IDENTIFYING FIXED POINTS AND SOLUTION OF NONLINEAR FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN INTUITIONISTIC MENGER PROBABILISTIC METRIC SPACES

In this paper, we extend the concept of compatible maps of type J, (J-1) and (J-2) and prove fixed point theorems for compatible mappings in Non-Archimedian Intuitionistic Menger Probabilistic Metric Space. An example and an application to functional equation is provided to support the theoretical results.

Keywords:
common fixed point; non-Archimedean intuitionistic Menger probabilistic metric spaces; compatible maps

Authorship

Ria Sharma

Department of Mathematics, Chandigarh University, Gharuan, Punjab, India - E-mail: riasharma4086@gmail.com Chandigarh UniversityIndiaGharuan, Punjab, IndiaDepartment of Mathematics, Chandigarh University, Gharuan, Punjab, India - E-mail: riasharma4086@gmail.com

https://orcid.org/0000-0002-7558-6304

Surjeet Singh Chauhan(Gonder)

Department of Mathematics, Chandigarh University, Gharuan, Punjab, India - E-mail: surjeetschauhan@yahoo.com Chandigarh UniversityIndiaGharuan, Punjab, IndiaDepartment of Mathematics, Chandigarh University, Gharuan, Punjab, India - E-mail: surjeetschauhan@yahoo.com

https://orcid.org/0000-0001-5500-9924

Vishal Gupta ^**Corresponding author

Department of Mathematics, MMEC, Maharishi Markandeshwar (Deemed to be University), Mullana, Haryana, India - E-mail: vgupta@mmumullana.org, vishal.gmn@gmail.com Maharishi Markandeshwar (Deemed to be University)IndiaMullana, Haryana, IndiaDepartment of Mathematics, MMEC, Maharishi Markandeshwar (Deemed to be University), Mullana, Haryana, India - E-mail: vgupta@mmumullana.org, vishal.gmn@gmail.com

https://orcid.org/0000-0001-9727-2827

*Corresponding author

SCIMAGO INSTITUTIONS RANKINGS

Formulas

Formulas (75)

(3.1.1)

(3.1.2)

(3.1.3)

(a) F_{x y} (t) + G_{x y} (t) \leq 1, (b) F_{x y} (t) = 1 i f a n d o n l y i f x = y, (c) F_{x y} (t) = F_{y x} (t), (d) F_{x y} (0) = 0, (e) F_{x y} (t) = 1, F_{y z} (s) = 1 \Rightarrow F_{x z} (m a x {t, s}) = 1, (f) G_{x y} (t) = 0 i f a n d o n l y i f x = y, (g) G_{x y} (t) = G_{y x} (t), (h) G_{x y} (0) = 1, (i) G_{x y} (t) = 0, G_{y z} (s) = 0 \Rightarrow G_{x z} (m i n {t, s}) = 1,

(j) F_{x z} (m a x {t, s}) \geq F_{x y} (t) * F_{y z} (s), (k) G_{x z} (m i n {t, s}) \geq G_{x y} (t) ◊ G_{y z} (s),

(a) F_{x y z} (t) + G_{x y z} (t) \leq 1, (b) F_{x y z} (t) = 1 i f a n d o n l y i f x = y = z, (c) F_{x y z} (t) = F_{z y x} (t), (d) F_{x y z} (0) = 0, (e) F_{x y} (t) = 1, F_{y z} (s) = 1 \Rightarrow F_{x z} (m a x {t, s}) = 1, (f) G_{x y z} (t) = 0 i f a n d o n l y i f x, y, z a r e p a i r w i s e e q u a l, (g) G_{x y z} (t) = G_{y x z} (t) = G_{z y x} (t), (h) G_{x y z} (0) = 1, (i) G_{x y} (t) = 0, G_{y z} (s) = 0 \Rightarrow G_{x z} (m i n {t, s}) = 1,

(j) F_{x z} (m a x {t, s}) \geq F_{x y} (t) * F_{y z} (s), (k) G_{x z} (m i n {t, s}) \geq G_{x y} (t) ⋄ G_{y z} (s),

C A x_{n} \to C q, C B x_{n} \to C q, C C x_{n} \to C q

g (F_{A B_{x_{n}}, B C_{x_{n}}, C A_{x_{n}}} (t)) \leq g (F_{A B_{x_{n}}, B C_{x_{n}}, C C_{x_{n}}} (t)) + g (F_{C C_{x_{n}}, C B_{x_{n}}, B A_{x_{n}}} (t)) + g (F_{B A_{x_{n}}, A C_{x_{n}}, C A_{x_{n}}} (t)) \to 0 + g (F_{C q, C q, C q} (t) + g (F_{B q, C q, C q} (t) a s (A C x_{n} = C A x_{n}), \to 0 + 0 + 0 a s g (F_{x y z} (t) = 0, i f a n y t w o o f x, y, z e q u a l . h (G_{A B_{x_{n}}, B C_{x_{n}}, C A_{x_{n}}} (t)) \geq h (G_{A B_{x_{n}}, B C_{x_{n}}, C C_{x_{n}}} (t)) + h (G_{C C_{x_{n}}, C B_{x_{n}}, B A_{x_{n}}} (t)) + h (G_{B A_{x_{n}}, A C_{x_{n}}, C A_{x_{n}}} (t)) \to 1 + h (G_{C q, C q, B q} (t)) + h (G_{B q, A q, A q} (t)) \to 1 + 0 + 0 = 1, a s (A C x_{n} = C A x_{n}) a n d g (F_{x y z} (t) = 0 .

g (F_{A B_{x_{n}}, B C_{x_{n}}, C A_{x_{n}}} (t)) \leq g (F_{A B_{x_{n}}, B C_{x_{n}}, C C_{x_{n}}} (t)) + g (F_{A B_{x_{n}} B C_{x_{n}}, C C_{x_{n}}} (t)) + g (F_{A B_{x_{n}}, C C_{x_{n}}, C A_{x_{n}}} (t)) \geq 1 + h (F_{C q, C q, C q} (t) + g (F_{C q, B q, C q}, a s A, C a r e c o n t i n u o u s \geq 1 + 0 + 0 + 0 = 1 a s g (F_{x y z} (t) = 0, i f a n y t w o o f x, y, z e q u a l . h (G_{A B_{x_{n}}, B C_{x_{n}}, C A_{x_{n}}} (t)) \geq h (G_{A B_{x_{n}}, B C_{x_{n}}, C C_{x_{n}}} (t)) + h (G_{C C_{x_{n}}, C B_{x_{n}}, B A_{x_{n}}} (t)) + h (G_{B A_{x_{n}}, A C_{x_{n}}, C A_{x_{n}}} (t)) \to 1 + h (G_{C q, C q, B q} (t)) + h (G_{B q, A q, A q} (t)) \to 1 + 0 + 0 = 1, a s (A C x_{n} = C A x_{n}) a n d g (F_{x y z} (t) = 0 .

h (G_{y_{2 n}, y_{2 n + 1}, y_{2 n + 2}} (t)) = h (G_{P x_{2 n}, Q y_{2 n + 1}, R z_{2 n + 2}} (t)) \geq φ (h (G_{y_{2 n}, y_{2 n + 1}, y_{2 n + 2}} (t))

h (G_{y_{2 n}, y_{2 n - 1}, y_{2 n - 2}} (t)) = h (G_{P x_{2 n}, Q y_{2 n - 1}, R z_{2 n - 2}} (t)) \geq φ (h (G_{y_{2 n - 2}, y_{2 n}, y_{2 n + 3}} (t))

h (G_{y_{n}, y_{n + 1}, y_{n + 2}} (t)) \geq φ (h (G_{y_{n - 2}, y_{n - 1}, y_{n}} (t)) f o r n = 0, 1, 2, . . . .

h (G_{y_{n}, y_{n + 1}, y_{n + 2}} (t)) \geq φ^{n} (h (G_{y_{0}, y_{1}, y_{2}} (t)) for n = 0, 1, 2, . . .

(3.1.1)

g (F_{y_{n}, y_{n + 1}, y_{n + 2}} (t)) \to 0 h (G_{y_{n}, y_{n + 1}, y_{n + 2}} (t)) \to 1, a s n \to \infty .

g (1 - ε_{0}) < g (F_{y_{m_{k}}, y_{n_{k}}, y_{p_{k}}} (t_{0})) \leq g (F_{y_{m_{k}}, y_{m_{k - 1}}, y_{m_{k - 2}}} (t_{0})) + g (F_{y_{m_{k - 2}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) \leq g (F_{y_{m_{k}}, y_{m_{k - 1}}, y_{m_{k - 2}}} (t_{0})) + g (1 - ε_{0})

h (ε_{0}) > h (G_{y_{m_{k}}, y_{n_{k}}, y_{p_{k}}} (t_{0})) \geq h (G_{y_{m_{k}}, y_{m_{k - 1}}, y_{m_{k - 2}}} (t_{0})) + h (G_{y_{m_{k - 2}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) \geq h (G_{y_{m_{k}}, y_{m_{k - 1}}, y_{m_{k - 2}}} (t_{0})) + h (ε_{0}) .

(3.1.2)

l i m_{n \to \infty} g (F_{y_{m_{k}}, y_{n_{k}}, y_{p_{k}}} (t_{0})) = g (1 - ε_{0}) a n d l i m_{n \to \infty} h (G_{y_{m_{k}}, y_{n_{k}}, y_{p_{k}}} (t_{0})) = h (ε_{0}) .

(3.1.3)

g (1 - ε_{0}) < g (F_{y_{m_{k}}, y_{n_{k}}, y_{p_{k}}} (t_{0})) \leq g (F_{y_{m_{k}}, y_{n_{k + 1}}, y_{p_{k + 2}}} (t_{0})) + g (F_{y_{m_{k}}, y_{p_{k + 1}}, y_{p_{k + 2}}} (t_{0})) + g (F_{y_{p_{k}}, y_{p_{k + 1}}, y_{p_{k + 2}}} (t_{0})) a n d h (ε_{0}) > h (G_{y_{m_{k}}, y_{n_{k}}, y_{p_{k}}} (t_{0})) \geq h (G_{y_{m_{k}}, y_{n_{k + 1}}, y_{p_{k + 2}}} (t_{0})) + h (G_{y_{m_{k}}, y_{p_{k + 1}}, y_{p_{k + 2}}} (t_{0})) + h (G_{y_{p_{k}}, y_{p_{k + 1}}, y_{p_{k + 2}}} (t_{0})) .

g (F_{y_{m_{k}}, y_{n_{k + 1}}, y_{p_{k + 2}}} (t_{0})) = g (F_{P x_{m_{k}}, Q x_{m_{k + 1}}, R x_{m_{k + 2}}} (t_{0})) \leq ϕ [m a x {g (F_{y_{m_{k - 2}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + g (F_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + g (F_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + g (F_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})), g (F_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + g (F_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + g (F_{y_{p_{k + 2}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})), g (F_{y_{m_{k}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + g (F_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + g (F_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})), g (F_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + g (F_{y_{n_{k + 1}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})) + g (F_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0}))}]

h (G_{y_{m_{k}}, y_{n_{k + 1}}, y_{p_{k + 2}}} (t_{0})) = h (G_{P x_{m_{k}}, Q x_{m_{k + 1}}, R x_{m_{k + 2}}} (t_{0})) \geq φ [m i n {h (G_{y_{m_{k - 2}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})), h (G_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})), h (G_{y_{m_{k}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})), h (G_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0}))}] \geq φ [m i n {h (ε_{0}) + h (G_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})), h (G_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})), h (G_{y_{m_{k}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{n_{k - 1}}, y_{p_{k}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})), h (G_{y_{m_{k}}, y_{m_{k - 2}}, y_{n_{k - 1}}} (t_{0})) + h (G_{y_{n_{k + 1}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0})) + h (G_{y_{p_{k + 2}}, y_{p_{k}}, y_{m_{k - 2}}} (t_{0}))}] .

g (1 - ε_{0}) \leq ϕ (g (1 - ε_{0})) < g (1 - ε_{0}) a n d h (ε_{0}) \geq φ (h (ε_{0})) > h (ε_{0}) .

g (F_{P P x_{2 n}, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \leq ϕ [m a x {g (F_{A B P x_{2 n}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{P P x_{2 n}, A B P X_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B P x_{2 n}} (t)), g (F_{P P x_{2 n}, A B P x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, A B P x_{2 n}, C D x_{2 n + 1}} (t)), g (F_{P P x_{2 n}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B P x_{2 n}} (t)), g (F_{P P x_{2 n}, A B P x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B P x_{2 n}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B P x_{2 n}} (t))}] .

h (G_{P P x_{2 n}, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \geq φ [m i n {h (G_{A B P x_{2 n}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{P P x_{2 n}, A B P X_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B P x_{2 n}} (t)), h (G_{P P x_{2 n}, A B P x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, A B P x_{2 n}, C D x_{2 n + 1}} (t)), h (G_{P P x_{2 n}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B P x_{2 n}} (t)), h (G_{P P x_{2 n}, A B P x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B P x_{2 n}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B P x_{2 n}} (t))}] .

g (F_{A B q, q, q} (t)) \leq ϕ [m a x {g (F_{A B q, q, q} (t)) + g (F_{q, A B q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, A B q} (t)), g (F_{q, A B q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, A B q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, A B q} (t)), g (F_{q, A B q, q} (t)) + g (F_{q, q, A B q} (t)) + g (F_{q, q, A B q} (t))}] = ϕ {g (F_{A B q, q, q} (t))}

h (G_{A B q, q, q} (t)) \geq φ [m i n {h (G_{A B q, q, q} (t)) + h (G_{q, A B q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, A B q} (t)), h (G_{q, A B q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, A B q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, A B q} (t)), h (G_{q, A B q, q} (t)) + h (G_{q, q, A B q} (t)) + h (G_{q, q, A B q} (t))}] = φ {h (G_{A B q, q, q} (t))} .

g (F_{P q, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \leq ϕ [m a x {g (F_{A B q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{P q, A B q, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B q} (t)), g (F_{P q, A B q, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, A B q, C D x_{2 n + 1}} (t)), g (F_{P q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B q} (t)), g (F_{P q, A B q, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B q} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B q} (t))}] .

h (G_{P q, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \geq φ [m i n {h (G_{A B q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{P q, A B q, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B q} (t)), h (G_{P q, A B q, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, A B q, C D x_{2 n + 1}} (t)), h (G_{P q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B q} (t)), h (G_{P q, A B q, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B q} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B q} (t))}] .

g (F_{q, q, q} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t))}] = ϕ {g (F_{q, q, q} (t))}

h (G_{q, q, q} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t))}] = φ {h (G_{q, q, q} (t))} .

g (F_{P B q, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \leq ϕ [m a x {g (F_{A B B q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{P B q, A B B q, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B B q} (t)), g (F_{P B q, A B B q, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, A B B q, C D x_{2 n + 1}} (t)), g (F_{P B q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B B q} (t)), g (F_{P B q, A B B q, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B B q} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B B q} (t))}] .

h (G_{P B q, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \geq φ [m i n {h (G_{A B B q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{P B q, A B B q, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B B q} (t)), h (G_{P B q, A B B q, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, A B B q, C D x_{2 n + 1}} (t)), h (G_{P B q, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B B q} (t)), h (G_{P B q, A B B q, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B B q} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B B q} (t)) .}]

g (F_{B q, q, q} (t)) \leq ϕ [m a x {g (F_{B q, q, q} (t)) + g (F_{q, B q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, B q} (t)), g (F_{q, B q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, B q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, B q} (t)), g (F_{q, B q, q} (t)) + g (F_{q, q, B q} (t)) + g (F_{q, q, B q} (t))}] = ϕ {g (F_{B q, q, q} (t))}

h (G_{B q, q, q} (t)) \geq φ [m i n {h (G_{B q, q, q} (t)) + h (G_{q, B q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, B q} (t)), h (G_{q, B q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, B q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, B q} (t)), h (G_{q, B q, q} (t)) + h (G_{q, q, B q} (t)) + h (G_{q, q, B q} (t))}] = φ {h (G_{B q, q, q} (t))} .

g (F_{P x_{2 n}, Q j, R x_{2 n + 1}} (t)) \leq ϕ [m a x {g (F_{A B x_{2 n}, C D j, U V x_{2 n + 1}} (t)) + g (F_{P x_{2 n}, A B x_{2 n}, C D j} (t)) + g (F_{Q j, C D j, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D j} (t)) + g (F_{Q j, C D j, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, A B x_{2 n}, C D j} (t)), g (F_{P x_{2 n}, C D j, U V x_{2 n + 1}} (t)) + g (F_{Q j, C D j, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D j} (t)) + g (F_{Q j, U V x_{2 n + 1}, A B x_{2 n}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t))}] .

h (G_{P x_{2 n}, Q j, R x_{2 n + 1}} (t)) \geq φ [m i n {h (G_{A B x_{2 n}, C D j, U V x_{2 n + 1}} (t)) + h (G_{P x_{2 n}, A B x_{2 n}, C D j} (t)) + h (G_{Q j, C D j, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D j} (t)) + h (G_{Q j, C D j, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, A B x_{2 n}, C D j} (t)), h (G_{P x_{2 n}, C D j, U V x_{2 n + 1}} (t)) + h (G_{Q j, C D j, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D j} (t)) + h (G_{Q j, U V x_{2 n + 1}, A B x_{2 n}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t))}] .

g (F_{q, Q j, q} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{Q j, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{Q j, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{Q j, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{Q j, q, q} (t)) + g (F_{q, q, q} (t))}] = ϕ {g (F_{q, Q j, q} (t))} .

h (G_{q, Q j, q} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{Q j, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{Q j, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{Q j, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{Q j, q, q} (t)) + h (G_{q, q, q} (t))}] = φ {h (G_{q, Q j, q} (t))} .

g (F_{P x_{2 n}, Q q, R x_{2 n + 1}} (t)) \leq ϕ [m a x {g (F_{A B x_{2 n}, C D q, U V x_{2 n + 1}} (t)) + g (F_{P x_{2 n}, A B x_{2 n}, C D q} (t)) + g (F_{Q q, C D q, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D q} (t)) + g (F_{Q q, C D q, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, A B x_{2 n}, C D q} (t)), g (F_{P x_{2 n}, C D q, U V x_{2 n + 1}} (t)) + g (F_{Q q, C D q, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D q} (t)) + g (F_{Q q, U V x_{2 n + 1}, A B x_{2 n}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t))}] .

h (G_{P x_{2 n}, Q q, R x_{2 n + 1}} (t)) \geq φ [m i n {h (G_{A B x_{2 n}, C D q, U V x_{2 n + 1}} (t)) + h (G_{P x_{2 n}, A B x_{2 n}, C D q} (t)) + h (G_{Q q, C D q, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D q} (t)) + h (G_{Q q, C D q, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, A B x_{2 n}, C D q} (t)), h (G_{P x_{2 n}, C D q, U V x_{2 n + 1}} (t)) + h (G_{Q q, C D q, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D q} (t)) + h (G_{Q q, U V x_{2 n + 1}, A B x_{2 n}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t))}] .

g (F_{q, Q q, q} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{Q q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{Q q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{Q q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{Q q, q, q} (t)) + g (F_{q, q, q} (t))}] = ϕ {g (F_{q, Q q, q} (t))} .

h (G_{q, Q q, q} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{Q q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{Q q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{Q q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{Q q, q, q} (t)) + h (G_{q, q, q} (t))}] = φ {h (G_{q, Q q, q} (t))} .

g (F_{P x_{2 n}, Q D q, R x_{2 n + 1}} (t)) \leq ϕ [m a x {g (F_{A B x_{2 n}, C D D q, U V x_{2 n + 1}} (t)) + g (F_{P x_{2 n}, A B x_{2 n}, C D D q} (t)) + g (F_{Q D q, C D D q, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D D q} (t)) + g (F_{Q D q, C D D q, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, A B x_{2 n}, C D D q} (t)), g (F_{P x_{2 n}, C D D q, U V x_{2 n + 1}} (t)) + g (F_{Q D q, C D D q, U V x_{2 n + 1}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D D q} (t)) + g (F_{Q D q, U V x_{2 n + 1}, A B x_{2 n}} (t)) + g (F_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t))}] .

h (G_{P x_{2 n}, Q D q, R x_{2 n + 1}} (t)) \geq φ [m i n {h (G_{A B x_{2 n}, C D D q, U V x_{2 n + 1}} (t)) + h (G_{P x_{2 n}, A B x_{2 n}, C D D q} (t)) + h (G_{Q D q, C D D q, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D D q} (t)) + h (G_{Q D q, C D D q, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, A B x_{2 n}, C D D q} (t)), h (G_{P x_{2 n}, C D D q, U V x_{2 n + 1}} (t)) + h (G_{Q D q, C D D q, U V x_{2 n + 1}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D D q} (t)) + h (G_{Q D q, U V x_{2 n + 1}, A B x_{2 n}} (t)) + h (G_{R x_{2 n + 1}, U V x_{2 n + 1}, A B x_{2 n}} (t))}] .

g (F_{q, D q, q} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{D q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{D q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{D q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{D q, q, q} (t)) + g (F_{q, q, q} (t))}] = ϕ {g (F_{q, D q, q} (t))} .

h (G_{q, D q, q} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{D q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{D q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{D q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{D q, q, q} (t)) + h (G_{q, q, q} (t))}] = φ {h (G_{q, D q, q} (t))} .

g (F_{P x_{2 n}, Q x_{2 n + 1}, R w} (t)) \leq ϕ [m a x {g (F_{A B x_{2 n}, C D x_{2 n + 1}, U V w} (t)) + g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V w} (t)) + g (F_{R w, U V w, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V w} (t)) + g (F_{R w, A B x_{2 n}, C D x_{2 n + 1}} (t)), g (F_{P x_{2 n}, C D x_{2 n + 1}, U V w} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V w} (t)) + g (F_{R w, U V w, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, U V w, A B x_{2 n}} (t)) + g (F_{R w, U V w, A B x_{2 n}} (t))}] .

h (G_{P x_{2 n}, Q x_{2 n + 1}, R w} (t)) \geq φ [m i n {h (G_{A B x_{2 n}, C D x_{2 n + 1}, U V w} (t)) + h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V w} (t)) + h (G_{R w, U V w, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V w} (t)) + h (G_{R w, A B x_{2 n}, C D x_{2 n + 1}} (t)), h (G_{P x_{2 n}, C D x_{2 n + 1}, U V w} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V w} (t)) + h (G_{R w, U V w, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, U V w, A B x_{2 n}} (t)) + h (G_{R w, U V w, A B x_{2 n}} (t))}] .

g (F_{q, q, R w} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{R w, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{R w, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{R w, q, q} (t))}] = ϕ {g (F_{q, q, R w} (t))}

h (G_{q, q, R w} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{R w, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{R w, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{R w, q, q} (t))}] = φ {h (G_{q, q, R w} (t))} .

g (F_{P x_{2 n}, Q x_{2 n + 1}, R q} (t)) \leq ϕ [m a x {g (F_{A B x_{2 n}, C D x_{2 n + 1}, U V q} (t)) + g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V q} (t)) + g (F_{R q, U V q, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V q} (t)) + g (F_{R q, A B x_{2 n}, C D x_{2 n + 1}} (t)), g (F_{P x_{2 n}, C D x_{2 n + 1}, U V q} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V q} (t)) + g (F_{R q, U V q, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, U V q, A B x_{2 n}} (t)) + g (F_{R q, U V q, A B x_{2 n}} (t))}] .

h (G_{P x_{2 n}, Q x_{2 n + 1}, R q} (t)) \geq φ [m i n {h (G_{A B x_{2 n}, C D x_{2 n + 1}, U V q} (t)) + h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V q} (t)) + h (G_{R q, U V q, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V q} (t)) + h (G_{R q, A B x_{2 n}, C D x_{2 n + 1}} (t)), h (G_{P x_{2 n}, C D x_{2 n + 1}, U V q} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V q} (t)) + h (G_{R q, U V q, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, U V q, A B x_{2 n}} (t)) + h (G_{R q, U V q, A B x_{2 n}} (t))}] .

g (F_{q, q, R q} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{R q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{R q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{R q, q, q} (t))}] = ϕ {g (F_{q, q, R q} (t))}

h (G_{q, q, R q} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{R q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{R q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{R q, q, q} (t))}] = φ {h (G_{q, q, R q} (t))} .

g (F_{P x_{2 n}, Q x_{2 n + 1}, R V q} (t)) \leq ϕ [m a x {g (F_{A B x_{2 n}, C D x_{2 n + 1}, U V V q} (t)) + g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V V q} (t)) + g (F_{R V q, U V V q, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V V q} (t)) + g (F_{R V q, A B x_{2 n}, C D x_{2 n + 1}} (t)), g (F_{P x_{2 n}, C D x_{2 n + 1}, U V V q} (t)) + g (F_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V V q} (t)) + g (F_{R V q, U V V q, A B x_{2 n}} (t)), g (F_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{V Q x_{2 n + 1}, U V V q, A B x_{2 n}} (t)) + g (F_{R V q, U V V q, A B x_{2 n}} (t))}] .

h (G_{P x_{2 n}, Q x_{2 n + 1}, R V q} (t)) \geq φ [m i n {h (G_{A B x_{2 n}, C D x_{2 n + 1}, U V V q} (t)) + h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V V q} (t)) + h (G_{R V q, U V V q, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V V q} (t)) + h (G_{R V q, A B x_{2 n}, C D x_{2 n + 1}} (t)), h (G_{P x_{2 n}, C D x_{2 n + 1}, U V V q} (t)) + h (G_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V V q} (t)) + h (G_{R V q, U V V q, A B x_{2 n}} (t)), h (G_{P x_{2 n}, A B x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{V Q x_{2 n + 1}, U V V q, A B x_{2 n}} (t)) + h (G_{R V q, U V V q, A B x_{2 n}} (t))}] .

g (F_{q, q, V q} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{V q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{V q, q, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{V q, q, q} (t))}] = ϕ {g (F_{q, q, V q} (t))} .

h (G_{q, q, V q} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{V q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{V q, q, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{V q, q, q} (t))}] = φ {h (G_{q, q, V q} (t))} .

g (F_{P (A B) x_{2 n}, Q x_{2 n + 1}, R x_{n + 2}} (t)) \leq ϕ [m a x {g (F_{A B (A B) x_{2 n}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + g (F_{P (A B) x_{2 n}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + g (F_{R x_{n + 2}, U V x_{n + 2}, A B (A B) x_{2 n}} (t)), g (F_{P (A B) x_{2 n}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + g (F_{R x_{n + 2}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)), g (F_{P (A B) x_{2 n}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + g (F_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + g (F_{R x_{n + 2}, U V x_{n + 2}, A B (A B) x_{2 n}} (t)), g (F_{P (A B) x_{2 n}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)) + g (F_{V Q x_{2 n + 1}, U V x_{n + 2}, A B (A B) x_{2 n}} (t)) + g (F_{R x_{n + 2}, U V x_{n + 2}, A B (A B) x_{2 n}} (t))}] .

h (G_{P (A B) x_{2 n}, Q x_{2 n + 1}, R x_{n + 2}} (t)) \geq φ [m i n {h (G_{A B (A B) x_{2 n}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + h (G_{P (A B) x_{2 n}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + h (G_{R x_{n + 2}, U V x_{n + 2}, A B (A B) x_{2 n}} (t)), h (G_{P (A B) x_{2 n}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + h (G_{R x_{n + 2}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)), h (G_{P (A B) x_{2 n}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + h (G_{V Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{n + 2}} (t)) + h (G_{R x_{n + 2}, U V x_{n + 2}, A B (A B) x_{2 n}} (t)), h (G_{P (A B) x_{2 n}, A B (A B) x_{2 n}, C D x_{2 n + 1}} (t)) + h (G_{V Q x_{2 n + 1}, U V x_{n + 2}, A B (A B) x_{2 n}} (t)) + h (G_{R x_{n + 2}, U V x_{n + 2}, A B (A B) x_{2 n}} (t))}] .

g (F_{P q, q, q} (t)) \leq ϕ [m a x {g (F_{q, q, q} (t)) + g (F_{P q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{P q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{P q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)), g (F_{P q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, q} (t))}] = ϕ {g (F_{P q, q, q} (t))} .

h (G_{P q, q, q} (t)) \geq φ [m i n {h (G_{q, q, q} (t)) + h (G_{P q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{P q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{P q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)), h (G_{P q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, q} (t))}] = φ {h (G_{P q, q, q} (t))} .

g (F_{P l, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \leq ϕ [m a x {g (F_{A B l, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{P l, A B l, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B l} (t)), g (F_{P l, A B l, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, A B l, C D x_{2 n + 1}} (t)), g (F_{P l, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B l} (t)), g (F_{P l, A B l, C D x_{2 n + 1}} (t)) + g (F_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B l} (t)) + g (F_{R x_{2 n + 2}, U V x_{2 n + 2}, A B l} (t))}] .

h (G_{P l, Q x_{2 n + 1}, R x_{2 n + 2}} (t)) \geq φ [m i n {h (G_{A B l, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{P l, A B l, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B l} (t)), h (G_{P l, A B l, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, A B l, C D x_{2 n + 1}} (t)), h (G_{P l, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{Q x_{2 n + 1}, C D x_{2 n + 1}, U V x_{2 n + 2}} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B l} (t)), h (G_{P l, A B l, C D x_{2 n + 1}} (t)) + h (G_{Q x_{2 n + 1}, U V x_{2 n + 2}, A B l} (t)) + h (G_{R x_{2 n + 2}, U V x_{2 n + 2}, A B l} (t))}] .

g (F_{P l, q, q} (t)) \leq ϕ [m a x {g (F_{P l, q, q} (t)) + g (F_{q, P l, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, P l} (t)), g (F_{q, P l, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, P l, q} (t)), g (F_{q, q, q} (t)) + g (F_{q, q, q} (t)) + g (F_{q, q, P l} (t)), g (F_{q, P l, q} (t)) + g (F_{q, q, P l} (t)) + g (F_{q, q, P l} (t))}] = ϕ {g (F_{P l, q, q} (t))

h (G_{P l, q, q} (t)) \geq φ [m i n {h (G_{P l, q, q} (t)) + h (G_{q, P l, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, P l} (t)), h (G_{q, P l, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, P l, q} (t)), h (G_{q, q, q} (t)) + h (G_{q, q, q} (t)) + h (G_{q, q, P l} (t)), h (G_{q, P l, q} (t)) + h (G_{q, q, P l} (t)) + h (G_{q, q, P l} (t))}] = φ {h (G_{P l, q, q} (t))} .

g (F_{P q, Q u, R v} (t)) \leq ϕ [m a x {g (F_{A B q, C D u, U V v} (t)) + g (F_{P q, A B q, C D u} (t)) + g (F_{Q u, C D u, U V v} (t)) + g (F_{R v, U V v, A B q} (t)), g (F_{P q, A B q, C D u} (t)) + g (F_{Q u, C D u, U V v} (t)) + g (F_{R v, A B q, C D u} (t)), g (F_{P q, C D u, U V v} (t)) + g (F_{Q u, C D u, U V v} (t)) + g (F_{R v, U V v, A B q} (t)), g (F_{P q, A B q, C D u} (t)) + g (F_{Q u, U V v, A B q} (t)) + g (F_{R v, U V v, A B q} (t))}] .

h (G_{P q, Q u, R v} (t)) \geq φ [m i n {h (G_{A B q, C D u, U V v} (t)) + h (G_{P q, A B q, C D u} (t)) + h (G_{Q u, C D u, U V v} (t)) + h (G_{R v, U V v, A B q} (t)), h (G_{P q, A B q, C D u} (t)) + h (G_{Q u, C D u, U V v} (t)) + h (G_{R v, A B q, C D u} (t)), h (G_{P q, C D u, U V v} (t)) + h (G_{Q u, C D u, U V v} (t)) + h (G_{R v, U V v, A B q} (t)), h (G_{P q, A B q, C D u} (t)) + h (G_{Q u, U V v, A B q} (t)) + h (G_{R v, U V v, A B q} (t))}] .

g (F_{q, u, v} (t)) \leq ϕ [m a x {g (F_{q, u, v} (t)) + g (F_{q, q, u} (t)) + g (F_{u, u, v} (t)) + g (F_{v, v, q} (t)), g (F_{q, q, u} (t)) + g (F_{u, u, v} (t)) + g (F_{v, q, u} (t)), g (F_{q, u, v} (t)) + g (F_{u, u, v} (t)) + g (F_{v, v, q} (t)), g (F_{q, q, u} (t)) + g (F_{u, v, q} (t)) + g (F_{v, v, q} (t))}] = ϕ {g (F_{q, u, v} (t))} .

h (G_{q, u, v} (t)) \geq φ [m i n {h (G_{q, u, v} (t)) + h (G_{q, q, u} (t)) + h (G_{u, u, v} (t)) + h (G_{v, v, q} (t)), h (G_{q, q, u} (t)) + h (G_{u, u, v} (t)) + h (G_{v, q, u} (t)), h (G_{q, u, v} (t)) + h (G_{u, u, v} (t)) + h (G_{v, v, q} (t)), h (G_{q, q, u} (t)) + h (G_{u, v, q} (t)) + h (G_{v, v, q} (t))} = φ {h (G_{q, u, v} (t))} .

g (F_{P x, Q y, R z} (t)) \leq ϕ [m a x {g (F_{x, y, z} (t)) + g (F_{x, x, y} (t)) + g (F_{y, y, z} (t)) + g (F_{z, z, x} (t)), g (F_{x, x, y} (t)) + g (F_{y, y, z} (t)) + g (F_{z, x, y} (t)), g (F_{x, y, z} (t)) + g (F_{y, y, z} (t)) + g (F_{z, z, x} (t)), g (F_{x, x, y} (t)) + g (F_{y, z, x} (t)) + g (F_{z, z, x} (t))}]

h (G_{P x, Q y, R z} (t)) \geq φ [m i n {h (G_{x, y, z} (t)) + h (G_{x, x, y} (t)) + h (G_{y, y, z} (t)) + h (G_{z, z, x} (t)), h (G_{x, x, y} (t)) + h (G_{y, y, z} (t)) + h (G_{z, x, y} (t)), h (G_{x, y, z} (t)) + h (G_{y, y, z} (t)) + h (G_{z, z, x} (t)), h (G_{x, x, y} (t)) + h (G_{y, z, x} (t)) + h (G_{z, z, x} (t))}],

g (F_{P x, Q y, R z} (t)) \leq ϕ (g (F_{A B_{x}, C D_{y}, U V_{z}} (t))) = \{\begin{cases} {(\frac{t}{t + 2})}^{∣ x - y + z ∣} & t > 0, \\ 0 & t \leq 0 \end{cases}

ϕ (g (F_{A B_{x, y, w} (t), C D_{l, w, y} (q), U V_{y, l, z}} (j))) = ϕ {m a x (\frac{t}{t + 2})^{∣ x - 3 y + w ∣} (\frac{q}{q + 2})^{∣ l - w + y ∣} (\frac{j}{j + 2})^{∣ y - l + z ∣}} \leq ϕ {\frac{m a x (t, q, j)}{m a x (t, q, j) + 2}}^{∣ x - 3 y + w ∣ + ∣ l - w + y ∣ + ∣ y - l + z ∣} \leq ϕ {\frac{m a x (t, q, j)}{m a x (t, q, j) + 2}}^{∣ x - 3 y + w + l - w + y + y - l + z ∣} \leq ϕ {\frac{m a x (t, q, j)}{m a x (t, q, j) + 2}}^{∣ x - 3 y + w + l - w + y + y - l + z ∣} \leq ϕ {\frac{m a x (t, q, j)}{m a x (t, q, j) + 2}}^{∣ x - y + z ∣} = g (F_{P x, Q y, R z} (m a x (t, q, j)) .

T (x) = \sup_{y, z \in N} {g (x, y, z) + K (x, y, z, T (τ (x, y, z)))},

h (t, q, j) = \sup_{x \in L} ∣ t (x) - q (x) + j (x) ∣

G_{t, q, j} (t) = \{\begin{cases} e^{- (\frac{h (t, q, j)}{t})} & t > 0 \\ 0 & t \leq 0, \end{cases}

h (G_{P x, Q y, R z} \geq ψ {h (G_{A B x, C D y, U V z} (t)} = h (G_{P x, Q y, R z} (m i n (t, q, j)),

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IDENTIFYING FIXED POINTS AND SOLUTION OF NONLINEAR FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN INTUITIONISTIC MENGER PROBABILISTIC METRIC SPACES

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