Classical stabilities of multiplicative inverse difference and adjoint functional equations

<jats:p>The aim of this present article is to investigate various classical stability results of the multiplicative inverse difference and adjoint functional equations
<jats:disp-formula><jats:alternatives><jats:tex-math>$$ m_{d} \biggl(\frac{rs}{r+s} \biggr)-m_{d} \biggl( \frac{2rs}{r+s} \biggr)= \frac{1}{2} \bigl[m_{d}(r)+m_{d}(s) \bigr] $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>m</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mi>r</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>r</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>]</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:disp-formula> and
<jats:disp-formula><jats:alternatives><jats:tex-math>$$ m_{a} \biggl(\frac{rs}{r+s} \biggr)+m_{a} \biggl( \frac{2rs}{r+s} \biggr)= \frac{3}{2} \bigl[m_{a}(r)+m_{a}(s) \bigr] $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mi>r</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>r</mml:mi><mml:mi>s</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi><mml:mo>+</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:mo>]</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:disp-formula> in the framework of non-zero real numbers. A proper counter-example is illustrated to prove the failure of the stability results for control cases. The relevance of these functional equations in optics is also discussed.</jats:p>


Introduction
The investigation of an approximate solution near to the exact solution of many mathematical equations such as functional, differential, difference and algebraic equations is a fascinating concept in the research domain of pure and applied mathematics. This theory emanated from an inquisitive query proposed in [24] and was developed through Stability (HURGS) by many mathematicians [1,9,10,13,14]. Recently, this theory has got a lot of momentum in dealing with different types of rational and multiplicative inverse functional equations with their applications in diversified fields, such as physics, electrical engineering, digital image processing. There are more instigating and novel results concerning this topic. For more information, refer to [2-7, 11, 12, 15, 18-22].
The rational form of the reciprocal, reciprocal difference and reciprocal adjoint functional equations and were introduced and obtained their classical stability results in [16,17]. The multiplicative inverse functional equation was dealt with in [23] to prove its non-Archimedean stabilities. For the first time in stability theory, a functional equation involving function of a rational argument is considered. It is easy to examine that the reciprocal mapping m(r) = 1 r is a solution of (1.4). Induced by the solution of (1.4) and since the power of a thin lens is the multiplicative inverse of its focal length, we focus on the following multiplicative inverse difference and adjoint functional equations, respectively, to investigate their solutions and analytical stabilities:

Stabilities of Eqs. (1.5) and (1.6)
In this fragment, we investigate various analytical stabilities associated with (1.5) and (1.6) in the domain of non-zero real numbers. To obtain the main results in a simple approach, consider the difference operators m d , m a : R × R − → R, defined respectively by for all r, s ∈ R .
Theorem 3.1 Assume a fixed constant μ = ±1 and a mapping m d : for all r, s ∈ R . Then a unique multiplicative inverse mapping M d : R − → R exists and satisfies (1.5) with the result that for all r ∈ R .
Proof Firstly, let us prove it for the case μ = 1. Switching (r, s) to (r, r) in (3.1), we obtain for all r ∈ R . Now, replacing r with r 2 μ in (3.4) and then multiplying by | 1 2 | μ , we get Letting → ∞ in (3.5) and using (3.2), we see that Now, we claim that M d satisfies (1.5). Plugging (r, s) into ( r 2 μ , s 2 μ ) in (3.1) and then in the resultant, dividing by 2 μ on its both sides, we obtain for all r, s ∈ R and for all positive integers . Now, using (3.2), (3.6) in (3.7), we find that M d satisfies (1.5) for all r, s ∈ R . For each r ∈ R and each integer , we have Applying (3.6) and letting → ∞, we obtain (3.3). Next we prove that M d is unique. For this, let us consider M d : R − → R to be an alternative multiplicative inverse mapping satisfying (1.5) and (3.3). Therefore for all r ∈ R . By the application of (3.3), we obtain for all r ∈ R . It can be noticed that M d is unique by allowing to ∞ in (3.8). This completes the proof for μ = 1. The proof for the case μ = -1 is similar to the above arguments.
The subsequent results are various stabilities pertinent to HUS, HUTRS and HUJRS of Eq. (1.5). The proofs directly follow from Theorem 3.1 for μ = -1. Hence, we omit the proofs. In the ensuing results, let m d : R − → R be a mapping.   Inspired by the counter-example in [8], we prove that Eq. (1.5) fails to be stable for θ = -1 in Corollary 3.3 in the domain of non-zero real numbers. For this, the function defined below is used to present a suitable counter-example.
Let a function ν : R * − → R be defined as for all r ∈ R . Then the function m d turns into a suitable example to illustrate that (1.5) is unstable for θ = -1 in Corollary 3.3 in the subsequent theorem.

Theorem 3.5
Assume that the function m d : R * − → R described in (3.9) satisfies the inequality for all r, s ∈ R . Then a multiplicative inverse mapping M d : R * − → R and a constant K > 0 do not exist such that for all r ∈ R * .

Occurrences of (1.5) and (1.6)
In manufacturing optical instruments, we come across compound lenses. Let the focal lengths of two thin lenses be r and s. Then the combined focal length F is given by F = rs r+s . But the power P of a lens is the reciprocal of the focal length F. Suppose p 1 = 1 r and p 2 = 1 s are powers of the above lenses, then the combined power P of the lenses is given by P = r + s. In view of these theories, we can associate Eqs.

Conclusion
In this study, we have proved that Eqs. (1.4), (1.5) and (1.6) are equivalent, which indicates that the multiplicative inverse mapping is a solution of Eqs. (1.5) and (1.6). In this investigation, zero is omitted to avoid singular cases in obtaining the stability results. We conclude that the stability results of the multiplicative inverse difference and adjoint functional equations hold good in the domain of non-zero real numbers. We have illustrated a suitable example that the stability result of (1.5) fails to hold good for a critical case. We have also discussed the occurrence of Eqs. (1.5) and (1.6) arising in various situations where focal lengths and powers of thin lenses in optics are related.