<p>This paper presents several sufficient conditions for the existence of at least one solution and at least one nonzero solution to the boundary value problem <Equation ID="Equ11"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2444_Article_Equ11.gif" Format="GIF" Height="69" Rendition="HTML" Resolution="72" Type="Linedraw" Width="390" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \Delta ^{n} (p(k) \Delta ^{n}x(k-n)) = \lambda f(k, x(k)), & k \in {\mathbb N}[n,T] \\ x(0)=x(1)=\ldots =x(n-1) =0, & \\ x(T+1)=x(T+2)= \ldots = x(T+n)=0, & \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi mathvariant="normal">Δ</mi> <mi>n</mi> </msup> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>-</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow /> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>…</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd /> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow /> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>…</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with a parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2444_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. The technical approach is based on variational methods and the Mountain pass lemma.</p>

Existence of Solutions to Nonlinear 2nth-order Discrete Boundary Value Problem with Parameter Dependence

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2025年6月14日

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Urszula Ostaszewska,
Ewa Schmeidel,
Malgorzata Zdanowicz

摘要

This paper presents several sufficient conditions for the existence of at least one solution and at least one nonzero solution to the boundary value problem $\begin{aligned} \Delta ^{n} (p(k) \Delta ^{n}x(k-n)) = \lambda f(k, x(k)), & k \in {\mathbb N}[n,T] \\ x(0)=x(1)=\ldots =x(n-1) =0, & \\ x(T+1)=x(T+2)= \ldots = x(T+n)=0, & \end{aligned}$ $\begin{matrix} Δ^{n} (p (k) Δ^{n} x (k - n)) = λ f (k, x (k)), & k \in N [n, T] \\ x (0) = x (1) = \dots = x (n - 1) = 0, \\ x (T + 1) = x (T + 2) = \dots = x (T + n) = 0, \end{matrix}$ with a parameter $\lambda $ $λ$ . The technical approach is based on variational methods and the Mountain pass lemma.