{
  "index": "1997-B-2",
  "type": "ANA",
  "tag": [
    "ANA"
  ],
  "difficulty": "",
  "question": "Let $f$ be a twice-differentiable real-valued function satisfying\n\\[f(x)+f''(x)=-xg(x)f'(x),\\]\nwhere $g(x)\\geq 0$ for all real $x$.  Prove that $|f(x)|$ is bounded.",
  "solution": "It suffices to show that $|f(x)|$ is bounded for $x \\geq 0$, since $f(-x)$\nsatisfies the same equation as $f(x)$.  But then\n\\begin{align*}\n\\frac{d}{dx}\\left(\n(f(x))^2 + (f'(x))^2 \\right) &= 2f'(x)(f(x)+f''(x)) \\\\\n&= -2xg(x)(f'(x))^2 \\leq 0,\n\\end{align*}\nso that $(f(x))^2 \\leq (f(0))^2 + (f'(0))^2$ for $x\\geq 0$.",
  "vars": [
    "x"
  ],
  "params": [
    "f",
    "g"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "realvar",
        "f": "mainfun",
        "g": "nonnegfun"
      },
      "question": "Let $mainfun$ be a twice-differentiable real-valued function satisfying\n\\[\nmainfun(realvar)+mainfun''(realvar)=-realvar nonnegfun(realvar) mainfun'(realvar),\n\\]\nwhere $nonnegfun(realvar)\\geq 0$ for all real $realvar$.  Prove that $|mainfun(realvar)|$ is bounded.",
      "solution": "It suffices to show that $|mainfun(realvar)|$ is bounded for $realvar \\geq 0$, since $mainfun(-realvar)$\nsatisfies the same equation as $mainfun(realvar)$.  But then\n\\begin{align*}\n\\frac{d}{d realvar}\\left(\n(mainfun(realvar))^2 + (mainfun'(realvar))^2 \\right) &= 2 mainfun'(realvar)(mainfun(realvar)+mainfun''(realvar)) \\\\\n&= -2 realvar nonnegfun(realvar)(mainfun'(realvar))^2 \\leq 0,\n\\end{align*}\nso that $(mainfun(realvar))^2 \\leq (mainfun(0))^2 + (mainfun'(0))^2$ for $realvar\\geq 0$. "
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "canvasrod",
        "f": "radiantlog",
        "g": "horizonwax"
      },
      "question": "Let $radiantlog$ be a twice-differentiable real-valued function satisfying\n\\[radiantlog(canvasrod)+radiantlog''(canvasrod)=-canvasrod\\,horizonwax(canvasrod)\\,radiantlog'(canvasrod),\\]\nwhere $horizonwax(canvasrod)\\geq 0$ for all real $canvasrod$.  Prove that $|radiantlog(canvasrod)|$ is bounded.",
      "solution": "It suffices to show that $|radiantlog(canvasrod)|$ is bounded for $canvasrod \\geq 0$, since $radiantlog(-canvasrod)$\nsatisfies the same equation as $radiantlog(canvasrod)$.  But then\n\\begin{align*}\n\\frac{d}{d canvasrod}\\left(\n(radiantlog(canvasrod))^2 + (radiantlog'(canvasrod))^2 \\right) &= 2\\,radiantlog'(canvasrod)\\bigl(radiantlog(canvasrod)+radiantlog''(canvasrod)\\bigr) \\\\\n&= -2 canvasrod\\,horizonwax(canvasrod)\\,(radiantlog'(canvasrod))^2 \\le 0,\n\\end{align*}\nso that $(radiantlog(canvasrod))^2 \\le (radiantlog(0))^2 + (radiantlog'(0))^2$ for $canvasrod\\ge 0$. Thus $|radiantlog(canvasrod)|$ is bounded."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantvalue",
        "f": "malfunction",
        "g": "negativity"
      },
      "question": "Let $malfunction$ be a twice-differentiable real-valued function satisfying\n\\[\nmalfunction(constantvalue)+malfunction''(constantvalue)=-constantvalue negativity(constantvalue) malfunction'(constantvalue),\n\\]\nwhere $negativity(constantvalue)\\geq 0$ for all real $constantvalue$.  Prove that $|malfunction(constantvalue)|$ is bounded.",
      "solution": "It suffices to show that $|malfunction(constantvalue)|$ is bounded for $constantvalue \\geq 0$, since $malfunction(-constantvalue)$ satisfies the same equation as $malfunction(constantvalue)$.  But then\n\\begin{align*}\n\\frac{d}{dconstantvalue}\\left(\n(malfunction(constantvalue))^2 + (malfunction'(constantvalue))^2 \\right) &= 2malfunction'(constantvalue)(malfunction(constantvalue)+malfunction''(constantvalue)) \\\\\n&= -2constantvalue negativity(constantvalue)(malfunction'(constantvalue))^2 \\leq 0,\n\\end{align*}\nso that $(malfunction(constantvalue))^2 \\leq (malfunction(0))^2 + (malfunction'(0))^2$ for $constantvalue\\geq 0$.}"
    },
    "garbled_string": {
      "map": {
        "x": "kblynsqe",
        "f": "crweoipd",
        "g": "zxfnuqma"
      },
      "question": "Let $crweoipd$ be a twice-differentiable real-valued function satisfying\n\\[\ncrweoipd(kblynsqe)+crweoipd''(kblynsqe)=-kblynsqezxfnuqma(kblynsqe)crweoipd'(kblynsqe),\n\\]\nwhere $zxfnuqma(kblynsqe)\\geq 0$ for all real $kblynsqe$.  Prove that $|crweoipd(kblynsqe)|$ is bounded.",
      "solution": "It suffices to show that $|crweoipd(kblynsqe)|$ is bounded for $kblynsqe \\geq 0$, since $crweoipd(-kblynsqe)$\nsatisfies the same equation as $crweoipd(kblynsqe)$.  But then\n\\begin{align*}\n\\frac{d}{dkblynsqe}\\left(\n(crweoipd(kblynsqe))^2 + (crweoipd'(kblynsqe))^2 \\right) &= 2crweoipd'(kblynsqe)(crweoipd(kblynsqe)+crweoipd''(kblynsqe)) \\\\\n&= -2kblynsqezxfnuqma(kblynsqe)(crweoipd'(kblynsqe))^2 \\leq 0,\n\\end{align*}\nso that $(crweoipd(kblynsqe))^2 \\leq (crweoipd(0))^2 + (crweoipd'(0))^2$ for $kblynsqe\\geq 0$. "
    },
    "kernel_variant": {
      "question": "Let $m\\in\\mathbb N$.  \nLet $F:\\mathbb R\\to\\mathbb R^{m}$ be a twice continuously differentiable ($C^{2}$) vector–valued function and let  \n$H:\\mathbb R\\to\\mathbb R^{m\\times m}$ be a continuous map such that every matrix $H(x)$ is symmetric and positive–semidefinite.\n\nAssume the following quantitative ellipticity away from the origin:\n\n$\\text{(H1)}\\;$ There exist numbers $R_{H}>0$ and $0<h_{\\min}\\le h_{\\max}<\\infty$ with  \n\\[\nh_{\\min} I_{m}\\preceq H(x)\\preceq h_{\\max} I_{m}\\qquad\\text{for all }|x|\\ge R_{H},\n\\]\nwhere “$\\preceq$’’ denotes the Loewner order.  \n(Continuity of $H$ then implies $h_{\\mathrm{int}}:=\\sup_{|x|\\le R_{H}}\\|H(x)\\|<\\infty$.)\n\nLet $\\psi:\\mathbb R\\to(0,\\infty)$ be a $C^{1}$–function which  \n$\\bullet$ attains its global minimum at $x=0$ (so $\\psi(0)=:\\psi_{0}>0$),  \n$\\bullet$ is non–decreasing on $[0,\\infty)$ and non–increasing on $(-\\infty,0]$, and  \n$\\bullet$ is bounded above: $0<\\psi(x)\\le\\psi_{\\max}$ for every $x\\in\\mathbb R$.\n\nDefine the rapidly growing, non–negative polynomial  \n\\[\nP(x):=(x^{4}-8x^{2}+15)^{2}\\qquad(\\text{hence }P(x)\\asymp x^{8}\\text{ and }P(x)\\ge 1\\text{ for }|x|\\ge 2).\n\\]\n\nSuppose that $F$ satisfies the second–order matrix ordinary differential equation  \n\\[\n\\psi(x)F(x)+F''(x)=-P(x)H(x)F'(x).\\tag{1}\n\\]\n\nProve that  \n(i)  $\\|F(x)\\|$ is bounded on $\\mathbb R$;  \n(ii) $\\|F'(x)\\|$ is bounded on $\\mathbb R$;  \n(iii) the improper integral $\\displaystyle\\int_{-\\infty}^{\\infty} F'(x)^{\\top}H(x)F'(x)\\,dx$ converges;  \n(iv) the one–sided limits  \n\n\\[\n\\Gamma_{+}:=\\lim_{x\\to\\infty}\\|F(x)\\|,\\qquad \n\\Gamma_{-}:=\\lim_{x\\to-\\infty}\\|F(x)\\|\n\\]\n\nboth exist and are finite (they need not coincide).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Notation.  For vectors and matrices we use the Euclidean norm $\\|\\cdot\\|$, the inner\nproduct “$\\cdot$”, and $\\lambda_{\\min}(M),\\lambda_{\\max}(M)$ for the extreme eigenvalues of a symmetric matrix $M$.\n\n\\textbf{Step 1.  A first–order “energy’’ identity.}  \nDefine the non–negative $C^{1}$–function\n\\[\nE(x):=\\psi(x)\\,\\|F(x)\\|^{2}+\\|F'(x)\\|^{2}.\n\\]\nDifferentiating and inserting (1) we obtain\n\\begin{align*}\nE'(x)\n&=\\psi'(x)\\|F\\|^{2}+2\\psi F\\!\\cdot\\! F'+2F'\\!\\cdot\\! F''\\\\\n&=\\psi'(x)\\|F\\|^{2}+2\\psi F\\!\\cdot\\! F'-2\\psi F\\!\\cdot\\! F'-2P\\,F'^{\\top}HF'\\\\\n&=\\psi'(x)\\|F(x)\\|^{2}-2P(x)F'(x)^{\\top}H(x)F'(x). \\tag{2}\n\\end{align*}\nHence\n\\[\nE'(x)+2P(x)F'(x)^{\\top}H(x)F'(x)=\\psi'(x)\\,\\|F(x)\\|^{2}. \\tag{3}\n\\]\n\n\\textbf{Step 2.  Uniform boundedness of $E$.}  \nBecause $\\psi$ is monotone on each half–line and bounded above,\n\\[\n\\int_{-\\infty}^{\\infty}|\\psi'(s)|\\,ds\n =\\int_{0}^{\\infty}\\psi'(s)\\,ds+\\int_{-\\infty}^{0}-\\psi'(s)\\,ds\n \\le 2(\\psi_{\\max}-\\psi_{0})<\\infty. \\tag{4}\n\\]\nDropping the non–negative second term in (3) gives\n\\[\nE'(x)\\le |\\psi'(x)|\\,\\|F(x)\\|^{2}\\le\\frac{|\\psi'(x)|}{\\psi_{0}}\\,E(x). \\tag{5}\n\\]\nSet $b(x):=\\dfrac{|\\psi'(x)|}{\\psi_{0}}\\in L^{1}(\\mathbb R)$.  \nBy Grönwall’s inequality,\n\\[\nE(x)\\le E(0)\\exp\\!\\Bigl(\\!\\int_{0}^{x}b(s)\\,ds\\Bigr)\n      \\le E(0)\\exp(\\|b\\|_{L^{1}})\\le E(0)\\exp\\!\\Bigl(\\tfrac{2(\\psi_{\\max}-\\psi_{0})}{\\psi_{0}}\\Bigr)\n      =:E_{*}. \\tag{6}\n\\]\n\n\\textbf{Step 3.  Proof of (i) and (ii).}  \nSince $\\psi(x)\\ge\\psi_{0}>0$,\n\\[\n\\|F(x)\\|^{2}\\le\\frac{E_{*}}{\\psi_{0}},\\qquad\n\\|F'(x)\\|^{2}\\le E_{*}\\quad\\text{for all }x\\in\\mathbb R. \\tag{7}\n\\]\nThus $F$ and $F'$ are globally bounded.\n\n\\textbf{Step 4.  Finiteness of $\\displaystyle\\int P\\,F'^{\\top}HF'\\,dx$.}  \nIntegrating (3) from $-R$ to $R$ yields\n\\[\nE(R)-E(-R)+2\\int_{-R}^{R}P\\,F'^{\\top}HF'\\,dx\n       =\\int_{-R}^{R}\\psi'(x)\\|F(x)\\|^{2}\\,dx. \\tag{8}\n\\]\nThe right–hand integral tends to a finite limit as $R\\to\\infty$ because $\\psi'\\in L^{1}(\\mathbb R)$ and $\\|F\\|$ is bounded; moreover $|E(\\pm R)|\\le E_{*}$.  \nHence the non–decreasing function $R\\mapsto\\int_{-R}^{R}P\\,F'^{\\top}HF'$ is bounded and therefore convergent:\n\\[\n\\int_{-\\infty}^{\\infty}P(x)\\,F'(x)^{\\top}H(x)F'(x)\\,dx<\\infty. \\tag{9}\n\\]\n\n\\textbf{Step 5.  Proof of (iii).}  \nChoose $N:=\\max\\{R_{H},2\\}$; note $P(x)\\ge 1$ for $|x|\\ge 2$.  Split the integral\n\\[\n\\int_{-\\infty}^{\\infty}F'^{\\top}HF'\\,dx\n=\\int_{|x|\\le N}F'^{\\top}HF'\\,dx+\\int_{|x|> N}F'^{\\top}HF'\\,dx.\n\\]\nOn $|x|\\le N$, continuity of $H$ gives $\\|H\\|\\le h_{\\mathrm{int}}$, so\n\\[\n\\int_{|x|\\le N}F'^{\\top}HF'\\,dx\n \\le 2N\\,h_{\\mathrm{int}}\\sup_{x\\in\\mathbb R}\\|F'(x)\\|^{2}<\\infty\\quad\\text{by }(7).\n\\]\nFor the tail $|x|> N$ we use $P\\ge 1$:\n\\[\nF'^{\\top}HF'\\le P\\,F'^{\\top}HF',\\qquad\n\\int_{|x|> N}F'^{\\top}HF'\\,dx\\le\\int_{|x|> N}P\\,F'^{\\top}HF'\\,dx<\\infty\\quad\\text{by }(9).\n\\]\nHence the integral in (iii) converges.\n\n\\textbf{Step 6.  Square–integrability with polynomial weight and $L^{1}$–integrability of $F'$.}  \nBecause $\\lambda_{\\min}(H(x))\\ge h_{\\min}$ for $|x|\\ge R_{H}$ and hence for $|x|> N$,\n\\[\nP(x)\\,\\|F'(x)\\|^{2}\\le\\frac{1}{h_{\\min}}\\,P(x)\\,F'(x)^{\\top}H(x)F'(x),\n\\]\nso from (9)\n\\[\n\\int_{|x|> N}P(x)\\,\\|F'(x)\\|^{2}\\,dx<\\infty. \\tag{10}\n\\]\nSince $P(x)\\asymp x^{8}$, there exists $c>0$ such that $P(x)\\ge c\\,x^{8}$ for $x\\ge N$, whence $P(x)^{-1}\\le C\\,x^{-8}$.  Applying Cauchy–Schwarz on $(N,\\infty)$:\n\\[\n\\int_{N}^{\\infty}\\|F'(t)\\|\\,dt\n   =\\int_{N}^{\\infty}P(t)^{-1/2}\\bigl[P(t)^{1/2}\\|F'(t)\\|\\bigr]\\,dt\n   \\le\\Bigl(\\int_{N}^{\\infty}P(t)^{-1}\\,dt\\Bigr)^{1/2}\n         \\Bigl(\\int_{N}^{\\infty}P(t)\\|F'(t)\\|^{2}\\,dt\\Bigr)^{1/2}<\\infty. \\tag{11}\n\\]\nAn identical calculation on $(-\\infty,-N)$ yields\n\\[\n\\int_{-\\infty}^{-N}\\|F'(t)\\|\\,dt<\\infty. \\tag{12}\n\\]\n\n\\textbf{Step 7.  Existence of the one–sided limits $\\Gamma_{+}$ and $\\Gamma_{-}$ (claim (iv)).}  \nWe treat the right half–line; the left is identical. For $X>Y\\ge N$,\n\\[\n\\bigl|\\|F(X)\\|-\\|F(Y)\\|\\bigr|\n  \\le\\int_{Y}^{X}\\|F'(t)\\|\\,dt, \\tag{13}\n\\]\nand the right–hand side tends to $0$ as $Y\\to\\infty$ by (11).  Hence $\\{\\|F(x)\\|\\}_{x\\ge N}$ is a Cauchy sequence and convergent:\n\\[\n\\Gamma_{+}:=\\lim_{x\\to\\infty}\\|F(x)\\|\\quad\\text{exists and is finite.} \\tag{14}\n\\]\nA symmetric argument on $(-\\infty,-N)$ furnishes $\\Gamma_{-}$.  \nNothing in the hypotheses forces $\\Gamma_{+}$ and $\\Gamma_{-}$ to coincide; item (iv) only asserts their individual existence, which is now established.\n\nThus all four statements (i)–(iv) are rigorously proved under assumptions (H1) and (1).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.755587",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensional structure: The unknown is now a vector-valued function\n   F : ℝ→ℝᵐ, and the coefficient H(x) is a symmetric positive-semidefinite\n   matrix, introducing linear-algebraic considerations absent from the scalar\n   original.\n\n2. Variable coefficient in front of F: The term ψ(x)F(x) replaces the constant\n   coefficient of the original problem.  Its boundedness, monotonicity and the\n   sign of ψ' play a crucial rôle and require careful control through an\n   integrated Grönwall argument.\n\n3. Rapidly growing weight P(x): Because P(x)→∞ like x⁸, the analysis of the\n   energy identity must cope with an unbounded, sign-controlled coefficient.\n   This growth is subsequently exploited to deduce L²–integrability of F'\n   despite having only weighted control from (16).\n\n4. Additional conclusions:  Beyond mere boundedness of |F| the problem\n   demands boundedness of F', convergence of an improper integral and the\n   existence of a limit of ‖F(x)‖.  Each of these requires an extra layer of\n   analysis (e.g. Cauchy criteria, splitting of integrals, subsequence\n   arguments) not needed in the original task.\n\n5. Multiple interacting concepts:  The solution combines differential-equation\n   manipulation, linear-algebraic positivity, an energy method, an\n   L¹-Grönwall estimate, and an L²–Cauchy argument, making it substantially\n   more sophisticated than both the original problem and the kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $m\\in\\mathbb N$.  \nLet $F:\\mathbb R\\to\\mathbb R^{m}$ be a twice continuously differentiable ($C^{2}$) vector–valued function and let  \n$H:\\mathbb R\\to\\mathbb R^{m\\times m}$ be a continuous map such that every matrix $H(x)$ is symmetric and positive–semidefinite.\n\nAssume the following quantitative ellipticity away from the origin:\n\n$\\text{(H1)}\\;$ There exist numbers $R_{H}>0$ and $0<h_{\\min}\\le h_{\\max}<\\infty$ with  \n\\[\nh_{\\min} I_{m}\\preceq H(x)\\preceq h_{\\max} I_{m}\\qquad\\text{for all }|x|\\ge R_{H},\n\\]\nwhere “$\\preceq$’’ denotes the Loewner order.  \n(Continuity of $H$ then implies $h_{\\mathrm{int}}:=\\sup_{|x|\\le R_{H}}\\|H(x)\\|<\\infty$.)\n\nLet $\\psi:\\mathbb R\\to(0,\\infty)$ be a $C^{1}$–function which  \n$\\bullet$ attains its global minimum at $x=0$ (so $\\psi(0)=:\\psi_{0}>0$),  \n$\\bullet$ is non–decreasing on $[0,\\infty)$ and non–increasing on $(-\\infty,0]$, and  \n$\\bullet$ is bounded above: $0<\\psi(x)\\le\\psi_{\\max}$ for every $x\\in\\mathbb R$.\n\nDefine the rapidly growing, non–negative polynomial  \n\\[\nP(x):=(x^{4}-8x^{2}+15)^{2}\\qquad(\\text{hence }P(x)\\asymp x^{8}\\text{ and }P(x)\\ge 1\\text{ for }|x|\\ge 2).\n\\]\n\nSuppose that $F$ satisfies the second–order matrix ordinary differential equation  \n\\[\n\\psi(x)F(x)+F''(x)=-P(x)H(x)F'(x).\\tag{1}\n\\]\n\nProve that  \n(i)  $\\|F(x)\\|$ is bounded on $\\mathbb R$;  \n(ii) $\\|F'(x)\\|$ is bounded on $\\mathbb R$;  \n(iii) the improper integral $\\displaystyle\\int_{-\\infty}^{\\infty} F'(x)^{\\top}H(x)F'(x)\\,dx$ converges;  \n(iv) the one–sided limits  \n\n\\[\n\\Gamma_{+}:=\\lim_{x\\to\\infty}\\|F(x)\\|,\\qquad \n\\Gamma_{-}:=\\lim_{x\\to-\\infty}\\|F(x)\\|\n\\]\n\nboth exist and are finite (they need not coincide).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "solution": "Notation.  For vectors and matrices we use the Euclidean norm $\\|\\cdot\\|$, the inner\nproduct “$\\cdot$”, and $\\lambda_{\\min}(M),\\lambda_{\\max}(M)$ for the extreme eigenvalues of a symmetric matrix $M$.\n\n\\textbf{Step 1.  A first–order “energy’’ identity.}  \nDefine the non–negative $C^{1}$–function\n\\[\nE(x):=\\psi(x)\\,\\|F(x)\\|^{2}+\\|F'(x)\\|^{2}.\n\\]\nDifferentiating and inserting (1) we obtain\n\\begin{align*}\nE'(x)\n&=\\psi'(x)\\|F\\|^{2}+2\\psi F\\!\\cdot\\! F'+2F'\\!\\cdot\\! F''\\\\\n&=\\psi'(x)\\|F\\|^{2}+2\\psi F\\!\\cdot\\! F'-2\\psi F\\!\\cdot\\! F'-2P\\,F'^{\\top}HF'\\\\\n&=\\psi'(x)\\|F(x)\\|^{2}-2P(x)F'(x)^{\\top}H(x)F'(x). \\tag{2}\n\\end{align*}\nHence\n\\[\nE'(x)+2P(x)F'(x)^{\\top}H(x)F'(x)=\\psi'(x)\\,\\|F(x)\\|^{2}. \\tag{3}\n\\]\n\n\\textbf{Step 2.  Uniform boundedness of $E$.}  \nBecause $\\psi$ is monotone on each half–line and bounded above,\n\\[\n\\int_{-\\infty}^{\\infty}|\\psi'(s)|\\,ds\n =\\int_{0}^{\\infty}\\psi'(s)\\,ds+\\int_{-\\infty}^{0}-\\psi'(s)\\,ds\n \\le 2(\\psi_{\\max}-\\psi_{0})<\\infty. \\tag{4}\n\\]\nDropping the non–negative second term in (3) gives\n\\[\nE'(x)\\le |\\psi'(x)|\\,\\|F(x)\\|^{2}\\le\\frac{|\\psi'(x)|}{\\psi_{0}}\\,E(x). \\tag{5}\n\\]\nSet $b(x):=\\dfrac{|\\psi'(x)|}{\\psi_{0}}\\in L^{1}(\\mathbb R)$.  \nBy Grönwall’s inequality,\n\\[\nE(x)\\le E(0)\\exp\\!\\Bigl(\\!\\int_{0}^{x}b(s)\\,ds\\Bigr)\n      \\le E(0)\\exp(\\|b\\|_{L^{1}})\\le E(0)\\exp\\!\\Bigl(\\tfrac{2(\\psi_{\\max}-\\psi_{0})}{\\psi_{0}}\\Bigr)\n      =:E_{*}. \\tag{6}\n\\]\n\n\\textbf{Step 3.  Proof of (i) and (ii).}  \nSince $\\psi(x)\\ge\\psi_{0}>0$,\n\\[\n\\|F(x)\\|^{2}\\le\\frac{E_{*}}{\\psi_{0}},\\qquad\n\\|F'(x)\\|^{2}\\le E_{*}\\quad\\text{for all }x\\in\\mathbb R. \\tag{7}\n\\]\nThus $F$ and $F'$ are globally bounded.\n\n\\textbf{Step 4.  Finiteness of $\\displaystyle\\int P\\,F'^{\\top}HF'\\,dx$.}  \nIntegrating (3) from $-R$ to $R$ yields\n\\[\nE(R)-E(-R)+2\\int_{-R}^{R}P\\,F'^{\\top}HF'\\,dx\n       =\\int_{-R}^{R}\\psi'(x)\\|F(x)\\|^{2}\\,dx. \\tag{8}\n\\]\nThe right–hand integral tends to a finite limit as $R\\to\\infty$ because $\\psi'\\in L^{1}(\\mathbb R)$ and $\\|F\\|$ is bounded; moreover $|E(\\pm R)|\\le E_{*}$.  \nHence the non–decreasing function $R\\mapsto\\int_{-R}^{R}P\\,F'^{\\top}HF'$ is bounded and therefore convergent:\n\\[\n\\int_{-\\infty}^{\\infty}P(x)\\,F'(x)^{\\top}H(x)F'(x)\\,dx<\\infty. \\tag{9}\n\\]\n\n\\textbf{Step 5.  Proof of (iii).}  \nChoose $N:=\\max\\{R_{H},2\\}$; note $P(x)\\ge 1$ for $|x|\\ge 2$.  Split the integral\n\\[\n\\int_{-\\infty}^{\\infty}F'^{\\top}HF'\\,dx\n=\\int_{|x|\\le N}F'^{\\top}HF'\\,dx+\\int_{|x|> N}F'^{\\top}HF'\\,dx.\n\\]\nOn $|x|\\le N$, continuity of $H$ gives $\\|H\\|\\le h_{\\mathrm{int}}$, so\n\\[\n\\int_{|x|\\le N}F'^{\\top}HF'\\,dx\n \\le 2N\\,h_{\\mathrm{int}}\\sup_{x\\in\\mathbb R}\\|F'(x)\\|^{2}<\\infty\\quad\\text{by }(7).\n\\]\nFor the tail $|x|> N$ we use $P\\ge 1$:\n\\[\nF'^{\\top}HF'\\le P\\,F'^{\\top}HF',\\qquad\n\\int_{|x|> N}F'^{\\top}HF'\\,dx\\le\\int_{|x|> N}P\\,F'^{\\top}HF'\\,dx<\\infty\\quad\\text{by }(9).\n\\]\nHence the integral in (iii) converges.\n\n\\textbf{Step 6.  Square–integrability with polynomial weight and $L^{1}$–integrability of $F'$.}  \nBecause $\\lambda_{\\min}(H(x))\\ge h_{\\min}$ for $|x|\\ge R_{H}$ and hence for $|x|> N$,\n\\[\nP(x)\\,\\|F'(x)\\|^{2}\\le\\frac{1}{h_{\\min}}\\,P(x)\\,F'(x)^{\\top}H(x)F'(x),\n\\]\nso from (9)\n\\[\n\\int_{|x|> N}P(x)\\,\\|F'(x)\\|^{2}\\,dx<\\infty. \\tag{10}\n\\]\nSince $P(x)\\asymp x^{8}$, there exists $c>0$ such that $P(x)\\ge c\\,x^{8}$ for $x\\ge N$, whence $P(x)^{-1}\\le C\\,x^{-8}$.  Applying Cauchy–Schwarz on $(N,\\infty)$:\n\\[\n\\int_{N}^{\\infty}\\|F'(t)\\|\\,dt\n   =\\int_{N}^{\\infty}P(t)^{-1/2}\\bigl[P(t)^{1/2}\\|F'(t)\\|\\bigr]\\,dt\n   \\le\\Bigl(\\int_{N}^{\\infty}P(t)^{-1}\\,dt\\Bigr)^{1/2}\n         \\Bigl(\\int_{N}^{\\infty}P(t)\\|F'(t)\\|^{2}\\,dt\\Bigr)^{1/2}<\\infty. \\tag{11}\n\\]\nAn identical calculation on $(-\\infty,-N)$ yields\n\\[\n\\int_{-\\infty}^{-N}\\|F'(t)\\|\\,dt<\\infty. \\tag{12}\n\\]\n\n\\textbf{Step 7.  Existence of the one–sided limits $\\Gamma_{+}$ and $\\Gamma_{-}$ (claim (iv)).}  \nWe treat the right half–line; the left is identical. For $X>Y\\ge N$,\n\\[\n\\bigl|\\|F(X)\\|-\\|F(Y)\\|\\bigr|\n  \\le\\int_{Y}^{X}\\|F'(t)\\|\\,dt, \\tag{13}\n\\]\nand the right–hand side tends to $0$ as $Y\\to\\infty$ by (11).  Hence $\\{\\|F(x)\\|\\}_{x\\ge N}$ is a Cauchy sequence and convergent:\n\\[\n\\Gamma_{+}:=\\lim_{x\\to\\infty}\\|F(x)\\|\\quad\\text{exists and is finite.} \\tag{14}\n\\]\nA symmetric argument on $(-\\infty,-N)$ furnishes $\\Gamma_{-}$.  \nNothing in the hypotheses forces $\\Gamma_{+}$ and $\\Gamma_{-}$ to coincide; item (iv) only asserts their individual existence, which is now established.\n\nThus all four statements (i)–(iv) are rigorously proved under assumptions (H1) and (1).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.581323",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensional structure: The unknown is now a vector-valued function\n   F : ℝ→ℝᵐ, and the coefficient H(x) is a symmetric positive-semidefinite\n   matrix, introducing linear-algebraic considerations absent from the scalar\n   original.\n\n2. Variable coefficient in front of F: The term ψ(x)F(x) replaces the constant\n   coefficient of the original problem.  Its boundedness, monotonicity and the\n   sign of ψ' play a crucial rôle and require careful control through an\n   integrated Grönwall argument.\n\n3. Rapidly growing weight P(x): Because P(x)→∞ like x⁸, the analysis of the\n   energy identity must cope with an unbounded, sign-controlled coefficient.\n   This growth is subsequently exploited to deduce L²–integrability of F'\n   despite having only weighted control from (16).\n\n4. Additional conclusions:  Beyond mere boundedness of |F| the problem\n   demands boundedness of F', convergence of an improper integral and the\n   existence of a limit of ‖F(x)‖.  Each of these requires an extra layer of\n   analysis (e.g. Cauchy criteria, splitting of integrals, subsequence\n   arguments) not needed in the original task.\n\n5. Multiple interacting concepts:  The solution combines differential-equation\n   manipulation, linear-algebraic positivity, an energy method, an\n   L¹-Grönwall estimate, and an L²–Cauchy argument, making it substantially\n   more sophisticated than both the original problem and the kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}