scholarly journals Positive solutions of a second-order nonlinear Robin problem involving the first-order derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhilin Yang
Keyword(s):  
Positive Solutions ◽  
A Priori Estimates ◽  
A Priori ◽  
Index Theory ◽  
Second Order ◽  
Iterative Sequence ◽  
Order Derivative ◽  
Robin Problem ◽  
First Order ◽  
Fixed Point Index Theory

AbstractThis paper is concerned with the second-order nonlinear Robin problem involving the first-order derivative: $$ \textstyle\begin{cases} u''+f(t,u,u^{\prime })=0, \\ u(0)=u'(1)-\alpha u(1)=0,\end{cases} $$ { u ″ + f ( t , u , u ′ ) = 0 , u ( 0 ) = u ′ ( 1 ) − α u ( 1 ) = 0 , where $f\in C([0,1]\times \mathbb{R}^{2}_{+},\mathbb{R}_{+})$ f ∈ C ( [ 0 , 1 ] × R + 2 , R + ) and $\alpha \in ]0,1[$ α ∈ ] 0 , 1 [ . Based on a priori estimates, we use fixed point index theory to establish some results on existence, multiplicity and uniqueness of positive solutions thereof, with the unique positive solution being the limit of of an iterative sequence. The results presented here generalize and extend the corresponding ones for nonlinearities independent of the first-order derivative.

scholarly journals Existence and Multiplicity of Positive Solutions for a System of Fourth-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Shoucheng Yu ◽  
Zhilin Yang
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Index Theory ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

We study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problems x(4)=ft,x,x′,-x′′,-x′′′,y,y′,-y′′,-y′′′,  y(4)=gt,x,x′,-x′′,-x′′′,y,y′,-y′′,-y′′′,  x(0)=x′(1)=x′′(0)=x′′′(1)=0, and y(0)=y′(1)=y′′(0)=y′′′(1)=0, where f,g∈C([0,1]×R+8,R+)  (R+:=[0,∞)). We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some integral identities and inequalities and R+2-monotone matrices.

scholarly journals Positive Solutions for a System of Fourth-Order p-Laplacian Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Lianlong Sun ◽  
Zhilin Yang
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Index Theory ◽  
Priori Estimates ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

We investigate the existence of positive solutions for the system of fourth-order p-Laplacian boundary value problems (|u′′|p-1u′′)′′=f1(t,u,v),  (|v′′|q-1v′′)′′=f2(t,u,v),  u(2i)(0)=u(2i)(1)=0,  i=0,1,  v(2i)(0)=v(2i)(1)=0,  i=0,1, where p,q>0 and f1,f2∈C([0,1]×ℝ+2,ℝ+)  (ℝ+:=[0,∞)). Based on a priori estimates achieved by utilizing Jensen’s integral inequalities and nonnegative matrices, we use fixed point index theory to establish our main results.

scholarly journals Positive Solutions for a Fourth-Order Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Kun Wang ◽  
Zhilin Yang
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Fourth Order ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Index Theory ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

This paper deals with the existence and multiplicity of positive solutions for the fourth-order boundary value problemu(4)=f(t,u,u′,−u′′, u′′′),u(0)=u′(1)=u′′′(0)=u′′(1)=0. Heref∈C([0,1]×ℝ+4,ℝ+)(ℝ+:=[0,+∞)). We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some integral identities and integral inequalities.

scholarly journals Positive solutions of a discrete second-order boundary value problems with fully nonlinear term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Liyun Jin ◽  
Hua Luo
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Fully Nonlinear ◽  
Fixed Point Index Theory ◽  
Weaker Conditions ◽  
Lower Limits

Abstract In this paper, we mainly consider a kind of discrete second-order boundary value problem with fully nonlinear term. By using the fixed-point index theory, we obtain some existence results of positive solutions of this kind of problems. Instead of the upper and lower limits condition on f, we may only impose some weaker conditions on f.

scholarly journals Positive Solutions for Neumann Boundary Value Problems of Second-Order Impulsive Differential Equations in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Xiaoya Liu ◽  
Yongxiang Li
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Boundary Value ◽  
Impulsive Differential Equations ◽  
Neumann Boundary ◽  
Index Theory ◽  
Second Order ◽  
Positive Elements ◽  
Fixed Point Index Theory ◽  
Neumann Boundary Value Problems

The existence of positive solutions for Neumann boundary value problem of second-order impulsive differential equations−u″(t)+Mu(t)=f(t,u(t),t∈J,t≠tk,-Δu'|t=tk=Ik(u(tk)),k=1,2,…,m,u'(0)=u'(1)=θ, in an ordered Banach spaceEwas discussed by employing the fixed point index theory of condensing mapping, whereM>0is a constant,J=[0,1],f∈C(J×K,K),Ik∈C(K,K),k=1,2,…,m, andKis the cone of positive elements inE. Moreover, an application is given to illustrate the main result.

scholarly journals Multiple positive solutions for nonlinear second-order m-point impulsive boundary value problems on time scales

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 817-827
Author(s):  
Ilkay Karaca ◽  
Fatma Fen
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Time Scales ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Multiple Positive Solutions ◽  
Impulsive Boundary Value Problems ◽  
Fixed Point Index Theory ◽  
Existence Of Positive Solutions

In this paper, by using fixed point index theory, we study the existence of positive solutions for nonlinear second-order m-point impulsive boundary value problems on time scales. As an application, we give an example to demonstrate our results.

scholarly journals Multiplicity of Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem with a Parameter

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jian Liu ◽  
Hanying Feng ◽  
Xingfang Feng
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solutions ◽  
Green's Function ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Point Boundary ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

This paper is concerned with the following second-order three-point boundary value problemu″t+β2ut+λqtft,ut=0,t∈0 , 1,u0=0,u(1)=δu(η), whereβ∈(0,π/2),δ>0,η∈(0,1), andλis a positive parameter. First, Green’s function for the associated linear boundary value problem is constructed, and then some useful properties of Green’s function are obtained. Finally, existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values ofλby means of the fixed point index theory.

scholarly journals Estimation of Tadalafil Using Derivative Spectrophotometry in Bulk Material and in Pharmaceutical Formulation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zamir G. Khan ◽  
Amod S. Patil ◽  
Atul A. Shirkhedkar
Keyword(s):  
Wavelength Range ◽  
Area Under Curve ◽  
Pharmaceutical Formulation ◽  
Second Order ◽  
Order Derivative ◽  
Uv Spectrophotometry ◽  
First Order ◽  
Beer's Law ◽  
Beer’S Law ◽  
Derivative Uv Spectrophotometry

Four simple, rapid, accurate, precise, reliable, and economical UV-spectrophotometric methods have been proposed for the determination of tadalafil in bulk and in pharmaceutical formulation. “Method A” is first order derivative UV spectrophotometry using amplitude, “method B” is first order derivative UV spectrophotometry using area under curve technique, “method C” is second order derivative UV spectrophotometry using amplitude, and “method D” is second order derivative UV spectrophotometry using area under curve technique. The developed methods have shown best results in terms of linearity, accuracy, precision, and LOD and LOQ for bulk drug and marketed formulation as well. In N,N-dimethylformamide, tadalafil showed maximum absorbance at 284 nm. For “method A” amplitude was recorded at 297 nm while for “method B” area under curve was integrated in the wavelength range of 290.60–304.40 nm. For “method C” amplitude was measured at 284 nm while for “method D” area under curve was selected in the wavelength range of 280.80–286.20 nm. For methods A and B, tadalafil obeyed Lambert-Beer’s law in the range of 05–50 μg/mL while for “methods C and D”, tadalafil obeyed Lambert-Beer’s law in the range of 20–70 μg/mL, and-for “methods A, B, C, and D” the correlation coefficients were found to be > than 0.999.

scholarly journals On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiang Li ◽  
Yongxiang Li
Keyword(s):  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Index Theory ◽  
Second Order ◽  
Multiple Delays ◽  
First Eigenvalue ◽  
Fixed Point Index Theory ◽  
Existence Conditions ◽  
Functional Differential ◽  
Periodic Boundary Problem

The existence results of positiveω-periodic solutions are obtained for the second-order functional differential equation with multiple delaysu″(t)+a(t)u(t)=f(t,u(t),u(t−τ1(t)),…,u(t−τn(t))), wherea(t)∈C(ℝ)is a positiveω-periodic function,f:ℝ×[0,+∞)n+1→[0,+∞)is a continuous function which isω-periodic int, andτ1(t),…,τn(t)∈C(ℝ,[0,+∞))areω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

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