# Positive solutions for an $$n$$ n th-order quasilinear boundary value problem

Positive solutions for an $$n$$ n th-order quasilinear boundary value problem This paper is concerned with the existence and multiplicity of positive solutions of the $$n$$ n th-order quasilinear boundary value problem \begin{aligned} {\left\{ \begin{array}{ll} -(\varphi (u^{(n-1)}))^\prime =f(t,u), \quad \text {a.e.}\ t\in [0,1],\\ u^{(i)}(0) = u^{(n-1)}(1)=0\ \quad (i=0, \ldots , n-2), \end{array}\right. } \end{aligned} - ( φ ( u ( n - 1 ) ) ) ′ = f ( t , u ) , a.e. t ∈ [ 0 , 1 ] , u ( i ) ( 0 ) = u ( n - 1 ) ( 1 ) = 0 ( i = 0 , … , n - 2 ) , where $$n\geqslant 2$$ n ⩾ 2 , $$\varphi : \mathbb R^+\rightarrow \mathbb R^+$$ φ : R + → R + is either a convex or concave homeomorphism, and $$f\in C([0,1]\times \mathbb R^+,\mathbb R^+)(\mathbb R^+:=[0,\infty ))$$ f ∈ C ( [ 0 , 1 ] × R + , R + ) ( R + : = [ 0 , ∞ ) ) . Based on a priori estimates achieved by utilizing Jensen’s inequalities for concave and convex functions, we use fixed point index theory to establish our main results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positive solutions for an $$n$$ n th-order quasilinear boundary value problem

, Volume 19 (1) – Mar 4, 2014
16 pages

/lp/springer_journal/positive-solutions-for-an-n-n-th-order-quasilinear-boundary-value-GCOAkpuex0
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0281-9
Publisher site
See Article on Publisher Site

### Abstract

This paper is concerned with the existence and multiplicity of positive solutions of the $$n$$ n th-order quasilinear boundary value problem \begin{aligned} {\left\{ \begin{array}{ll} -(\varphi (u^{(n-1)}))^\prime =f(t,u), \quad \text {a.e.}\ t\in [0,1],\\ u^{(i)}(0) = u^{(n-1)}(1)=0\ \quad (i=0, \ldots , n-2), \end{array}\right. } \end{aligned} - ( φ ( u ( n - 1 ) ) ) ′ = f ( t , u ) , a.e. t ∈ [ 0 , 1 ] , u ( i ) ( 0 ) = u ( n - 1 ) ( 1 ) = 0 ( i = 0 , … , n - 2 ) , where $$n\geqslant 2$$ n ⩾ 2 , $$\varphi : \mathbb R^+\rightarrow \mathbb R^+$$ φ : R + → R + is either a convex or concave homeomorphism, and $$f\in C([0,1]\times \mathbb R^+,\mathbb R^+)(\mathbb R^+:=[0,\infty ))$$ f ∈ C ( [ 0 , 1 ] × R + , R + ) ( R + : = [ 0 , ∞ ) ) . Based on a priori estimates achieved by utilizing Jensen’s inequalities for concave and convex functions, we use fixed point index theory to establish our main results.

### Journal

PositivitySpringer Journals

Published: Mar 4, 2014

### References

• Multiple positive solutions for some p-Laplacian boundary value problems
Ji, D; Ge, W
• Nonlocal conjugate type boundary value problems of higher order
Webb, JRL

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