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Spectral theory for nonconservative transmission line networks

  • Received: 01 August 2010 Revised: 01 April 2011
  • Primary: 34B45.

  • The global theory of transmission line networks with nonconservative junction conditions is developed from a spectral theoretic viewpoint. The rather general junction conditions lead to spectral problems for nonnormal operators. The theory of analytic functions which are almost periodic in a strip is used to establish the existence of an infinite sequence of eigenvalues and the completeness of generalized eigenfunctions. Simple eigenvalues are generic. The asymptotic behavior of an eigenvalue counting function is determined. Specialized results are developed for rational graphs.

    Citation: Robert Carlson. Spectral theory for nonconservative transmission line networks[J]. Networks and Heterogeneous Media, 2011, 6(2): 257-277. doi: 10.3934/nhm.2011.6.257

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  • The global theory of transmission line networks with nonconservative junction conditions is developed from a spectral theoretic viewpoint. The rather general junction conditions lead to spectral problems for nonnormal operators. The theory of analytic functions which are almost periodic in a strip is used to establish the existence of an infinite sequence of eigenvalues and the completeness of generalized eigenfunctions. Simple eigenvalues are generic. The asymptotic behavior of an eigenvalue counting function is determined. Specialized results are developed for rational graphs.


    P.L Čebyšev in the year 1882 has proved the following interesting inequality:

    $ |1babaf(x)g(x)dx(1babaf(x)dx)(1babag(x)dx)|112(ba)2fg. $

    where $ f, g $ are absolutely continuous functions defined on $ [a, b] $ and $ f', g' \in L_{\infty}[a, b] $. The left hand side of the above equation is denoted by $ T(f, g) $ is called Cebysev Functional if the integral exists. The applications of above type of inequalities can be found in the field of coding theory, statistics and other branches of mathematics.

    In last few decades many researchers have obtained various extensions and generalizations of above inequalities using various techniques see [1,2]. Study of inequalities have attracted the attention of researchers from various fields due to its wide applications in various fields [3,4].

    During last few years the subject of Fractional Calculus has been developed rapidly due to the applications in various fields of science and engineering. Various new definitions of fractional derivatives and integrals have been obtained by various researchers depending on the applications such as Riemann liouville, Caputo, Saigo, Hilfer, Hadmard, Katugampola and others See [5,6,7,8]. Many results on study of mathematical inequalities using various new fractional definitions such as Conformable and generalized fractional integral were obtained in [9,10]. Recently in [11,12,13,14,15] the authors have obtained the results on Cebysev inequalities using various fractional integral and derivatives definitions.

    In [7] authors have given definations of fractional derivative and integrals of a functions with respect to another functions. Recently in [16,17] authors have studied the $ \psi $ Caputo and $ \psi $ Hilfer fractional derivative of a function with respect to another functions and its applications. The $ \psi $ fractional and integral definations are more generalized and it reduces to Riemann Liouville, Hadmard and Erdelyi-Kober fractional definitions for different values of $ \psi $.

    Motivated from the above mentioned literature the aim of this paper is to obtain $ \psi $ Caputo fractional Čebyšev inequalities involving functions of two and three variables.

    Now in this section we give some basic definitions and properties which are useful in our subsequent discussions. In [7,8] the authors have defined the fractional integrals and fractional derivative of a function with respect to another function as follows.

    Definition 2.1 [7,16]. Let $ I = [a, b] $ be an interval, $ \alpha > 0 $, $ f $ is an integrable function defined on $ I $ and $ \psi \in C^1(I) $ an increasing function such that $ \psi '\left(x \right) \ne 0 $ for all $ x \in I $ then fractional derivative and integral of $ f $ is given by

    $ I_{a + }^{\alpha , \psi } f(x) = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( x \right) - \psi \left( t \right)} \right)^{\alpha - 1} f\left( t \right)dt} $

    and

    $ Dα,ψa+f(x)=(1ψ(x)ddx)nInα,ψa+f(x)=1Γ(nα)(1ψ(x)ddx)nxaψ(t)(ψ(x)ψ(t))nα1f(t)dt, $

    respectively. Similarly right fractional integral and right fractional derivative are given by

    $ I_{b - }^{\alpha , \psi } f(x) = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( t \right) - \psi \left( x \right)} \right)^{\alpha - 1} f\left( t \right)dt} $

    and

    $ Dα,ψbf(x)=(1ψ(x)ddx)nInα,ψbf(x)=1Γ(nα)(1ψ(x)ddx)nxaψ(t)(ψ(t)ψ(x))nα1f(t)dt. $

    In [16] Almedia has considered a Caputo type fractional derivative with respect to another function.

    Definition 2.2 [16] Let $ \alpha > 0 $, $ n \in \mathbb{N} $, $ I $ is the interval $ -\infty \le a < b \le \infty $, $ f, \psi \in C^n (I) $ two functions such that $ \psi $ is increasing and $ \psi '\left(x \right) \ne 0 $ for all $ x \in I $. The left $ \psi $-Caputo fractional derivative of $ f $ of order $ \alpha $ is given by

    $ {}^CD_{a + }^{\alpha , \psi } f\left( x \right) = I_{a + }^{n - \alpha , \psi } \left( {\frac{1}{{\psi '\left( x \right)}}\frac{d}{{dx}}} \right)^n f\left( x \right), $

    and the right $ \psi $-Caputo fractional derivative of $ f $ is given by

    $ {}^CD_{b - }^{\alpha , \psi } f\left( x \right) = I_{b - }^{n - \alpha , \psi } \left( {-\frac{1}{{\psi '\left( x \right)}}\frac{d}{{dx}}} \right)^n f\left( x \right). $

    For given $ \alpha \notin\mathbb{N} $

    $ {}^CD_{a + }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {n - \alpha } \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( x \right) - \psi \left( t \right)} \right)^{n - \alpha - 1} f_\psi ^{\left[ n \right]} \left( t \right)dt} $

    and

    $ {}^CD_{b - }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {n - \alpha } \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( t \right) - \psi \left( x \right)} \right)^{n - \alpha - 1} (-1)^n f_\psi ^{\left[ n \right]} \left( t \right)dt}. $

    In particular when $ \alpha \in (0, 1) $ then

    $ {}^CD_{a + }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {1 - \alpha } \right)}}\int\limits_a^x {\left( {\psi \left( x \right) - \psi \left( t \right)} \right)^{ - \alpha } f'\left( t \right)dt} $

    and

    $ {}^CD_{b - }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {1 - \alpha } \right)}}\int\limits_a^x {\left( {\psi \left( t \right) - \psi \left( x \right)} \right)^{ - \alpha } f'\left( t \right)dt}. $

    In [18] the author has defined the $ \psi $ fractional partial integral with respect to another functions as

    Definition 2.3 Let $ \theta = (a, b) $ and $ \alpha = (\alpha_1, \alpha_2) $ where $ 0 \le \alpha_1, \alpha_2 \le 1 $. Also put $ I = [a, k] \times [b, m] $ where $ a, b $ and $ k, m $ are positive constants. Also let $ \psi(.) $ be an increasing positive monotone function on $ (a, k] \times (b, m] $ having continuous derivative $ \psi'(.) $ on $ (a, k] \times (b, m] $. Then the fractional partial integral is

    $ Iα;ψθu(x,y)=1Γ(α1)Γ(α2)xaybψ(s)ψ(t)(ψ(x)ψ(s))α11(ψ(y)ψ(t))α21f(s,t)dtds. $

    The Caputo fractional partial derivative is defined as follows

    Definition 2.4 Let $ \theta = (a, b) $ and $ \alpha = (\alpha_1, \alpha_2) $ where $ 0 \le \alpha_1, \alpha_2 \le 1 $. Also put $ I = [a, k] \times [b, m] $ where $ a, b $ and $ a, b $ are positive constants. Also let $ \psi(.) $ be an increasing function on $ (a, k] \times (b, m] $ and $ \psi'(.) \neq 0 $ on $ (a, k] \times (b, m] $. The $ \psi $ Caputo fractional partial derivative of functions of two variables of order $ \alpha $ is given by

    $ {}^CD_\theta ^{\alpha ;\psi } u\left( {x, y} \right) = I_\theta ^{2 - \alpha ;\psi } \left( {\frac{1}{{\psi '(s)\psi '(t)}}\frac{{\partial ^2\alpha }}{{\partial y\partial x}}} \right)u\left( {x, y} \right). $

    We use the following notation:

    $ {}^CD_\theta ^{\alpha ;\psi } u\left( {x, y} \right) = \frac{{\partial _\psi ^{2\alpha } u}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}\left( {x, y} \right). $

    We define the norm for a function of two variables as follows

    $ \left\| {{}^CD_\theta ^{\alpha ;\psi } f} \right\|_\infty = \sup \left| {{}^CD_\theta ^{\alpha ;\psi } f\left( {x, y} \right)} \right|. $

    Similarly as in Definition (2.3) and (2.4) we define the $ \psi $ fractional partial integral with respect to another functions and $ \psi $ Caputo fractional partial derivative of functions of three variables as follows:

    Definition 2.5 Let $ \Theta = (a, b, c) $ and $ \alpha = (\alpha_1, \alpha_2, \alpha_3) $ where $ 0 \le \alpha_1, \alpha_2, \alpha_3 \le 1 $. Also put $ I = [a, k] \times [b, m] \times [c, n] $ where $ a, b, c $ and $ k, m, n $ are positive constants. Also let $ \psi(.) $ be an increasing positive monotone function on $ (a, k] \times (b, m] \times [c, n] $ having continuous derivative $ \psi'(.) $ on $ (a, k] \times (b, m] \times (c, n] $.

    Then the fractional partial integral is

    $ Iα;ψΘu(x,y,z)=1Γ(α1)Γ(α2)xaybzcψ(s)ψ(t)ψ(r)×(ψ(x)ψ(s))α11(ψ(y)ψ(t))α21(ψ(z)ψ(r))α31f(s,t,r)drdtds. $

    Definition 2.6 Let $ \theta = (a, b, c) $ and $ \alpha = (\alpha_1, \alpha_2, \alpha_3) $ where $ 0 \le \alpha_1, \alpha_2, \alpha_3\le 1 $. Also put $ I = [a, k] \times [b, m] \times [c, n] $ where $ a, b, c $ and $ k, m, n $ are positive constants. Also let $ \psi(.) $ be an increasing function on $ (a, k] \times (b, m] \times (c, n] $ and $ \psi'(.) \neq 0 $ on $ (a, k] \times (b, m] \times (c, n] $. The $ \psi $ Caputo fractional partial derivative of functions of three variables of order $ \alpha $ is given by

    $ {}^CD_\Theta ^{\alpha ;\psi } u\left( {x, y, z} \right) = I_\Theta ^{3 - \alpha ;\psi } \left( {\frac{1}{{\psi '(s)\psi '(t)\psi '(r)}}\frac{{\partial ^3 }}{{\partial z\partial y\partial x}}} \right)u\left( {x, y, z} \right). $

    We use the following notation:

    $ {}^CD_\Theta ^{\alpha ;\psi } u\left( {x, y, z} \right) = \frac{{\partial _\psi ^{3\alpha } u}}{{\partial _\psi z^\alpha \partial _\psi y^\alpha x^\alpha }}\left( {x, y, z} \right). $

    We define the norm for a function of three variables as follows

    $ \left\| {{}^CD_\Theta ^{\alpha ;\psi } f} \right\|_\infty = \sup \left| {{}^CD_\Theta ^{\alpha ;\psi } f\left( {x, y, z} \right)} \right|. $

    Now we give the $ \psi $ Caputo fractional Čebyšev inequality involving functions of two variables as follows:

    Theorem 3.1 Let $ f, g:[a, l] \times [b, m] \rightarrow R $ be a continuous function on $ [a, l] \times [b, m] $ and $ \frac{{\partial ^{2\alpha } f}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }} $, $ \frac{{\partial ^{2\alpha } g}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }} $ exists continuous and bounded on $ [a, l] \times [b, m] $ and $ \alpha = (\alpha_1, \alpha_2) $. Then

    $ |lamb[f(x,y)g(x,y)12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]dydx]|18(ψ(l)ψ(a))(ψ(m)ψ(b))lamb[|g(x,y)|Dα;ψθf+g(x,y)Dα;ψθg]dydx, $ (3.1)

    where

    $ G(f(x,y))=12[f(a,y)+f(x,m)+f(x,b)+f(l,y)]14[f(a,b)+f(a,m)+f(l,b)+f(l,m)] $

    and

    $ H(2αfψyαψxα(x,y))=1Γ(α1)Γ(α2)××[xaybψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdtxamyψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdtlxybψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt+lxmyψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt]. $

    Proof. From the given hypotheses for $ (x, y) \in [a, l] \times [b, m] $ we have

    $ 1Γ(α1)Γ(α2)xaybψ(t)ψ(s)×(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt=1Γ(α1)xaψ(s)(ψ(x)ψ(t))α11[αfψsα(s,t)|yc]=1Γ(α1)xaψ(s)(ψ(y)ψ(t))α11[αfψsα(t,y)αfψsα(t,b)]=f(t,y)|xaf(t,b)|xa=f(x,y)f(a,y)f(x,b)+f(a,b). $ (3.2)

    Similarly we have

    $ 1Γ(α1)Γ(α2)xamyψ(t)ψ(s)×(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt=f(x,y)f(a,m)+f(x,m)+f(a,y), $ (3.3)
    $ 1Γ(α1)Γ(α2)lxybψ(t)ψ(s)×(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt=f(x,y)f(l,b)+f(x,b)+f(l,y), $ (3.4)
    $ 1Γ(α1)Γ(α2)lxmyψ(t)ψ(s)×(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(s,t)dsdt=f(x,y)+f(l,b)f(x,b)f(l,y). $ (3.5)

    Adding the above identities we have

    $ 4f(x,y)2[f(a,y)+f(x,m)+f(x,b)+f(l,y)]+[f(a,b)+f(a,m)+f(l,b)+f(l,m)]=1Γ(α1)Γ(α2)[xaybψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdtxadyψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdtlxybψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt+lxmyψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt]. $ (3.6)

    From $ (3.6) $ we have

    $ f(x, y) - G(f(x, y)) = \frac{1}{4}H\left( {\frac{{\partial ^{2\alpha } f}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}(x, y)} \right), $ (3.7)

    for $ (x, y) \in [a, l] \times [b, m] $. Similarly we have

    $ g(x, y) - G(g(x, y)) = \frac{1}{4}H\left( {\frac{{\partial ^{2\alpha } g}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}(x, y)} \right), $ (3.8)

    for $ (x, y) \in [a, l] \times [b, m] $.

    Multiplying $ (3.7) $ by $ g(x, y) $, $ (3.8) $ by $ f(x, y) $ adding them and Integrating over $ (x, y) \in [a, l] \times [b, m] $ we get

    $ lamb[2f(x,y)g(x,y)g(x,y)G(f(x,y))f(x,y)G(g(x,y))]dydx=18lamb[H(2αfψyαψxα(x,y))g(x,y)+14f(x,y)H(2αgψyαψxα(x,y))]. $ (3.9)

    From the properties of modulus we have

    $ |H(2αfψyαψxα(x,y))|1Γ(α1)Γ(α2)lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αfψsαψtα(t,s)|dsdt(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2cDα;ψθf, $ (3.10)
    $ |H(2αgψyαψxα(x,y))|1Γ(α1)Γ(α2)lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αgψsαψtα(t,s)|dsdt(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2cDα;ψθg. $ (3.11)

    From $ (3.9) $, $ (3.10) $ and $ (3.11) $ we have

    $ |lamb[f(x,y)g(x,y)12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]]dydx|18lamb[|H(2αfψyαψxα(x,y))||g(x,y)|+|H(2αgψyαψxα(x,y))||f(x,y)|]18lamb{|g(x,y)|[1Γ(α1)Γ(α2)×[lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αfψsαψtα(t,s)|dsdt]+|f(x,y)|×[lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αgψsαψtα(t,s)|dsdt]}dydx18(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2×lamb[|g(x,y)|cDα;ψθf+|f(x,y)|cDα;ψθg]dydx, $ (3.12)

    which is required inequality.

    Theorem 3.2 Let $ f, g, G(f(x, y)), G(g(f(x, y)), {\frac{{\partial ^{2\alpha } f}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}}, {\frac{{\partial ^{2\alpha } g}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}} $ be as in Theorem 3.1 then

    $ |lamb{f(x,y)g(x,y)[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)G(f(x,y))G(g(x,y))]}dydx116{(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2}2cDα;ψθfcDα;ψθg, $ (3.13)

    for $ (x, y) \in [a, l] \times [b, m] $.

    Proof. Multiplying left hand side and right hand side of $ (3.7) $ and $ (3.8) $ we have

    $ f(x,y)g(x,y)[f(x,y)G(g(x,y))+g(x,y)G(f(x,y))]=116H(2αfψyαψxα(x,y))H(2αgψyαψxα(x,y)). $ (3.14)

    Integrating $ (3.14) $ over $ [a, l] \times [b, m] $ and from the properties of modulus we get

    $ |lamb{f(x,y)g(x,y)[G(g(x,y))f(x,y)+G(f(x,y))g(x,y)]G(f(x,y))G(g(x,y))}dydx|116lamb|H(2αfψyαψxα(x,y))||H(2αgψyαψxα(x,y))|dydx. $ (3.15)

    Now using $ (3.13), (3.14) $ in $ (3.19) $ we get required inequality $ (3.13) $.

    Now in our result we give the $ \psi $ Caputo fractional Čebyšev inequality involving functions of three variables. We use some notations as follows:

    $ A(p(u,v,w))=18[p(a,b,c)+p(k,m,n)]14[p(u,b,c)+p(u,m,n)+p(u,m,c)+p(u,b,n)]14[p(a,v,c)+p(k,v,n)+p(a,v,n)+p(k,v,c)]14[p(a,b,w)+p(k,m,w)+p(k,b,w)+p(a,m,w)]+12[p(a,v,w)+p(k,v,w)]+12[p(u,b,w)+p(u,m,w)]+12[p(u,v,c)+p(u,v,n)] $ (4.1)

    and

    $ B(3αpψwαψvαψuα(u,v,w))=1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)uavbncψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)uamvwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)kuvbwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(u)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)uamrnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)kumvwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)kuvbnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)kumvnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr. $ (4.2)

    Now we give our next result as

    Theorem 4.1 Let $ f, g:[a, k] \times [b, m] \times [c, n] \rightarrow R $ be a continuous function on $ [a, l] \times [b, m] $ and $ \frac{{\partial ^{3\alpha } f}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $, $ \frac{{\partial ^{3\alpha } g}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $ exists and continuous and bounded on $ [a, k] \times [b, m] \times [c, n] $. Then

    $ kambnc[f(u,v,w)g(u,v,w)12[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))]]dwdvdu116(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3×kambnc[|g(u,v,w)|cDα;ψΘf+|f(u,v,w)|cDα;ψΘg]dwdvdu, $ (4.3)

    where $ A, B $ are as given in $ (4.1), (4.2) $.

    Proof. From the hypotheses we have for $ u, v, w \in [a, k] \times [b, m] \times [c, n] $

    $ 1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr=1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,t)|wcdsdr=1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,w)dsdr1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,c)dsdr=1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,s,w)|vbdr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,s,c)|vbdr=1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,v,w)dr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,b,w)dr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,v,c)dr+1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,b,c)dr=f(r,v,w)|uaf(r,b,w)|uaf(r,v,c)|ua+f(r,b,c)|ua=f(u,v,w)f(a,v,w)f(u,b,w)+f(a,b,w)f(u,v,c)+f(a,v,c)+f(u,b,c)+f(a,b,c). $

    Thus we have

    $ f(u,v,w)=f(a,v,w)+f(u,b,w)f(a,b,w)+f(u,v,c)f(a,v,c)f(u,b,c)f(a,b,c)1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.4)

    Similarly we have

    $ f(u,v,w)=f(u,v,n)+f(a,v,w)+f(u,b,w)+f(a,b,n)f(a,b,w)f(a,v,n)f(v,b,n)1Γ(α1)Γ(α2)Γ(α3)uavbnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.5)
    $ f(u,v,w)=f(u,m,w)+f(u,v,c)+f(a,m,c)+f(a,v,w)f(u,m,c)f(a,m,w)f(a,v,c)1Γ(α1)Γ(α2)Γ(α3)uamvwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.6)
    $ f(u,v,w)=f(k,s,t)+f(k,b,c)+f(u,v,c)+f(u,b,w)f(k,v,c)f(k,b,w)f(u,b,c)1Γ(α1)Γ(α2)Γ(α3)kuvbwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.7)
    $ f(u,v,w)=f(u,m,w)+f(u,v,n)+f(a,m,n)+f(a,v,w)f(u,m,n)f(a,m,w)f(a,v,n)+1Γ(α1)Γ(α2)Γ(α3)uamvnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.8)
    $ f(u,v,w)=f(r,m,t)+f(u,v,c)+f(k,s,t)+f(k,m,c)f(k,m,w)f(k,v,c)f(u,m,c)+1Γ(α1)Γ(α2)Γ(α3)kumvwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.9)
    $ f(u,v,w)=f(k,v,w)+f(k,b,n)+f(u,v,n)+f(u,b,t)f(k,v,n)f(k,b,w)f(u,b,n)+1Γ(α1)Γ(α2)Γ(α3)kuvbnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr $ (4.10)

    and

    $ f(u,v,w)=f(k,m,n)+f(k,v,w)+f(u,m,w)+f(u,v,n)f(k,m,w)f(k,v,n)f(u,m,n)+1Γ(α1)Γ(α2)Γ(α3)kumvnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr. $ (4.11)

    Adding the above identities we have

    $ f\left( {u, v, w} \right) - A\left( {f\left( {u, v, w} \right)} \right) = \frac{1}{8}B\left( {\frac{{\partial ^{3\alpha } f}}{{\partial _\psi w^\alpha \partial _\psi v^\alpha \partial _\psi u^\alpha }}(u, v, w)} \right), $ (4.12)

    for $ (u, v, w) \in [a, k] \times [b, m] \times [c, n] $.

    Similarly we have

    $ g\left( {u, v, w} \right) - A\left( {g\left( {u, v, w} \right)} \right) = \frac{1}{8}B\left( {\frac{{\partial ^{3\alpha } g}}{{\partial _\psi w^\alpha \partial _\psi v^\alpha \partial _\psi u^\alpha }}(u, v, w)} \right), $ (4.13)

    for $ (u, v, w) \in [a, k] \times [b, m] \times [c, n] $.

    Now multiplying $ (4.12) $ and $ (4.13) $ by $ g(u, v, w) $ and $ f(u, v, w) $ respectively, adding them and Integrating over $ [a, k] \times [b, m] \times [c, n] $ we have

    $ kambnc[f(u,v,w)g(u,v,w)12[g(u,v,w)A(f(u,v,w))g(u,v,w)A(f(u,v,w))]]dwdvdu=116kambnc[g(u,v,w)B(3αfψwαψvαψuα(u,v,w))+f(u,v,w)B(3αgψwαψvαψuα(u,v,w))]. $ (4.14)

    From the properties of modulus we have

    $ |B(3αfψwαψvαψuα(u,v,w))|kambncψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21×(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3CDα;ψΘf, $ (4.15)
    $ |B(3αgψwαψvαψuα(u,v,w))|kambncψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21×(ψ(n)ψ(t))α313αgψtαψsαψrα(r,s,t)dtdsdr(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3CDα;ψΘg. $ (4.16)

    Now by substituting the values from equation $ (4.15) $ and $ (4.16) $ in $ (4.14) $ we get the required inequality $ (4.3) $.

    Theorem 4.2 Let $ f, g $, $ \frac{{\partial ^{3\alpha } f}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $ and $ \frac{{\partial ^{3\alpha } g}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $ be as in Theorem $ 4.1 $. Then

    $ |kambnc[f(u,v,w)g(u,v,w)[A(f(u,v,w))g(u,v,w)A(g(u,v,w))f(u,v,w)A(f(u,v,w))A(g(u,v,w))|dwdvdu164{(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3}2CDα;ψΘfCDα;ψΘg, $ (4.17)

    for $ (r, s, t) \in [a, k] \times [b, m] \times [c, n] $ and $ A, B $ are as given in $ (4.1), (4.2) $.

    Proof. Multiplying left hand and right hand side of equation $ (4.12) $ and $ (4.13) $ we have

    $ f(u,v,w)g(u,v,w)[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))A(f(u,v,w))A(g(u,v,w))]=164B(3αfψwαψvαψuα(u,v,w))B(3αgψwαψvαψuα(u,v,w)). $ (4.18)

    Integrating over $ [a, k] \times [b, m] \times [c, n] $ and from the properties of modulus we have

    $ |kambnc[f(u,v,w)g(u,v,w)[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))A(f(u,v,w))A(g(u,v,w))]]|dwdvdu164kambnc|B(3αfψwαψvαψuα(u,v,w))B(3αfψwαψvαψuα(u,v,w))|dwdvdu. $ (4.19)

    Using $ (4.15) $ and $ (4.16) $ in $ (4.19) $ we get the required inequality $ (4.17) $.

    Remark: If we put different values for $ \psi(x) $ as $ x, ln x, x^{\sigma} $then it reduces to various types of fractional Čebyšev inequalities such as Riemann Liouville fractional, Hadmard Fractional and Erdelyi-Kober fractional inequalities respectively.

    In this paper, we studied Čebyšev like inequalities. We proved some new $ \psi $ Caputo fractional Čebyšev type inequalities involving functions of two and three variables.

    All authors declare no conflict of interest in this paper.

    [1] A. Agarwal, S. Das and D. Sen, Power dissipation for systems with junctions of multiple quantum wires, Physical Review B, 81 (2010).
    [2] L. Ahlfors, "Complex Analysis," McGraw-Hill, New York, 1966.
    [3] F. Ali Mehmeti, "Nonlinear Waves in Networks," Akademie Verlag, Berlin, 1994.
    [4] R. Carlson, Inverse eigenvalue problems on directed graphs, Transactions of the American Mathematical Society, 351 (1999), 4069-4088. doi: 10.1090/S0002-9947-99-02175-3
    [5] R. Carlson, Linear network models related to blood flow, in Quantum Graphs and Their Applications, Contemporary Mathematics, 415 (2006), 65-80.
    [6] C. Cattaneo and L. Fontana, D'Alembert formula on finite one-dimensional networks, J. Math. Anal. Appl., 284 (2003), 403-424. doi: 10.1016/S0022-247X(02)00392-X
    [7] G. Chen, S. Krantz, D. Russell, C. Wayne, H. West and M. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams, SIAM Journal on Applied Mathematics., 49 (1989), 1665-1693. doi: 10.1137/0149101
    [8] S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (1995), 545-573. doi: 10.1512/iumj.1995.44.2001
    [9] E. B. Davies, Eigenvalues of an elliptic system, Math. Z., 243 (2003), 719-743. doi: 10.1007/s00209-002-0464-0
    [10] E. B. Davies, P. Exner and J. Lipovsky, Non-Weyl asymptotics for quantum graphs with general coupling conditions, J. Phys. A, 43 (2010).
    [11] Y. Fung, "Biomechanics," Springer, New York, 1997.
    [12] I. Herstein, "Topics in Algebra," Xerox College Publishing, Waltham, 1964.
    [13] B. Jessen and H. Tornhave, Mean motions and zeros of almost periodic functions, Acta Math., 77 (1945), 137-279. doi: 10.1007/BF02392225
    [14] T. Kato, "Perturbation Theory for Linear Operators," Springer-Verlag, New York, 1995.
    [15] V. Kostrykin, J. Potthoff and R. Schrader, Contraction semigroups on metric graphs, in Analysis on Graphs and Its Applications, PSUM, 77 (2008), 423-458.
    [16] T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Phys., 274 (1999), 76-124. doi: 10.1006/aphy.1999.5904
    [17] M. Kramar and E. Sikolya, Spectral properties and asymptotic periodicity of flows in networks, Math. Z., 249 (2005), 139-162. doi: 10.1007/s00209-004-0695-3
    [18] M. Kramar Fijavz, D. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusions in networks, Appl. Math. Optim., 55 (2007), 219-240. doi: 10.1007/s00245-006-0887-9
    [19] M. Krein and A. Nudelman, Some spectral properties of a nonhomogeneous string with a dissipative boundary condition, J. Operator Theory, 22 (1989), 369-395.
    [20] B. Levin, "Distribution of Zeros of Entire Functions," American Mathematical Society, Providence, 1980.
    [21] S. Lang, "Algebra," Addison-Wesley, 1984.
    [22] G. Lumer, "Equations de Diffusion Generales sur des Reseaux Infinis," Seminar Goulaouic-Schwartz, 1980.
    [23] G. Lumer, "Connecting of Local Operators and Evolution Equations on Networks," Lecture Notes in Math., 787, Springer, 1980.
    [24] P. Magnusson, G. Alexander, V. Tripathi and A. Weisshaar, "Transmission Lines and Wave Propagation," CRC Press, Boca Raton, 2001.
    [25] G. Miano and A. Maffucci, "Transmission Lines and Lumped Circuits," Academic Press, San Diego, 2001.
    [26] L. J. Myers and W. L. Capper, A transmission line model of the human foetal circulatory system, Medical Engineering and Physics, 24 (2002), 285-294. doi: 10.1016/S1350-4533(02)00019-X
    [27] S. Nicaise, Spectre des reseaux topologiques finis, Bulletin des sciences mathematique, 111 (1987), 401-413.
    [28] J. Ottesen, M. Olufsen and J. Larsen, "Applied Mathematical Models in Human Physiology," SIAM, 2004. doi: 10.1137/1.9780898718287
    [29] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1983.
    [30] S. Sherwin, V. Franke, J. Peiro and K. Parker, One-dimensional modeling of a vascular network in space-time variables, Journal of Engineering Mathematics, 47 (2003), 217-250. doi: 10.1023/B:ENGI.0000007979.32871.e2
    [31] J. von Below, A characteristic equation associated to an eigenvalue problem on C2 networks, Lin. Alg. Appl., 71 (1985), 309-325. doi: 10.1016/0024-3795(85)90258-7
    [32] L. Zhou and G. Kriegsmann, A simple derivation of microstrip transmission line equations, SIAM J. Appl. Math., 70 (2009), 353-367. doi: 10.1137/080737563
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