Utafiti huu unahamisha masharti muhimu ya optimality ya Euler–Lagrange na Legendre kutoka calculus of variations ya classical kwenda fractional calculus of variations inayojumuisha katika tatizo moja Caputo fractional derivative na Riemann–Liouville-type fractional integral weight. Watafiti wanajenga framework inayoonyesha hasa kwamba ukubwa wa jamaa wa parameters mbili — order ya Caputo derivative \(0<\alpha\leq1\) na parameter \(\beta>0\) inayobainisha Riemann–Liouville structure katika functional — hubadilisha moja kwa moja optimality conditions.

Chombo kikuu cha kihisabati cha utafiti ni fractional generalization ya classical Du Bois–Reymond lemma. Kupitia lemma hii, watafiti wanatoa Euler–Lagrange conditions katika integral form kwa weak local minimum chini ya tofauti mbalimbali za initial na terminal boundary conditions, bila kutumia integration-by-parts formula kama hatua kuu. Conditions zilizopatikana zinachukua miundo tofauti katika regimes za \(\beta>\alpha\) na \(0<\beta\leq\alpha\leq1\).

Matokeo ya pili makuu yanahusu Legendre necessary condition. Katika literature ya awali, ilijadiliwa kwamba kwa \(0<\alpha<1\), function na Caputo derivative yake haziwezi kwa wakati mmoja kuwa na compact support, na kwa sababu hiyo classical Legendre proof strategy haiwezi kutumika chini ya endpoint constraints. Waandishi wa kazi hii wanapendekeza kwamba kwa kuchagua variation maalum, classical second-variation approach inaweza kurekebishwa kwa fractional setting kwa fixed-end problem na free-initial–fixed-terminal problem.

Mfano unaochunguzwa kwa fixed-end problem rahisi zaidi unaonyesha kwamba uhusiano kati ya \(\alpha\) na \(\beta\) unaweza kuathiri si tu form ya equation bali pia existence ya solution. Katika Example 4.1 ya watafiti, wakati \(\beta>\alpha\) hakuna solution inayokidhi simultaneously boundary conditions mbili zilizotolewa, wakati \(0<\beta\leq\alpha\leq1\) explicit extremal hupatikana. Hasa, \(\beta=\alpha<1\) ikichaguliwa solution huwa \(x(t)=t^\alpha\), na katika classical limit \(\alpha=\beta=1\), huwa \(x(t)=t\).

Matokeo si experimental data, physical-system measurement, wala numerical performance comparison. Utafiti ni theoretical mathematics research unaoendelea kupitia function spaces, fractional integral na derivative operators, variations na analytical proofs. Kwa hiyo nguvu ya matokeo imewekewa mipaka na mathematical structure ya necessary optimality conditions zilizotolewa chini ya assumptions husika; hakuna sufficiency theorem inayohakikisha minimum ya general problem kwa kutimizwa kwa Legendre condition pekee.

Ina maana gani kwa Uturuki?

Matokeo ya utafiti hayategemei nchi au industrial system maalum. Kwa mtazamo wa Uturuki, umuhimu wa kisayansi unaowezekana ni kutoa analytical method kwa watafiti wanaofanya kazi kwenye fractional differential equations, optimal control, dynamic systems zenye memory effects na fractional mathematical modeling kuhusu jinsi ya kujenga Euler–Lagrange na Legendre conditions chini ya boundary conditions. Hata hivyo, kwa kuwa utafiti haujajaribu Turkish engineering system yoyote, physical process, control device au real dataset, performance au accuracy result kwa application maalum haiwezi kutolewa.

Tatizo msingi katika fractional calculus of variations

Katika classical calculus of variations, lengo kwa kawaida ni kutambua functions zinazotoa minimum au maximum ya functional. Mojawapo ya necessary conditions kuu za matatizo haya ni Euler–Lagrange equation. Utafiti unachunguza jinsi framework hii ya classical inavyobadilika katika fractional mathematics ambapo derivatives zinapanuliwa hadi non-integer orders.

Main functional inayochunguzwa na watafiti ni:

\[ J(x(\cdot)) = \int_{t_0}^{t_1} (t_1-t)^{\beta-1} L\!\left( t, x(t), ({}^{c}D_{t_0+}^{\alpha}x)(t) \right) dt. \]

Hapa:

• \(0<\alpha\leq1\), ni order ya Caputo fractional derivative.
• \(\beta>0\), inaamua order ya Riemann–Liouville-type weight katika integral functional.
• \(L(t,x,y)\), ni integrand au Lagrange function katika variation problem.
• \({}^{c}D_{t_0+}^{\alpha}x\), ni left Caputo fractional derivative.

Classical case hupatikana wakati \(\alpha=\beta=1\).

Riemann–Liouville fractional integral

Katika utafiti, left Riemann–Liouville fractional integral inafafanuliwa kama:

\[ (I_{t_0+}^{\alpha}\varphi)(t) = \frac{1}{\Gamma(\alpha)} \int_{t_0}^{t} (t-\tau)^{\alpha-1} \varphi(\tau)\,d\tau \]

na right Riemann–Liouville fractional integral kama:

\[ (I_{t_1-}^{\alpha}\varphi)(t) = \frac{1}{\Gamma(\alpha)} \int_t^{t_1} (\tau-t)^{\alpha-1} \varphi(\tau)\,d\tau \]

.

\(\Gamma(\alpha)\) ni Gamma function. Sifa kuu ya fractional integral ni kwamba inategemea si tu value ya function katika point moja bali pia history yake katika interval fulani ya time au variable. Nonlocal structure hii ni mojawapo ya sababu kuu zinazofanya local derivative techniques za classical calculus of variations zisiweze kuhamishwa moja kwa moja kwenye fractional problems.

Caputo fractional derivative

Kwa \(0<\alpha\leq1\), left Caputo derivative katika source inafafanuliwa kama:

\[ ({}^{c}D_{t_0+}^{\alpha}\varphi)(t) = \frac{d}{dt} \left[ I_{t_0+}^{1-\alpha} \bigl(\varphi(\cdot)-\varphi(t_0)\bigr) \right](t) \]

.

Watafiti wanazingatia solutions katika \(C^\alpha\) space, ambako function na Caputo derivative yake ni continuous, au katika piecewise-continuous \(PC^\alpha\) space, inayoruhusu Caputo derivative kuwa na finite number ya first-kind discontinuities.

Generalized Du Bois–Reymond lemma

Mwanzo wa main proof chain ya utafiti ni Lemma 3.1. Watafiti wanagawanya structure inayotokea wakati:

\[ \int_{t_0}^{t_1} (t_1-t)^{\beta-1} f(t) ({}^{c}D_{t_0+}^{\alpha}h)(t)\,dt = 0 \]

inashikilia kwa suitable variations zote \(h\) ambazo ni zero kwenye endpoints, katika parameter regimes mbili.

Regime ya kwanza: \(\beta>\alpha>0\)

\[ (t_1-t)^{\beta-\alpha}f(t)=0. \]

Regime ya pili: \(0<\beta\leq\alpha\leq1\)

Source inatoa:

\[ f(t) = \frac{k}{\Gamma(\alpha)} (t_1-t)^{\alpha-\beta} \]

.

Mgawanyo huu unaeleza kwa nini Euler–Lagrange conditions katika sehemu iliyobaki ya paper zinagawanyika katika structures mbili tofauti. \(\alpha\) na \(\beta\) hazibaki technical parameters pekee; zinakuwa factors zinazoamua admissible form ya solution.

Dokezo kuhusu kauli ya Lemma 3.1 katika source v1

Katika sehemu ya pili ya Lemma 3.1, source inaandika constant kama \(k\neq0\). Hata hivyo, mwanzoni mwa proof hiyo hiyo inaelezwa kwamba \(f(t)=0\) pia inatimiza integral equality moja kwa moja. Kwa hiyo kuna small scope inconsistency kuhusu zero constant kati ya theorem statement na proof text katika uploaded v1. Detail hii haijasahihishwa kimya kimya hapa.

Euler–Lagrange condition inapatikanaje?

Ikichukuliwa kwamba function \(x^0\) inatoa weak local minimum:

\[ \phi(\lambda) = J(x^0+\lambda h) \]

single-variable function huwa na minimum katika \(\lambda=0\). Kutoka Fermat condition:

\[ \phi'(0)=0 \]

hupatikana.

First variation hii inakuwa:

\[ \int_{t_0}^{t_1} (t_1-t)^{\beta-1} \left[ \left\langle L_x(t),h(t) \right\rangle + \left\langle L_y(t), ({}^{c}D_{t_0+}^{\alpha}h)(t) \right\rangle \right]dt = 0 \]

.

Baada ya hapo watafiti hutumia generalized Du Bois–Reymond lemma kupata integral-form Euler–Lagrange conditions.

Euler–Lagrange conditions kwa fixed-end problem

\[ b(t) = (t_1-t)^{\beta-1} L_x \left( t,x^0(t), ({}^{c}D_{t_0+}^{\alpha}x^0)(t) \right) \]

kwa \(\beta>\alpha\), Theorem 4.1 inatoa:

\[ (t_1-t)^{1-\alpha} (I_{t_1-}^{\alpha}b)(t) + (t_1-t)^{\beta-\alpha} L_y \left( t,x^0(t), ({}^{c}D_{t_0+}^{\alpha}x^0)(t) \right) = 0 \]

.

Kwa \(0<\beta\leq\alpha\leq1\), regime ya pili ni:

\[ (t_1-t)^{1-\beta} (I_{t_1-}^{\alpha}b)(t) + L_y \left( t,x^0(t), ({}^{c}D_{t_0+}^{\alpha}x^0)(t) \right) = \frac{k}{\Gamma(\alpha)} (t_1-t)^{\alpha-\beta} \]

.

Kwa nini uhusiano wa α–β ni muhimu kweli?

Example 4.1: solution existence inabadilikaje?

Watafiti wanachunguza functional:

\[ J(x(\cdot)) = \int_0^1 (1-t)^{\beta-1} \left[ ({}^{c}D_{0+}^{\alpha}x)(t) \right]^2 dt \]

chini ya boundary conditions:

\[ x(0)=0, \qquad x(1)=1 \]

.

Kwa \(\beta>\alpha\), Euler–Lagrange condition inapungua kuwa:

\[ ({}^{c}D_{0+}^{\alpha}x)(t)=0 \]

. Hii inatoa \(x(t)=x(0)=0\), hivyo condition \(x(1)=1\) haiwezi kutimizwa. Watafiti kwa hiyo wanasema hakuna solution katika regime hii.

Kwa \(0<\beta\leq\alpha\leq1\), extremal ni:

\[ x(t) = (2\alpha-\beta) \int_0^t (t-\tau)^{\alpha-1} (1-\tau)^{\alpha-\beta} d\tau \]

.

Wakati \(\beta=\alpha<1\):

\[ x(t)=t^\alpha \]

na katika classical limit \(\alpha=\beta=1\):

\[ x(t)=t \]

hupatikana.

Watafiti pia hutumia convexity ya \(L(y)=y^2\) kuonyesha tofauti kwamba extremal iliyopatikana katika mfano huu maalum ni minimum kweli.

Fractional counterpart ya Legendre condition

Kwa second variation, source hutumia:

\[ \delta^2J(x^0,h) = \int_{t_0}^{t_1} (t_1-t)^{\beta-1} \Big[ \langle P(t){}^{c}D^\alpha h, {}^{c}D^\alpha h \rangle + 2\langle Q(t)h, {}^{c}D^\alpha h \rangle + \langle R(t)h,h \rangle \Big]dt \geq0 \]

.

Hapa:

\[ P(t)=L_{yy}, \qquad Q(t)=L_{xy}, \qquad R(t)=L_{xx} \]

zinatathminiwa along corresponding extremal.

Legendre necessary condition ya Theorem 4.2 ni:

\[ \langle P(t)r,r\rangle\geq0 \]

na inapendekezwa kuwa necessary katika points zote \(t\) ambako Caputo derivative ni continuous na kwa kila \(r\in\mathbb{R}^n\).

Kwa maneno mengine:

\[ L_{yy}(t,x^0,{}^{c}D^\alpha x^0) \]

matrix lazima iwe positive semidefinite katika points husika. Hii ndiyo fractional counterpart ya classical Legendre condition.

Ugumu wa classical proof strategy unajaribiwaje kushindwa?

Kwa sababu ya nonlocal structure ya Caputo fractional derivative, standard variations zinazotumiwa katika classical calculus of variations, ambapo function yenyewe na derivative yake ni zero nje ya small interval, haziwezi kujengwa moja kwa moja.

Badala yake, watafiti wanajenga special variation ambayo Caputo derivative yake imefafanuliwa mapema:

\[ h(t) = \frac{1}{\Gamma(\alpha)} \int_{t_0}^{t} (t-\tau)^{\alpha-1}g(\tau)d\tau. \]

Hapa \(g\) inafafanuliwa kuwa zero nje ya small interval iliyochaguliwa \([\sigma-\varepsilon,\sigma+\varepsilon]\), na constant ya ziada \(k\) inatumika kuhakikisha boundary condition \(h(t_1)=0\).

Kwa structure hii, hata kama \(h\) yenyewe inaenea nje ya interval kwa sababu ya Caputo nonlocality, \({}^{c}D^\alpha h=g\) inaweza kulocalized katika region iliyochaguliwa. Original strategy ya Legendre proof katika utafiti inategemea hasa wazo hili.

Checkpoint kuhusu Legendre proof katika uploaded v1

Katika proof ya Theorem 4.2, baada ya kufafanua \(M=\max|f|\), upper bound ya order \(-8M^2\gamma\varepsilon\) inatumika kwa negative second-variation term kuelekea Equation (19). Katika uploaded v1 text, haijaonyeshwa wazi jinsi, kutoka kwenye definition \(M=\max|f|\) pekee, positive lower bound inayohitajika kwa \(\int(f-f(c))^2dt\) inavyopatikana. Kwa hiyo step hii haijawasilishwa hapa kama independent newly verified result; proof strategy ya waandishi na condition waliyofikia imewasilishwa kwa source fidelity.

Madarasa manne ya boundary conditions

Matokeo ya kuvutia kwa free right endpoint

Watafiti wanasisitiza kutoka Theorem 4.5 na Theorem 4.6 kwamba wakati right endpoint \(x(t_1)\) ni free na:

\[ \beta>\alpha>0 \]

kwa existence ya extremal, derivative ya endpoint cost kwa final state variable lazima iwe:

\[ l_{x_1}=0 \]

.

Je, classical Euler–Lagrange equation inarecovered?

Ndiyo. Katika Remark 4.2, \(\alpha=\beta=1\) inachaguliwa. Integral conditions kwa Bolza problem zinakuwa:

\[ \int_t^{t_1}L_x(\tau)d\tau + L_y(t) + l_{x_1} = 0 \]

na:

\[ \int_{t_0}^{t_1}L_x(t)dt + l_{x_0} + l_{x_1} = 0 \]

.

Derivative ya equation ya kwanza kwa time inaongoza kwenye classical:

\[ -\frac{d}{dt}L_y+L_x=0 \]

Euler–Lagrange equation. Endpoint conditions pia hupatikana kama:

\[ L_y(t_0)=l_{x_0}, \qquad L_y(t_1)=-l_{x_1} \]

.

Hivyo watafiti wanaonyesha kwamba hata katika classical case, Euler–Lagrange na transversality conditions zinaweza kutolewa katika approach yao kupitia Du Bois–Reymond lemma bila kurejea moja kwa moja integration by parts.

Mbinu na Matokeo ya Utafiti

Aina ya utafiti

Utafiti si experimental, observational, simulation wala machine-learning research. Hakuna dataset, experimental group, sample size, physical device, laboratory measurement au statistical significance test. Method inajumuisha kabisa analytical mathematics, functional analysis, fractional operators na calculus-of-variations proofs.

Mathematical assumptions

• \(0<\alpha\leq1\).
• \(\beta>0\).
• \([t_0,t_1]\) ni fixed na finite interval.
• Functions zinazingatiwa katika \(C^\alpha\) au, katika Legendre section, \(PC^\alpha\) classes.
• Katika theorems husika, continuity assumptions zinazohitajika zinawekwa kwa \(L\), \(L_x\), \(L_y\) na kwa second variation \(L_{xx}\), \(L_{xy}\), \(L_{yy}\).
• Katika problems zenye endpoint cost, \(l\) inatakiwa kuwa continuously differentiable hadi order husika.

Proof chain

1. Riemann–Liouville integrals na derivatives pamoja na Caputo derivative zinafafanuliwa.
2. \(C^\alpha\) na \(PC^\alpha\) function spaces zinajengwa.
3. Lemma 3.1 inatoa main fractional Du Bois–Reymond result.
4. Lemma 3.2 inaonyesha continuity ya \(S\) operator inayotumika katika subsequent Euler–Lagrange proof.
5. Lemma 3.3 inatoa vector form inayoweza kutumika kwa first variation expression yenye \(h\) na \({}^{c}D^\alpha h\).
6. Theorem 4.1 inatoa Euler–Lagrange necessary conditions kwa fixed-end problem.
7. Example 4.1 inaonyesha impact ya \(\alpha\)–\(\beta\) relationship kwenye solution existence.
8. Second variation inajengwa na Legendre condition inatolewa katika Theorem 4.2 kwa kutumia special variation.
9. Theorems 4.3 na 4.4 zinapatikana kwa free-initial–fixed-terminal problem.
10. Fractional Bolza problem yenye free endpoints mbili inachunguzwa katika Theorem 4.5.
11. Fixed-initial–free-terminal problem inashughulikiwa katika Theorem 4.6.
12. Classical Euler–Lagrange na transversality conditions zinarecovered katika limit \(\alpha=\beta=1\).

Matokeo ya msingi ya Theorem 4.2 na 4.4

Common necessary condition ya Legendre section ni:

\[ \left\langle L_{yy} \left( t,x^0(t), ({}^{c}D_{t_0+}^{\alpha}x^0)(t) \right)r, r \right\rangle \geq0 \]

.

Condition hii inatolewa katika Theorem 4.2 kwa fixed-end problem na katika Theorem 4.4 kwa free-initial–fixed-terminal problem.

Condition ni necessary tu. Kwa general case, yenyewe pekee haithibitishi kwamba function inatoa minimum.

Example 4.2

Katika free-initial–fixed-terminal problem:

\[ J(x) = \int_0^1 (1-t)^{\beta-1} ({}^{c}D_{0+}^{\alpha}x)^2dt + x^2(0), \qquad x(1)=1 \]

inazingatiwa.

Kwa \(0<\beta\leq\alpha\leq1\), source inapata explicit extremal; kwa \(\beta>\alpha\), necessary conditions zinakuwa:

\[ {}^{c}D_{0+}^{\alpha}x=0, \qquad x(0)=0, \qquad x(1)=1 \]

ambazo haziendani, na watafiti wanasema hakuna solution.

Example 4.3

Kwa free-endpoint Bolza problem:

\[ B(x) = \int_0^1 (1-t)^{\beta-1} ({}^{c}D_{0+}^{\alpha}x)^2dt + x^2(0)+x^2(1) \]

katika regime \(0<\beta\leq\alpha\leq1\), watafiti hatimaye wanapata:

\[ x(0)=0 \]

na hivyo:

\[ x(t)=0 \]

extremal.

Example 4.4

Kwa \(\beta>\alpha\):

\[ B(x) = \int_0^1 (1-t)^{\beta-1} \left[ x(t)+({}^{c}D_{0+}^{\alpha}x)^2 \right]dt + x^2(0) \]

functional inatoa:

\[ ({}^{c}D_{0+}^{\alpha}x)(t) = -\frac{\Gamma(\beta)} {2\Gamma(\alpha+\beta)} (1-t)^\alpha \]

na:

\[ x(0)=-\frac{1}{2\beta} \]

.

Ujumbe wa Remark 4.1

Watafiti wanalinganisha Euler–Lagrange formulas zao na differential-form Euler–Lagrange condition ya kazi ya awali ili kusisitiza umuhimu wa formulas zao kubeba wazi uhusiano kati ya \(\alpha\) na \(\beta\).

Katika mfano wao, formulation yao inatoa \(x(t)=0\) optimum, huku previous formulation inayolinganishwa isitoe suitable solution katika same \(C^\alpha\) solution space. Comparison hii inalenga kuonyesha kwamba \(\alpha\)–\(\beta\) distinction si symbolic rewriting tu.

Main contributions za utafiti

• Fractional analogue ya classical Du Bois–Reymond lemma inajengwa kwa \(0<\alpha\leq1\), \(\beta>0\).
• Euler–Lagrange necessary conditions zinatolewa katika integral form ambapo uhusiano wa \(\alpha\) na \(\beta\) unaonekana wazi.
• Boundary configurations nne tofauti za initial–terminal zinachunguzwa.
• Classical second-variation strategy yenye special variation inatumika kwa Legendre condition katika fixed-end problem na free-initial–fixed-terminal problem.
• Katika free-right-end problems zenye \(\beta>\alpha\), necessary condition \(l_{x_1}=0\) inaonyeshwa.
• Kurudi kwenye classical Euler–Lagrange na transversality conditions kunaonyeshwa kwa \(\alpha=\beta=1\).
• Example problems zinaonyesha concretely impact ya \(\alpha\)–\(\beta\) relationship kwenye extremal na solution existence.

Matokeo ambayo utafiti hauonyeshi

• Haijaonyeshwa kwamba kila fractional variation problem ina solution.
• Haijaonyeshwa kwamba Legendre condition ni sufficient kwa minimum katika general case.
• Theorems hazitumiki automatically kwa fractional derivative definitions zote; utafiti umejengwa juu ya Caputo derivative.
• Hakuna experimental validation kwenye real physical au engineering system.
• Hakuna numerical algorithm iliyotengenezwa wala computation-time comparison.
• Haijatest kwamba Riemann–Liouville/Caputo models ni accurate zaidi kuliko models nyingine kwa real system maalum.
• Industrial, economic au clinical performance claim haiwezi kutolewa kutoka kwenye matokeo.

Dokezo la Chanzo na Mbinu

Author institutions katika uploaded v1

1. Shikhi Sh. Yusubov: Department of Mathematics, Shanghai University, Shanghai, China.
2. Shakir Sh. Yusubov: Baku State University, Department of Mechanics and Mathematics, Baku, Azerbaijan.
3. Elimhan N. Mahmudov: Azerbaijan National Aviation Academy, Baku, Azerbaijan.
4. Elimhan N. Mahmudov: Azerbaijan National Academy of Sciences, Institute of Control Systems, Baku, Azerbaijan.
5. Elimhan N. Mahmudov: Research Center for Mathematical Modeling and Optimization, UNEC, Baku, Azerbaijan.

Corresponding author: Katika uploaded v1, Elimhan N. Mahmudov ametajwa kama corresponding author na ORCID yake ni 0000-0003-2879-6154.

Affiliation difference kati ya publication versions

Katika uploaded v1 ya tarehe 7 Juni 2025, Elimhan N. Mahmudov anaorodheshwa na Azerbaijan National Aviation Academy, Azerbaijan National Academy of Sciences Institute of Control Systems, na UNEC Research Center for Mathematical Modeling and Optimization. Katika final Springer publication record, affiliations ni Azerbaijan National Aviation Academy na Azerbaijan University of Architecture and Construction. Version difference hii haijaunganishwa kimya kimya; institution list hapo juu inaakisi uploaded v1 ambayo ndiyo msingi wa scientific content.

Funding na conflict of interest

Katika uploaded source, waandishi wanasema hawakupokea fund, grant au financial support nyingine.

Source inasema hakuna conflict of interest.

Data, experiment na code status

Kazi ni theoretical mathematics article. Hakuna human, animal, clinical sample, experimental measurement wala observational dataset iliyotumika. Uploaded source haina separate data-availability section wala code repository section. Theorems na examples zimetengenezwa analytically.

Publication rights na license note

Uploaded arXiv version iko chini ya arXiv non-exclusive distribution license inayompa arXiv haki ya kusambaza kazi permanently na non-exclusively. License hii haitoi third parties automatic Creative Commons derivative au republication rights. Final Journal of Optimization Theory and Applications record inasema Springer Nature au relevant rightsholder ana exclusive publication rights juu ya article.

Kwa hiyo Verianla content hainakili pages za source, typesetting appearance ya equations au publisher design. Mathematical equations zimeandikwa upya kwa madhumuni ya kuwasilisha scientific ideas huku source fidelity ikihifadhiwa.

Preprint na peer-reviewed version distinction

Scientific content ya makala hii ya Verianla inategemea uploaded arXiv:2506.06736v1 text. Imehakikishwa bibliographically kwamba kazi baadaye ilikuwa peer-reviewed version of record katika Journal of Optimization Theory and Applications. Hata hivyo, kwa kuwa full final publication text haikutumika kama primary scientific source hapa, mathematical au editorial changes zinazoweza kuwa zimefanywa baada ya v1 hazijahamishwa kimya kimya kwenye uploaded text.

Mathematical editorial notes kuhusu uploaded v1

Point ya kwanza: Katika statement ya Lemma 3.1 kwa \(0<\beta\leq\alpha\leq1\), constant imeandikwa \(k\neq0\). Hata hivyo, katika “sufficiency” section ya proof imeandikwa pia kwamba \(f(t)=0\) inatimiza initial integral equality. Hivyo kuna visible inconsistency kati ya theorem statement na proof text kuhusu inclusion ya zero solution.

Point ya pili: Katika proof ya Legendre condition ya Theorem 4.2, baada ya definition \(M=\max|f|\), required direction ya estimate \(-8M^2\gamma\varepsilon\) inayotumika kwa negative main term kuelekea Equation (19) haijaelezwa kwa undani katika uploaded v1 text. Kwa hiyo Verianla haihardeni step hii kama independently verified new theorem; inaiwasilisha kama “proof presented by the authors”. Ukweli kwamba kazi baadaye ilikubaliwa kwa peer-reviewed publication unaonyesha current publication status, lakini kwa kuwa mathematical text inayotumika hapa ni v1, editorial-check note hii imehifadhiwa.

Boundary ya scientific interpretation

Main claim ya utafiti ni kwamba katika fractional calculus of variations, Euler–Lagrange necessary conditions zinaweza kutolewa kupitia generalized Du Bois–Reymond lemma, na chini ya endpoint constraints fulani classical second-variation approach inaweza kutumika kwa Legendre condition kwa kutumia special variations. Matokeo haya yanapaswa kutafsiriwa ndani ya specified function spaces, continuity conditions na parameter framework \(0<\alpha\leq1,\ \beta>0\).

Utafiti hauwezi kutatua general fractional optimization problems zote, hautoi theorem kwa fractional derivative types zote, na haubadilishi Legendre condition kuwa general sufficiency condition. Pia hakuna validation kwenye real system au numerical algorithm.