The attention of surgeons to pilonidal sinus disease is increasing. We aimed to estimate the incidence of Pilonidal sinus disease, and verify the employed management and its outcome in term of surgical site infection, recurrence and patients' satisfaction.
A cohort study included 224 patients with pilonidal sinus disease (Jan 2014 to April 2020).
Mean age was 23.83 ± 4.9 years, with male predominance (male to female ratio of 2.25:1). Incidence of pilonidal sinus disease was 2.9% of all surgical clinic population. Mean duration of symptoms was 65.8 ± 50.7 days. Majority (80.8%) had chronic pilonidal sinus, whereas the remainder 19.2% had acute onset with abscess. In pilonidal sinus the surgical modality was fashioned according to the extent of the disease keeping in mind the number of sinus opening in form of Limberg flap (44.2%), primary closure (19.6%), or laid open to heal by secondary intention (17%). Recurrence of pilonidal sinus was seen in 2.2% and not affected by the procedure employed (P = 0.4). Whereas, in cases of pilonidal abscess the recurrence rate was 27.9%. The difference was significant (p = 0.00001). Over all patients' satisfaction was very good/excellent in 187 (83.4%).
To reduce congestion of the operating lists in central hospitals, one can use a clear criterion based on the extent of the disease and the number of sinus openings, this will facilitate the management of pilonidal sinus disease in peripheral hospital settings, and comparable results can be achieved in terms of recurrence rate and patient satisfaction.
Citation: Fauwaz Fahad Alrashid, Saadeldin Ahmed Idris, Abdul Ghani Qureshi. Current trends in the management of pilonidal sinus disease and its outcome in a periphery hospital[J]. AIMS Medical Science, 2021, 8(1): 70-79. doi: 10.3934/medsci.2021008
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The attention of surgeons to pilonidal sinus disease is increasing. We aimed to estimate the incidence of Pilonidal sinus disease, and verify the employed management and its outcome in term of surgical site infection, recurrence and patients' satisfaction.
A cohort study included 224 patients with pilonidal sinus disease (Jan 2014 to April 2020).
Mean age was 23.83 ± 4.9 years, with male predominance (male to female ratio of 2.25:1). Incidence of pilonidal sinus disease was 2.9% of all surgical clinic population. Mean duration of symptoms was 65.8 ± 50.7 days. Majority (80.8%) had chronic pilonidal sinus, whereas the remainder 19.2% had acute onset with abscess. In pilonidal sinus the surgical modality was fashioned according to the extent of the disease keeping in mind the number of sinus opening in form of Limberg flap (44.2%), primary closure (19.6%), or laid open to heal by secondary intention (17%). Recurrence of pilonidal sinus was seen in 2.2% and not affected by the procedure employed (P = 0.4). Whereas, in cases of pilonidal abscess the recurrence rate was 27.9%. The difference was significant (p = 0.00001). Over all patients' satisfaction was very good/excellent in 187 (83.4%).
To reduce congestion of the operating lists in central hospitals, one can use a clear criterion based on the extent of the disease and the number of sinus openings, this will facilitate the management of pilonidal sinus disease in peripheral hospital settings, and comparable results can be achieved in terms of recurrence rate and patient satisfaction.
Hidradenitis suppurativa;
Almikhwah General Hospital;
Pilonidal sinus disease;
Surgical site infection;
Pilonidal sinus;
Pilonidal abscess
In 1940, Ulam [25] proposed the following question concerning the stability of group homomorphisms: Under what condition does there is an additive mapping near an approximately additive mapping between a group and a metric group? In the next year, Hyers [7,8] answered the problem of Ulam under the assumption that the groups are Banach spaces. A generalized version of the theorem of Hyers for approximately linear mappings was given by Rassias [22]. Since then, the stability problems of various functional equation have been extensively investigated by a number of authors (see [3,6,13,14,16,17,22,24,26,27,28]). By regarding a large influence of Ulam, Hyers and Rassias on the investigation of stability problems of functional equations the stability phenomenon that was introduced and proved by Rassias [22] in the year 1978 is called the Hyers-Ulam-Rassias stability.
Consider the functional equation
f(x+y)+f(x−y)=2f(x)+2f(y). | (1.1) |
The quadratic function f(x)=cx2 is a solution of this functional Equation (1.1), and so one usually is said the above functional equation to be quadratic [5,10,11,12]. The Hyers-Ulam stability problem of the quadratic functional equation was first proved by Skof [24] for functions between a normed space and a Banach space. Afterwards, the result was extended by Cholewa [2] and Czerwik [4].
Now, we consider the following functional equation:
f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y). | (1.2) |
It is easy to see that the function f(x)=cx4 satisfies the functional equation (1.2). Hence, it is natural that Eq (1.2) is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping (see [15,19]).
The theory of random normed spaces (briefly, RN-spaces) is important as a generalization of deterministic result of normed spaces and also in the study of random operator equations. The notion of an RN-space corresponds to the situations when we do not know exactly the norm of the point and we know only probabilities of possible values of this norm. Random theory is a setting in which uncertainty arising from problems in various fields of science, can be modelled. It is a practical tool for handling situations where classical theories fail to explain. Random theory has many application in several fields, for example, population dynamics, computer programming, nonlinear dynamical system, nonlinear operators, statistical convergence and so forth.
In 2008, Mihet and Radu [18] applied fixed point alternative method to prove the stability theorems of the Cauchy functional equation:
f(x+y)−f(x)−f(y)=0 |
in random normed spaces. In 2008, Najati and Moghimi [20] obtained a stability of the functional equation deriving from quadratic and additive function:
f(2x+y)+f(2x−y)+2f(x)−f(x+y)−f(x−y)−2f(2x)=0 | (1.3) |
by using the direct method. After that, Jin and Lee [9] proved the stability of the above mentioned functional equation in random normed spaces.
In 2011, Saadati et al. [21] proved the nonlinear stability of the quartic functional equation of the form
16f(x+4y)+f(4x−y)=306[9f(x+y3)+f(x+2y)]+136f(x−y)−1394f(x+y)+425f(y)−1530f(x) |
in the setting of random normed spaces. Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, the fixed point theory, the theory of intuitionistic spaces and the theory of functional equations were also presented. Azadi Kenary [1] investigated the Ulam stability of the following nonlinear function equation
f(f(x)−f(y))+f(x)+f(y)=f(x+y)+f(x−y), |
in random normed spaces.
In this note, we investigate the general solution for the quartic functional equation of the form
(3n+4)f(n∑i=1xi)+n∑j=1f(−nxj+n∑i=1,i≠jxi)=(n2+2n+1)n∑i=1,i≠j≠kf(xi+xj+xk)−12(3n3−2n2−13n−8)n∑i=1,i≠jf(xi+xj)+12(n3+2n2+n)n∑i=1,i≠jf(xi−xj)+12(3n4−5n3−7n2+13n+12)n∑i=1f(xi) | (1.4) |
(n∈N,n>4) and also investigate the Hyers-Ulam stability of the quartic functional equation in random normed spaces by using the direct approach and the fixed point approach.
In this part, we make some notations and basic definitions used in this article.
Definition 2.1. A function T:[0,1]×[0,1]→[0,1] is called a continuous triangular norm, if T satisfies the following condition
(a) T is commutative and associative;
(b) T is continuous;
(c) T(a,1)=a for all a∈[0,1];
(d) T(a,b)≤T(c,d) when a≤c and b≤d for all a,b,c,d∈[0,1].
Typical examples of continuous t−norms are Tp(a,b)=ab,Tm(a,b)=min(a,b) and TL(a,b)=max(a+b−1,0) (The Lukasiewicz t−norm). Recall [23] that if T is a t−norm and {xn} is a given sequence of numbers in [0,1], then Tni=1xn+i is defined recurrently by T′i=1xi=xi and Tni=1xi=T(Tn−1i=1xi,xn) for n≥2,T∞i=1xi is defined as T∞i=1xn+i. It is known that, for the Lukasiewicz t−norm, the following implication holds:
limn→∞(TL)∞i=1xn+i=1⇔∞∑n=1(1−xn)<∞. |
Definition 2.2. A random normed space (briefly, RN−space) is a triple (X,μ,T), where X is a vector space. T is a continuous t−norm and μ is a mapping from X into D+ satisfying the following conditions:
(RN1) μx(t)=ε0(t) for all t>0 if and only if x=0;
(RN2) μαx(t)=μx(t|α|) for all x∈X, and alpha∈ℜ with α≠0;
(RN3) μx+y(t+s)≥T(μx(t),μy(s)) for all x,y∈X and t,s≥0.
Definition 2.3. Let (X,μ,T) be an RN−space.
1). A sequence {xn} in X is said to be convergent to a point x∈X if, for any ε>0 and λ>0, there exists a positive integer N such that μxn−x(ε)>1−λ for all n>N.
2). A sequence {xn} in X is called a Cauchy sequence if, for any ε>0andλ>0, there exists a positive integer N such that μxn−xm(ε)>1−λ for all n≥m≥N.
3). An RN−space (X,μ,T) is said to be complete, if every Cauchy sequence in X is convergent to a point in X.
Throughout this paper, we use the following notation for a given mapping f:X→Y as
Df(x1,⋯,xn):=(3n+4)f(n∑i=1xi)+n∑j=1f(−nxj+n∑i=1,i≠jxi)−(n2+2n+1)n∑i=1,i≠j≠kf(xi+xj+xk)−12(3n3−2n2−13n−8)n∑i=1,i≠jf(xi+xj)+12(n3+2n2+n)n∑i=1,i≠jf(xi−xj)+12(3n4−5n3−7n2+13n+12)n∑i=1f(xi) |
for all x1,x2,⋯,xn∈X.
In this section we investigate the general solution of the n-variable quartic functional equation (1.4).
Theorem 3.1. Let X and Y be real vector spaces. If a mapping f:X→Y satisfies the functional equation (1.4) for all x1,⋯,xn∈X, then f:X→Y satisfies the functional equation (1.2) for all x,y∈X.
Proof. Assume that f satisfies the functional equation (1.4). Putting x1=x2=⋯=xn=0 in (1.4), we get
(3n+4)f(0)=(n2+2n+1){6n−20+4(n−4)(n−5)2+(n−4)(n−5)(n−6)6}f(0)−12(3n3−2n2−13n−8){4n−10+(n2−9n+20)2}f(0)+12(n3+2n2+n){4n−10+(n2−9n+20)2}f(0)−n2(3n4−5n3−7n2+13n+12)f(0) | (3.1) |
It follows from (3.1) that
f(0)=0. | (3.2) |
Replacing (x1,x2,⋯,xn) by (x,0,⋯,0⏟(n−1)−times) in (1.4), we have
(3n+4)f(x)+n4f(−x)+(n−1)f(x)=(n4−n3−3n2+n+22)f(x)−12(3n4−5n3−11n2+5n+8)f(x)+12(n4+n3+n2−n)f(x)+12(3n4−5n3−7n2+13n+12)f(x) | (3.3) |
for all x∈X. It follows from (3.3) that
f(−x)=f(x) |
for all x∈X. Setting x1=x2=⋯=xn=x in (1.4), we obtain
(3n+4)f(x)+nf(x)=(n2+2n+1){6n−20+4(27)(n−4)(n−5)2+(n−4)(n−5)(n−6)6}f(x)−12(3n3−2n2−13n−8){4n−10+(n2−9n+20)2}f(2x)+12(n3+2n2+n){4n−10+(n2−9n+20)2}f(0)+n2(3n4−5n3−7n2+13n+12)f(x) | (3.4) |
for all x∈X. It follows from (3.4) and (3.2) that
f(2x)=16f(x) |
for all x∈X. Setting x1=x2=⋯=xn=x in (1.4), we get
(3n+4)f(x)+nf(x)=(n2+2n+1){6n−20+4(n2−9n+20)2+(n2−9n+20)(n−6)6}f(3x)−12(3n3−2n2−13n−8){4n−10+16(n2−9n+20)2}f(x)+12(n3+2n2+n){4n−10+(n2−9n+20)2}f(0)+n2(3n4−5n3−7n2+13n+12)f(x) | (3.5) |
for all x∈X. It follows from (3.5) and (3.2) that
f(3x)=81f(x) |
for all x∈X. In general for any positive m, we get
f(mx)=m4f(x) |
for all x∈X. Replacing (x1,x2,⋯,xn) by (x,x,⋯,x⏟(n−1)−times,y) in (1.4), we get
(3n+4)(f((n−1)x+y)+(n−1)f(−2x+y))+f((n−1)x−ny)=(n2+2n+1)(3n−9+(n2−3n+2)2)f(2x+y)(n2+2n+1)(3n−11+(3n2−27n+602)+((n−4)(n−5)(n−6)6))f(3x)−12(3n3−2n2−13n−8){3n−9+(n2−9n+20)2}f(2x)−(n−1)2(3n3−2n2−13n−8)f(x+y)+12(n3+2n2+n)(n−1)f(x−y)+12(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.6) |
for all x,y∈X. It follows from (3.6) that
(3n+4)(f((n−1)x+y)+(n−1)f(−2x+y))+f((n−1)x−ny)=(n2+2n+1)(n2−3n+22)f(2x+y)+81(n2+2n+1)(n3−5n2+11n−66)f(x)−16(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)f(x+y)+12(n3+2n2+n)(n−1)f(x−y)+12(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.7) |
for all x,y∈X. Replacing y by −y in (3.7), we get
(3n+4)(f((n−1)x−y)+(n−1)f(2x+y))+f(()x+y)=(n2+2n+1)(n2−3n+22)f(2x−y)+81(n2+2n+1)(n3−5n2+11n−66)f(x)−16(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)f(x−y)+12(n3+2n2+n)(n−1)f(x+y)+12(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.8) |
for all x,y∈X. Adding (3.7) and (3.8), we get
(3n+4)((n−1)2{f(x+y)+f(x−y)}+2{f((n−1)x)−(n−1)2f(x)}+2{f(y)−(n−1)2f(y)})+(n−1)(f(−2x+y)+f(2x+y))+(f((n−1)x−ny)+f((n−1)x+ny))=(n2+2n+1)(n2−3n+22)(f(2x−y)+f(2x+y))+162(n2+2n+1)(n3−5n2+11n−66)f(x)−32(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)(f(x−y)+f(x+y))+12(n3+2n2+n)(n−1)(f(x+y)+f(x−y))+(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.9) |
for all x,y∈X. It follows from (3.9) that
(3n+4)((n−1)2{f(x+y)+f(x−y)}+2{f((n−1)x)−(n−1)2f(x)}+2{f(y)−(n−1)2f(y)})+(n−1)(f(2x−y)+f(2x+y))+n2(n−1)2(f(x+y)+f(x−y))+2{f((n−1)x)−n2(n−1)2f(x)+n4f(y)−n2(n−1)2f(y)}=(n2+2n+1)(n2−3n+22)(f(2x−y)+f(2x+y))+162(n2+2n+1)(n3−5n2+11n−66)f(x)−32(3n3−2n2−13n−8)(n2−3n+24)f(x)−(n−1)2(3n3−2n2−13n−8)(f(x−y)+f(x+y))+12(n3+2n2+n)(n−1)(f(x+y)+f(x−y))+(3n4−5n3−7n2+13n+12)(f(y)+(n−1)f(x)) | (3.10) |
for all x,y∈X. It follows from (3.10) that
−2f(2x+y)−2f(2x−y)=−8f(x+y)−8f(x−y)−48f(x)+12f(y) | (3.11) |
for all x,y∈X. From (3.11), we get
(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y) |
for all x,y∈X. Thus the mapping f:X→Y is quartic.
In this section, the Ulam-Hyers stability of the quartic functional equation (1.4) in RN−space is provided. Throughout this part, let X be a linear space and (Y,μ,T) be a complete RN−space.
Theorem 4.1. Let j=±1, f:X→Y be a mapping for which there exists a mapping η:Xn→D+ satisfying
limk→∞T∞i=0(η2(k+i)x1,2(k+i)x2,...,2(k+i)xn(24(k+i+1)jt))=1=limk→∞η2kjx1,2kjx2,..,2kjxn(24kjt) |
such that f(0)=0 and
μDf(x1,x2,...,xn)(t)≥η(x1,x2,...,xn)(t) | (4.1) |
for all x1,x2,...,xn∈X and all t>0. Then there exists a unique quartic mapping Q:X→Y satisfying the functional equation (1.4) and
μQ(x)−f(x)(t)≥T∞i=0(η2(i+1)jx,2(i+1)jx,,...,2(i+1)jx(14(3n5−5n4−11n3+5n2+8n)24(i+1)jt)) |
for all x∈X and all t>0. The mapping Q(x) is defined by
μC(x)(t)=limk→∞μf(2kjx)24kj(t) | (4.2) |
for all x∈X and all t>0.
Proof. Assume j=1. Setting (x1,x2,...,xn) by (x,x,...,x⏟n−times) in (4.1), we obtain
μ(3n5−5n4−11n3+5n2+8n)4f(2x)−4(3n5−5n4−11n3+5n2+8n)f(x)(t)≥η(x,x,...,x⏟n−times)(t) | (4.3) |
for all x∈X and all t>0. It follows from (4.2) and (RN2) that
μf(2x)16−f(x)(t)≥η(x,x,...,x⏟n−times)(24(3n5−5n4−11n3+5n2+8n)t) |
for all x∈X and all t>0. Replacing x by 2kx in (4.3), we have
μf(2k+1x)24(k+1)−f(2kx)24k(t)≥η(2kx,2kx,...,2kx⏟n−times)(24k(3n5−5n4−11n3+5n2+8n)16t)≥η(x,x,...,x⏟n−times)(24k(3n5−5n4−11n3+5n2+8n)16αkt) | (4.4) |
for all x∈X and all t>0. It follows from f(2nx)24n−f(x)=n−1∑k=0f(2k+1x)24(k+1)−f(2kx)24k and (4.4) that
μf(2nx)24n−f(x)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)16)≥Tn−1k=0(ηx,x,0,...,0(t))=η(x,x,...,x⏟n−times)(t), |
μf(2nx)24n−f(x)(t)≥η(x,x,...,x⏟n−times)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)24) | (4.5) |
for all x∈X and all t>0. Replacing x by 2mx in (4.5), we get
μf(2n+mx)24(n+m)−f(2mx)24m(t)≥η(x,x,...,x⏟n−times)(tn+m∑k=mαk24k(3n5−5n4−11n3+5n2+8n)16). | (4.6) |
Since η(x,x,...,x⏟n−times)(tn+m∑k=mαk24k(3n5−5n4−11n3+5n2+8n)16)→1 as m,n→∞, {f(2nx)24n} is a Cauchy sequence in (Y,μ,T). Since (Y,μ,T) is a complete RN-space, this sequence converges to some point C(x)∈Y. Fix x∈X and put m=0 in (4.6). Then we have
μf(2nx)24n−f(x)(t)≥η(x,x,...,x⏟n−times)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)16) |
and so, for every δ>0, we get
μC(x)−f(x)(t+δ)≥T(μQ(x)−f(2nx)24n(δ),μf(2nx)24n−f(x)(t))≥T(μQ(x)−f(2nx)24n(δ),η(x,x,...,x⏟n−times)(tn−1∑k=0αk24k(3n5−5n4−11n3+5n2+8n)16)). | (4.7) |
Taking the limit as n→∞ and using (4.7), we have
μC(x)−f(x)(t+δ)≥η(x,x,...,x⏟n−times)((3n5−5n4−11n3+5n2+8n)(24−α)t). | (4.8) |
Since δ is arbitrary, by taking δ→0 in (4.8), we have
μQ(x)−f(x)(t)≥η(x,x,...,x⏟n−times)((3n5−5n4−11n3+5n2+8n)(24−α)t). | (4.9) |
Replacing (x1,x2,...,xn) by (2nx1,2nx2,...,2nxn) in (4.1), respectively, we obtain
μDf(2nx1,2nx2,...,2nxn)(t)≥η2nx1,2nx2,...,2nxn(24nt) |
for all x1,x2,...,xn∈X and for all t>0. Since
limk→∞T∞i=0(η2k+ix1,2k+ix2,...,2k+ixn(24(k+i+1)jt))=1, |
we conclude that Q fulfils (1.1). To prove the uniqueness of the quartic mapping Q, assume that there exists another quartic mapping D from X to Y, which satisfies (4.9). Fix x∈X. Clearly, Q(2nx)=24nQ(x) and D(2nx)=24nD(x) for all x∈X. It follows from (4.9) that
μQ(x)−D(x)(t)=limn→∞μQ(2nx)24nD(2nx)24n(t)μQ(2nx24n(t)≥min{μQ(2nx)24nf(2nx)24n(t2),μD(2nx)24nf(2nx)24n(t2)} |
≥η(2nx,2nx,...,2nx⏟n−times)(24n(3n5−5n4−11n3+5n2+8n)(24−α)t)≥η(x,x,...,x⏟n−times)(24n(3n5−5n4−11n3+5n2+8n)(24−α)tαn). |
Since limn→∞(24n(3n5−5n4−11n3+5n2+8n)(24−α)tαn)=∞, we get
limn→∞ηx,x,0,...,0(24n(3n5−5n4−11n3+5n2+8n)(24−α)tαn)=1.
Therefore, it follows that μQ(x)−D(x)(t)=1 for all t>0 and so Q(x)=D(x).
This completes the proof.
The following corollary is an immediate consequence of Theorem 4.1, concerning the stability of (1.4).
Corollary 4.2. Let Ω and ℧ be nonnegative real numbers. Let f:X→Y satisfy the inequality
μDf(x1,x2,...,xn)(t)≥{ηΩ(t),ηΩn∑i=1‖xi‖℧(t)ηΩ(n∏i=1‖xi‖n℧)(t),ηΩ(n∏i=1‖xi‖℧+n∑i=1‖xi‖n℧)(t) |
for all x1,x2,x3,...,xn∈X and all t>0, then there exists a unique quartic mapping Q:X→Y such that
μf(x)−Q(x)(t)≥{ηΩ|60|(3n5−5n4−11n3+5n2+8n)(t)ηΩ‖x‖℧|4|(3n5−5n4−11n3+5n2+8n)|24−2℧|(t),℧≠4ηΩ‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4nη(n+1)Ω‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4n |
for all x∈X and all t>0.
In this section, we prove the Ulam-Hyers stability of the functional equation (1.4) in random normed spaces by the using fixed point method.
Theorem 5.1. Let f:X→Y be a mapping for which there exists a mapping η:Xn→D+ with the condition
limk→∞ηδkix1,δkix2,...,δkixn(δkit)=1 |
for all x1,x2,x3,...,xn∈X and all 0, where
δi={2i=0;12i=1; |
and satisfy the functional inequality
μDf(x1,x2,...,xn)(t)≥ηx1,x2,...,xn(t) |
for all x1,x2,x3,...,xn∈X and all 0. If there exists L=L(i) such that the function x→β(x,t)=η(x2,x2,...,x2⏟n−times)((3n5−5n4−11n3+5n2+8n)t) has the property that
β(x,t)≤L1δ4iβ(δix,t) | (5.1) |
for all x∈X and t>0, then there exists a unique quartic mapping Q:X→Y satisfying the functional equation (1.4) and
μQ(x)−f(x)(L1−i1−Lt)≥β(x,t) |
for all x∈X and t>0.
Proof. Let d be a general metric on Ω such that
d(p,q)=inf{k∈(0,∞)/μ(p(x)−q(x))(kt)≥β(x,t),x∈X,t>0}. |
It is easy to see that (Ω,d) is complete. Define T:Ω→Ω by Tp(x)=1δ4ip(δix) for all x∈X. Now assume that for p,q∈Ω, we have d(p,q)≤K. Then
μ(p(x)−q(x))(kt)≥β(x,t)⇒μ(p(x)−q(x))(Ktδ4i)≥β(x,t)⇒d(Tp(x),Tq(x))≤KL⇒d(Tp,Tq)≤Ld(p,q) | (5.2) |
for all p,q∈Ω. Therefore, T is a strictly contractive mapping on Ω with Lipschitz constant L. It follows from (5.2) that
μ(3n5−5n4−11n3+5n2+8n)4f(2x)−4(n3n5−5n4−11n3+5n2+8n)f(x)(t)≥η(x,x,...,x⏟n−times)(t) | (5.3) |
for all x∈X. It follows from (5.3) that
μf(2x)16−f(x)(t)≥η(x,x,...,x⏟n−times)((3n5−5n4−11n3+5n2+8n)16t) | (5.4) |
for all x∈X. By using (5.1) for the case i=0, it reduce to
μf(2x)16−f(x)(t)≥Lβ(x,t) |
for all x∈X. Hence we obtain
d(μTf,f)≤L=L1−i<∞ | (5.5) |
for all x∈X. Replacing x by x2 in (5.4), we get
μf(x)16−f(x2)(t)≥η(x2,x2,...,x2⏟n−times)((3n5−5n4−11n3+5n2+8n)16t) |
for all x∈X. By using (5.1) for the case i=1, it reduce to
μ16f(x2)−f(x)(t)≥β(x,t)⇒μTf(x)−f(x)(t)≥β(x,t) |
for all x∈X. Hence we get
d(μTf,f)≤L=L1−i<∞ | (5.6) |
for all x∈X. From (5.5) and (5.6), we can conclude
d(μTf,f)≤L=L1−i<∞ |
for all x∈X.
The remaining proof is similar to the proof of Theorem 4.1. So Q is a unique fixed point of T in the set such that
μ(f(x)−Q(x))(L1−i1−Lt)≥β(x,t) |
for all x∈X and t>0. This completes the proof of the theorem.
From Theorem 4.1, we obtain the following corollary concerning the stability for the functional equation (1.4).
Corollary 5.2. Suppose that a mapping f:X→Y satisfies the inequality
μDf(x1,x2,...,xn)(t)≥{ηΩ(t),ηΩn∑i=1‖xi‖℧(t)ηΩ(n∏i=1‖xi‖℧)(t),ηΩ(n∏i=1‖xi‖℧+n∑i=1‖xi‖n℧)(t), |
for all x1,x2,...,xn∈X and all t>0, where Ω,℧ are constants with Ω>0. Then there exists a unique quartic mapping Q:X→Y such that
μf(x)−Q(x)(t)≥{ηΩ|60|(3n5−5n4−11n3+5n2+8n)(t)ηΩ‖x‖℧|4|(3n5−5n4−11n3+5n2+8n)|24−2℧|(t),℧≠4ηΩ‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4nη(n+1)Ω‖x‖n℧|4|(3n5−5n4−11n3+5n2+8n)|24−2n℧|(t),℧≠4n |
for all x∈X and all t>0.
Proof. Set
μDf(x1,x2,...,xn)(t)≥{ηΩ(t),ηΩn∑i=1‖xi‖℧(t),ηΩ(n∏i=1‖xi‖℧)(t),ηΩ(n∏i=1‖xi‖℧+n∑i=1‖xi‖n℧)(t) |
for all x1,x2,...,xn∈X and all t>0. Then
η(δkix1,δkix2,...,δkixn)(δ4kit)={ηΩδ4ki(t),ηΩn∑i=1‖xi‖℧δ(4−n℧)ki(t),ηΩ(n∏i=1‖xi‖℧δ(4−n℧)ki)(t),ηΩ(n∏i=1‖xi‖℧δ(4−n℧)ki+n∑i=1‖xi‖n℧)(t), |
={→ 1 as k→∞→ 1 as k→∞→ 1 as k→∞→ 1 as k→∞. |
But we have β(x,t)=η(x2,x2,...,x2⏟n−times)(14(3n5−5n4−11n3+5n2+8n)t) has the property L1δ4iβ(δix,t) for all x∈X and t>0. Now
β(x,t)={η4Ω3n5−5n4−11n3+5n2+8n(t)η4Ω‖x‖℧24℧(3n5−5n4−11n3+5n2+8n)(t)η4Ω‖x‖n℧24℧(3n5−5n4−11n3+5n2+8n)(t)η4Ω‖x‖ns2n℧(3n5−5n4−11n3+5n2+8n)(t), |
L1δ4iβ(δix,t)={ηδ−4iβ(x)(t)ηδ℧−4iβ(x)(t)ηδ℧−4iβ(x)(t)ηδn℧−4iβ(x)(t). |
By (4.1), we prove the following eight cases:
L=2−4 ifi=0 and L=24 if i=1L=2℧−4 for ℧<4 ifi=0 and L=24−℧ for ℧>4 if i=1L=2n℧−4 for ℧<4n ifi=0 and L=24−n℧ for ℧>4n if i=1L=2n℧−4 for ℧<4n ifi=0 and L=24−n℧ for ℧>4n if i=1 |
Case 1: L=2−4 if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω60(3n5−5n4−11n3+5n2+8n))(t). |
Case 2: L=24 if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω−60(3n5−5n4−11n3+5n2+8n))(t). |
Case 3: L=2℧−4 for ℧<4 if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ωn‖x‖℧4(3n5−5n4−11n3+5n2+8n)(24−2℧))(t). |
Case 4: L=24−℧ for ℧>4 if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(nΩ‖x‖s4(3n5−5n4−11n3+5n2+8n)(2℧−24))(t). |
Case 5: L=2n℧−4 for ℧<4n if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω‖x‖n℧4(3n5−5n4−11n3+5n2+8n)(24−2n℧))(t). |
Case 6: L=24−n℧ for ℧>4n if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η(Ω‖x‖n℧−4(3n5−5n4−11n3+5n2+8n)(2n℧−24))(t). |
Case 7: L=2n℧−4 for ℧<4n if i=0
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η((n+1)‖x‖n℧4(3n5−5n4−11n3+5n2+8n)(24−2n℧))(t). |
Case 8 : L=24−n℧ for ℧>4n if i=1
μf(x)−Q(x)(t)≥L1δ4iβ(δix,t)(t)≥η((n+1)Ω‖x‖n℧−4(3n5−5n4−11n3+5n2+8n)(2n℧−24))(t). |
Hence the proof is complete.
In this note we investigated the general solution for the quartic functional equation (1.4) and also investigated the Hyers-Ulam stability of the quartic functional equation (1.4) in random normed space using the direct approach and the fixed point approach. This work can be applied to study the stability in various spaces such as intuitionistic random normed spaces, quasi-Banach spaces and fuzzy normed spaces. Moreover, the results can be applied to investigate quartic homomorphisms and quratic derivations in Banach algebras, random normed algebras, fuzzy Banach algebras and C∗-ternary algebras.
This work was supported by Incheon National University Research Grant 2020-2021. We would like to express our sincere gratitude to the anonymous referee for his/her helpful comments that will help to improve the quality of the manuscript.
The authors declare that they have no competing interests.
[1] | Lim J, Shabbir J (2019) Pilonidal sinus disease—a literature review. World J Surg Surgical Res 2: 1117. |
[2] |
Karandikar S, Naik NG (2017) Retrospective analysis of 30 cases of pilonidal sinus by various techniques of conservative and surgical management and review of literature. Int Surg J 4: 291-295. doi: 10.18203/2349-2902.isj20164457
![]() |
[3] |
Stauffer VK, Luedi MM, Kauf P, et al. (2018) Common surgical procedures in pilonidal sinus disease: a meta-analysis, merged data analysis, and comprehensive study on recurrence. Sci Rep 8: 1-28. doi: 10.1038/s41598-017-17765-5
![]() |
[4] |
Doll D, Orlik A, Maier K, et al. (2019) Impact of geography and surgical approach on recurrence in global pilonidal sinus disease. Sci Rep 9: 1-24. doi: 10.1038/s41598-019-51159-z
![]() |
[5] |
Darwish AA, Eskandaros MS, Hegab A (2017) Sacrococcygeal pilonidal sinus: modified sinotomy versus lay-open, limited excision, and primary closure. Egypt J Surg 36: 13-19. doi: 10.4103/1110-1121.199901
![]() |
[6] |
Mahmood F, Hussain A, Akingboye A (2020) Pilonidal sinus disease: review of current practice and prospects for endoscopic treatment. Ann Med Surg 57: 212-217. doi: 10.1016/j.amsu.2020.07.050
![]() |
[7] |
Lamdark T, Vuille-dit-Bille RN, Bielicki IN, et al. (2020) Treatment strategies for pilonidal sinus disease in Switzerland and Austria. Medicina 56: 341. doi: 10.3390/medicina56070341
![]() |
[8] |
Duman K, Gırgın M, Harlak A (2017) Prevalence of sacrococcygeal pilonidal disease in Turkey. Asian J Surg 40: 434-437. doi: 10.1016/j.asjsur.2016.04.001
![]() |
[9] |
Rao J, Deora H, Mandia R (2015) A retrospective study of 40 cases of pilonidal sinus with excision of tract and Z-plasty as treatment of choice for both primary and recurrent cases. Indian J Surg 77: 691-693. doi: 10.1007/s12262-013-0983-4
![]() |
[10] | Mohamed Abd-Elfattah AM, Elsayed Fahmi KS, Eltih OA, et al. (2020) The Karydakis Flap Versus the Limberg Flap in the treatment of pilonidal sinus disease. Zagazig Univ Med J 26: 900-907. |
[11] |
Aysan E, Ilhan M, Bektas H, et al. (2013) Prevalence of sacrococcygeal pilonidal sinus as a silent disease. Surg Today 43: 1286-1289. doi: 10.1007/s00595-012-0433-0
![]() |
[12] | Burnett D, Smith SR, Young CJ (2018) The surgical management of pilonidal disease is uncertain because of high recurrence rates. Cureus 10: e2625. |
[13] |
Bascom J (2008) Surgical treatment of pilonidal disease. BMJ 336: 842-843. doi: 10.1136/bmj.39535.397292.BE
![]() |
[14] |
Iesalnieks I, Ommer A, Petersen S, et al. (2016) German national guideline on the management of pilonidal disease. Langenbecks Arch Surg 401: 599-609. doi: 10.1007/s00423-016-1463-7
![]() |
[15] |
Brown SR, Lund JN (2019) The evidence base for pilonidal sinus surgery is the pits. Tech Coloproctol 23: 1173-1175. doi: 10.1007/s10151-019-02116-5
![]() |
[16] |
Søndenaa K, Nesvik I, Andersen E, et al. (1995) Bacteriology and complications of chronic pilonidal sinus treated with excision and primary suture. Int J Colorectal Dis 10: 161-166. doi: 10.1007/BF00298540
![]() |
[17] |
Chaudhuri A, Bekdash BA (2005) Single-dose metronidazole versus 5-day multidrug antibiotic regimen in excision of pilonidal sinuses with primary closure: a prospective randomized controlled double-blinded study. Int J Colorectal Dis 17: 355-358. doi: 10.1007/s00384-002-0416-5
![]() |
[18] | Miocinović M, Horzić M, Bunoza D (2001) The prevalence of anaerobic infection in pilonidal sinus of the sacrococcygeal region and its effect on the complications. Acta Med Croatica 55: 87-90. |
[19] |
Awad MMS, Saad KM (2006) Does closure of chronic pilonidal sinus still remain a matter of debate after bilateral rotation flap? (N-shaped closure technique). Indian J Plast Surg 39: 157-162. doi: 10.4103/0970-0358.29545
![]() |
[20] | Badr ML, Mohammed MA, Zahran SM (2018) Prospective randomized comparative study of a Karydakis flap versus ordinary midline closure for the treatment of primary pilonidal sinus. Menoufia Med J 31: 102-107. |
[21] |
Azab AS, Kamal MS, Saad RA, et al. (1984) Radical cure of pilonidal sinus by a transposition rhomboid flap. Br J Surg 71: 154-155. doi: 10.1002/bjs.1800710227
![]() |
[22] |
Mahdy T, Mahdy T, Gaertner WB, et al. (2008) Surgical treatment of the pilonidal disease: primary closure or flap reconstruction after excision. Dis Colon Rectum 51: 1816-1822. doi: 10.1007/s10350-008-9436-8
![]() |
[23] | Akca T, Colak T, Ustunsoy B, et al. (2005) Randomized clinical trial comparing primary closure with the Limberg flap in the treatment of primary sacrococcygeal pilonidal disease. BMJ 92: 1081-1084. |
[24] |
Al-Khayat H, Al-Khayat H, Sadeq A, et al. (2007) Risk factors for wound complication in pilonidal sinus procedures. J Am Coll Surg 205: 439-444. doi: 10.1016/j.jamcollsurg.2007.04.034
![]() |
[25] | Kanlioz M, Ekici U (2019) Complications during the recovery period after pilonidal sinus surgery. Cureus 11: e4501. |
[26] | Iesalnieks I, Ommer A (2019) The management of pilonidal sinus. Dtsch Arztebl Int 116: 12-21. |
[27] | Clothier PR, Haywood IR (1984) The natural history of the post anal (pilonidal) sinus. Ann R Coll Surg Engl 66: 201-203. |
[28] |
Mustafi N, Engels P (2016) Post-surgical wound management of pilonidal cysts with a haemoglobin spray: a case series. J Wound Care 25: 191-198. doi: 10.12968/jowc.2016.25.4.191
![]() |
[29] |
Kayaalp C, Olmez A, Aydin C, et al. (2010) Investigation of a one-time phenol application for pilonidal disease. Med Princ Pract 19: 212-215. doi: 10.1159/000285291
![]() |