New Stability Results of Multiplicative Inverse Quartic Functional Equations

: The purpose of this investigation is to introduce different forms of multiplicative inverse functional equations, to solve them and to establish the stability results of them in the framework of matrix normed spaces. A suitable counter-example is also provided to prove the instability of the results for a singular case. The applications of the equations dealt in this study are discussed with the fluid resistances of blood vessels and also an important concept in Raman spectroscopy.


Introduction & Preliminaries
Various linear spaces of bounded Hilbert space operators such as mapping spaces, tensor products of operator spaces, quotients spaces in operator theory are abstractly characterized through matrix normed spaces [19].The characterization of these spaces indicates that they further be considered as spaces of operator.Due to this result, the operator spaces theory has noteworthy application in operator algebra theory [5].The result obtained in [19] is invoked to the theory of ordered operator spaces [19].The proof given in [6] is achieved by the technique applied in [13].
Here, we evoke the fundamental ideas of matrix normed spaces.We utilize the ensuing notions: •   () is the set of all square matrices of order  in a normed space ; •   ∈  1, (ℂ) denotes  th element is 1, and the other elements are 0; •   ∈   (ℂ) means (, ) th -element is 1, and the other elements are 0; •   ⊗  ∈   () indicates (, ) th -element is , and the other elements are 0.
Suppose  1 and  2 are vector spaces.Then for a given mapping :  1 ⟶  2 and for an integer  > 0, define   :   ( 1 ) ⟶   ( 2 ) by   ([  ]) = [(  )] for all [  ] ∈   ( 1 ).More information pertinent to matrix normed spaces are available in [12,27] The inspiration for the stability theory of mathematical equations is due to the question raised in [28] regarding homomorphisms in group theory.There are various responses provided in [1,8,11,14,15] to the question posed in [28].For the first time, the functional equation , where (≠ 0) ∈ ℝ and  is any real constant.
The non-Archimedean stability of the multiplicative inverse fourth power functional equation is obtained in [10].It can be easily verified that the multiplicative inverse fourth power mapping () = 1  4 satisfies equation (1.2).Later, there are several published papers on solutions, stability results and applications of various forms of multiplicative inverse or rational type or reciprocal functional equations in the literature.For further details, one can refer to [3,9,16,18,20,21,22,23,24,25,26].
In order to explore applications further, we extend equation (1.2) to new forms as, multiplicative inverse fourth power difference functional equation  ( and a multiplicative inverse fourth power adjoint functional equation  ( We prove the equivalency of equations (1.3) and (1.4) to achieve their solutions.The stability results of (1.3) and (1.4) are investigated via direct and fixed point techniques in the domain of matrix normed spaces.An apt instance is demonstrated to substantiate the non-stability result.The inferences of equations (1.3) and (1.4) are acquired by employing them in certain occurences in fluid dynamics and Raman spectroscopy.In the entire study, let us assume that  2 ≠ − 1 to avoid singularity in the main results.Also, unless or otherwise specified, we consider  to be a matrix normed space containing non-singular square matrices of  with a norm ‖⋅‖ so that ‖‖ ≤ 1 for all  ∈  and ℬ to be a matrix complete normed space, respectively with norm ‖⋅‖  .Then, applying Taylor's series expansion, we can find  1  after truncating to ( + 1) terms [2].Thus, the rational powers of  can be computed for all  ∈ .For a mapping :  ⟶ ℬ and for easy computation, let the difference operators :  ×  ⟶ ℬ and   :   ( × ) ⟶   (ℬ) be defined by

Identicalness of equations (1.3) and (1.4)
In the ensuing theorem, we prove that equations (1.3) and (1.4) are equivalent to each other.Theorem 2.1 Let : ℝ ⋆ ⟶ ℝ be a mapping.Then, the following statements are equivalent.
3. Suppose  is a solution of (1.4).Then it satisfies (1.4).By the analogous reasoning stated above, when ) in (1.4) and simplified additionally, we have Hence  is a multiplicative inverse fourth power mapping.

Stabilities of equation (1.3) via direct technique
In this part, we determine the stabilities of equation ( 1.3) in the domain of matrix normed spaces.The following lemma is a key element to achieve our major results.Lemma 3.1 [4] The following assertions are true:  (𝒜).Therefore, :  ⟶ ℬ is a unique solution of (3.3) and hence it is multiplicative inverse fourth power mapping, as required.This completes the proof.Proof.The proof goes through the same way as in Theorem 3.2, and so it is excluded.Corollary 3.5 Suppose  < −4 and (≥ 0) ∈ ℝ.Let a mapping :  ⟶ ℬ satisfies (3.4).Then, a solution :  ⟶ ℬ of (1.3) (𝒜).Here ϒ is defined as in Theorem 3.4.Proof.The required result follows via the proof of Theorem 3.7, and so the details are neglected.Corollary 3.10 Let ℬ be a  ∞ -complete normed space.Let  < −4 and (≥ 0) ∈ ℝ.Let a mapping :  ⟶ ℬ satisfies (3.9).Then, a solution :  ⟶ ℬ of (1.3) exists whihc is unique with the result that Proof.The similar arguments as in the proof of Theorem 3.9 will lead to the proof of this corollary by taking ( 1  ,  2  ) = (‖ 1  ‖  + ‖ 2  ‖  ).

Pertinence of equations (1.3) and (1.4) in fluid dynamics and Raman Spectroscopy
In this section, we present the pertinence of equations (1.3) and (1.4) in various field such as fluid dynamics and Raman spectroscopy.

Fluid Dynamics
The fundamental factors that are necessary to find the blood flow resistance () within a single vessel are the radius of the vessel (), the length of the vessel () and the viscoscity of the blood ().Among the above three factors, the most signifcant factor is the radius of the vessel in terms of quantity and physiology.Since the vascular smooth muscle in the wall of the blood vessel contracts and expands, the radius of the blood vessel is a primary factor.Also, a very small change in the radius of vessel leads to large change in resistance, where as length of vessel does not change significantly and viscosity of blood normally stays within a small range (except when hematocrit changes).Then the fluid resistance is directionary proportional to  and  and inversely proportional to  4 , which is given by  =   4 where  is constant of proportionality.Suppose  and  are kept constant, then the fluid resistance is given by  =   4 .where  is a constant.If  1 and  2 are the radii of two blood vessels, then the fluid resistances in those blood vessels can be considered as ( 1 ) and ( 2 ), respectively, where  is a reciprocal fourth power mapping.Also,  (

Raman Spectroscopy
In Raman spectroscopy, the solution of equations (1.3) and (1.4) can be applied in studying the nongraphite samples with distinct crystallite sizes and laser energies.The disorder-induced Raman bands  and  are denoted by   and   .Then the intensity ratio of these disorder-induced Raman bands   /  is proportional to the multiplicative inverse fourth power of the laser energies.

Discussion and Conclusions
As of our knowledge, our findings in this study are novel in the field of stability theory.This is our antecedent endevavour to deal with new type of reciprocal fourth power functional equations.It is shown that the equations (1.3) and (1.4) are equivalent to each other to conclude that their solution is reciprocal fourth power mapping.The stability results of different forms of reciprocal functional equations are obtained by many mathematicians in various spaces.But, in this work, we have considered matrix normed spaces to analyze the results of equations (1.3) and (1.4) and they are found to be stable except for a singular case.For the failure of stability result when critical case arises, we have illustrated an apt example.At the end of this study, we have presented pertinency of equations (1.3) and (1.4) in fluid dynamics and Raman spectroscopy.