^{a + 1}} t] .

Definition 6

([30]). The

r

-th order

(p; q)

-integral of the Riemann-Liouville-like type for the function

y \in C ([0, T], R)

is defined by

{}^{R}I_{(p; q)}^{r} y (t) = \{\begin{matrix} \frac{1}{Γ_{(p; q)} (r) p^{(\binom{r}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r - 1)} y [\frac{v}{p^{r - 1}}] d_{(p; q)} v, & r > 0, \\ y (t), & r = 0, \end{matrix}

(15)

if the integral converges.

If

p = 1

, then

{}^{R}I_{(1; q)}^{r} y (t) = {}^{R}I_{(q)}^{r} y (t)

which has been defined in Definition 2.

Definition 7

([30]). Let

λ = [r] + 1

. The

r

-th order

(p; q)

-derivative of the Caputo-like type for

y \in C^{(λ)} ([0, T], R)

is defined by

{}^{c}D_{(p; q)}^{r} y (t) = {}^{R}I_{(p; q)}^{λ - r} D_{(p; q)}^{λ} y (t) = \frac{1}{Γ_{(p; q)} (λ - r) p^{(\binom{λ - r}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(λ - r - 1)} D_{(p; q)}^{λ} y [\frac{v}{p^{λ - r - 1}}] d_{(p; q)} v,

(16)

if the integral converges.

If

p = 1

, then

{}^{c}D_{(1; q)}^{r} y (t) = {}^{c}D_{(q)}^{r} y (t)

which has been defined in Definition 3.

In the following, some important properties are recalled taken from [30].

Lemma 1

([30]). Let

r, \tilde{r} > 0

. Then

$(A (p; q)) {}^{R}I_{(p; q)}^{r} [{}^{R}I_{(p; q)}^{\tilde{r}} y (t)] = {}^{R}I_{(p; q)}^{\tilde{r}} [{}^{R}I_{(p; q)}^{r} y (t)] = {}^{R}I_{(p; q)}^{r + \tilde{r}} y (t)$ .
$(B (p; q)) {}^{c}D_{(p; q)}^{r} [{}^{R}I_{(p; q)}^{r} y (t)] = y (t)$ .

Lemma 2

([30]). Let

r, \tilde{r} > 0

and

y (t) = t^{\tilde{r}}

. Then

$(C (p; q)) {}^{R}I_{(p; q)}^{r} y (t) = \frac{Γ_{(p; q)} (\tilde{r} + 1)}{Γ_{(p; q)} (\tilde{r} + r + 1)} t^{\tilde{r} + r}$ .
$(D (p; q)) {}^{c}D_{(p; q)}^{r} y (t) = p^{r} \frac{Γ_{(p; q)} (\tilde{r} + 1)}{Γ_{(p; q)} (\tilde{r} - r + 1)} t^{\tilde{r} - r}, \tilde{r} > r$ .
$(E (p; q)) \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r - 1)} v^{\tilde{r}} d_{(p; q)} v = B_{(p; q)} (\tilde{r} + 1, r) t^{r + \tilde{r}} .$

Theorem 1

([30]). Let

λ = [r] + 1

. Then

{}^{R}I_{(p; q)}^{r} [{}^{c}D_{(p; q)}^{r} y (t)] = y (t) - \sum_{a = 0}^{λ - 1} \frac{D_{(p; q)}^{a} y (0)}{Γ_{(p; q)} (a + 1) p^{(\binom{r}{2})}} t^{a} .

In the simplified form, we have

{}^{R}I_{(p; q)}^{r} [{}^{c}D_{(p; q)}^{r} y (t)] = y (t) + k_{0}^{*} + k_{1}^{*} t + \dots + k_{λ - 1}^{*} t^{λ - 1},

where

k_{a}^{*} \in R; a = 0, 1, \dots, λ - 1

.

2.3. Fixed Point Theory

Now, we continue our preliminaries based on fixed point theory. We first begin by introducing some needed collections of sets.

Remark 1.

Let

(X_{*}, ∥ \cdot ∥_{X_{*}})

be a normed space. The collections

P_{B} (X_{*}), P_{CL} (X_{*}), P_{CM} (X_{*})

and

P_{CX} (X_{*})

contain all bounded, closed, compact and convex sets in

X_{*}

, respectively.

By

Θ

, we consider a subcollection of all operators

θ : [0, \infty) \to [0, \infty)

(which are non-decreasing) with

\sum_{a = 1}^{\infty} θ^{a} (t) < \infty, θ (t) < t, for all t > 0 .

Definition 8

([50]). Let

E : X_{*} \to X_{*}

and

ρ : X_{*}^{2} \to R_{\geq 0}

. Then

(a): E is $ρ$ - $θ$ -contraction if

$ρ (y_{1}, y_{2}) d (E y_{1}, E y_{2}) \leq θ (d (y_{1}, y_{2})), \forall y_{1}, y_{2} \in X_{*} .$
(b): E is $ρ$ -admissible if $ρ (E y_{1}, E y_{2}) \geq 1$ whenever $ρ (y_{1}, y_{2}) \geq 1$ .

Definition 9

([56]). Let

E : X_{*} \to P (X_{*})

be a multi-valued function.

(c): $y \in X_{*}$ is an endpoint of $E$ if $E (y) = {y}$ .
(d): $E$ has an approximate endpoint property if

$inf_{y_{1} \in X_{*}} [sup_{y_{2} \in E (y_{1})} d (y_{1}, y_{2})] = 0 .$

Definition 10

([51]). Assume

E : X_{★} \to P_{CL, B} (X_{★})

,

ρ : X_{★}^{2} \to [0, + \infty)

and

θ \in Θ

. Let

H_{d}

be the Pompeiu-Hausdorff metric. Then

(e): $E$ is $ρ$ -admissible if for every $y_{1} \in X_{★}$ and $y_{2} \in E y_{1}$ ,

$ρ (y_{1}, y_{2}) \geq 1 \Rightarrow ρ (y_{2}, y_{3}) \geq 1, for all y_{3} \in E y_{2} .$
(f): $E$ is an $ρ$ - $θ$ -contraction if

$ρ (y_{1}, y_{2}) H_{d} (E y_{1}, E y_{2}) \leq θ (d (y_{1}, y_{2})), for all y_{1}, y_{2} \in X_{★} .$

In the following, we recall needed fixed point and endpoint theorems as a reminder.

Theorem 2

([50]). Let a metric space

(X_{*}, d)

be complete,

ρ : X_{*} \times X_{*} \to R

, and

θ \in Θ

. Let

E : X_{*} \to X_{*}

be

ρ

-

θ

-contraction. Moreover,

(1): E is $ρ$ -admissible on $X_{*}$ ;
(2): $\exists y_{0} \in X_{*}$ s.t. $ρ (y_{0}, E y_{0}) \geq 1$ ;
(3): For every sequence ${y_{i}}$ in $X_{*}$ with $y_{i} \to y$ , if $ρ (y_{i}, y_{i + 1}) \geq 1$ for all $i \geq 1$ , then $ρ (y_{i}, y) \geq 1$ for each $i \geq 1$ .

Then E has a fixed point.

Theorem 3

([49]). Let

Y \subseteq X_{*}

be a non-empty bounded, closed, convex set, and

E_{1}

and

E_{2}

be defined on Y so that

(1): $E_{1} y_{1} + E_{2} y_{2} \in Y, for all y_{1}, y_{2} \in Y$ ;
(2): The continuous function $E_{1}$ is compact;
(3): $E_{2}$ is contraction.

Then,

\exists y \in Y

so that

y = E_{1} y + E_{2} y

(Krasnoselskii’s fixed point theorem).

Theorem 4

([51]). Let a metric space

(X_{*}, d)

be complete,

θ \in Θ

, and

ρ : X_{*} \times A_{*} \to [0, \infty)

be strictly increasing. Let

E : X_{*} \to P_{CL, B} (X_{*})

be an

ρ

-

θ

-contraction. Moreover,

1.: $E$ is $ρ$ -admissible;
2.: $ρ (y_{0}, y_{1}) \geq 1$ for some $y_{0} \in A_{*}$ and $y_{1} \in E y_{0}$ ;
3.: For every sequence ${y_{i}}$ in $X_{*}$ with $ρ (y_{i}, y_{i + 1}) \geq 1$ and $y_{i} \to y$ for all $i \in N$ , there is a subsequence ${y_{i_{r}}}$ of ${y_{i}}$ so that $ρ (y_{i_{r}}, y) \geq 1$ for each $r \in N$ .

Then,

E

has a fixed point.

Theorem 5

([56]). Let a metric space

(X_{*}, d)

be complete. Moreover,

1.: The upper semi-continuous function $θ : [0, \infty) \to [0, \infty)$ is so that $\underset{t \to \infty}{lim inf} (t - θ (t)) > 0$ and $θ (t) < t$ for all $t > 0$ ;
2.: $E : X_{*} \to P_{CL, B} (X_{*})$ is so that $H_{d} (E y_{1}, E y_{2}) \leq θ (d (y_{1}, y_{2}))$ for each $y_{1}, y_{2} \in X_{*}$ .

Then

E

has a unique endpoint if and only if it has an approximate endpoint property.

3. On the Generalized $(p; q)$ -Difference Navier Problem (4)

To conduct an analysis on the solutions of the generalized sequential

(p; q)

-difference Navier problem (4), we introduce

X_{*} = \{y (t) : y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t) \in C_{R} (I_{(p; q)}^{T})\}

as a spece containing all real-valued functions on

I_{(p; q)}^{T}

, which is a Banach space with the norm

{∥ y ∥}_{X_{*}} = sup_{t \in I_{(p; q)}^{T}} | y (t) | + sup_{t \in I_{(p; q)}^{T}} | {}^{c}D_{(p; q)}^{r_{2}} y (t) |,

for each

y \in X_{*}

.

In the following, we provide a lemma under which the structure of the solutions related to the proposed generalized sequential

(p; q)

-difference Navier problem (4) is determined in the context of an integral equation.

Lemma 3.

Let

g \in C_{R} (I_{(p; q)}^{T})

,

r_{1} \in (1, 2]

,

r_{2} \in (1, 2]

and

β, λ, δ, γ \in R^{+}

. Then

y^{*}

is a solution to the generalized

(p; q)

-difference Navier problem

\{\begin{matrix} {}^{c}D_{(p; q)}^{r_{1}} ({}^{c}D_{(p; q)}^{r_{2}} y) (t) = g (t), (t \in [0, \frac{T}{p}], 0 < q < p \leq 1, T > 0), \\ β y (t) |_{t = 0} = λ y (t) |_{t = \frac{T}{p}} = δ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = 0} = γ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = \frac{T}{p}} = 0, \end{matrix}

(17)

if and only if it is satisfied the

(p; q)

-integral equation

\begin{matrix} y (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} g [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} g [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v . \end{matrix}

(18)

Proof.

Let

y *

be a solution of the generalized

(p; q)

-Navier problem (17). Simply, we see that

{}^{c}D_{(p; q)}^{r_{1}} ({}^{c}D_{(p; q)}^{r_{2}} y^{*}) (t) = g (t) .

Since

r_{1} \in (1, 2]

, by applying the

r_{1}

-th order

(p; q)

-integral of the Riemann-Liouville-like type, we get

{}^{c}D_{(p; q)}^{r_{2}} y * (t) = \frac{1}{Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v + c_{0}^{*} + c_{1}^{*} t,

for each the constants

c_{0}^{*}, c_{1}^{*} \in R

. The third condition, i.e.,

δ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = 0} = 0

, gives

c_{0}^{*} = 0

immediately. Thus, we have

{}^{c}D_{(p; q)}^{r_{2}} y * (t) = \frac{1}{Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v + c_{1}^{*} t .

(19)

Now, in view of (19) and by the fourth condition, i.e.,

γ {}^{c}D_{(p; q)}^{r_{2}} y (t) |_{t = \frac{T}{p}} = 0

, we obtain

\frac{γ}{Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v + γ c_{1}^{*} \frac{T}{p} = 0,

and accordingly,

c_{1}^{*} = - \frac{p}{T Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v .

(20)

In the following, by (20), the Equation (19) becomes

\begin{matrix} {}^{c}D_{(p; q)}^{r_{2}} y^{*} (t) & = \frac{1}{Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v . \end{matrix}

(21)

Similarly, since

r_{2} \in (1, 2]

, by taking the

r_{2}

-th order

(p; q)

-integral of the Riemann-Liouville-like type on (21), we may write

\begin{matrix} y * (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} g [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{p t^{1 + r_{2}}}{T Γ_{(p; q)} (2 + r_{2}) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v + c_{0}^{* *} + c_{1}^{* *} t, \end{matrix}

for each the constants

c_{0}^{* *}, c_{1}^{* *} \in R

. The first condition, i.e.,

β y (t) |_{t = 0} = 0

, yields

c_{0}^{* *} = 0

immediately. Hence,

\begin{matrix} y * (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} g [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{p t^{1 + r_{2}}}{T Γ_{(p; q)} (2 + r_{2}) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v + c_{1}^{* *} t . \end{matrix}

(22)

Next, the second condition, i.e.,

λ y (t) |_{t = \frac{T}{p}} = 0

, gives

\begin{matrix} c_{1}^{* *} & = - \frac{p}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} g [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{p^{2} {(\frac{T}{p})}^{1 + r_{2}}}{T^{2} Γ_{(p; q)} (2 + r_{2}) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v . \end{matrix}

Put

c_{1}^{* *}

into (22). Therefore,

\begin{matrix} y * (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} g [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} g [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} g [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

and this shows that

y *

is satisfied

(p; q)

-integral Equation (18). We can prove the converse by direct computation and this completes the proof. □

Based on the previous lemma, a new operator

G : X_{*} \to X_{*}

is defined by

\begin{matrix} (G y) & (t) = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} E [\frac{v}{p^{r_{1} + r_{2} - 1}}, y (\frac{v}{p^{r_{1} + r_{2} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y (\frac{v}{p^{r_{1} + r_{2} - 1}})] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} E [\frac{v}{p^{r_{1} + r_{2} - 1}}, y (\frac{v}{p^{r_{1} + r_{2} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y (\frac{v}{p^{r_{1} + r_{2} - 1}})] d_{(p; q)} v \\ + \frac{p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} E [\frac{v}{p^{r_{1} - 1}}, y (\frac{v}{p^{r_{1} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y (\frac{v}{p^{r_{1} - 1}})] d_{(p; q)} v . \end{matrix}

The function

y *

as a solution of the generalized

(p; q)

-difference Navier problem (4) is a fixed point of the newly defined operator G.

Before proving the main results, we recall a double

(p; q)

-integral in the following proposition which is used later.

Proposition 1

([30]). Let

r_{1}, r_{2} \in R

. Then

\begin{matrix} I & = \int_{0}^{t} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} {(t - qv)}_{(p; q)}^{(1 - r_{2})} {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} d_{(p; q)} w d_{(p; q)} v \\ = p^{(\binom{2 - r_{2}}{2}) + (\binom{r_{1} + r_{2} - 4}{2})} \frac{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{1} + r_{2} - 4)}{Γ_{(p; q)} (r_{1} - 1)} t^{r_{1} - 2} . \end{matrix}

Also, keep in mind the following notations for simplicity:

\begin{matrix} L_{1} & = \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} + \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + 1) Γ_{(p; q)} (r_{2} + 2)}, \\ L_{2} & = \frac{{(\frac{T}{p})}^{r_{1} - 2}}{Γ_{(p; q)} (r_{1} - 1)} + \frac{p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{1} + r_{2} + 1)} + \frac{[1 + Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2)] p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1} + 1)}, \end{matrix}

(23)

and

\begin{matrix} L_{3} & = \frac{{(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} + \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + 1) Γ_{(p; q)} (r_{2} + 2)}, \\ L_{4} & = \frac{p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{1} + r_{2} + 1)} + \frac{[1 + Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2)] p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1} + 1)} . \end{matrix}

(24)

We start our first existence theorem as follows, with the help of the

ρ

-

θ

-contractions and

ρ

-admissible functions.

Theorem 6.

Let

Λ : R^{2} \times R^{2} \to R

,

E \in C ([0, \frac{T}{p}] \times X_{*}^{2}, X_{*})

and

θ \in Θ

. Moreover,

$(P 1)$: For each $y_{1}, y_{2}, x_{1}, x_{2} \in X_{*}$ and $t \in [0, \frac{T}{p}]$ , we have

$| E (t, y_{1}, x_{1}) - E (t, y_{2}, x_{2}) | \leq \tilde{L} θ (| y_{1} - y_{2} | + | x_{1} - x_{2} |),$

and

$Λ ((y_{1} (t), x_{1} (t)), (y_{2} (t), x_{2} (t))) \geq 0,$

so that $\tilde{L} = \frac{1}{L_{1} + L_{2}}$ ;
$(P 2)$: Some $y_{0} \in X_{*}$ exists s.t. $\forall t \in [0, \frac{T}{p}]$ ,

$Λ ((y_{0} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{0} (t)), (G y_{0} (t), {}^{c}D_{(p; q)}^{r_{2}} (G y_{0} (t)))) \geq 0,$

and also, we have

$Λ ((y_{1} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{1} (t)), (y_{2} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{2} (t))) \geq 0,$

which gives

$Λ ((G y_{1} (t), {}^{c}D_{(p; q)}^{r_{2}} (G y_{1} (t))), (G y_{2} (t), {}^{c}D_{(p; q)}^{r_{2}} (G y_{2} (t)))) \geq 0,$

for each $y_{1}, y_{2} \in X_{*}$ and all $t \in [0, \frac{T}{p}]$ ;
$(P 3)$: For every sequence ${y_{i}}_{i \geq 1} \subseteq X_{*}$ converging to $y$ , the inequality

$Λ ((y_{i} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i} (t)), (y_{i + 1} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i + 1} (t))) \geq 0,$

gives

$Λ ((y_{i} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i} (t)), (y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))) \geq 0,$

for each i and $t \in [0, \frac{T}{p}]$ .

Then, the generalized sequential

(p; q)

-difference Navier problem (4) has a solution on

[0, \frac{T}{p}]

.

Proof.

Let

y_{1}, y_{2} \in X_{*}

be arbitrary so that

Λ ((y_{1} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{1} (t)), (y_{2} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{2} (t))) \geq 0,

for all

t \in [0, \frac{T}{p}]

. For simplicity in the computations, we put

\{\begin{matrix} E_{t, p}^{r_{1}, r_{2}} (y) = E [\frac{t}{p^{r_{1} + r_{2} - 1}}, y (\frac{t}{p^{r_{1} + r_{2} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y (\frac{t}{p^{r_{1} + r_{2} - 1}})], (t \in [0, \frac{T}{p}]), \\ E_{t, p}^{r_{1}} (y) = E [\frac{t}{p^{r_{1} - 1}}, y (\frac{t}{p^{r_{1} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y (\frac{t}{p^{r_{1} - 1}})], (t \in [0, \frac{T}{p}]) . \end{matrix}

(25)

By the hypotheses, we clearly have

\begin{matrix} | E_{t, p}^{r_{1}, r_{2}} & (y_{1}) - E_{t, p}^{r_{1}, r_{2}} (y_{2}) | = | E [\frac{t}{p^{r_{1} + r_{2} - 1}}, y_{1} (\frac{t}{p^{r_{1} + r_{2} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y_{1} (\frac{t}{p^{r_{1} + r_{2} - 1}})] \\ - E [\frac{t}{p^{r_{1} + r_{2} - 1}}, y_{2} (\frac{t}{p^{r_{1} + r_{2} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y_{2} (\frac{t}{p^{r_{1} + r_{2} - 1}})] | \\ \leq \tilde{L} θ (| y_{1} (\frac{t}{p^{r_{1} + r_{2} - 1}}) - y_{2} (\frac{t}{p^{r_{1} + r_{2} - 1}}) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{1} (\frac{t}{p^{r_{1} + r_{2} - 1}}) - {}^{c}D_{(p; q)}^{r_{2}} y_{2} (\frac{t}{p^{r_{1} + r_{2} - 1}}) |) \\ \leq \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) . \end{matrix}

(26)

Similarly,

\begin{matrix} | E_{t, p}^{r_{1}} (y_{1}) - E_{t, p}^{r_{1}} (y_{2}) | & \leq \tilde{L} θ (| y_{1} (\frac{t}{p^{r_{1} - 1}}) - y_{2} (\frac{t}{p^{r_{1} - 1}}) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{1} (\frac{t}{p^{r_{1} - 1}}) - {}^{c}D_{(p; q)}^{r_{2}} y_{2} (\frac{t}{p^{r_{1} - 1}}) |) \\ \leq \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) . \end{matrix}

Now, by (26), we may write

\begin{matrix} | G y_{1} (t) & - G y_{2} (t) | \\ \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | E_{v, p}^{r_{1}, r_{2}} (y_{1}) - E_{v, p}^{r_{1}, r_{2}} (y_{2}) | d_{(p; q)} v \\ + \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | E_{v, p}^{r_{1}, r_{2}} (y_{1}) - E_{v, p}^{r_{1}, r_{2}} (y_{2}) | d_{(p; q)} v \\ + \frac{| p^{2} {(\frac{T}{p})}^{r_{2} + 1} t | + | p T t^{r_{2} + 1} |}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} | E_{v, p}^{r_{1}} (y_{1}) - E_{v, p}^{r_{1}} (y_{2}) | d_{(p; q)} v \\ \leq \frac{\tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}})}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} d_{(p; q)} v \\ + \frac{pt \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}})}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} d_{(p; q)} v \\ + \frac{(| p^{2} {(\frac{T}{p})}^{r_{2} + 1} t | + | p T t^{r_{2} + 1} |) \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}})}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} d_{(p; q)} v \\ \leq \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) + \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + 1) Γ_{(p; q)} (r_{2} + 2)} \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) \\ = \tilde{L} L_{1} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) . \end{matrix}

Also, we have

\begin{matrix} | ({}^{c}D_{(p; q)}^{r_{2}} & G y_{1}) (t) - ({}^{c}D_{(p; q)}^{r_{2}} G y_{2}) (t) | \\ \leq \frac{{(\frac{T}{p})}^{r_{1} - 2}}{Γ_{(p; q)} (r_{1} - 1)} \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) + \frac{p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{1} + r_{2} + 1)} \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) \\ + \frac{[1 + Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2)] p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1} + 1)} \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) \\ = \tilde{L} L_{2} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) . \end{matrix}

These give

∥ G y_{1} - G y_{2} ∥_{X_{*}} \leq (L_{1} + L_{2}) \tilde{L} θ (∥ y_{1} - y_{2} ∥_{X_{*}}) = θ (∥ y_{1} - y_{2} ∥_{X_{*}})

. Now, define

ρ : X_{*} \times X_{*} \to [0, \infty)

by

ρ (y_{1}, y_{2}) = \{\begin{matrix} 1, if Λ ((y_{1} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{1} (t)), (y_{2} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{2} (t))) \geq 0, \\ 0, otherwise, \end{matrix}

for each

y_{1}, y_{2} \in X_{*}

. Then, by using these arbitrary members

y_{1}, y_{2} \in X_{*}

, we get

ρ (y_{1}, y_{2}) d (G y_{1}, G y_{2}) \leq θ (d (y_{1}, y_{2})) .

This inequality confirms that G is an

ρ

-

θ

-contraction. Moreover, G can be proved easily to be

ρ

-admissible and

ρ (y_{0}, G y_{0}) \geq 1

. Finally, consider

{y_{i}}_{i \geq 1} \subseteq X_{*}

converging to

y

and let

ρ (y_{i}, y_{i + 1}) \geq 1

for all i. Based on the definition of the non-negative function

ρ

, one can write

Λ ((y_{i} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i} (t)), (y_{i + 1} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i + 1} (t))) \geq 0 .

So the hypothesis of theorem implies that

Λ ((y_{i} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i} (t)), (y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))) \geq 0 .

Therefore,

ρ (y_{i}, y) \geq 1

for all i. Since all hypotheses of Theorem 2 are fulfilled, so

G (y * *) = y * * \in X_{*} .

That means that

y * *

is a solution of the generalized sequential

(p; q)

-difference Navier problem (4). □

The next existence theorem gets help from the standard contractions along with the compact operators used in the Krasnoselskii’s fixed point theorem.

Theorem 7.

Let

E \in C ([0, \frac{T}{p}] \times X_{*}^{2}, X_{*})

. Moreover,

$(P 4)$: $\exists k \in C ([0, \frac{T}{p}], R)$ s.t. $\forall t \in [0, \frac{T}{p}]$ and $y_{1}, y_{2}, x_{1}, x_{2} \in X_{*}$ ,

$| E (t, y_{1}, x_{1}) - E (t, y_{2}, x_{2}) | \leq k (t) (| y_{1} - y_{2} | + | x_{1} - x_{2} |);$
$(P 5)$: $\exists f \in C ([0, \frac{T}{p}], R^{+})$ and there is a non-decreasing function $θ \in C (R^{+}, R^{+})$ s.t. for all $t \in [0, \frac{T}{p}]$ and each $y_{1}, y_{2} \in X_{*}$ ,

$| E (t, y_{1}, y_{2}) | \leq f (t) θ (| y_{1} | + | y_{2} |) .$

Then, the generalized sequential

(p; q)

-difference Navier problem (4) has at least one solution if

L = ∥ k ∥ (L_{3} + L_{4}) < 1,

(27)

where

∥ k ∥ = {sup}_{t \in [0, \frac{T}{p}]} | k (t) |

and

L_{3}, L_{4}

are supposed in (24).

Proof.

Set

∥ f ∥ = {sup}_{t \in [0, \frac{T}{p}]} | f (t) |

and choose an approximate value

ε > 0

so that

ε \geq {θ (∥ y ∥}_{X_{*}}) ∥ f ∥ (L_{1} + L_{2}),

(28)

where

L_{1}

and

L_{2}

are assumed in (23), and also, define

Y_{ε} = {y \in X_{*} {: ∥ y ∥}_{X_{*}} \leq ε}

. This non-empty set is bounded, closed and convex in

X_{*}

. Two operators

G_{1}

and

G_{2}

can be defined on

Y_{ε}

as

(G_{1} y) (t) = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} E_{v, p}^{r_{1}, r_{2}} (y) d_{(p; q)} v,

and

\begin{matrix} (G_{2} y) (t) = & - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} E_{v, p}^{r_{1}, r_{2}} (y) d_{(p; q)} v \\ + \frac{p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} E_{v, p}^{r_{1}} (y) d_{(p; q)} v, \end{matrix}

for all

t \in [0, \frac{T}{p}]

. Put

\hat{θ} = {sup}_{y \in X_{*}} {θ (∥ y ∥}_{X_{*}})

. For each

y_{1}, y_{2} \in Y_{ε}

, the following inequalities are satisfied as follow

\begin{matrix} | (G_{1} y_{1} & + G_{2} y_{2}) (t) | \\ \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | E_{v, p}^{r_{1}, r_{2}} (y_{1}) | d_{(p; q)} v \\ + \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | E_{v, p}^{r_{1}, r_{2}} (y_{2}) | d_{(p; q)} v \\ + \frac{| p^{2} {(\frac{T}{p})}^{r_{2} + 1} t | + | p T t^{r_{2} + 1} |}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} | E_{v, p}^{r_{1}} (y_{2}) | d_{(p; q)} v \\ \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} f (v) θ (| y_{1} (v) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{1} (v) |) d_{(p; q)} v \\ + \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} f (v) θ (| y_{2} (v) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{2} (v) |) d_{(p; q)} v \\ + \frac{| p^{2} {(\frac{T}{p})}^{r_{2} + 1} t | + | p T t^{r_{2} + 1} |}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} f (v) θ (| y_{2} (v) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{2} (v) |) d_{(p; q)} v \\ \leq \hat{θ} ∥ f ∥ [\frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} + \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + 1) Γ_{(p; q)} (r_{2} + 2)}] = \hat{θ} ∥ f ∥ L_{1}, \end{matrix}

and

\begin{matrix} | ({}^{c}D_{(p; q)}^{r_{2}} G_{1} y_{1} + {}^{c}D_{(p; q)}^{r_{2}} G_{2} y_{2}) (t) | & \leq \hat{θ} ∥ f ∥ [\frac{{(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (r_{1} - 1)} + \frac{p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{1} + r_{2} + 1)} \\ + \frac{[1 + Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2)] p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1} + 1)}] \\ = \hat{θ} ∥ f ∥ L_{2} . \end{matrix}

Therefore,

∥ G_{1} y_{1} + G_{2} y_{2} ∥_{X_{*}} \leq \hat{θ} ∥ f ∥ (L_{1} + L_{2}) \leq ε

. That is,

(G_{1} y_{1} + G_{2} y_{2}) \in Y_{ε} .

The continuity property for the single-valued function E follows that

G_{1}

is continuous too. So, on the set

Y_{ε}

and for each member

y

in it, we estimate

\begin{matrix} | (G_{1} y) (t) | & \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \\ \times | E [\frac{v}{p^{r_{1} + r_{2} - 1}}, y (\frac{v}{p^{r_{1} + r_{2} - 1}}), {}^{c}D_{(p; q)}^{r_{2}} y (\frac{v}{p^{r_{1} + r_{2} - 1}})] | d_{(p; q)} v \\ \leq \frac{{(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} {∥ f ∥ θ (∥ y ∥}_{X_{*}}), \end{matrix}

and

\begin{matrix} | ({}^{c}D_{(p; q)}^{r_{2}} G_{1} y) (t) | & \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2} - 4) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2} - 4}{2}) + (\binom{2 - r_{2}}{2})}} \\ \times \int_{0}^{t} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} {(t - qv)}_{(p; q)}^{(1 - r_{2})} {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} \\ \times | E [\frac{w}{p^{r_{1} + r_{2} - 5}}, y (\frac{w}{p^{r_{1} + r_{2} - 5}}), {}^{c}D_{(p; q)}^{r_{2}} y (\frac{w}{p^{r_{1} + r_{2} - 5}})] | d_{(p; q)} w d_{(p; q)} v \\ \leq \frac{{(\frac{T}{p})}^{r_{1} - 2}}{Γ_{(p; q)} (r_{1} - 1)} {∥ f ∥ θ (∥ y ∥}_{X_{*}}) . \end{matrix}

Thus,

∥ G_{1} {y ∥}_{X_{*}} \leq [\frac{{(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} + \frac{{(\frac{T}{p})}^{r_{1} - 2}}{Γ_{(p; q)} (r_{1} - 1)}] ∥ f ∥ θ (ε) .

Immediately, and from the above inequality, one can find the uniform boundedness of

G_{1}

on

Y_{ε}

. To survey the compactness of

G_{1}

on

Y_{ε}

, let

t_{1}, t_{2} \in [0, \frac{T}{p}]

be chosen arbitrarily so that

t_{1} < t_{2}

. Then, by the notation (25), estimate

\begin{matrix} | (G_{1} y) & (t_{2}) - (G_{1} y) (t_{1}) | \\ = | \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t_{2}} {(t_{2} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} E_{v, p}^{r_{1}, r_{2}} (y) d_{(p; q)} v \\ - \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{1}}{2})}} \int_{0}^{t_{1}} {(t_{1} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} E_{v, p}^{r_{1}, r_{2}} (y) d_{(p; q)} v | \\ \leq | \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t_{1}} [{(t_{2} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} - {(t_{1} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)}] E_{v, p}^{r_{1}, r_{2}} (y) d_{(p; q)} v | \\ + | \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{1}}{2})}} \int_{t_{1}}^{t_{2}} {(t_{2} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} E_{v, p}^{r_{1}, r_{2}} (y) d_{(p; q)} v | \\ \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t_{1}} [{(t_{2} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} - {(t_{1} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)}] | E_{v, p}^{r_{1}, r_{2}} (y) | d_{(p; q)} v \\ + \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{1}}{2})}} \int_{t_{1}}^{t_{2}} {(t_{2} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | E_{v, p}^{r_{1}, r_{2}} (y) | d_{(p; q)} v \\ \leq ∥ f ∥ θ (ε) [\frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t_{1}} [{(t_{2} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} - {(t_{1} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)}] d_{(p; q)} v \\ + \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{1}}{2})}} \int_{t_{1}}^{t_{2}} {(t_{2} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} d_{(p; q)} v] . \end{matrix}

The last inequality (its right-hand side) is independent of

y

. So

| (G_{1} y) (t_{2}) - (G_{1} y) (t_{1}) |

tends to zero as

t_{1}

tends to

t_{2}

. Also,

\begin{matrix} | ({}^{c}D_{(p; q)}^{r_{2}} G_{1} y) & (t_{2}) - ({}^{c}D_{(p; q)}^{r_{2}} G_{1} y) (t_{1}) | \leq | \frac{1}{Γ_{(p; q)} (r_{1} + r_{2} - 4) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2} - 4}{2}) + (\binom{2 - r_{2}}{2})}} \\ \times \int_{0}^{t_{2}} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} {(t_{2} - qv)}_{(p; q)}^{(1 - r_{2})} {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} E_{w, p}^{r_{1}, r_{2}} (y) d_{(p; q)} w d_{(p; q)} v \\ - \frac{1}{Γ_{(p; q)} (r_{1} + r_{2} - 4) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2} - 4}{2}) + (\binom{2 - r_{2}}{2})}} \\ \times \int_{0}^{t_{1}} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} {(t_{1} - qv)}_{(p; q)}^{(1 - r_{2})} {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} E_{w, p}^{r_{1}, r_{2}} (y) d_{(p; q)} w d_{(p; q)} v | \\ \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2} - 4) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2} - 4}{2}) + (\binom{2 - r_{2}}{2})}} \\ \times \int_{0}^{t_{1}} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} [{(t_{2} - qv)}_{(p; q)}^{(1 - r_{2})} - {(t_{1} - qv)}_{(p; q)}^{(1 - r_{2})}] \\ \times {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} | E_{w, p}^{r_{1}, r_{2}} (y) | d_{(p; q)} w d_{(p; q)} v \\ + \frac{1}{Γ_{(p; q)} (r_{1} + r_{2} - 4) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2} - 4}{2}) + (\binom{2 - r_{2}}{2})}} \\ \times \int_{t_{1}}^{t_{2}} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} {(t_{2} - qv)}_{(p; q)}^{(1 - r_{2})} {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} | E_{w, p}^{r_{1}, r_{2}} (y) | d_{(p; q)} w d_{(p; q)} v \\ \leq ∥ f ∥ θ (ε) [\frac{1}{Γ_{(p; q)} (r_{1} + r_{2} - 4) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2} - 4}{2}) + (\binom{2 - r_{2}}{2})}} \\ \times \int_{0}^{t_{1}} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} [{(t_{2} - qv)}_{(p; q)}^{(1 - r_{2})} - {(t_{1} - qv)}_{(p; q)}^{(1 - r_{2})}] {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} d_{(p; q)} w d_{(p; q)} v \\ + \frac{1}{Γ_{(p; q)} (r_{1} + r_{2} - 4) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2} - 4}{2}) + (\binom{2 - r_{2}}{2})}} \\ \times \int_{t_{1}}^{t_{2}} \int_{0}^{\frac{v}{p^{1 - r_{2}}}} {(t_{2} - qv)}_{(p; q)}^{(1 - r_{2})} {(\frac{v}{p^{(1 - r_{2})}} - q w)}_{(p; q)}^{(r_{1} + r_{2} - 5)} d_{(p; q)} w d_{(p; q)} v] . \end{matrix}

Once again, the last inequality (its right-hand side) is independent of

y

and so,

| ({}^{c}D_{(p; q)}^{r_{2}} G_{1} y) (t_{2}) - ({}^{c}D_{(p; q)}^{r_{2}} G_{1} y) (t_{1}) | \to 0,

as

t_{1} \to t_{2}

. Therefore,

∥ (G_{1} y) (t_{2}) - (G_{1} y) (t_{1}) ∥_{X_{*}} \to 0,

as

t_{1} \to t_{2}

. Hence, the equi-continuity of the operator

G_{1}

is proved. By the Arzelá-Ascoli theorem, the compactness of

G_{1}

is to be held on

Y_{ε}

. Lastly, it is showed that

G_{2}

is a contraction. For each

y_{1}, y_{2} \in Y_{ε}

, we have

\begin{matrix} | (G_{2} & y_{1}) (t) - (G_{2} y_{2}) (t) | \\ \leq \frac{p | t |}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | E_{v, p}^{r_{1}, r_{2}} (y_{1}) - E_{v, p}^{r_{1}, r_{2}} (y_{2}) | d_{(p; q)} v \\ + \frac{p^{2} {(\frac{T}{p})}^{r_{2} + 1} | t | + p T | t^{r_{2} + 1} |}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} | E_{v, p}^{r_{1}} (y_{1}) - E_{v, p}^{r_{1}} (y_{2}) | d_{(p; q)} v \\ \leq \frac{p | t |}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \\ \times k (v) θ (| y_{1} (v) - y_{2} (v) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{1} (v) - {}^{c}D_{(p; q)}^{r_{2}} y_{2} (v) |) d_{(p; q)} v \\ + \frac{p^{2} {(\frac{T}{p})}^{r_{2} + 1} | t | + p T | t^{r_{2} + 1} |}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} \\ \times k (v) θ (| y_{1} (v) - y_{2} (v) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{1} (v) - {}^{c}D_{(p; q)}^{r_{2}} y_{2} (v) |) d_{(p; q)} v \\ \leq \frac{{(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} ∥ k ∥ ∥ y_{1} - y_{2} ∥_{X_{*}} + \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + 1) Γ_{(p; q)} (r_{2} + 2)} ∥ k ∥ ∥ y_{1} - y_{2} ∥_{X_{*}} \\ = ∥ k ∥ L_{3} {∥ y_{1} - y_{2} ∥}_{X_{*}}, \end{matrix}

and

\begin{matrix} | ({}^{c}D_{(p; q)}^{r_{2}} & G_{2} y_{1}) (t) - ({}^{c}D_{(p; q)}^{r_{2}} G_{2} y_{2}) (t) | \\ \leq \frac{p^{1 + r_{2}} | t^{1 - r_{2}} |}{T Γ_{(p; q)} (r_{1} + r_{2}) Γ_{(p; q)} (2 - r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \\ \times k (v) θ (| y_{1} (v) - y_{2} (v) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{1} (v) - {}^{c}D_{(p; q)}^{r_{2}} y_{2} (v) |) d_{(p; q)} v \\ + \frac{[p^{2 + r_{2}} {(\frac{T}{p})}^{r_{2} + 1} | t^{1 - r_{2}} | + p^{1 + r_{2}} T Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (2 - r_{2}) | t |]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} \\ \times k (v) θ (| y_{1} (v) - y_{2} (v) | + | {}^{c}D_{(p; q)}^{r_{2}} y_{1} (v) - {}^{c}D_{(p; q)}^{r_{2}} y_{2} (v) |) d_{(p; q)} v \\ \leq \frac{p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1) Γ_{(p; q)} (2 - r_{2})} ∥ k ∥ ∥ y_{1} - y_{2} ∥_{X_{*}} \\ + \frac{[1 + Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2)] p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1} + 1)} ∥ k ∥ ∥ y_{1} - y_{2} ∥_{X_{*}} \\ = ∥ k ∥ L_{4} {∥ y_{1} - y_{2} ∥}_{X_{*}} . \end{matrix}

Thus,

∥ G_{2} y_{1} - G_{2} y_{2} ∥_{X_{*}} \leq ∥ k ∥ (L_{3} + L_{4}) ∥ y_{1} - y_{2} ∥_{X_{*}} = L {∥ y_{1} - y_{2} ∥}_{X_{*}},

where

L < 1

, and under this Lipschitz constant,

G_{2}

is a contraction on

Y_{ε}

. The conclusion of Theorem 3 implies the existence of solution for the generalized sequential

(p; q)

-difference Navier problem (4). □

4. On the Generalized $(p; q)$ -Difference Navier Problem (5)

The existence theorems for the generalized sequential

(p; q)

-difference Navier inclusion problem (5) are established in this section.

For the generalized sequential

(p; q)

-difference Navier inclusion problem (5), we call

y \in C_{X_{*}} (I_{(p; q)}^{T}, X_{*})

as a solution if the given boundary conditions are satisfied for

y

and also, there is some

\bar{F} \in L^{1} (I_{(p; q)}^{T})

so that

\bar{F} (t) \in E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))

for almost all

t \in I_{(p; q)}^{T}

and

\begin{matrix} y (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \bar{F} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \bar{F} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} \bar{F} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

for all

t \in I_{(p; q)}^{T}

. The set of all selections of the multi-valued function

E

is given by

{(S)}_{E, y} = \{\bar{F} \in L^{1} (I_{(p; q)}^{T}) : \bar{F} (t) \in E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)) for all t \in I_{(p; q)}^{T}\},

for each

y \in X_{*}

. Moreover, the operator

F : X_{*} \to P (X_{*})

is considered as

F (y) = {ℏ \in X_{*} : there exists \bar{F} \in {(S)}_{E, y} : ℏ (t) = \overset{ˇ}{g} (t) for all t \in I_{(p; q)}^{T}},

(29)

where

\begin{matrix} \overset{ˇ}{g} (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \bar{F} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \bar{F} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} \bar{F} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v . \end{matrix}

For simplicity, put

α_{1} = ∥ α ∥ L_{1} and α_{2} = ∥ α ∥ L_{2} .

(30)

In this step, we are going to use the special multi-valued functions of the

ρ

-

θ

-contractive type and

ρ

-admissible type for proving the existence result related to solutions of the generalized sequential

(p; q)

-difference Navier inclusion problem (5).

Theorem 8.

Let

E : I_{(p; q)}^{T} \times X_{*}^{2} \to P_{CM} (X_{*})

. Moreover,

$(P 6)$: $E$ is bounded and integrable so that $E (\cdot, y_{1}, y_{2}) : I_{(p; q)}^{T} \to P_{CM}$ is measurable for each $y_{1}, y_{2} \in X_{*}$ ;
$(P 7)$: There are $α \in C (I_{(p; q)}^{T}, [0, \infty))$ and $θ \in Θ$ so that

$H_{d} (E (t, y_{1}, y_{2}), E (t, \tilde{y_{1}}, \tilde{y_{2}})) \leq α (t) (\frac{\tilde{L}}{∥ α ∥}) θ (| y_{1} - \tilde{y_{1}} | + | y_{2} - \tilde{y_{2}} |),$

(31)

for all $t \in I_{(p; q)}^{T}$ and each $y_{1}, y_{2}, \tilde{y_{1}}, \tilde{y_{2}} \in X_{*}$ , where ${sup}_{t \in I_{(p; q)}^{T}} | α (t) | = ∥ α ∥$ , $\tilde{L} = \frac{1}{L_{1} + L_{2}}$ and $L_{1}, L_{2}$ are the constants assumed in (23);
$(P 8)$: There is $Λ : R^{2} \times R^{2} \to R$ so that $Λ ((y_{1}, y_{2}), (\tilde{y_{1}}, \tilde{y_{2}})) \geq 0$ for each $y_{1}, y_{2}, \tilde{y_{1}}, \tilde{y_{2}} \in X_{*}$ ;
$(P 9)$: There is a sequence ${y_{i}}_{i \geq 1} \subseteq X_{*}$ converging to $y$ so that the inequality

$Λ ((y_{i} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i} (t)), (y_{i + 1} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i + 1} (t))) \geq 0, \forall t \in I_{(p; q)}^{T}, i \geq 1,$

implies the existence of a subsequence ${y_{i_{r}}}_{r \geq 1}$ of ${y_{i}}$ with

$Λ ((y_{i_{r}} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i_{r}} (t)), (y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))) \geq 0,$

for all $t \in I_{(p; q)}^{T}$ and $r \geq 1$ ;
$(P 10)$: There are $y_{0} \in X_{*}$ and $ℏ \in F (y_{0})$ so that

$Λ ((y_{0} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{0} (t)), (ℏ (t), {}^{c}D_{(p; q)}^{r_{2}} ℏ (t))) \geq 0,$

for all $t \in I_{(p; q)}^{T}$ , where $F : X_{*} \to P (X_{*})$ is defined by (29);
$(P 11)$: For each $y \in X_{*}$ and $ℏ \in F (y)$ such that

$Λ ((y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)), (ℏ (t), {}^{c}D_{(p; q)}^{r_{2}} ℏ (t))) \geq 0,$

there is some $\overset{ˇ}{g} \in F (y)$ so that

$Λ ((ℏ (t), {}^{c}D_{(p; q)}^{r_{2}} ℏ (t)), (\overset{ˇ}{g} (t), {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} (t))) \geq 0,$

for all $t \in I_{(p; q)}^{T}$ .

Then, the generalized sequential

(p; q)

-difference Navier inclusion problem (5) has a solution.

Proof.

It is known that a fixed point of

F : X_{*} \to P (X_{*})

is the solution of the generalized sequential

(p; q)

-difference Navier inclusion problem (5). The closed-valued multi-valued function

t \to E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))

is measurable for each

y \in X_{*}

. So,

E

has also a measurable selection; that is,

{(S)}_{E, y} \neq \emptyset

.

In the next step, we establish that

F (y) \subseteq X_{*}

is closed for each

y \in X_{*}

. Let

{y_{i}}_{i \geq 1}

be a sequence in

F (y)

so that

y_{i} \to y

. For every i,

\exists {\bar{F}}_{i} \in {(S)}_{E, y}

so that

\begin{matrix} y_{i} (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{i} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{i} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} {\bar{F}}_{i} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

for almost all

t \in I_{(p; q)}^{T}

. Note that

E

is compact-valued. Hence, a subsequence

{{\bar{F}}_{i}}_{i \geq 1}

exists so that converges to some

\bar{F} \in L^{1} (I_{(p; q)}^{T})

. Immediately, we find that

\bar{F} \in {(S)}_{E, y}

and

\begin{matrix} y_{i} (t) & \to y (t) \\ = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \bar{F} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} \bar{F} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} \bar{F} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

for all

t \in I_{(p; q)}^{T}

. Consequently,

y \in F (y)

and

F

possesses the closed values. Again, since

E

has the compact values, clearly one can show that

F (y)

is bounded for every

y \in X_{*}

.

In this step, we establish that

F

is an

ρ

-

θ

-contraction. First, define a non-negative function

ρ

on

X_{*} \times X_{*}

as

ρ (y, \tilde{y}) = \{\begin{matrix} 1 if Λ ((y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)), (\tilde{y} (t), {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t))) \geq 0, \\ 0 otherwise, \end{matrix}

for each

y, \tilde{y} \in X_{*}

. Let

y, \tilde{y} \in X_{*}

and

ℏ_{1} \in F (\tilde{y})

. Choose

{\bar{F}}_{1} \in {(S)}_{E, \tilde{y}}

so that

\begin{matrix} ℏ_{1} (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{1} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{1} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} {\bar{F}}_{1} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

for all

t \in I_{(p; q)}^{T}

. From (31), one may write

H_{d} (E (t, y, {}^{c}D_{(p; q)}^{r_{2}} y), E (t, \tilde{y}, {}^{c}D_{(p; q)}^{r_{2}} \tilde{y})) \leq α (t) (\frac{\tilde{L}}{∥ α ∥}) θ (| y - \tilde{y} | + | {}^{c}D_{(p; q)}^{r_{2}} y - {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} |),

for every

y, \tilde{y} \in X_{*}

with

Λ ((y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)), (\tilde{y} (t), {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t))) \geq 0,

for almost all

t \in I_{(p; q)}^{T}

. There is some

\overset{ˇ}{g} \in E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))

so that

| {\bar{F}}_{1} (t) - \overset{ˇ}{g} | \leq α (t) (\frac{\tilde{L}}{∥ α ∥}) θ (| y (t) - \tilde{y} (t) | + | {}^{c}D_{(p; q)}^{r_{2}} y (t) - {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t) |) .

Define

F : I_{(p; q)}^{T} \to P (X_{*})

by

F (t) = \{\overset{ˇ}{g} \in X_{*} : | {\bar{F}}_{1} (t) - \overset{ˇ}{g} | \leq α (t) (\frac{\tilde{L}}{∥ α ∥}) θ (| y (t) - \tilde{y} (t) | + | {}^{c}D_{(p; q)}^{r_{2}} y (t) - {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t) |)\},

for all

t \in I_{(p; q)}^{T}

. Note that

{\bar{F}}_{1}

and

ω = α (\frac{\tilde{L}}{∥ α ∥}) θ (| y - \tilde{y} | + | {}^{c}D_{(p; q)}^{r_{2}} y - {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} |)

are measurable. Therefore,

F (\cdot) \cap E (\cdot, y (\cdot), {}^{c}D_{(p; q)}^{r_{2}} y (\cdot))

is measurable. This time, we choose

{\bar{F}}_{2} \in E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))

so that for all

t \in I_{(p; q)}^{T}

,

| {\bar{F}}_{1} (t) - {\bar{F}}_{2} (t) | \leq α (t) (\frac{\tilde{L}}{∥ α ∥}) θ (| y (t) - \tilde{y} (t) | + | {}^{c}D_{(p; q)}^{r_{2}} y (t) - {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t) |) .

Consider

ℏ_{2} \in F (y)

as

\begin{matrix} ℏ_{2} (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{2} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{2} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} {\bar{F}}_{2} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

for all

t \in I_{(p; q)}^{T}

. Then,

\begin{matrix} | ℏ_{1} (t) & - ℏ_{2} (t) | \\ \leq \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | {\bar{F}}_{1} [\frac{v}{p^{r_{1} + r_{2} - 1}}] - {\bar{F}}_{2} [\frac{v}{p^{r_{1} + r_{2} - 1}}] | d_{(p; q)} v \\ + \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} | {\bar{F}}_{1} [\frac{v}{p^{r_{1} + r_{2} - 1}}] - {\bar{F}}_{2} [\frac{v}{p^{r_{1} + r_{2} - 1}}] | d_{(p; q)} v \\ + \frac{| p^{2} {(\frac{T}{p})}^{r_{2} + 1} t | + | p T t^{r_{2} + 1} |}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} | {\bar{F}}_{1} [\frac{v}{p^{r_{1} - 1}}] - {\bar{F}}_{2} [\frac{v}{p^{r_{1} - 1}}] | d_{(p; q)} v \\ \leq \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} ∥ α ∥ (\frac{\tilde{L}}{∥ α ∥}) θ (∥ y - \tilde{y} ∥_{X_{*}}) \\ + \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + 1) Γ_{(p; q)} (r_{2} + 2)} ∥ α ∥ (\frac{\tilde{L}}{∥ α ∥}) θ (∥ y - \tilde{y} ∥_{X_{*}}) \\ \leq [\frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + r_{2} + 1)} + \frac{2 {(\frac{T}{p})}^{r_{1} + r_{2}}}{Γ_{(p; q)} (r_{1} + 1) Γ_{(p; q)} (r_{2} + 2)}] ∥ α ∥ (\frac{\tilde{L}}{∥ α ∥}) θ (∥ y - \tilde{y} ∥_{X_{*}}) \\ = \tilde{L} L_{1} θ (∥ y - \tilde{y} ∥_{X_{*}}), \end{matrix}

and

\begin{matrix} | {}^{c}D_{(p; q)}^{r_{2}} ℏ_{1} (t) - {}^{c}D_{(p; q)}^{r_{2}} ℏ_{2} (t) | & \leq [\frac{{(\frac{T}{p})}^{r_{1} - 2}}{Γ_{(p; q)} (r_{1} - 1)} + \frac{p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{1} + r_{2} + 1)} \\ + \frac{[1 + Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2)] p^{r_{2}} {(\frac{T}{p})}^{r_{1}}}{Γ_{(p; q)} (2 - r_{2}) Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1} + 1)}] \\ \times ∥ α ∥ (\frac{\tilde{L}}{∥ α ∥}) θ (∥ y - \tilde{y} ∥_{X_{*}}) \\ = \tilde{L} L_{2} θ (∥ y - \tilde{y} ∥_{X_{*}}), \end{matrix}

for all

t \in I_{(p; q)}^{T}

. Thus,

\begin{matrix} ∥ ℏ_{1} - ℏ_{2} ∥_{X_{*}} = & sup_{t \in I_{(p; q)}^{T}} | ℏ_{1} (t) - ℏ_{2} (t) | + sup_{t \in I_{(p; q)}^{T}} | {}^{c}D_{(p; q)}^{r_{2}} ℏ_{1} (t) - {}^{c}D_{(p; q)}^{r_{2}} ℏ_{2} (t) | \\ \leq & (L_{1} + L_{2}) \tilde{L} θ (∥ y - \tilde{y} ∥_{X_{*}}) = θ (∥ y - \tilde{y} ∥_{X_{*}}) . \end{matrix}

Hence,

ρ (y, \tilde{y}) H_{d} (F (y) - F (\tilde{y})) \leq θ (∥ y - \tilde{y} ∥_{X_{*}}),

is fulfilled for each

y, \tilde{y} \in X_{*}

. This means that

F

is an

ρ

-

θ

-contraction.

Now, let

y \in X_{*}

and

\tilde{y} \in F (y)

so that

ρ (y, \tilde{y}) \geq 1

and

Λ ((y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)), (\tilde{y} (t), {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t))) \geq 0 .

There is some

\overset{ˇ}{g} \in F (\tilde{y})

so that

Λ ((\tilde{y} (t), {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t)), (\overset{ˇ}{g} (t), {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} (t))) \geq 0 .

This implies that

ρ (\tilde{y}, \overset{ˇ}{g}) \geq 1

establishing the fact that

F

is

ρ

-admissible. Next, consider

y_{0} \in X_{*}

and

\tilde{y} \in F (y_{0})

so that

\forall t \in I_{(p; q)}^{T}

,

Λ ((y_{0} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{0} (t)), (\tilde{y} (t), {}^{c}D_{(p; q)}^{r_{2}} \tilde{y} (t))) \geq 0 .

Clearly,

ρ (y_{0}, \tilde{y}) \geq 1

. Let

{y_{i}}_{i \geq 1} \subseteq X_{*}

be a sequence converging to

y

and let, for all i,

ρ (y_{i}, y_{i + 1}) \geq 1

. Then,

Λ ((y_{i} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i} (t)), (y_{i + 1} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i + 1} (t))) \geq 0 .

By

(P 9)

, there is a subsequence

{y_{i_{r}}}_{r \geq 1}

of

{y_{i}}

so that

Λ ((y_{i_{r}} (t), {}^{c}D_{(p; q)}^{r_{2}} y_{i_{r}} (t)), (y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))) \geq 0

for all

t \in I_{(p; q)}^{T}

. We find that

ρ (y_{i_{r}}, y) \geq 1

for all r. By considering the above statements, all hypotheses of Theorem 4 are fulfilled. The existence of some fixed point for

F

is proved, and so, the generalized sequential

(p; q)

-difference Navier inclusion problem (5) involves a solution. □

The last existence theorem is related to the existence of an endpoint for the given multi-valued function by using the approximate endpoint property.

Theorem 9.

Let

E : I_{(p; q)}^{T} \times X_{*}^{2} \to P_{CM} (X_{*})

. Moreover,

$(P 12)$: There is $θ : [0, \infty) \to [0, \infty)$ so that ${lim inf}_{t \to \infty} (t - θ (t)) \geq 0$ and $θ (t) \leq t, \forall t > 0$ . Here, $θ$ is non-decreasing and upper semi-continuous;
$(P 13)$: $E : I_{(p; q)}^{T} \times X_{*}^{2} \to P_{CM} (X_{*})$ is bounded and integrable such that $E (\cdot, y_{1}, y_{2}) : I_{(p; q)}^{T} \to P_{CM} (X_{*})$ is measurable for each $y_{1}, y_{2} \in X_{*}$ ;
$(P 14)$: There is some $α \in C (I_{(p; q)}^{T}, [0, \infty))$ so that

$H_{d} (E (t, y_{1}, y_{2}), E (t, \tilde{y_{1}}, \tilde{y_{2}})) \leq α (t) {\tilde{L}}_{★} θ (| y_{1} - \tilde{y_{1}} | + | y_{2} - \tilde{y_{2}} |)$

(32)

holds for each $t \in I_{(p; q)}^{T}$ and $y_{1}, y_{2}, \tilde{y_{1}}, \tilde{y_{2}} \in X_{*}$ , where ${\tilde{L}}_{★} = \frac{1}{α_{1} + α_{2}}$ and $α_{1}, α_{2}$ are assumed in (30);
$(P 15)$: $F$ , given by (29), has the approximate endpoint property.

Then, the generalized sequential

(p; q)

-difference Navier inclusion problem (5) has a solution.

Proof.

Here, we shall prove that

F : X_{*} \to P (X_{*})

has an endpoint. As the first step, we establish that

F (y) \subseteq X_{*}

is closed for each

y \in X_{*}

. Note that the multi-valued function

t \to E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))

possesses the closed values and is measurable for every

y \in X_{*}

. Hence, there is a measurable selection for

E

, and accordingly,

{(S)}_{E, y} \neq \emptyset

. Immediately, similar to the proof of Theorem 8, one can easily prove that

F (y)

admits the closed values. Also,

F (y)

is bounded for every

y \in X_{*}

because

E

has the compact values.

In the following, we focus on the establishment of the inequality

H_{d} (F (y), F (\overset{ˇ}{g})) \leq θ (∥ y - \overset{ˇ}{g} ∥_{X_{*}}) .

For finding this purpose, let

y, \overset{ˇ}{g} \in X_{*}

,

ℏ_{1} \in F (\overset{ˇ}{g})

and select

{\bar{F}}_{1} \in {(S)}_{E, \overset{ˇ}{g}}

so that

\begin{matrix} ℏ_{1} (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{1} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{1} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} {\bar{F}}_{1} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

for almost all

t \in I_{(p; q)}^{T}

. From (32), we get

\begin{matrix} H_{d} (E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t)), & E (t, \overset{ˇ}{g} (t), {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} (t))) \\ \leq α (t) {\tilde{L}}_{★} θ (| y (t) - \overset{ˇ}{g} (t) | + | {}^{c}D_{(p; q)}^{r_{2}} y (t) - {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} (t) |), \end{matrix}

for all

t \in I_{(p; q)}^{T}

. There is some

\bar{μ} \in E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))

so that

| {\bar{F}}_{1} (t) - \bar{μ} | \leq α (t) {\tilde{L}}_{★} θ (| y (t) - \overset{ˇ}{g} (t) | + | {}^{c}D_{(p; q)}^{r_{2}} y (t) - {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} (t) |) .

Define

σ : I_{(p; q)}^{T} \to P (X_{*})

by

σ (t) = \{\bar{μ} \in X_{*} : | {\bar{F}}_{1} (t) - \bar{μ} | \leq α (t) {\tilde{L}}_{★} θ (| y (t) - \overset{ˇ}{g} (t) | + | {}^{c}D_{(p; q)}^{r_{2}} y (t) - {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} (t) |)\},

for all

t \in I_{(p; q)}^{T}

. Notice that

{\bar{F}}_{1}

and

b_{*} = α {\tilde{L}}_{★} θ (| y - \overset{ˇ}{g} | + | {}^{c}D_{(p; q)}^{r_{2}} y - {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} |)

are measurable. Therefore,

σ (\cdot) \cap E (\cdot, y (\cdot), {}^{c}D_{(p; q)}^{r_{2}} y (\cdot))

is measurable.

This time, take

{\bar{F}}_{2} \in E (t, y (t), {}^{c}D_{(p; q)}^{r_{2}} y (t))

so that

| {\bar{F}}_{1} (t) - {\bar{F}}_{2} (t) | \leq α (t) {\tilde{L}}_{★} θ (| y (t) - \overset{ˇ}{g} (t) | + | {}^{c}D_{(p; q)}^{r_{2}} y (t) - {}^{c}D_{(p; q)}^{r_{2}} \overset{ˇ}{g} (t) |),

for all

t \in I_{(p; q)}^{T}

. Also, take

ℏ_{2} \in F (y)

so that

\begin{matrix} ℏ_{2} (t) & = \frac{1}{Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{t} {(t - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{2} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ - \frac{pt}{T Γ_{(p; q)} (r_{1} + r_{2}) p^{(\binom{r_{1} + r_{2}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} + r_{2} - 1)} {\bar{F}}_{2} [\frac{v}{p^{r_{1} + r_{2} - 1}}] d_{(p; q)} v \\ + \frac{[p^{2} {(\frac{T}{p})}^{r_{2} + 1} t - p T t^{r_{2} + 1}]}{T^{2} Γ_{(p; q)} (r_{2} + 2) Γ_{(p; q)} (r_{1}) p^{(\binom{r_{1}}{2})}} \int_{0}^{\frac{T}{p}} {(\frac{T}{p} - qv)}_{(p; q)}^{(r_{1} - 1)} {\bar{F}}_{2} [\frac{v}{p^{r_{1} - 1}}] d_{(p; q)} v, \end{matrix}

for all

t \in I_{(p; q)}^{T}

. If we continue the implemented steps in Theorem 8, we have

\begin{matrix} ∥ ℏ_{1} - ℏ_{2} ∥_{X_{*}} & = sup_{t \in I_{(p; q)}^{T}} | ℏ_{1} (t) - ℏ_{2} (t) | + sup_{t \in I_{(p; q)}^{T}} | {}^{c}D_{(p; q)}^{r_{2}} ℏ_{1} (t) - {}^{c}D_{(p; q)}^{r_{2}} ℏ_{2} (t) | \\ \leq (α_{1} + α_{2}) {\tilde{L}}_{★} θ (∥ y - \overset{ˇ}{g} ∥_{X_{*}}) = θ (∥ y - \overset{ˇ}{g} ∥_{X_{*}}) . \end{matrix}

Therefore,

H_{d} (F (y), F (\overset{ˇ}{g})) \leq θ (∥ y - \overset{ˇ}{g} ∥_{X_{*}}),

for every

y, \overset{ˇ}{g} \in X_{*}

. On the other hand,

F

has the approximate endpoint property by the hypothesis

(P 15)

. Accordingly, the existence of

y * * \in X_{*}

is established by Theorem 5 so that

F (y * *) = {y * *}

; that is, as we expected,

y * *

is an endpoint for

F

. Therefore, the generalized sequential

(p; q)

-difference Navier inclusion problem (5) has a solution

y * *

. □

5. Examples

Analysis of some numerical examples are conducted here, to validate the theoretical results in the previous sections.

Example 1.

We model the following generalized sequential

(p; q)

-difference Navier problem of the elastic beam as

\{\begin{matrix} {}^{c}D_{(0.5; 0.45)}^{1.56} ({}^{c}D_{(0.5; 0.45)}^{1.79} y) (t) = \frac{t | {tan}^{- 1} (y (t)) |}{40 + 40 | {tan}^{- 1} (y (t)) |} + 0.025 t {}^{c}D_{(0.5; 0.45)}^{1.79} y (t), \\ 3.5 y (0) = 8.2 y (0.5) = 12.24 {}^{c}D_{(0.5; 0.45)}^{1.79} y (0) = 2.2 {}^{c}D_{(0.5; 0.45)}^{1.79} y (0.5) = 0, \end{matrix}

(33)

where

p = 0.5

,

q = 0.45

,

r_{1} = 1.56

,

r_{2} = 1.79

,

β = 3.5

,

λ = 8.2

,

δ = 12.24

,

γ = 2.2

, and

T = 0.25

with

\frac{T}{p} = 0.5

and

t \in I_{(0.5; 0.45)}^{0.25} = [0, 0.5]

. Moreover, the continuous nonlinear function

E : [0, 0.5] \times R^{2} \to R

is defined by

E (t, y (t), w (t)) = \frac{t | {tan}^{- 1} (y (t)) |}{40 + 40 | {tan}^{- 1} (y (t)) |} + 0.025 t w (t) .

For each

y_{1}, y_{2}, w_{1}, w_{2} \in R

, we can estimate

\begin{matrix} | E (t, y_{1} (t), w_{1} (t)) - E (t, y_{2} (t), w_{2} (t)) | & \leq 0.025 t (| {tan}^{- 1} (y_{1} (t)) - {tan}^{- 1} (y_{2} (t)) | + | w_{1} (t) - w_{2} (t) |) \\ \leq 0.025 t (| y_{1} (t) - y_{2} (t) | + | w_{1} (t) - w_{2} (t) |) . \end{matrix}

Put

k (t) = 0.025 t

for all

t

. Then,

∥ k ∥ = {sup}_{t \in [0, 0.5]} | 0.025 t | = 0.0125

. Now, the nondecreasing function

θ : R^{+} \to R^{+}

is defined by

θ (ς) = ς

for each

ς \in R^{+}

. Then,

\begin{matrix} | E (t, y (t), {}^{c}D_{(0.5; 0.45)}^{1.79} y (t)) | & \leq 0.025 t (| y (t) | + | {}^{c}D_{(0.5; 0.45)}^{1.79} y (t) |) \\ = 0.025 t θ (| y (t) | + | {}^{c}D_{0.57 (0.5; 0.45)}^{1.79} y (t) |) . \end{matrix}

Obviously,

f : I_{(0.5; 0.45)}^{0.25} = [0, 0.5] \to R^{+}

given by

f (t) = 0.025 t

is continuous. By (24), we compute

\begin{matrix} L_{3} & = \frac{0.098}{Γ_{(0.5; 0.45)} (4.35)} + \frac{0.196}{Γ_{(0.5; 0.45)} (2.56) Γ_{(0.5; 0.45)} (3.79)} \approx 0.019204, \\ L_{4} & = \frac{0.098}{Γ_{(0.5; 0.45)} (0.21) Γ_{(0.5; 0.45)} (4.35)} + \frac{[1 + Γ_{(0.5; 0.45)} (0.21) Γ_{(0.5; 0.45)} (3.79)] 0.098}{Γ_{(0.5; 0.45)} (0.21) Γ_{(0.5; 0.45)} (3.79) Γ_{(0.5; 0.45)} (2.56)} \approx 0.058137 . \end{matrix}

By (27), we have

L \approx 0.0009667625 < 1

. Hence, Theorem 7 concludes that the generalized sequential

(p; q)

-difference Navier problem (33) has at least one solution on

I_{(0.5; 0.45)}^{0.25}

.

The next example deals with the inclusion version of the Navier

(p; q)

-difference equation.

Example 2.

By using the given parameters in the previous example, i.e.,

p = 0.5

,

q = 0.45

,

r_{1} = 1.56

,

r_{2} = 1.79

,

β = 3.5

,

λ = 8.2

,

δ = 12.24

,

γ = 2.2

, and

T = 0.25

with

\frac{T}{p} = 0.5

and

t \in I_{(0.5; 0.45)}^{0.25} = [0, 0.5]

, we model the generalized sequential

(p; q)

-difference Navier inclusion problem of the elastic beam as

\{\begin{matrix} {}^{c}D_{(0.5; 0.45)}^{1.56} ({}^{c}D_{(0.5; 0.45)}^{1.79} y) (t) = [0, \frac{t | sin (y (t)) |}{48 (1 + t^{2}) (1 + | sin (y (t)) |)} + \frac{2 t | {}^{c}D_{(0.5; 0.45)}^{1.79} y (t) |}{96 (1 + 2 t) (1 + | {}^{c}D_{(0.5; 0.45)}^{1.79} y (t) |)}], \\ 3.5 y (0) = 8.2 y (0.5) = 12.24 {}^{c}D_{(0.5; 0.45)}^{1.79} y (0) = 2.2 {}^{c}D_{(0.5; 0.45)}^{1.79} y (0.5) = 0, \end{matrix}

(34)

Define

E : I_{(0.5; 0.45)}^{0.25} = [0, 0.5] \times R^{2} \to P (R)

by

E (t, y_{1} (t), y_{2} (t)) = [0, \frac{t | sin (y_{1} (t)) |}{48 (1 + t^{2}) (1 + | sin (y_{1} (t)) |)} + \frac{2 t | y_{2} (t) |}{96 (1 + 2 t) (1 + | y_{2} (t) |)}],

for each

t \in [0, 0.5]

. The function

α \in C ([0, 0.5], [0, \infty))

is chosen so that

α (t) = \frac{t}{8}

for each

t \in [0, 0.5]

. So,

∥ α ∥ = {sup}_{t \in [0, 0.5]} | \frac{t}{8} | = \frac{0.5}{8} = 0.0625

. Also, choose the function

θ : [0, \infty) \to [0, \infty)

as

θ (t) = \frac{t}{6}

for almost all

t > 0

. Note that

θ

is non-decreasing and upper semi-continuous. In this case,

{lim inf}_{t \to \infty} (t - θ (t)) > 0

and

θ (t) < t

for all

t > 0

. Now, (23) and (30) give

\begin{matrix} L_{1} & = \frac{0.196}{Γ_{(0.5; 0.45)} (4.35)} + \frac{0.196}{Γ_{(0.5; 0.45)} (2.56) Γ_{(0.5; 0.45)} (3.79)} \approx 0.028804, \\ L_{2} & = \frac{1.3568}{Γ_{(0.5; 0.45)} (0.56)} + \frac{0.098}{Γ_{(0.5; 0.45)} (0.21) Γ_{(0.5; 0.45)} (4.35)} \\ + \frac{0.098 [1 + Γ_{(0.5; 0.45)} (0.21) Γ_{(0.5; 0.45)} (3.79)]}{Γ_{(0.5; 0.45)} (0.21) Γ_{(0.5; 0.45)} (3.79) Γ_{(0.5; 0.45)} (2.56)} \approx 1.898857, \end{matrix}

and

α_{1} = ∥ α ∥ L_{1} \approx 0.00180025 and α_{2} = ∥ α ∥ L_{2} \approx 0.1186785625 .

For every

y_{1}, y_{2}, \tilde{y_{1}}, \tilde{y_{2}} \in R

, we get

\begin{matrix} H_{d} (E (t, y_{1} (t), y_{2} (t)) & , E (t, \tilde{y_{1}} (t), \tilde{y_{2}} (t))) \leq \frac{t}{8} . \frac{1}{6} (| y_{1} (t) - \tilde{y_{1}} (t) | + | y_{2} (t) - \tilde{y_{2}} (t) |) \\ = \frac{t}{8} θ (| y_{1} (t) - \tilde{y_{1}} (t) | + | y_{2} (t) - \tilde{y_{2}} (t) |) \\ \leq α (t) θ (| y_{1} (t) - \tilde{y_{1}} (t) | + | y_{2} (t) - \tilde{y_{2}} (t) |) [\frac{1}{α_{1} + α_{2}}] . \end{matrix}

In the last step, a set-valued map

F : X_{*} \to P (X_{*})

is defined as

F (y) = \{ℏ \in X_{*} : \exists \bar{F} \in {(S)}_{E, y} so that ℏ (t) = \overset{ˇ}{g} (t) \forall t \in I_{(0.5; 0.45)}^{0.25} = [0 . 0.5]\},

where

\begin{matrix} \overset{ˇ}{g} (t) & = \frac{1}{Γ_{(0.5; 0.45)} (3.35) 0 . 5^{(\binom{3.35}{2})}} \int_{0}^{t} {(t - 0.45 v)}_{(0.5; 0.45)}^{(2.35)} \bar{F} [\frac{v}{0 . 5^{2.35}}] d_{(p; q)} v \\ - \frac{0.5 t}{0.25 Γ_{(0.5; 0.45)} (3.35) 0 . 5^{(\binom{3.35}{2})}} \int_{0}^{0.5} {(0.5 - 0.45 v)}_{(0.5; 0.45)}^{2.35} \bar{F} [\frac{v}{0 . 5^{2.35}}] d_{(p; q)} v \\ + \frac{[0 . 5^{2} {(0.5)}^{2.79} t - 0.125 t^{2.79}]}{0.0625 Γ_{(0.5; 0.45)} (3.79) Γ_{(0.5; 0.45)} (1.56) 0 . 5^{(\binom{1.56}{2})}} \int_{0}^{0.5} {(0.5 - 0.45 v)}_{(0.5; 0.45)}^{0.56} \bar{F} [\frac{v}{0 . 5^{0.56}}] d_{(p; q)} v . \end{matrix}

Finally, Theorem 9 implies that the generalized sequential

(p; q)

-difference Navier inclusion problem (34) of the elastic beam has a solution.

6. Conclusions

In this paper, we dealt with two different cases of an elastic beam modeling in the context of the notions of

(p; q)

-calculus. In fact, in the framework of the existing mathematical definitions related to the single-valued and multi-valued functions, we generalized the fourth-order differential equation of the elastic beam changes to the two

(p; q)

-difference Navier equation and

(p; q)

-difference Navier inclusion, separately. Of course, we emphasize that the main goal of this study is to investigate the existence results of the solutions for both systems, not to obtain a new method for finding numerical

(p; q)

-solutions. We recalled the main definitions about the

ρ

-

θ

-contractions and

ρ

-admissible functions, and then, by using the fixed point theorems and endpoint theorems, proved our desired theorems about the existence property of the solutions. Based on these results, one can extend the studies later for defining new

(p; q)

-integral transforms or other new

(p; q)

-algorithms to approximate the numerical solutions with the help of the real numerical data.

Author Contributions

Conceptualization: S.E. and S.K.N.; Formal Analysis: S.E., S.K.N., I.S. and J.T.; Investigation: S.E. and S.K.N.; Methodology: I.S. and J.T.; Software: S.E.; Writing—Original Draft: S.E. and S.K.N.; Writing—Review & Editing: I.S. and J.T. All authors have read and agreed to the last version of the manuscript.

Funding

This research budget was allocated by National Science, Research and Innovation Fund (NSRF) and King Mongkut’s University of Technology North Bangkok with Contract no. KMUTNB-FF-67-B-01.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The first author would like to thank Azarbaijan Shahid Madani University.

Conflicts of Interest

The authors declare no conflict of interest.

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Etemad, S.; Ntouyas, S.K.; Stamova, I.; Tariboon, J. On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam. Fractal Fract. 2024, 8, 236. https://doi.org/10.3390/fractalfract8040236

AMA Style

Etemad S, Ntouyas SK, Stamova I, Tariboon J. On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam. Fractal and Fractional. 2024; 8(4):236. https://doi.org/10.3390/fractalfract8040236

Chicago/Turabian Style

Etemad, Sina, Sotiris K. Ntouyas, Ivanka Stamova, and Jessada Tariboon. 2024. "On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam" Fractal and Fractional 8, no. 4: 236. https://doi.org/10.3390/fractalfract8040236

APA Style

Etemad, S., Ntouyas, S. K., Stamova, I., & Tariboon, J. (2024). On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam. Fractal and Fractional, 8(4), 236. https://doi.org/10.3390/fractalfract8040236

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On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

Abstract

1. Introduction

2. Preliminaries

2.1. `q`-Calculus

2.2. (`p`; `q`)-Calculus

2.3. Fixed Point Theory

3. On the Generalized $(p; q)$ -Difference Navier Problem (4)

4. On the Generalized $(p; q)$ -Difference Navier Problem (5)

5. Examples

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam

Abstract

1. Introduction

2. Preliminaries

2.1. q-Calculus

2.2. (p; q)-Calculus

2.3. Fixed Point Theory

3. On the Generalized ( p ; q ) -Difference Navier Problem (4)

4. On the Generalized ( p ; q ) -Difference Navier Problem (5)

5. Examples

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Share and Cite

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Further Information

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2.1. `q`-Calculus

2.2. (`p`; `q`)-Calculus

3. On the Generalized $(p; q)$ -Difference Navier Problem (4)

4. On the Generalized $(p; q)$ -Difference Navier Problem (5)