Nonlinear Sturm-Liouville Boundary Value Problems Research Articles

Positive solutions of sub-superlinear Sturm–Liouville problems

In this paper, we introduce the notion of a strict lower/upper solution to nonlinear Sturm–Liouville boundary value problems. Based on the maximum principles, we establish a result of Leray–Schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions. Applying the result and the fixed point index theory in cones, we obtain the global existence results of positive solutions for sub-superlinear Sturm–Liouville problems.

Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

This paper is devoted to prove the existence of solutions of the nonlinear Sturm-Liouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of solutions in terms of Kuratowski measure of noncompactness. Mönch's fixed point theorem is used to prove the main result. By the unification property of time scales, our result is valid for Sturm-Liouville differential equations and difference equations, but more interestingly by the extension property, it is also valid for Sturm-Liouville $q$-difference equation.

Global bifurcation and multiple results for Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville boundary value problem { ( L u ) ( t ) = λ a ( t ) f ( u ( t ) ) , 0 < t < 1 , R 1 ( u ) = α 1 u ( 0 ) + β 1 u ′ ( 0 ) = 0 , R 2 ( u ) = α 2 u ( 1 ) + β 2 u ′ ( 1 ) = 0 , where L is the linear Sturm–Liouville operator ( L u ) ( t ) = − ( p ( t ) u ′ ( t ) ) ′ + q ( t ) u ( t ) . We obtain a global bifurcation result for a related bifurcation problem. We then use this to obtain multiple (at least eight) solutions of the Sturm–Liouville problem having specified nodal properties.

Positive solutions of a nonlinear Sturm–Liouville boundary-value problem

AbstractWe establish the existence of positive solutions of the Sturm–Liouville problemwhereWe assume g and $\hat{q}$ to be non-negative, continuous functions, a(s) is a positive continuous function, c≥0, p&gt;1, and the function h is sub-quadratic with respect to u′. We combine a priori estimates with a fixed-point result of Krasnosel'skii to obtain the existence of a positive solution.

On the existence and nonexistence of positive solutions for nonlinear Sturm–Liouville boundary value problems

In this paper the existence and nonexistence results of positive solutions are obtained for Sturm–Liouville boundary value problem − ( p ( x ) u ′ ) ′ + q ( x ) u = f ( x , u ) , x ∈ ( 0 , 1 ) , a u ( 0 ) − b p ( 0 ) u ′ ( 0 ) = 0 , c u ( 1 ) + d p ( 1 ) u ′ ( 1 ) = 0 , where p ∈ C 1 [ 0 , 1 ] , q ∈ C [ 0 , 1 ] , p ( x ) > 0 , q ( x ) ⩾ 0 for x ∈ [ 0 , 1 ] , f ∈ C ( [ 0 , 1 ] × R + ) , a , b , c , d ⩾ 0 are constants and satisfy ( a + b ) ( c + d ) > 0 . The discussion is based on the positivity estimation for the Green's function of associated linear boundary value problem and the fixed point index theory in cones.

THE RELATION BETWEEN SIGN-CHANGING SOLUTION AND POSITIVE-NEGATIVE SOLUTIONS FOR NONLINEAR OPERATOR EQUATIONS AND ITS APPLICATIONS

By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.

Global Bifurcation on Time Scales

We consider the structure of the solution set of a nonlinear Sturm–Liouville boundary value problem defined on a general time scale. Using global bifurcation theory we show that unbounded continua of nontrivial solutions bifurcate from the trivial solution at the eigenvalues of the linearization, and we show that certain nodal properties of the solutions are preserved along these continua. These results extend the well-known results of Rabinowitz for the case of Sturm–Liouville ordinary differential equations.

A Switch in Nodal Structure in Coupled Systems of Nonlinear Sturm-Liouville Boundary Value Problems

A Switch in Nodal Structure in Coupled Systems of Nonlinear Sturm-Liouville Boundary Value Problems

Parameter ranges for the existence of solutions whose state components have specified nodal structure in coupled multiparameter systems of nonlinear Sturm–Liouville boundary value problems

SynopsisThe set of solutions to the two-parameter systemhas been shown in a preceding paper of the author to exhibit a topological-functional analytic structure analogous to the structure of solution sets for nonlinear Sturm–Liouville boundary value problems. As the parameter λ and µ are varied, transitions in the solution set occur, first from trivial solutions to solutions (u, 0) with u having n nodes on (a, b) or solutions (0, v) with v having m nodes on (a, b), and then to solutions of the form (u, v), where u has n nodes on (a, b) and v has m nodes on (a, b), with n possibly different from m. Moreover, each transition is global in an appropriate bifurcation theoretic sense, with preservation of nodal structure. This paper explores these phenomena more closely, focusing on the range of parameters (λ, µ) for the existence of solutions (u, v) with u having n nodes on (a, b) and v having m nodes on (a, b) and its dependence on the assumptions placed on the coupling functions f and g. The principal tools of the analysis are the Alexander–Antman Bifurcation Theorem and a priori estimate techniques based on the maximum principle.

Global Preservation of Nodal Structure in Coupled Systems of Nonlinear Sturm-Liouville Boundary Value Problems

In this paper, we examine the solution set to the coupled system ($*$) \[ \left \{ {_{ - ({p_2}(x)\upsilon ’(x))’ + {q_2}(x)\upsilon (x) = \mu \upsilon (x) + \upsilon (x) \cdot g(u(x),\upsilon (x)),}^{ - ({p_1}(x)u’(x))’ + {q_1}(x)u(x) = \lambda u(x) + u(x) \cdot f(u(x),\upsilon (x))}} \right .\] where $\lambda ,\mu \in R,x \in [a,b]$, and the system ($*$) is subject to zero Dirichlet boundary data on $u$ and $\upsilon$. We determine conditions on $f$ and $g$ which permit us to assert the existence of continua of solutions to ($*$) characterized by $u$ having $n - 1$ simple zeros in $(a,b),\upsilon$ having $m - 1$ simple zeros in $(a,b)$, where $n$ and $m$ are positive but not necessarily equal integers. Moreover, we also determine conditions under which these continua link solutions to ($*$) of the form $(\lambda ,\mu ,u,0)$ with $u$ having $n - 1$ simple zeros in $(a,b)$ to solutions of ($*$) of the form $(\lambda ,\mu ,0,\upsilon )$ with $\upsilon$ having $m - 1$ simple zeros in $(a,b)$.

The shooting method for some nonlinear Sturm-Liouville boundary value problems

The shooting method is used to prove existence and uniqueness of the solution for a semilinear Sturm-Liouville boundary value problem (N).\(\frac{\partial }{{\partial u}}f(x,u)\) “lies between two consecutive eigenvalues” of the related linear problem, the shooting function turns out to be strongly monotone.

Global preservation of nodal structure in coupled systems of nonlinear Sturm-Liouville boundary value problems

In this paper, we examine the solution set to the coupled system ( ∗ * ) \[ { − ( p 2 ( x ) υ ′ ( x ) ) ′ + q 2 ( x ) υ ( x ) = μ υ ( x ) + υ ( x ) ⋅ g ( u ( x ) , υ ( x ) ) , − ( p 1 ( x ) u ′ ( x ) ) ′ + q 1 ( x ) u ( x ) = λ u ( x ) + u ( x ) ⋅ f ( u ( x ) , υ ( x ) ) \left \{ {_{ - ({p_2}(x)\upsilon ’(x))’ + {q_2}(x)\upsilon (x) = \mu \upsilon (x) + \upsilon (x) \cdot g(u(x),\upsilon (x)),}^{ - ({p_1}(x)u’(x))’ + {q_1}(x)u(x) = \lambda u(x) + u(x) \cdot f(u(x),\upsilon (x))}} \right . \] where λ , μ ∈ R , x ∈ [ a , b ] \lambda ,\mu \in R,x \in [a,b] , and the system ( ∗ * ) is subject to zero Dirichlet boundary data on u u and υ \upsilon . We determine conditions on f f and g g which permit us to assert the existence of continua of solutions to ( ∗ * ) characterized by u u having n − 1 n - 1 simple zeros in ( a , b ) , υ (a,b),\upsilon having m − 1 m - 1 simple zeros in ( a , b ) (a,b) , where n n and m m are positive but not necessarily equal integers. Moreover, we also determine conditions under which these continua link solutions to ( ∗ * ) of the form ( λ , μ , u , 0 ) (\lambda ,\mu ,u,0) with u u having n − 1 n - 1 simple zeros in ( a , b ) (a,b) to solutions of ( ∗ * ) of the form ( λ , μ , 0 , υ ) (\lambda ,\mu ,0,\upsilon ) with υ \upsilon having m − 1 m - 1 simple zeros in ( a , b ) (a,b) .

Solvability of a nonlinear Sturm-Liouville boundary-value problem for a second-order integrodifferential equation with one-sided restrictions on the growth of the right side with respect to the first derivative

Solvability of a nonlinear Sturm-Liouville boundary-value problem for a second-order integrodifferential equation with one-sided restrictions on the growth of the right side with respect to the first derivative