scholarly journals Existence Theorems on Solvability of Constrained Inclusion Problems and Applications

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Teffera M. Asfaw
Keyword(s):  
Reflexive Banach Space ◽  
Existence Theorems ◽  
Maximal Monotone ◽  
Existence Of Solution ◽  
Existence Results ◽  
Uniformly Convex ◽  
Homotopy Invariance ◽  
Inclusion Problems ◽  
Compact Resolvents ◽  
Nonlinear Variational Inequality

LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual spaceX⁎. LetT:X⊇D(T)→2X⁎be a maximal monotone operator andC:X⊇D(C)→X⁎be bounded and continuous withD(T)⊆D(C). The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the typeT+Cprovided thatCis compact orTis of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition onT+C. The operatorCis neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.

scholarly journals Split variational inclusions for Bregman multivalued maximal monotone operators

2020 ◽  
Author(s):  
Mujahid Abbas ◽  
Faik Gürsoy ◽  
Yusuf Ibrahim ◽  
Abdul Rahim Khan
Keyword(s):  
Strong Convergence ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Strong Convergence Theorem ◽  
Variational Inclusions ◽  
Uniformly Convex ◽  
Inclusion Problems ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

We introduce a new algorithm to approximate a solution of split variational inclusion problems of multivalued maximal monotone operators in uniformly convex and uniformly smooth Banach spaces under the Bregman distance. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. As application, we solve a split minimization problem and provide a numerical example to support better findings of our result.

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
In-Sook Kim ◽  
Jung-Hyun Bae
Keyword(s):  
Reflexive Banach Space ◽  
Degree Theory ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Dense Subspace ◽  
Uniformly Convex ◽  
Infinite Dimensional ◽  
Surjectivity Result ◽  
Densely Defined

AbstractLet X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X⊃D(T) → 2X* is a maximal monotone multi-valued operator and C: X⊃D(C) → X* is a generalized pseudomonotone quasibounded operator with L ⊂ D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T x + λ C x, with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.

scholarly journals A Degree Theory for Compact Perturbations of Monotone Type Operators and Application to Nonlinear Parabolic Problem

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Teffera M. Asfaw
Keyword(s):  
Parabolic Problem ◽  
Existence Theorems ◽  
Degree Theory ◽  
Maximal Monotone ◽  
Uniformly Convex ◽  
Topological Degree Theory ◽  
Nonlinear Parabolic Problem ◽  
Compact Perturbations ◽  
Nonlinear Parabolic ◽  
Monotone Type

Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X⁎. Let T:X⊇D(T)→2X⁎ be maximal monotone, S:X→2X⁎ be bounded and of type (S+), and C:D(C)→X⁎ be compact with D(T)⊆D(C) such that C lies in Γστ (i.e., there exist σ≥0 and τ≥0 such that Cx≤τx+σ for all x∈D(C)). A new topological degree theory is developed for operators of the type T+S+C. The theory is essential because no degree theory and/or existence result is available to address solvability of operator inclusions involving operators of the type T+S+C, where C is not defined everywhere. Consequently, new existence theorems are provided. The existence theorem due to Asfaw and Kartsatos is improved. The theory is applied to prove existence of weak solution (s) for a nonlinear parabolic problem in appropriate Sobolev spaces.

scholarly journals Existence results for first derivative dependent ϕ-Laplacian boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imran Talib ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Existence Results ◽  
Main Concern ◽  
First Derivative ◽  
Boundary Functions ◽  
Carathéodory Function

Abstract Our main concern in this article is to investigate the existence of solution for the boundary-value problem $$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$ ( ϕ ( x ′ ( t ) ) ′ = g 1 ( t , x ( t ) , x ′ ( t ) ) , ∀ t ∈ [ 0 , 1 ] , ϒ 1 ( x ( 0 ) , x ( 1 ) , x ′ ( 0 ) ) = 0 , ϒ 2 ( x ( 0 ) , x ( 1 ) , x ′ ( 1 ) ) = 0 , where $g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ g 1 : [ 0 , 1 ] × R 2 → R is an $L^{1}$ L 1 -Carathéodory function, $\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $ ϒ i : R 3 → R are continuous functions, $i=1,2$ i = 1 , 2 , and $\phi :(-a,a)\rightarrow \mathbb{R}$ ϕ : ( − a , a ) → R is an increasing homeomorphism such that $\phi (0)=0$ ϕ ( 0 ) = 0 , for $0< a< \infty $ 0 < a < ∞ . We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.

scholarly journals Inertial Krasnoselskii–Mann Method in Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 638
Author(s):  
Yekini Shehu ◽  
Aviv Gibali
Keyword(s):  
Weak Convergence ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Splitting Method ◽  
Uniformly Convex ◽  
Inclusion Problems ◽  
Uniformly Smooth ◽  
Mann Algorithm ◽  
Uniformly Smooth Banach Spaces

In this paper, we give a general inertial Krasnoselskii–Mann algorithm for solving inclusion problems in Banach Spaces. First, we establish a weak convergence in real uniformly convex and q-uniformly smooth Banach spaces for finding fixed points of nonexpansive mappings. Then, a strong convergence is obtained for the inertial generalized forward-backward splitting method for the inclusion. Our results extend many recent and related results obtained in real Hilbert spaces.

scholarly journals Existence Results for a Fully Fourth-Order Boundary Value Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yongxiang Li ◽  
Qiuyan Liang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fourier Analysis ◽  
Growth Condition ◽  
Fourth Order ◽  
Boundary Value ◽  
Existence Of Solution ◽  
Existence Results ◽  
Analysis Method

We discuss the existence of solution for the fully fourth-order boundary value problemu(4)=f(t,u,u′,u′′,u′′′),0≤t≤1,u(0)=u(1)=u′′(0)=u′′(1)=0. A growth condition onfguaranteeing the existence of solution is presented. The discussion is based on the Fourier analysis method and Leray-Schauder fixed point theorem.

scholarly journals An Approximate Proximal Point Algorithm for Maximal Monotone Inclusion Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Lingling Huang ◽  
Sanyang Liu ◽  
Weifeng Gao
Keyword(s):  
Hilbert Spaces ◽  
Proximal Point Algorithm ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Computational Results ◽  
Monotone Inclusion ◽  
Proximal Point ◽  
Inclusion Problems ◽  
Correction Step

This paper presents and analyzes a strongly convergent approximate proximal point algorithm for finding zeros of maximal monotone operators in Hilbert spaces. The proposed method combines the proximal subproblem with a more general correction step which takes advantage of more information on the existing iterations. As applications, convex programming problems and generalized variational inequalities are considered. Some preliminary computational results are reported.

scholarly journals Existence Results for Some Equilibrium Problems Involvingα-Monotone Bifunction

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Ayed E. Hashoosh ◽  
Mohsen Alimohammady ◽  
M. K. Kalleji
Keyword(s):  
Banach Spaces ◽  
Differential Inclusion ◽  
Equilibrium Problems ◽  
Existence Of Solution ◽  
Existence Results ◽  
Closed Sets ◽  
Monotone Bifunction

This paper deals with some existence results of equilibrium problems(EPΨ)on convex and closed sets (either bounded or unbounded) in Banach spaces. Moreover, an application to the existence of solution for a differential inclusion is given.

scholarly journals Best proximity point theorems for weak cyclic Kannan contractions

Filomat ◽  
2011 ◽  
Vol 25 (1) ◽  
pp. 145-154 ◽  
Author(s):  
Ancuţa Petric
Keyword(s):  
Banach Space ◽  
Best Proximity Point ◽  
Convex Banach Space ◽  
Existence Results ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Best Proximity Points ◽  
Kannan Contraction

In this paper we introduce the notion of weak cyclic Kannan contraction. We give some convergence and existence results for best proximity points for weak cyclic Kannan contractions in the setting of a uniformly convex Banach space.

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