Teffera Asfaw - Academia.edu (original) (raw)
Papers by Teffera Asfaw
Journal of function spaces, 2016
Let be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual ... more Let be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space *. Let : ⊇ () → 2 * be maximal monotone of type Γ (i.e., there exist ≥ 0 and a nondecreasing function : [0, ∞) → [0, ∞) with (0) = 0 such that ⟨V * , − ⟩ ≥ − ‖ ‖ − (‖ ‖) for all ∈ (), V * ∈ , and ∈), : ⊃ () → * be linear, surjective, and closed such that −1 : * → is compact, and : → * be a bounded demicontinuous operator. A new degree theory is developed for operators of the type + +. The surjectivity of can be omitted provided that () is closed, is densely defined and self-adjoint, and = , a real Hilbert space. The theory improves the degree theory of Berkovits and Mustonen for + , where is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when is monotone, a maximality result is included for and +. The theory is applied to prove existence of weak solutions in = 2 (0, ; 1 0 (Ω)) of the nonlinear equation given by / −∑ =1 ((/) (
Abstract and Applied Analysis, 2017
Nonlinear Analysis: Theory, Methods & Applications, 2011
Let X be an infinite dimensional real reflexive Banach space with dual space X * and G ⊂ X , open... more Let X be an infinite dimensional real reflexive Banach space with dual space X * and G ⊂ X , open and bounded. Assume that X and X * are locally uniformly convex. Let T : X ⊃ D(T) → 2 X * be maximal monotone and strongly quasibounded, S : X ⊃ D(S) → X * maximal monotone, and C : X ⊃ D(C) → X * strongly quasibounded w.r.t. S and such that it satisfies a generalized (S +)-condition w.r.t. S. Assume that D(S) = L ⊂ D(T) ∩ D(C), where L is a dense subspace of X , and 0 ∈ T (0), S(0) = 0. A new topological degree theory is introduced for the sum T + S + C , with degree mapping d(T + S + C , G, 0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S + C , as above.
Mediterranean Journal of Mathematics
Journal of Mathematical Analysis and Applications, 2022
Journal of Mathematical Analysis and Applications, 2021
In this paper we give a new wavefront estimation technique that overcomes the main disadvantages ... more In this paper we give a new wavefront estimation technique that overcomes the main disadvantages of the phase diversity (PD) algorithms, namely the large computational complexity and the fact that the solutions can get stuck in a local minima. Our approach gives a good starting point for an iterative algorithm based on solving a linear system, but it can also be used as a new wavefront estimation method. The method is based on the Born approximation of the wavefront for small phase aberrations which leads to a quadratic point-spread function (PSF), and it requires two diversity images. First we take the differences between the focal plane image and each of the two diversity images, and then we eliminate the constant object, element-wise, from the two equations. The result is an overdetermined set of linear equations for which we give three solutions using linear least squares (LS), truncated total least squares (TTLS) and bounded data uncertainty (BDU). The last two approaches are suited when considering measurements affected by noise. Simulation results show that the estimation is faster than conventional PD algorithms.
Abstract and Applied Analysis, 2015
LetXbe a real locally uniformly convex reflexive separable Banach space with locally uniformly co... more LetXbe a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual spaceX∗. LetT:X⊇D(T)→2X∗be maximal monotone andS:X⊇D(S)→X∗quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach spaceW⊂D(S), dense and continuously embedded inX. Assume, further, that there existsd≥0such that〈v∗+Sx,x〉≥-dx2for allx∈D(T)∩D(S)andv∗∈Tx. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the typeT+S. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operatorL:X⊇D(L)→X∗is given as a result of surjectivity ofL+S, whereSis of type(M)with respect toL. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the ...
Abstract and Applied Analysis, 2022
Nirenberg proposed a problem as to whether or not a continuous and expansive operator T : X ⟶ X (... more Nirenberg proposed a problem as to whether or not a continuous and expansive operator T : X ⟶ X (where X is a Hilbert space) is surjective if R T ∘ ≠ ∅ . I shall give a positive answer for the problem provided that R T ∘ is unbounded. For contents related to this paper, the reader is referred to the remarks and the study of Asfaw (2021). The present paper gives a complete answer for the problem that has been open for about 47 years.
In this paper, we consider the system of nonlinear ordinary differential equation ˙ x = x(t)(1 − ... more In this paper, we consider the system of nonlinear ordinary differential equation ˙ x = x(t)(1 − x(t)) − a(t)f (x(t) ,y (t)) − h(t)γ(x(t)) ˙ y = y(t)(−d(t )+ b(t)g(x(t) ,y (t))) − k(t)ρ(y(t)) with t ≥ 0 and initial conditions x(0) = x0 > 0, y(0) = y0 > 0 and a, b, c, d, h, k and f, g are continuous from [0, ∞ )t o (0, ∞), ρ, γ are continuous from [0, ∞ )t o [0, ∞). Boundedness of solution(often called permanence) of this system is proved under suitable assumptions on the functions involved. Different examples of predator prey models are discussed as an application of the result. At last equilibria of some models are computed and their stability is analyzed.
Communications in Mathematical Analysis, 2015
Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dua... more Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X *. Let T : X → X * be demicontinuous, quasimonotone and α-expansive, and C : X → X * be compact such that either (i) T x + Cx, x ≥ −d x for all x ∈ X or (ii) T x + Cx, x ≥ −d x 2 for all x ∈ X and some suitable positive constants α and d. New surjectivity results are given for the operator T + C. The results are new even for C = {0}, which gives a partial positive answer for Nirenberg's problem for demicontinuous, quasimonotone and α-expansive mapping. Existence result on the surjectivity of quasimonotone perturbations of multivalued maximal monotone operator is included. The theory is applied to prove existence of generalized solution in H 1 0 (Ω) of nonlinear elliptic equation of the type
Partial Differential Equations in Applied Mathematics, 2021
Abstract Let X be a real locally uniformly convex reflexive Banach space with locally uniformly c... more Abstract Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X ∗ . Let T : X ⊇ D ( T ) → 2 X ∗ be maximal monotone, S : X → 2 X ∗ be bounded of type ( S + ) and C : X ⊇ D ( C ) → 2 X ∗ be compact with D ( T ) ⊆ D ( C ) . A degree theory is developed for operators of the type T + S + C . In general, the operator S + C : D ( C ) → 2 X ∗ is not necessarily an operator of type ( S + ) because D ( C ) is neither convex nor closed. Thus developing a degree theory for these class of operators is relevant as it generalizes the theories for operators of the type T + S . A new existence theorem is given for the solvability of the constrained problem T u + S u + C u ∋ 0 u ∈ D ( T ) . The case C = { 0 } gives those theories and existence theorems for operators of the type T + S . An application on the existence of solution for a nonlinear parabolic variational inequality problem is provided.
Advances in Operator Theory, 2019
Let X be a real locally uniformly convex reflexive Banach space with the locally uniformly convex... more Let X be a real locally uniformly convex reflexive Banach space with the locally uniformly convex dual space X * , and let K be a nonempty, closed, and convex subset of X. Let T : X ⊇ D(T) → 2 X * be maximal monotone, let S : K → 2 X * be bounded and of type (S +), and let C : X ⊇ D(C) → X * with D(T) ∩ D(∂ϕ) ∩ K ⊆ D(C). Let ϕ : X → (−∞, ∞] be a proper, convex, and lower semicontinuous function. New existence theorems are proved for solvability of variational inequality problems of the type VIP(T + S + C, K, ϕ, f *) if C is compact and VIP(T + C, K, ϕ, f *) if T is of compact resolvent and C is bounded and continuous. Various improvements and generalizations of the existing results for T + S and ϕ are obtained. The theory is applied to prove existence of solution for nonlinear constrained variational inequality problems.
Abstract and Applied Analysis, 2017
Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dua... more 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.
G. Kartsatos, for supervising me with extremely valuable guidelines, advice, remarks and suggesti... more G. Kartsatos, for supervising me with extremely valuable guidelines, advice, remarks and suggestions during the preparation of this dissertation. I wish to express my sincere thanks to Professor Yuncheng You, Professor Wen-Xiu Ma and Professor Boris Shekhtman for their interest to be members of my Ph.D. advisory committee. My sincere thanks go to Professor Witanachchi Sarath for chairing my dissertation defence. My appreciation go to the entire faculty and staff of the USF mathematics department for their kind hospitality during my study at USF. Many thanks go to my wife and family for their patience, interest and support throughout my student career. My thanks go to my friends for their initiatives.
Journal of Mathematical Analysis and Applications, 2016
Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly c... more Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X ⁎ and G be a nonempty, bounded and open subset of X. Let T : X ⊇ D ( T ) → 2 X ⁎ and A : X ⊇ D ( A ) → 2 X ⁎ be maximal monotone operators. Assume, further, that, for each y ∈ X , there exists a real number β ( y ) and there exists a strictly increasing function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) with ϕ ( 0 ) = 0 , ϕ ( t ) → ∞ as t → ∞ satisfying 〈 w ⁎ , x − y 〉 ≥ − ϕ ( ‖ x ‖ ) ‖ x ‖ − β ( y ) for all x ∈ D ( A ) , w ⁎ ∈ A x , and S : X → 2 X ⁎ is bounded of type ( S + ) or bounded pseudomonotone such that 0 ∉ ( T + A + S ) ( D ( T ) ∩ D ( A ) ∩ ∂ G ) or 0 ∉ ( T + A + S ) ( D ( T ) ∩ D ( A ) ∩ ∂ G ) ‾ , respectively. New degree theory is developed for operators of the type T + A + S with degree mapping d ( T + A + S , G , 0 ) . The degree is shown to be unique invariant under suitable homotopies. The theory developed herein generalizes the Asfaw and Kartsatos degree theory for operators of the type T + S . New results on surjectivity and solvability of variational inequality problems are obtained. The mapping theorems extend the corresponding results for operators of type T + S . The degree theory developed herein is used to show existence of weak solution of nonlinear parabolic problem in appropriate Sobolev spaces.
Nonlinear Analysis: Theory, Methods & Applications, 2015
Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly c... more Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X ∗ . Let T : X ⊇ D ( T ) → 2 X ∗ be maximal monotone and S : X → 2 X ∗ be bounded pseudomonotone. Let p be a nonnegative integer. Assume, further, that there exist nonnegative constants a i ( i = 1 , 2 , … , p ) such that 〈 v ∗ + w ∗ , x 〉 ≥ − ∑ i = 1 p a i ‖ x ‖ i − α ( ‖ x ‖ ) ‖ x ‖ p + 2 for all x ∈ D ( T ) with sufficiently large ‖ x ‖ , v ∗ ∈ T x and w ∗ ∈ S x , where α : [ 0 , ∞ ) → [ 0 , ∞ ) such that α ( t ) → 0 as t → ∞ . New surjectivity results are given for the operator T + S along with weakly coercive type hypothesis on T + S . The results are new and improve the corresponding theory for coercive operators of monotone type. The theory developed herein can be suitably applied in the study of partial differential equations, variational and hemi-variational inequality problems in appropriate Sobolev spaces. To demonstrate the applicability of the theory, an example of time periodic parabolic partial differential equation, which models nonmonotone semipermeability problem, is provided.
Abstract and Applied Analysis, 2018
LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual ... more 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.
arXiv (Cornell University), Nov 20, 2021
Let X be a real Banach space with its dual X * and G be a nonempty, bounded and open subset of X ... more Let X be a real Banach space with its dual X * and G be a nonempty, bounded and open subset of X with 0 ∈ G. Let T : X ⊇ D(T) → 2 X be an m-accretive operator with 0 ∈ D(T) and 0 ∈ T (0), and let C be a compact operator from X into X with D(T) ⊆ D(C). We prove that f ∈ R(T) + R(C) if C is multivalued and f ∈ R(T + C) if C is single-valued, provided T x + Cx + εx ∋ f for all x ∈ D(T) ∩ ∂G and ε > 0. The surjectivity of T + C is proved if T is expansive and T + C is weakly coercive. Analogous results are given if T has compact resolvents and C is continuous and bounded. Various results by Kartsatos, and Kartsatos and Liu are improved, and a result by Morales is generalized.
Journal of Mathematical Analysis and Applications, 2021
Abstract Let X be a real reflexive Banach space with X ⁎ its dual space and G be a nonempty and o... more Abstract Let X be a real reflexive Banach space with X ⁎ its dual space and G be a nonempty and open subset of X. Let A : X ⊇ D ( A ) → 2 X ⁎ be a strongly quasibounded maximal monotone operator and T : X ⊇ D ( T ) → 2 X ⁎ be an operator of class A G ( S + ) introduced by Kittila. We develop a topological degree theory for the operator A + T . The theory generalizes the Browder degree theory for operators of type ( S + ) and extends the Kittila degree theory for operators of class A G ( S + ) . New existence results are established. The existence results give generalizations of similar known results for operators of type ( S + ) . Applications to strongly nonlinear problems are included.
Journal of function spaces, 2016
Let be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual ... more Let be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space *. Let : ⊇ () → 2 * be maximal monotone of type Γ (i.e., there exist ≥ 0 and a nondecreasing function : [0, ∞) → [0, ∞) with (0) = 0 such that ⟨V * , − ⟩ ≥ − ‖ ‖ − (‖ ‖) for all ∈ (), V * ∈ , and ∈), : ⊃ () → * be linear, surjective, and closed such that −1 : * → is compact, and : → * be a bounded demicontinuous operator. A new degree theory is developed for operators of the type + +. The surjectivity of can be omitted provided that () is closed, is densely defined and self-adjoint, and = , a real Hilbert space. The theory improves the degree theory of Berkovits and Mustonen for + , where is bounded demicontinuous pseudomonotone. New existence theorems are provided. In the case when is monotone, a maximality result is included for and +. The theory is applied to prove existence of weak solutions in = 2 (0, ; 1 0 (Ω)) of the nonlinear equation given by / −∑ =1 ((/) (
Abstract and Applied Analysis, 2017
Nonlinear Analysis: Theory, Methods & Applications, 2011
Let X be an infinite dimensional real reflexive Banach space with dual space X * and G ⊂ X , open... more Let X be an infinite dimensional real reflexive Banach space with dual space X * and G ⊂ X , open and bounded. Assume that X and X * are locally uniformly convex. Let T : X ⊃ D(T) → 2 X * be maximal monotone and strongly quasibounded, S : X ⊃ D(S) → X * maximal monotone, and C : X ⊃ D(C) → X * strongly quasibounded w.r.t. S and such that it satisfies a generalized (S +)-condition w.r.t. S. Assume that D(S) = L ⊂ D(T) ∩ D(C), where L is a dense subspace of X , and 0 ∈ T (0), S(0) = 0. A new topological degree theory is introduced for the sum T + S + C , with degree mapping d(T + S + C , G, 0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S + C , as above.
Mediterranean Journal of Mathematics
Journal of Mathematical Analysis and Applications, 2022
Journal of Mathematical Analysis and Applications, 2021
In this paper we give a new wavefront estimation technique that overcomes the main disadvantages ... more In this paper we give a new wavefront estimation technique that overcomes the main disadvantages of the phase diversity (PD) algorithms, namely the large computational complexity and the fact that the solutions can get stuck in a local minima. Our approach gives a good starting point for an iterative algorithm based on solving a linear system, but it can also be used as a new wavefront estimation method. The method is based on the Born approximation of the wavefront for small phase aberrations which leads to a quadratic point-spread function (PSF), and it requires two diversity images. First we take the differences between the focal plane image and each of the two diversity images, and then we eliminate the constant object, element-wise, from the two equations. The result is an overdetermined set of linear equations for which we give three solutions using linear least squares (LS), truncated total least squares (TTLS) and bounded data uncertainty (BDU). The last two approaches are suited when considering measurements affected by noise. Simulation results show that the estimation is faster than conventional PD algorithms.
Abstract and Applied Analysis, 2015
LetXbe a real locally uniformly convex reflexive separable Banach space with locally uniformly co... more LetXbe a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual spaceX∗. LetT:X⊇D(T)→2X∗be maximal monotone andS:X⊇D(S)→X∗quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach spaceW⊂D(S), dense and continuously embedded inX. Assume, further, that there existsd≥0such that〈v∗+Sx,x〉≥-dx2for allx∈D(T)∩D(S)andv∗∈Tx. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the typeT+S. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operatorL:X⊇D(L)→X∗is given as a result of surjectivity ofL+S, whereSis of type(M)with respect toL. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the ...
Abstract and Applied Analysis, 2022
Nirenberg proposed a problem as to whether or not a continuous and expansive operator T : X ⟶ X (... more Nirenberg proposed a problem as to whether or not a continuous and expansive operator T : X ⟶ X (where X is a Hilbert space) is surjective if R T ∘ ≠ ∅ . I shall give a positive answer for the problem provided that R T ∘ is unbounded. For contents related to this paper, the reader is referred to the remarks and the study of Asfaw (2021). The present paper gives a complete answer for the problem that has been open for about 47 years.
In this paper, we consider the system of nonlinear ordinary differential equation ˙ x = x(t)(1 − ... more In this paper, we consider the system of nonlinear ordinary differential equation ˙ x = x(t)(1 − x(t)) − a(t)f (x(t) ,y (t)) − h(t)γ(x(t)) ˙ y = y(t)(−d(t )+ b(t)g(x(t) ,y (t))) − k(t)ρ(y(t)) with t ≥ 0 and initial conditions x(0) = x0 > 0, y(0) = y0 > 0 and a, b, c, d, h, k and f, g are continuous from [0, ∞ )t o (0, ∞), ρ, γ are continuous from [0, ∞ )t o [0, ∞). Boundedness of solution(often called permanence) of this system is proved under suitable assumptions on the functions involved. Different examples of predator prey models are discussed as an application of the result. At last equilibria of some models are computed and their stability is analyzed.
Communications in Mathematical Analysis, 2015
Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dua... more Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X *. Let T : X → X * be demicontinuous, quasimonotone and α-expansive, and C : X → X * be compact such that either (i) T x + Cx, x ≥ −d x for all x ∈ X or (ii) T x + Cx, x ≥ −d x 2 for all x ∈ X and some suitable positive constants α and d. New surjectivity results are given for the operator T + C. The results are new even for C = {0}, which gives a partial positive answer for Nirenberg's problem for demicontinuous, quasimonotone and α-expansive mapping. Existence result on the surjectivity of quasimonotone perturbations of multivalued maximal monotone operator is included. The theory is applied to prove existence of generalized solution in H 1 0 (Ω) of nonlinear elliptic equation of the type
Partial Differential Equations in Applied Mathematics, 2021
Abstract Let X be a real locally uniformly convex reflexive Banach space with locally uniformly c... more Abstract Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space X ∗ . Let T : X ⊇ D ( T ) → 2 X ∗ be maximal monotone, S : X → 2 X ∗ be bounded of type ( S + ) and C : X ⊇ D ( C ) → 2 X ∗ be compact with D ( T ) ⊆ D ( C ) . A degree theory is developed for operators of the type T + S + C . In general, the operator S + C : D ( C ) → 2 X ∗ is not necessarily an operator of type ( S + ) because D ( C ) is neither convex nor closed. Thus developing a degree theory for these class of operators is relevant as it generalizes the theories for operators of the type T + S . A new existence theorem is given for the solvability of the constrained problem T u + S u + C u ∋ 0 u ∈ D ( T ) . The case C = { 0 } gives those theories and existence theorems for operators of the type T + S . An application on the existence of solution for a nonlinear parabolic variational inequality problem is provided.
Advances in Operator Theory, 2019
Let X be a real locally uniformly convex reflexive Banach space with the locally uniformly convex... more Let X be a real locally uniformly convex reflexive Banach space with the locally uniformly convex dual space X * , and let K be a nonempty, closed, and convex subset of X. Let T : X ⊇ D(T) → 2 X * be maximal monotone, let S : K → 2 X * be bounded and of type (S +), and let C : X ⊇ D(C) → X * with D(T) ∩ D(∂ϕ) ∩ K ⊆ D(C). Let ϕ : X → (−∞, ∞] be a proper, convex, and lower semicontinuous function. New existence theorems are proved for solvability of variational inequality problems of the type VIP(T + S + C, K, ϕ, f *) if C is compact and VIP(T + C, K, ϕ, f *) if T is of compact resolvent and C is bounded and continuous. Various improvements and generalizations of the existing results for T + S and ϕ are obtained. The theory is applied to prove existence of solution for nonlinear constrained variational inequality problems.
Abstract and Applied Analysis, 2017
Let X be a real locally uniformly convex reflexive Banach space with locally uniformly convex dua... more 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.
G. Kartsatos, for supervising me with extremely valuable guidelines, advice, remarks and suggesti... more G. Kartsatos, for supervising me with extremely valuable guidelines, advice, remarks and suggestions during the preparation of this dissertation. I wish to express my sincere thanks to Professor Yuncheng You, Professor Wen-Xiu Ma and Professor Boris Shekhtman for their interest to be members of my Ph.D. advisory committee. My sincere thanks go to Professor Witanachchi Sarath for chairing my dissertation defence. My appreciation go to the entire faculty and staff of the USF mathematics department for their kind hospitality during my study at USF. Many thanks go to my wife and family for their patience, interest and support throughout my student career. My thanks go to my friends for their initiatives.
Journal of Mathematical Analysis and Applications, 2016
Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly c... more Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X ⁎ and G be a nonempty, bounded and open subset of X. Let T : X ⊇ D ( T ) → 2 X ⁎ and A : X ⊇ D ( A ) → 2 X ⁎ be maximal monotone operators. Assume, further, that, for each y ∈ X , there exists a real number β ( y ) and there exists a strictly increasing function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) with ϕ ( 0 ) = 0 , ϕ ( t ) → ∞ as t → ∞ satisfying 〈 w ⁎ , x − y 〉 ≥ − ϕ ( ‖ x ‖ ) ‖ x ‖ − β ( y ) for all x ∈ D ( A ) , w ⁎ ∈ A x , and S : X → 2 X ⁎ is bounded of type ( S + ) or bounded pseudomonotone such that 0 ∉ ( T + A + S ) ( D ( T ) ∩ D ( A ) ∩ ∂ G ) or 0 ∉ ( T + A + S ) ( D ( T ) ∩ D ( A ) ∩ ∂ G ) ‾ , respectively. New degree theory is developed for operators of the type T + A + S with degree mapping d ( T + A + S , G , 0 ) . The degree is shown to be unique invariant under suitable homotopies. The theory developed herein generalizes the Asfaw and Kartsatos degree theory for operators of the type T + S . New results on surjectivity and solvability of variational inequality problems are obtained. The mapping theorems extend the corresponding results for operators of type T + S . The degree theory developed herein is used to show existence of weak solution of nonlinear parabolic problem in appropriate Sobolev spaces.
Nonlinear Analysis: Theory, Methods & Applications, 2015
Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly c... more Abstract Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X ∗ . Let T : X ⊇ D ( T ) → 2 X ∗ be maximal monotone and S : X → 2 X ∗ be bounded pseudomonotone. Let p be a nonnegative integer. Assume, further, that there exist nonnegative constants a i ( i = 1 , 2 , … , p ) such that 〈 v ∗ + w ∗ , x 〉 ≥ − ∑ i = 1 p a i ‖ x ‖ i − α ( ‖ x ‖ ) ‖ x ‖ p + 2 for all x ∈ D ( T ) with sufficiently large ‖ x ‖ , v ∗ ∈ T x and w ∗ ∈ S x , where α : [ 0 , ∞ ) → [ 0 , ∞ ) such that α ( t ) → 0 as t → ∞ . New surjectivity results are given for the operator T + S along with weakly coercive type hypothesis on T + S . The results are new and improve the corresponding theory for coercive operators of monotone type. The theory developed herein can be suitably applied in the study of partial differential equations, variational and hemi-variational inequality problems in appropriate Sobolev spaces. To demonstrate the applicability of the theory, an example of time periodic parabolic partial differential equation, which models nonmonotone semipermeability problem, is provided.
Abstract and Applied Analysis, 2018
LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual ... more 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.
arXiv (Cornell University), Nov 20, 2021
Let X be a real Banach space with its dual X * and G be a nonempty, bounded and open subset of X ... more Let X be a real Banach space with its dual X * and G be a nonempty, bounded and open subset of X with 0 ∈ G. Let T : X ⊇ D(T) → 2 X be an m-accretive operator with 0 ∈ D(T) and 0 ∈ T (0), and let C be a compact operator from X into X with D(T) ⊆ D(C). We prove that f ∈ R(T) + R(C) if C is multivalued and f ∈ R(T + C) if C is single-valued, provided T x + Cx + εx ∋ f for all x ∈ D(T) ∩ ∂G and ε > 0. The surjectivity of T + C is proved if T is expansive and T + C is weakly coercive. Analogous results are given if T has compact resolvents and C is continuous and bounded. Various results by Kartsatos, and Kartsatos and Liu are improved, and a result by Morales is generalized.
Journal of Mathematical Analysis and Applications, 2021
Abstract Let X be a real reflexive Banach space with X ⁎ its dual space and G be a nonempty and o... more Abstract Let X be a real reflexive Banach space with X ⁎ its dual space and G be a nonempty and open subset of X. Let A : X ⊇ D ( A ) → 2 X ⁎ be a strongly quasibounded maximal monotone operator and T : X ⊇ D ( T ) → 2 X ⁎ be an operator of class A G ( S + ) introduced by Kittila. We develop a topological degree theory for the operator A + T . The theory generalizes the Browder degree theory for operators of type ( S + ) and extends the Kittila degree theory for operators of class A G ( S + ) . New existence results are established. The existence results give generalizations of similar known results for operators of type ( S + ) . Applications to strongly nonlinear problems are included.