Triple Positive Solutions Research Articles

Triple positive solutions of a boundary value problem for nonlinear singular second-order differential equations of mixed type withp-Laplacian

In this paper, we establish the existence of triple positive solutions of a two-point boundary value problem for the nonlinear singular second-order differential equations of mixed type with a p-Laplacian operator. We also demonstrate that the results obtained can be applied to study certain higher order mixed boundary value problems. Finally, an example is given to demonstrate the use of the main results of this paper.

Triple positive solutions to third order three-point BVP with increasing homeomorphism and positive homomorphism

In this paper, we investigate the existence of triple positive solutions for nonlinear differential equations boundary value problems with increasing homeomorphism and positive homomorphism operator. By using fixed point theorems in cones, we establish results on the existence of three positive solutions with suitable growth conditions imposed on the nonlinear term. As applications, two examples are given to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.

The existence of three positive solutions of [formula omitted]-point boundary value problems for some dynamic equations on time scales

In this paper, we define a new operator which improves and generalizes a p -Laplacian operator for some p > 1 . By using this operator, we consider the existence of triple positive solutions of m -point boundary value problems for some dynamic equations on time scales [ φ ( p ( t ) u Δ ( t ) ) ] ∇ + a ( t ) f ( u ( t ) ) = 0 , t ∈ [ 0 , T ] T κ ∩ T k , u ( 0 ) − B 0 ( ∑ i = 1 m − 2 α i u Δ ( ξ i ) ) = 0 , u Δ ( T ) = 0 , where φ : R → R is an increasing homeomorphism and positive homomorphism and φ ( 0 ) = 0 . We show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term by using a new fixed-point theorem.

Multiple Positive Symmetric Solutions top-Laplacian Dynamic Equations on Time Scales

This paper makes a study on the existence of positive solution top-Laplacian dynamic equations on time scales𝕋. Some new sufficient conditions are obtained for the existence of at least single or twin positive solutions by using Krasnosel'skii's fixed point theorem and new sufficient conditions are also obtained for the existence of at least triple or arbitrary odd number positive solutions by using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem. As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations, as well as in the general time-scale setting.

Three positive solutions to initial-boundary value problems of nonlinear delay differential equations

In this paper, we consider the existence of triple positive solutions to the boundary value problem of nonlinear delay differential equation (�(x 0 (t))) 0 +a(t)f(t,x(t),x 0 (t),xt) = 0, 0 < t < 1, x0 = 0, x(1) = 0, where � : R ! R is an increasing homeomorphism and positive homomorphism with �(0) = 0, and xt is a function in C(( �,0),R) defined by xt(�) = x(t + �) for � � � � 0. By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.

Multiple Positive Solutions of a Nonlinear Four-Point Singular Boundary Value Problem with a p-Laplacian Operator on Time Scales

We present sufficient conditions for the existence of at least twin or triple positive solutions of a nonlinear four-point singular boundary value problem with a -Laplacian dynamic equation on a time scale. Our results are obtained via some new multiple fixed point theorems.

The Existence of Triple Positive Solutions of Nonlinear Four-point Boundary Value Problem with p-Laplacian

This paper deals with a class of singular p-Laplacian equation (φ_p(u′(t)))′+a(t)f(t,u(t),u′(t)) = 0,0t1 subject to the following nonlinear boundary conditionsαφ_p(u′(0)) -βφ_p(u′(e)) = 0,γφ_p(u(1)) +δφ_p(u′(η)) = 0, whereφ_P(x) = |x|~(p-2)x,p1.By using the Avery-Peterson fixed point theorem,sufficient conditions for the existence of at least three positive solutions to the boundary value problem mentioned above are established.The recent results are generalized and improved.

Triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities

In this paper, we study the existence of triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities. We first study the associated Green’s function and obtain some useful properties. Our main tool is the fixed point theorem due to Avery and Peterson. The results of this paper are new and extent previously known results.

Multiple positive pseudo-symmetric solutions of p-Laplacian dynamic equations on time scales

Let T be a pseudo-symmetric time scale such that 0 , T ∈ T . We consider a three-point boundary value problem for p -Laplacian dynamic equations on time scales T . Some new sufficient conditions are obtained for the existence of at least single, twin, triple or arbitrary odd positive pseudo-symmetric solutions of this problem by using pseudo-symmetric technique and fixed-point theorems in cone. Our results generalize and improve the results in paper by Su et al. [Y.H. Su, W.T. Li, H.R. Sun, Triple positive pseudo-symmetric solutions of three-point BVPs for p -Laplacian dynamic equations on time scales, Nonlinear Anal. TMA 68 (2008) 1442–1452]. As applications, three examples are given to illustrate the main results and their differences. These results are new for the special cases of continuous and discrete equations, as well as in the general time scale setting.

Multiple positive solutions for nonlinear singular m-point boundary value problem

In this paper, we study the existence of multiple positive solutions for the following nonlinear m-point boundary value problem (BVP) ( φ ( u ′ ) ) ′ + a ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , u ′ ( 0 ) = ∑ i = 1 m - 2 a i u ′ ( ξ i ) , u ( 1 ) = ∑ i = 1 k b i u ( ξ i ) - ∑ i = k + 1 s b i u ( ξ i ) - ∑ i = s + 1 m - 2 b i u ′ ( ξ i ) , where φ : R → R is an increasing homeomorphism and homomorphism and φ ( 0 ) = 0 , 1 ⩽ k ⩽ s ⩽ m - 2 , a i , b i ∈ ( 0 , + ∞ ) with 0 < ∑ i = 1 k b i - ∑ i = k + 1 s b i < 1 , 0 < ∑ i = 1 m - 2 a i < 1 , 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 , a ( t ) ∈ C ( ( 0 , 1 ) , [ 0 , + ∞ ) ) , f ∈ C ( [ 0 , + ∞ ) , [ 0 , + ∞ ) ) . Some new results are obtained for the existence of at least twin or triple positive solutions of the above problem by applying Avery–Henderson and a new fixed-point theorems, respectively. As an application, some examples are included to illustrate the main results. In particular, our results extend and improve some known results.

Triple positive solutions of three-point boundary value problems for fourth-order differential equations

Using the Avery and Peterson fixed point theorem, we investigate the existence of three positive solutions for the fourth order three-point boundary value problem u(4)(t)=h(t)f(t,u(t),u″(t)),0<t<1,u(0)=u(1)=0,c1u″(ξ)−c2u‴(ξ)=0,c3u″(1)+c4u‴(1)=0, where c1,c2,c3 and c4 are nonnegative constants satisfying c1c4+c2c3+c1c3>0, 0<ξ<1 and h(t) is sign-changing on [0,1]. The interesting point is that the nonlinear term is allowed to depend on u″.

Existence of triple symmetric positive solutions for four-point boundary-value problem with one-dimensional p-Laplacian

In this paper, we consider the four-point boundary value problem for one-dimensional p-Laplacian $$\bigl(\phi_{p}(u'(t))\bigr)'+q(t)f\bigl(t,u(t),u'(t)\bigr)=0,\quad t\in(0,1),$$ subject to the boundary conditions $$u(0)-\beta u'(\xi)=0,\qquad u(1)+\beta u'(\eta)=0,$$ where φ p (s)=|s| p−2 s. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three symmetric positive solutions to the above boundary value problem. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly.

Triple positive solutions for a class of [formula omitted]-point dynamic equations on time scales with [formula omitted]-Laplacian

This paper deals with a class of second-order nonlinear m -point dynamic equation on time scales with one-dimensional p -Laplacian. Using a fixed point theorem for operators on a cone, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is that the nonlinear term f is involved with the first-order delta derivative explicitly. Meanwhile, an example is worked out to demonstrate the main results.

Triple positive solutions for a boundary value problem of nonlinear fractional differential equation

In this paper, we investigate the existence of three positiv e solutions for the nonlinear fractional boundary value problem D �+ u(t) + a(t) f (t, u(t), u 00 (t)) = 0, 0 < t < 1, 3 < � � 4, u(0) = u 0 (0) = u 00 (0) = u 00 (1) = 0, where D �+ is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point l ies in the fact that the nonlinear term is allowed to depend on the second order derivative u 00 .

Triple Positive Solutions of Fourth-Order Four-Point Boundary Value Problems for -Laplacian Dynamic Equations on Time Scales

Recently, there have beenmany papersworking on the existence of positive solutions to boundary value problems for differential equations on time scales see, e.g., 1–4 and the references therein . This has been mainly due to their unification of the theory of differential and difference equations. An introduction to this unification is given in 5 . Now, this study is still a new area of fairly theoretical exploration in mathematics. However, it has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models see, e.g., 1, 5 . We let T be any time scale nonempty closed subset of R and let a, b be subset of T such that a, b {t ∈ T : a ≤ t ≤ b}. Thus, R, Z, N, No, that is, the real numbers, the integers, the natural numbers, and the nonnegative integers, are examples of time scales. In this paper, we study the existence of multiple positive solutions for the fourth-order four-point nonlinear dynamic equation on time scales with p-Laplacian:

Existence of triple positive solutions for a third-order three-point boundary value problem

In this paper we investigate the existence of triple positive solutions for the nonlinear third-order three-point boundary value problem u ‴ ( t ) = a ( t ) f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , 0 < t < 1 , u ( 0 ) = δ u ( η ) , u ′ ( η ) = 0 , u ″ ( 1 ) = 0 , where δ ∈ ( 0 , 1 ) , η ∈ [ 1 / 2 , 1 ) are constants. f : [ 0 , 1 ] × [ 0 , ∞ ) × R 2 → [ 0 , ∞ ) , q : ( 0 , 1 ) → [ 0 , ∞ ) are continuous. First, Green’s function for the associated linear boundary value problem is constructed, and then, by using a fixed-point theorem due to Avery and Peterson, we establish results on the existence of triple positive solutions to the boundary value problem.

Some new results on the existence of positive solutions for the one-dimensional [formula omitted]-Laplacian boundary value problems on time scales

This paper is devoted to using the Leggett–Williams fixed point theorem to derive some sufficient conditions for the existence of triple positive solutions to a type of p -Laplacian boundary value problems on time scales. The interesting point is that the nonlinear term f depends on the first-order delta derivative explicitly.

Existence theory for positive solutions to one-dimensional p-Laplacian boundary value problems on time scales

In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales ( φ p ( u Δ ( t ) ) ) Δ + h ( t ) f ( u σ ( t ) ) = 0 , t ∈ [ a , b ] , u ( a ) − B 0 ( u Δ ( a ) ) = 0 , u Δ ( σ ( b ) ) = 0 , where φ p ( u ) is p-Laplacian operator, i.e., φ p ( u ) = | u | p − 2 u , p > 1 . Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel'skii's fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett–Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied.

Triple positive pseudo-symmetric solutions to a four-point boundary value problem with p-Laplacian

This work deals with the existence of triple positive pseudo-symmetric solutions for the one-dimensional p -Laplacian ( ϕ p ( u ′ ) ) ′ ( t ) + q ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) − β u ′ ( ξ ) = 0 , u ( ξ ) − δ u ′ ( η ) = u ( 1 ) + δ u ′ ( 1 + ξ − η ) , where ϕ p ( s ) = | s | p − 2 ⋅ s , p > 1 . By means of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three positive pseudo-symmetric solutions to the above boundary value problem. The interesting point is that the nonlinear term is involved with the first-order derivative explicitly.

Existence of triple positive solutions for multi-point boundary value problems with a one dimensional p-Laplacian

In this paper, we consider the existence of multiple positive solutions for multi-point boundary value problems with a one dimensional p -Laplacian { ( ϕ p ( u ′ ( t ) ) ) ′ + q ( t ) f ( t , u ( t ) , u ′ ( t ) ) = 0 , 0 < t < 1 u ( 0 ) = ∑ i = 1 m − 2 α i u ( ξ i ) , u ( 1 ) = ∑ i = 1 m − 2 β i u ( ξ i ) , where ϕ p ( s ) = | s | p − 2 s , p > 1 , α i ≤ β i ( 1 ≤ i ≤ m − 2 ) ∈ [ 0 , ∞ ) , ∑ i = 1 m − 2 α i , ∑ i = 1 m − 2 β i ∈ ( 0 , 1 ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 . By the fixed point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.